A TREATISE ON THE CALCULUS OF FINITE DIFFERENCES. A TREATISE ON THE CALCULUS OF FINITE DIFFERENCES, BY GEOEGE B;OOLE, D.C.L. 1ATE HONORARY MEMBER OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY AND PROFESSOR OF MATHEMATICS IN THE QUEEN'S UNIVERSITY, IRELAND. EDITED BY J. F. MOULTON, FELLOW AND ASSISTANT TUTOR OF CHRIST'S COLLEGE, CAMBRIDGE. THIRD EDITION. Honfcon : MACMILLAN A1S1D CO. 1880 \Tlie Right of Translation is reserved.] CTambrfofje: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. PREFACE TO THE FIRST EDITION. IN the following exposition of the Calculus of Finite Dif- ferences, particular attention has been paid to the connexion of its methods with those of the Differential Calculus — a connexion which in some instances involves far more than a merely formal analogy. Indeed the work is in some measure designed as a sequel to my Treatise on Differential Equations. And it has been •*•'« ^ i : composed on the same plan. Mr Stirling, of Trinity College, Cambridge, has rendered me much valuable assistance in the revision of the proof- sheets. In offering him my best thanks for his kind aid, I am led to express a hope that the work will be found to be free from important errors. QUEEN'S COLLEGE, CORK, April 18, 1860. GEORGE BOOLE. PREFACE TO THE SECOND EDITION. WHEN I commenced to prepare for the press a Second Edition of the late Dr Boole's Treatise on Finite Differ- ences, my intention was to leave the work unchanged save by the insertion of sundry additions in the shape of para- graphs marked off from the rest of the text. But I soon found that adherence to such a principle would greatly lessen the value of the book as a Text-book, since it would be impossible to avoid confused arrangement and even much repetition. I have therefore allowed myself considerable freedom as regards the form and arrangement of those parts where the additions are considerable, but I have strictly adhered to the principle of inserting all that was contained in the First Edition. As such Treatises as the present are in close connexion with the course of Mathematical Study at the University of Cambridge, there is considerable difficulty in deciding the question how far they should aim at being exhaustive. I have held it best not to insert investigations that involve complicated analysis unless they possess great suggestiveness or are the bases of important developments of the subject. Under the present system the premium on wide superficial reading is so great that such investigations, if inserted, would seldom be read. But though this is at present the case, PREFACE TO THE SECOND EDITION. Vll there is every reason to hope that it will not continue to be so; and in view of a time when students will aim at an exhaustive study of a few subjects in preference to a super- ficial acquaintance with the whole range of Mathematical research, I have added brief notes referring to most of the papers on the subjects of this Treatise that have appeared in the Mathematical Serials, and to other original sources. In virtue of such references, and the brief indication of the subject of the paper that accompanies each, it is hoped that this work may serve as a handbook to students who wish to read the subject more . thoroughly than they could do by confining themselves to an Educational Text-book. The latter part of the book has been left untouched. Much of it I hold to be unsuited to a work like the present, partly for reasons similar to those given above, and partly because it treats in a brief and necessarily imperfect manner subjects that had better be left to separate treatises. It is impossible within the limits of the present work to treat adequately the Calculus of Operations and the Calculus of Functions, and I should have preferred leaving them wholly to such treatises as those of Lagrange, Babbage, Carmichael, De Morgan, &c. I have therefore abstained from making any additions to these portions of the book, and have made it my chief aim to render more evident the remarkable analogy between the Calculus of Finite Differences and the Differential Calculus. With this view I have suffered myself to digress into the subject of the Singular Solutions of Differ- ential Equations, to a much greater extent than Dr Boole had done. But I trust that the advantage of rendering the Vlll PREFACE TO THE SECOND EDITION. investigation a complete one will be held to justify the irrelevance of much of it to that which is nominally the subject of the book. It is partly from similar considerations that I have adopted a nomenclature slightly differing from that commonly used (e.g. Partial Difference-Equations for Equations of Partial Differences). I am greatly indebted to Mr R T. Wright of Christ's College for his kind assistance. He has revised the proofs for me, and throughout the work has given me valuable suggestions of which I have made free use. JOHN F. MOULTOK CHRIST'S COLLEGE, Oct. 1872. CONTENTS. DIFFERENCE- AND SUM-CALCULUS. CHAPTER I. PAGE NATURE OF THE CALCULUS OF FINITE DIFFERENCES . 1 CHAPTER II. DIRECT THEOREMS OF FINITE DIFFERENCES . 4 Differences of Elementary Functions, 6. Expansion in factorials, 11. Generating Functions, 14. Laws and relations of E, A and — , 16. Secondary form of Maclaurin's Theorem, 22. Herschel's Theo- rem, 24. Miscellaneous Expansions, 25. Exercises, 28. CHAPTER III. ON INTERPOLATION, AND MECHANICAL QUADRATURE . 33 Nature of the Problem, 33. Given values equidistant, 34. Not equi- distant— Lagrange's Method, 38. Gauss' Method, 42. Cauchy's Method, 43. Application to Statistics, 43. Areas of Curves, 46. Weddle's rule, 48. Gauss' Theorem on the best position of the given ordinates, 51. Laplace's method of Quadratures, 53. Refer- ences on Interpolation, &c. 55. Connexion between Gauss' Theo- rem and Laplace's Coefficients, 57. Exercises, 57. B. F. D. b X CONTENTS. CHAPTER IV. PAGE FINITE INTEGRATION, AND THE SUMMATION OF SERIES 62 Meaning of Integration, 62. Nature of the constant of Integration, 64. Definite and Indefinite Integrals, 65. Integrable forms and Summation of series— Factorials, 65. Inverse Factorials, 66. Eational and integral Functions, 68. Integrable Fractions, 70. Functions of the form ax(x), 72. Miscellaneous Forms, 75. Eepeated Integration, 77. Conditions of extension of direct to inverse forms, 78. Periodical constants, 80. Analogy between the Integral and Sum-Calculus, 81. Beferences, 83. Exercises, 83. CHAPTER Y. THE APPROXIMATE SUMMATION OF SERIES . 87 Development of 2, 87. Analogy with the methods adopted for the development of /, 87 (note). Division of the problem, 88. De- velopment of 2 in powers of D (Euler-Maclaurin Formula), 89. Values of Bernoulli's Numbers, 90. Applications, 91. Deter- mination of Constant, 95. Development of 2n, 96. Development of Swj; and 2nux in differences of a factor of ux, 99. Method of in- creasing the degree of approximation obtained by Maclaurin Theorem, 100. Expansion in inverse factorials, 102, Beferences, 103. Exercises, 103. CHAPTER YI. BERNOULLI'S NUMBERS, AND FACTORIAL COEFFICIENTS . 107 Various expressions for Bernoulli's Numbers— De Moivre's, 107. In terms of 2-^, 109. Baabe's (in factors), 109. As definite inte- Mr grals, 110. Euler's Numbers, 110. Bauer's Theorem, 112. Fac- torial Coefficients, 113. Beferences, 116. Exercises, 117. CHAPTER VII. CONVERGENCY AND DIVERGENCY OF SERIES . 123 Definitions, 123. Case in which ux has aperiodic factor, 124. Cauchy's Proposition, 126. First derived Criterion, 129. Supplemental Criteria — Bertrand's Form, 132. De Morgan's Form, 134. Third Form, 135. Theory of Degree, 136. Application of Tests to the Euler-Maclaurin Formula, 139. Order of Zeros, 139. Befer- ences, 140. Exercises, 140. CONTENTS. XI CHAPTER YJII. PAGE EXACT THEOREMS . , , .145 Necessity for finding the limits of error in our expansions, 145. Ee- mainder in the Generalized Form of Taylor's Theorem, 146. Ee- mainder in the Maclaurin Sum-Formula, 149. Eeferences, 152. Boole's Limit of the Kemainder of the Series for SM^ 154. DIFFERENCE- AND FUNCTIONAL EQUATIONS. CHAPTER IX. DIFFERENCE-EQUATIONS OF THE FIRST ORDER . 157 Definitions, 157. Genesis, 158. Existence of a complete Primitive, 160. Linear Equations of the First Order, 161. Difference-equa- tions of the first order but not of the first degree— Clairault's Form, 167. One variable absent, 167. Equations Homogeneous in w, 168.— Exercises, 169. CHAPTER X. GENERAL THEORY OF THE SOLUTIONS OF DIFFERENCE- AND DIFFERENTIAL EQUATIONS OF THE FIRST ORDER . 171 Difference-Equations — their solutions, 171. Derived Primitives, 172. Solutions derived from the Variation of a Constant, 174. Analo- gous method in Differential Equations, 177. Comparison between the solutions of Differential and Difference-Equations, 179. As- sociated primitives, 182. Possible non-existence of Complete Inte- gral, 183. Detailed solution of ux — xAux + (AuxY, 185. Origin of singular solutions of Differential Equations, 189. Their ana- logues in Difference- Equations, 190. Eemarks on the complete curves that satisfy a Differential Equation, 191. Anomalies of Singular Solutions, 193. Explanation of the same, 194. Prin- ciple of Continuity, 198. Eecapitulation of the classes of solu- tions that a Difference-Equation may possess, 204. Exercises, 205. CHAPTER XI. LINEAR DIFFERENCE- EQUATIONS WITH CONSTANT COEFFICIENTS . . . . 208 Introductory remarks, 208. Solution of/ (E)ux = 0, 209. Solution of f(E)ux = X, 213. Examination of Symbolical methods, 215. Spe- cial forms of X, 218. Exercises, 219. Xll CONTENTS. CHAPTER XII. PAGE MISCELLANEOUS PROPOSITIONS AND EQUATIONS. SIMULTANEOUS EQUATIONS . . .221 Equations reducible to Linear Equations with Constant Coefficients, 221. Binomial Equations, 222. Depression of Linear Equa- tions, 224. Generalization of the above, 225. Equations solved by performance of A71, 228. Sylvester's Forms, 229. Simultane- ous Equations, 231. Exercises, 232. CHAPTER XIH. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS. SYMBOLICAL AND GENERAL METHODS . . 236 Symbolical Methods, 236. Solution of Linear Difference-Equations in Series, 243. Finite Solution of Difference-Equations, 246. Binomial Equations, 248. Exercises, 263. CHAPTER XIV. MIXED AND PARTIAL DIFFERENCE-EQUATIONS . 264 Definitions, 264. Partial Difference-Equations, 266. Method of Generating Functions, 275. Mixed Difference-Equations, 277. Exercises, 289. CHAPTER XV. OF THE CALCULUS OF FUNCTIONS . .291 Definitions, 291. Direct Problems, 292. Periodical Functions, 298. Functional Equations, 301. Exercises, 312. CHAPTER XVI. GEOMETRICAL APPLICATIONS . . .316 Nature of the problems, 316. Miscellaneous instances, 317. Exer- cises, 325. ANSWERS TO THE EXERCISES 326 FINITE DIFFEBENCES. CHAPTER I. NATUKE OF THE CALCULUS OF FINITE DIFFERENCES. -1. THE Calculus of Finite Differences may be strictly defined as the science which is occupied about the ratios of the simultaneous increments of quantities mutually depen- dent. The Differential Calculus is occupied about the limits to which such ratios approach as the increments are indefi- nitely diminished. In the latter branch of analysis if we represent the inde- pendent variable by x, any dependent -variable considered as a function of x is represented primarily indeed by $ (x), but, when the rules of differentiation founded on its functional character are established, by a single letter, as u. In the notation of the Calculus of Finite Differences these modes of expression seem to be in some measure blended. The de- pendent function of x is represented by ux, the suffix taking the place of the symbol which in the former mode of notation is enclosed in brackets. Thus, if ux = (/>(#), then and so on. But this mode of expression rests only on a con- vention, and as it was adopted for convenience, so when con- venience demands it is laid aside. The step of transition from a function of x to its increment, and still further to the ratio which that increment bears to the increment of x, may be contemplated apart from its sub- B. F. D. 1 2 NATUEE OF THE CALCULUS [CH. I. ject, and it is often important that it should be so contem- plated, as an operation governed by laws. Let then A, pre- fixed to the expression of any function of x, denote the operation of taking the increment of that function correspond- ing to a given constant increment Ace of the variable x. Then, representing as above the proposed function of x by ux, we have and Here then we might say that as -7- is the fundamental ope- ration of the Differential Calculus, so -r - is the fundamental A# operation of the Calculus of Finite Differences. But there is a difference between the two cases which ought to be noted. In the Differential Calculus ^- is not a true fraction, nor have du and doc any distinct meaning as symbols of quantity. The fractional form is adopted to express the limit to which a true fraction approaches. Hence -j- , and not d, there represents a real operation. But in the Calculus of Finite Differences -r— - is a true fraction. Its nu- A# merator &ux stands for an actual magnitude. Hence A might itself be taken as the fundamental operation of this Calculus, always supposing the actual value of Arc to be given ; and the Calculus of Finite Differences might, in its symbolical charac- ter, be defined either as the science of the laws of the operation A, the value of A# being supposed given, or as the science of the laws of the operation -r— . In consequence of the funda- mental difference above noted between the Differential Calcu- lus and the Calculus of Finite Differences, the term Finite ceases to be necessary as a mark of distinction. The former is a calculus of limits, not of differences. ART. 2.] OF FINITE DIFFERENCES. 3 *2. Though Aa; admits of any constant value, the value usually given to it is unity. There are two reasons for this. First. The Calculus of Finite Differences has for its chief subject of application the terms of series. Now the law of a series, however expressed, has for its ultimate object the deter- mination of the values of the successive terms as dependent upon their numerical order and position. Explicitly or im- plicitly, each term is a function of the integer which ex- presses its position in the series. And thus, to revert to language familiar in the Differential Calculus, the inde- pendent variable admits only of integral values whose com- mon difference is unity. For instance, in the series of terms •12 02 02 A? J- , * > «> ? * , . . . the general or #th term is #2. It is an explicit function of x, but the values of x are the series of natural numbers, and A0»l. Secondly. When the general term of a series is a function of an independent variable t whose successive differences are constant but not equal to unity, it is always possible to replace that independent variable by another, x, whose com- mon difference shall be unity. Let <£ (t) be the general term of the series, and let A£ = h ; then assuming t = hx we have whence A# = 1. Thus it suffices to establish the rules of the Calculus on the assumption that the finite difference of the independent variable is unity. At the same time it will be noted that this assumption reduces to equivalence the symbols -r— and A. We shall therefore in the following chapters de.velope the theory of the operation denoted by A and defined by the equation &ux = ux+l-ux. But we shall, where convenience suggests, consider the more general operation where A# = h. 1—2 CHAPTER II. DIRECT THEOREMS OF FINITE DIFFERENCES. vl. THE operation denoted by A is capable of repetition. For the difference of a function of x, being itself a function of x, is subject to operations of the same kind. In accordance with the algebraic notation of indices, the difference of the difference of a function of x, usually called the second difference, is expressed by attaching the index 2 to the symbol A. Thus In like manner and generally sAX ........................ (1), the last member being termed the nih difference of the function ux. If we suppose ux = x3, the successive values of ux with their successive differences of the first, second, and third orders will be represented in the following scheme : Values of x 1 23 4 5 ux 1 8 27 64 125 216... Aw. 7 19 37 61 91 ... AX 12 18 24 30... A3'W 6 6 6.. It may be observed that each set of differences may either be formed from the preceding set by successive subtractions in accordance with the definition of the symbol A, or calcu- lated from the general expressions for Aw, A*w, &c. by assign- ART. 2.] DIRECT THEOREMS OF FINITE DIFFERENCES. 5 ing to x the successive values 1, 2, 3, &c. Since ux = x*, we shall have A%, = (x + I)3 -x* = 3x? + 3a? + 1, AX = A (3#2 -f a» + 1) = 6x + 6, AX =6. It may also be noted that the third differences are here constant. And generally if ux be a rational and integral function of x of the nth degree, its nth differences will be constant. For let ux = axn + bxn~l -f &c., then AH, = a (x + l)n + b (x + l)n~l + &c. — axn — bxn~l — &c. &c.', 5j, 52, &c., being constant coefficients. Hence AM^ is a rational and integral function of x of the degree n — l. Repeating the process, we have AX = an (n - 1) aT2 + c.aT8 + c/1'4 + &c., a rational and integral function of the degree n — 2; and so on. Finally we shall have AX = <™>-l)(rc-2)...l, a constant quantity. Hence also we have AV = 1.2...n ..................... (2). -2. While the operation or series of operations denoted by A, A2, ... An are always possible when the subject-function ux is given, there are certain elementary cases in which the forms of the results are deserving of particular attention, and these we shall next consider. 6 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. Differences of Elementary Functions. 1st. Let ux=x (x -V)(x- 2) ... (a? - m+ 1). Then by definition, = ra#(#-l) (#-2) ... (#-w + 2), When the factors of a continued product increase or de- crease by a constant difference, or when they are similar functions of a variable which, in passing from one to the other, increases or decreases by a constant difference, as in the expression sin x sin (x + h) sin (x + 2h) ... sin [x + (m — 1) h], the factors are usually called factorials, and the term in which they are involved is called a factorial term. For the particular kind of factorials illustrated in the above example it is com- mon to employ the notation x(x-l)...(x-m + l}=x^ ............ (1), doing which, we have - (2). Hence, x(>n~1} being also a factorial term, and generally AV"} = w» (m - 1) ... (m - n + 1) x(m~n) ...... (3). Sndly. Let ux = — -. — -^r - -, — - - ^ . x (x + 1) . . . (x + m — 1) Then by definition, > + 2)... — m (4) ART. 2.] DIRECT THEOREMS OF FINITE DIFFERENCES. 7 Hence, adopting the notation we have ^x^=-mx^'l) ..................... (5). Hence by successive repetitions of the operation A, AV*) = - m (- ra - 1) ... (-m - ?i + 1) af™> = (- l)n m (m+ 1) ... (m + n- 1) #(~m~M) ...... (6), and this may be regarded as an extension of (3). * Srdly. Employing the most general form of factorials, we find x wA_t . . . u^i ...... (7), A l- = _J^V^ (8)f and in particular if ux = ace + b, . ux_m^ (9), A- -=- - (10). uxux^ . . . u^^ uxux+l . . . ux+m In like manner we have A log ux = log ux+1 - log u, = log ^ . ux To this result we may give the form + ^r) (")• So also (ufu^...u,_M) = \og^ (12). 4thly. To find the successive differences of a. 8 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. We have = (a-l)a" .................. (13). Hence AV=(a-l)V, and generally, AV=(a-l)"a* .................. (14). Hence also, since amx = (am}x, we have AV* = (am -1)" amx .................. (15). 5thly. To deduce the successive differences of sin (ax + b) and cos (ax + b). A sin (ax + b) = sin (ax + b + a) — sin (ax + b) = 2 sin ^ cos (ax + b + ~ 4 * o • a ' ( — 2 sin ^ sin ax + 2 V By inspection of the form of this result we see that A2 sin (ax + b) = ^2 sin |J sin (ax+ b + a + TT) (16). And generally, A'sin (ax + V) = (2 sin |)"sin |aa! + 6 + In the same way it will be found that A* cos (ax + b) = (2 sin ^"cos Lx+ b + These results might also be deduced by substituting for the sines and cosines their exponential values and applying (15). 3. The above are the most important forms. The follow- ing are added merely for the sake of exercise. ART. 4.] DIRECT THEOREMS OF FINITE DIFFERENCES. To find the differences of tan ux and A tan ux = tan ux+l — tan u2 sin ux+l _ sin cos u sin AM ~v Next, A tan"1!*,,. = tan'X — tan~X (2). From the above, or independently, it is easily shewn that sin a ,ON A tan CM? = - —f - -r ............ (3), cos ax cos a (x + 1) A tan'1 ax = tan'1 - —^ — - — 5-= ............ (4). 1 + cfx + c?x* Additional examples will be found in the exercises at the end of this chapter. 4. When the increment of x is indeterminate, the opera- tion denoted by — merges, on supposing A# to become tAX infinitesimal but the subject-function to remain unchanged, into the operation denoted by -j- . The following are illus- trations of the mode in which some of the general theorems of the Calculus of Finite Differences thus merge into theorems of the Differential Calculus. 10 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. Ex. We have A sin x _ sin (x + A#) — sin x 2 sin A# sin in f x H -- - — J And, repeating the operation n times, (2 sin J Aa)n sin (a> + n An sin a v 2 y V 2 . It is easy to- see that the limiting form of this equation is dn sin x dxn a known theorem of the Differential Calculus. Again, we have A XM-P ,r*^"A^' r*3* Acz> d — a (2), And hence, generally, *-! Supposing Ao? to become infinitesimal, this gives by the ordinary rule for vanishing fractions (4). But it is not from examples like these to be inferred that the Differential Calculus is merely a particular case of the Calculus of Finite Differences. The true nature of their con- ! nexion will be developed in a future chapter. AKT. 5.] DIRECT THEOREMS OF FINITE DIFFERENCES. 11 Expansion by factorials. 5. Attention has been directed to the formal analogy between the differences of factorials and the differential coefficients of powers. This analogy is further developed in the following proposition. To develope (x), a given rational and integral function of x of the mth degree, in a series of factorials. Assume 0 (»)=« + Ix + cx(*) + dx(5)...+hx (x). And the actual values of the former might be determined by expressing both mem- bers of the equation in ascending powers of x, equating coeffi- cients, and solving the linear equations which result. Instead of doing this, let us take the successive differences of (1). We find by (2), Art. 2, A$(x)=b + 2cx + 3daP...+mhaF*-v (2), \ A2 (a?) = 2c + 3 . A"0(a?)=OT(0»-l)...U ....................... (4). And now making x— 0 in the series of equations (1)...(4), and representing by A0 (0), A2<£ (0), &c. what A<£ (x), A2<£ (x), &c. become when x = 0, we have = a, Whence determining a, &, c, ... h, we have If with greater generality we assume <£ (a?) = a + bx + ex (x - h) + dx (x -Ii)(x- 2Ji) + &c., DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. we shall find by proceeding as before, (except in the employ- ing of — for A, where A# = h,) where the brackets { } denote that in the enclosed function, after reduction, x is to be made equal to 0. Maclaurin's theorem is the limiting form to which the above theorem approaches when the increment Aic is inde- finitely diminished. General theorems expressing relations between the successive values, successive differences, and successive differential coef- ficients of functions. 6. In the equation of definition we have the fundamental relation connecting the first differ- ence of a function with two successive values of that function. Taylor's theorem gives us, if h be put equal to unity, dux Id*ux 1 d3i which is the fundamental relation connecting the first differ- ence of a function with its successive differential coefficients. From these fundamental relations spring many general theo- rems expressing derived relations between the differences of the higher orders, the successive values, and the differential coefficients of functions. As concerns the history of such theorems it may be ob- served that they appear to have been first suggested by par- ticular instances, and then established, either by that kind of proof which consists in shewing that if a theorem is true for any particular integer value of an index n, it is true for the next greater value, and therefore for all succeeding values ; or else by a peculiar method, hereafter to be explained, called the method of Generating Functions. But having AET. 7.] DIRECT THEOREMS OF FINITE DIFFERENCES. 13 been once established, the very forms of the theorems led to a deeper conception of their real nature, and it came to be understood that they were consequences of the formal laws of combination of those operations by which from a given function its succeeding values, its differences, and its differ- ential coefficients are derived. 7. These progressive methods will be illustrated in the following example. Ex. Required to express ux+n in terms of ux and its suc- cessive differences. We have AX- Hence proceeding as before we find 3AX These special results suggest, by the agreement of their coefficients with those of the successive powers of a binomial, the general theorem Suppose then this theorem true for a particular value of n, then for the next greater value we have n (n — 1) A , x + ^ 2 AX , n(n- 1) (n - 2) A + ~ ^ AX ?/, + n AX DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. the form of which shews that the theorem remains true for the next greater value of n, therefore for the value of n still succeeding, and so on ad infinitum. But it is true- for n=\, and therefore for all positive integer values of n whatever. 8. We proceed to demonstrate the same theorem by the method of generating functions. Definition. If (f> (t) is capable of being developed in a series of powers of t, the general term of the expansion being represented by uxtx, then (t) is said to be the generating function of u£. And this relation is expressed in the form ,. Thus we have since - — ~ is the coefficient of f in the development of ef\ In like manner j. "1 O since n — » — =^ is the coefficient of f in the development 1 ,2*..(# + 1) of the first member. And generally, if Gux = ^> (£), then ^M*n — '"y" ^w*+« = -^r- (2). Hence therefore But the first member is obviously equal to G&ux, therefore /o\ (*)• ART. 8.] DIRECT THEOREMS OF FINITE DIFFERENCES. 15 And generally To apply these theorems to the problem under considera- tion we have, supposing still Gux = <£ (t), = ff Hence ff AV. + &c. X + &c.J . n (n — 1) A n v 2 y AX + &c. which agrees with (1). <» Although on account of the extensive use which has been made of the method of generating functions, especially by the older analysts, we have thought it right to illustrate its general principles, it is proper to notice that there exists an objection in point of scientific order to the employment of the method for the demonstration of the direct theorems of the Calculus of Finite Differences ; viz. that Q- is, from its very nature, a symbol of inversion (Diff. Equations, p. 375, 1st Ed.). In applying it, we do not perform a direct and definite ope- ration, but seek the answer to a question, viz. What is that function which, on performing the direct operation of deve- lopment, produces terms possessing coefficients of a certain form ? and this is a question which admits of an infinite variety of answers according to the extent of the development and the kind of indices supposed admissible. Hence the distributive property of the symbol G, as virtually employed 16 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. in the above example, supposes limitations which are not implied in the mere definition of the symbol. It must be supposed to have reference to the same system of indices in the one member as in the other; and though, such conven- tions being supplied, it becomes a strict method of proof, its indirect character still remains*. 9. We proceed to the last of the methods referred to in Art. 6, viz. that which is founded upon the study of the ulti- mate laws of the operations involved. In addition to the symbol A, we shall introduce a symbol E to denote the ope- ration of giving to a; in a proposed subject function the incre- ment unity; — its definition being Laws and Relations of the symbols E, A and -j- . 1st. The symbol A is distributive in its operation. Thus For In like manner we have &(ux-vx + &c...)=Awa.-Ava;+&c .................. (3). 2ndly. The symbol A is commutative with respect to any constant coefficients in the terms of the subject to which it is applied. Thus a being constant, (4). And from this law in combination with the preceding one, we have, a, 6,... being constants, &c ............... (5). : * The student can find instances of the use of Generating Functions in Lacroix, Diff. and Int. Gal. in. 322. Examples of a fourth method, at once elegant and powerful, due originally to Abel, are given in Grunert's Archiv. XYIII. 381. ART. 9.] DIRECT THEOREMS OF FINITE DIFFERENCES. 17 Srdly. The symbol A obeys the index law expressed by the equation AWAX = ATO+X .................. (6), m and n being positive indices. For, by the implied definition of the index m, A^A"^ = (A A. . .m times) (AA. . .n times) ux = {AA. . . (m + n) times} ux = Am+X. These are the primary laws of combination of the symbol A. It will be seen from these that A combines with A and with constant quantities, as symbols of quantity combine with each other. Thus, (A -f a) u denoting Aw + au, we should have, in virtue of the first two of the above laws, (A + a) (A + b) u = {A2 + (a + 6) A + ab] u = A2w + (a + 6) &u+ab'u ..................... (7), the developed result of the combination (A + a) (A + 6) being in form the same as if A were a symbol of quantity. The index law (6) is virtually an expression of the formal consequences of the truth that A denotes an operation which, performed upon any function of x, converts it into another function of x upon which the same operation may be repeated. Perhaps it might with propriety be termed the law of repe- tition; — as such it is common to all symbols of operation, except such, if such there be, as so alter the nature of the subject to which they are applied, as to be incapable of repetition*. It was however necessary that it should be dis- tinctly noticed, because it constitutes a part of the formal ground of the general theorems of the calculus. The laws which have been established for the symbol A are even more obviously true for the symbol E. The two symbols are connected by the equation * For instance, if $ denote an operation which, when performed on two quantities x, y, gives a single function A', it is an operation incapable of repe- tition in the sense of the text, since 02 (x, ?/) = (X) is unmeaning. But if it be taken to represent an operation which when performed on x, y, pives the two functions A', Y, it is capable of repetition since 02(», y) = (X, Y), which has a definite meaning. In this case it obeys the index law. B. F. D. 2 18 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. since (8), and they are connected with -v- by the relation -E-^V: ............................ (9), founded on the symbolical form of Taylor's theorem. For du . IdV . 1 d*u It thus appears that E, A, and -7- , are connected by the two equations J£=l+A = e^ ....(10), and from the fact that E and A are thus both expressible by means of -r- we might have inferred that the symbols E, A, and v- * combine each with itself, with constant quantities, CtiX and with each other, as if they were individually symbols of quantity. (Differential Equations, Chapter XVI.) 10. In the following section these principles will be applied to the demonstration of what may be termed the direct general theorems of the Calculus of Differences. The conditions of their inversion, i. e. of their extension to cases in which symbols of operation occur under negative indices, will * In place of -=- we shall often use the symbol Z>. The equations will then be E = l + A = tD, a form which has the advantage of not assuming that the independent variable has been denoted by x. ART. 10.] DIRECT THEOREMS OF FINITE DIFFERENCES. 19 be considered, so far as may be necessary, in subsequent chapters. Ex. 1. To develope ux+n in a series consisting of ux and its successive differences (Ex. of Art. 7, resumed). By definition ux+l = Eux, ux^ = E*ux, &c. Therefore (1), Ex. 2. To express A'Xc in terms of ^ and its successive values. Since &ux = wx+1 — w^ = Eux — w^, we have and as, the operations being performed, each side remains a function of x, Hence, interpreting the successive terms, Of particular applications of this theorem those are the most important which result from supposing ux = x™. We have 20 DIRECT THEOEEMS OF FINITE DIFFERENCES. [CH. ft. Now let the notation A"0m be adopted to express what the first member of the above equation becomes when x — 0 ; then &nQm = nm-n(n-l)m n(n-l)(n-2r n (*- 1) (n-2)'(n -3)" 1.2 1.2.3 • The systems of numbers expressed by AnOm are of frequent occurrence in the theory of series*. From (2) Art. 1, we have A"0" = 1.2... n, and, equating this with the corresponding value given by (5), we have 1 . 2 ... n = nn -n (n - l)w + ^"^ (n - 2)" - &c....(6)f. Ex. 3. To obtain developed expressions for the nib differ- ence of the product of two functions ux and v^ Since Auxvx = ux+1.vx+l-uj;vx = Eux.E'vx-uxvx, where E applies to ux alone, and E' to vx alone, we have and generally l)nuxvx ................. (7). It now only remains to transform, if needful, and to de- velope the operative function in the second member according to the nature of the expansion required. Thus if it be required to express &nuxvx in ascending diifer- * A very simple method of calculating their values will be given in Ex. 8 of this chapter. f This formula is of use in demonstrating Wilson's Theorem, that 1 + 1 n - 1 is divisible by n when n is a prune number. ART. 10.] DIRECT THEOREMS OF FINITE DIFFERENCES. 21 ences of vxt we must change E' into A' + 1, regarding A' as operating only on VK. We then have Remembering then that A and E operate only on ux and A' only on vx, and that the accent on the latter symbol may be dropped when that symbol only precedes vx> we have + 5jLl A-X, . AX + &c ...... (8), the expansion required. As a particular illustration, suppose ux = ax. Then, since An-ri^ = An~V^ = ar A^a* = a3*' (a- l)n~r, by (14), Art. 2, we have A"a% = a" {(a - 1)X + n (a - IJ^aAv, + TO (n~ 1} (a - l)n-2a2A2^ + &c.} ........ (9). z Again, ifr the expansion is to be ordered according to suc- cessive values of vxt it is necessary to expand the untrans- formed operative function in the second member of (7) in ascending powers of E' and develope the result. We find AX», = (-!)• {«,*„ - «wmv, + ^^ »^V2 - &c.) ... -(10). Lastly, if the expansion is to involve only the differences of ux and vx) then, changing E into 1 4- A, and E' into 1 -f A', we have ^ ................. (11), and the symbolic trinomial in the second member is now to be developed and the result interpreted. 22 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. Ex. 4. To express A*X. in terms of the differential co- efficients of u x. By (10), Art. 9, A = c* - 1. Hence AX = (^-1)X .................... (12). Now t being a symbol of quantity, we have ......... (13), on expansion, ^41? <42, being numerical coefficients. Hence and therefore The coefficients AJ} A2,...&c. may be determined in various ways, the simplest in principle being perhaps to de- velope the right-hand member of (13) by the polynomial theorem, and then seek the aggregate coefficients of the suc- cessive powers of t. But the expansion may also be effected with complete determination of the constants by a remarkable secondary form of Maclaurin's theorem, which we shall pro- ceed to demonstrate. Secondary form of Maclaurin's Theorem. PROP. The development of (t) in positive and integral powers of t, when such development is possible, may be expressed in the form c/> f-^r J (T denotes what $ f -r-J where c> f-r (T denotes what f -r- xm becomes when x = 0. ART. 10.] DIRECT THEOREMS OF FINITE DIFFERENCES. 23 First, we shall shew that if (f> (a?) and ^ (x) are any two functions of x admitting of development in the form a + bx + ex2 + &c., then #f«-*#W .................. (15), provided that x be made equal to 0, after the implied opera- tions are performed. For, developing all the functions, each member of the above equation is resolved into a series of terms of the form / d\m A I -=- j xn, while in corresponding terms of the two members the order of the indices ra and n will be reversed. (d \m -7-1 xn is equal to 0 if ra is greater than n, to 1 . 2...n if ra is equal to n, and again to 0 if ra is less than n and at the same time as equal to 0 ; for in this case of1""1 is a factor. Hence if x = 0, and therefore under the same condition the equation (15) is true, or, adopting the notation above explained, Now by Maclaurin's theorem in its known form Hence, applying the above theorem of reciprocity, the secondary form in question. The two forms of Mac- laurin's theoism (17), (18) may with propriety be termed conjugate. 2i DIEECT THEOREMS OF FINITE DIFFERENCES. [CH. II. A simpler proof of the above theorem (which may be more shortly written <£(£) = <£ (D) e0'') is obtained by regarding it as a particular case of Herschel's theorem, viz. or, symbolically written, (e<) = (E) e°'*.* The truth of the last theorem is at once rendered evident by assuming An€nt to be any term in the expansion of $ (e*) in powers of e*. Then since Anent ~ AnEn eQtt the identity of the two series is evident. But £(*) = * (^g e<) = (log E) e° • * (by Herschel's theorem) which is the secondary form of Maclaurin's theorem. As a particular illustration suppose (t) = (e*-l)re, then by means of either of the above theorems we easily deduce (e* - l)n = AW0 .t + AM02. ~ + A*03 . j-^ + &c. But A*0m is equal to 0 if m is less than n and to 1 . 2 . 3. . .71 if m is equal to %, (Art. 1). Hence Hence therefore since Aww = (e*8 — l)*w we have =^ , AMOra+1 d*»t* AMOra+2 (T^M ^Jl + 1.2...(w+l)'^w+"i + 1.2...(n+2)^n+2+ the theorem sought. . The reasoning employed in the above investigation pro- ceeds upon the assumption that n is a positive integer. The * Since both A and D performed on a constant produce ac, result zero, it is obvious that 0 (D) 6'= 0(0) C= (A) C, and (E)C=<}> 1) C. It is of course assumed throughout that the coefficients in $ are constants. AET. 11.] DIRECT THEOREMS OF FINITE DIFFERENCES. 25 very important case in which n = — 1 will be considered in another chapter of this work. dnu Ex. 5. To express -j—n in terms of the successive differences of u. A Since edx = 1 + A, we have therefore QgY = jlog (1 + A)]" .............. (22), and the right-hand member must now be developed in as- cending powers of A. In the particular case of n = 1, we have du A2w , A3u A4w p - = ±u--- + — . --+&C ........ (23). 11. It would be easy, but it is needless, to multiply these general theorems, some of those above given being valuable rather as an illustration of principles than for their intrinsic importance. We shall, however, subjoin two general theo- rems, of which (21) and (23) are particular cases, as they serve to shew how striking is the analogy between the parts played by factorials in the Calculus of Differences and powers in the Differential Calculus. By Differential Calculus we have du t2 d\ Perform <£(A) on both sides (A having reference to t alone), and subsequently put t = 0. This gives of which (21) is a particular case. 26 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. By (2) we have Perform f-^-J on each side, and subsequently put t = 0 ; of which (23) is a particular case. 12. We have seen in Art. 9 that the symbols A, E and -7- or D have, with certain restrictions, the same laws of corn- eta? bination as . constants. It is easy to see that, in general, these laws will hold good when they combine with other symbols of operation provided that these latter also obey the above-mentioned laws. By these means the Calculus of Finite Differences may be made to render considerable assist- ance to the Infinitesimal Calculus, especially in the evaluation of Definite Integrals. We subjoin two examples of this; further applications of this method may be seen in a Memoire by Cauchy (Journal Poly technique, Vol. xvii.). Ex. 6. To shew that B(m+I,n)= (- l)mAw - , where m is a positive integer. 1 [m We have - = I e~nxdx ; 1 r00 r . • . Am - = A1" e~nxdx = n J« J = 1 zn * (z — I)"1 dz (assuming z — e~*) * 0 = (-l)w.5(m + l,n). ART. 12.] DIRECT THEOREMS OF FINITE DIFFERENCES. 27 Ex. 7. Evaluate u=l Am^ *&> m being a positive integer greater than a ; A relating to n alone. Let 2/c be the even integer next greater than a + 1, then Now the first member of the right-hand side of (26) is a rational integral function of n of an order lower than m. It therefore vanishes when the operation A™ is performed on it We have therefore f =\ J0 * + nz jr« 'y (assuming 7T (Tod. Int. Cal. Art. 255, 3rd Ed. C17T This example illustrates strikingly the nature and limits of the commutability of order of the operations I and A. Had we changed the order (as in (27)) without previously preparing the quantity under the sign of integration, we should have had which is infinite if a be positive. The explanation of this singularity is as follows : — If we write for Aw its equivalent (E—l)m and expand /oo Am<£ (#, n) dx expresses the integral 28 EXERCISES. [CH. II. of a quantity of m + 1 terms of the form Ap (sc, n +p), while r00 Aml <£ (x, n) dx expresses the sum of m + 1 separate inte- •' 0 grals, each having under the integral sign one of the terms of the above quantity. Where each term separately integrated gives a finite result, it is of course indifferent which form is used, but where, as in the case before us, two or more would give infinity as result the second form cannot be used. 13. Ex. 8. To shew that 0 (E) On = E$ (E) (F-1 (28). Let ArErQn and ErArE^On~l be corresponding terms of the two expansions in (28). Then, since each of them equals Arrn, the identity of the two series is manifest. Since E= 1 + A the theorem may also be written <£ (A) On = E$ (A) Qn~\ and under this form it affords the simplest mode of calcu- lating the successive values of Am(T. Putting <£ (A) = Am, we have AMOn = E . wA^CT1 = m ( A^O71'1 + AmOn"1), and the differences of O*1 can be at once calculated from those of O"-1. Other theorems about the properties of the remarkable set of numbers of the form AmOw will be found in the accom- panying exercises. Those desirous of further information on the subject may consult the papers of Mr J. Blissard and M. Worontzof in the Quarterly Journal of Mathematics, Vols. vill. and IX. EXERCISES. 1. Find the first differences of the following functions : 2-sinJ, tanj, cot (2" a). EX. 2.] EXEECISES. 29 J'2. Shew that * 3. Prove the following theorems : AO* ~0* + &c. = 0 (x > 1) ... . 1.2 w-m m+I 4. Shew that, if m be less than r, 5. Express the differential coefficient of a factorial in factorials. Ex. x(m\ 6. Shew that AnOn, AnOn+1 ...... form a recurring series, and find its scale of relation. 7. If P/^ shew that 8. Shew that What class of series would the above theorem enable us to convert from a slow to a rapid convergence ? 30 EXERCISES. [EX. 9. 9. Shew that and hence calculate the first four terms of the expression. (P^2 - P2A3 + &c.) Om = 0 if m > 2. Prove that unless m = n when it is equal to \n. 11. Prove that - - n) of8* (1 -w) (2 - n) af*+ &c. 12. If a; = ed, prove that AO" c? A A30 13. If ^u^y^ux^y^ — uXiy B,nd. if A"^,, be expanded in a series of differential coefficients of uKt u> shew that the general term will be 14. Express Aw#w in a series of terms proceeding by powers of x by means of the differences of the powers of 0. By means of the same differences, find a finite expression for the infinite series 'I! EX. 15.] EXERCISES. 31 where m is a positive integer, and reduce the result when m = 4. 15. Prove that = (x + n - V)(n]kux) m) w and find the analogous theorems in the Infinitesimal Calculus. 16. Find ux from the equations l-Vl-4*2 (1) GUX= -g- -; (2) <**.=/#). 17. Find a symbolical expression for the wth difference of the product of any number of functions in terms of the differences of the separate functions, and deduce Leibnitz's theorem therefrom. 18. If Pn be the number of ways in which a polygon of n sides can be divided into triangles by its diagonals, and fcf) (t) = GPn, shew that *19. Shew that f •^ n and a being positive quantities. *20. Shew that 00 sin 2nx sinma; _ _ wAm (2n — m) ' ~ if 2w > m > a all being positive. * In Questions 19 and 20 A acts on n alone. 32 EXERCISES. [EX. 21. /OO • j)l sin 2rc# . ^J? . j^ jg constant for o # 777 all values of n between — and oo . 21. Shew that if p be a positive integer r 1.2.3.. ..2» e**.sm2^.cfo = ^ Jo (Bertrand, Cal. Int. p. 185.) 22. Shew that \n -ip+1 , \n-l -ip A n-1 G)P _ ** ± "t" ^ LA ~ — — — , 23. Demonstrate the formula An^^fn + DA^P and apply it to construct a table of the differences of the powers of unity up to the fifth power. CHAPTER III. ON INTERPOLATION, AND MECHANICAL QUADRATURE. 1. THE word interpolate has been adopted in analysis to denote primarily the interposing of missing terms in a series of quantities supposed subject to a determinate law of mag- nitude, but secondarily and more generally to denote the calculating, under some hypothesis of law or continuity, of any term of a series from the values of any other terms sup- posed given. As no series of particular values can determine a law, the problem of interpolation is an indeterminate one. To find an analytical expression of a function from a limited number of its numerical values corresponding to given values of its independent variable x is, in Analysis, what in Geometry it would be to draw a continuous curve through a number of given points. And as in the latter case the number of pos- sible curves, so in the former the number of analytical ex- pressions satisfying the given conditions, is infinite. Thus the form of the function — the species of the curve — must be assumed a priori. It may be that the evident character of succession in the values observed indicates what kind of assumption is best. If for instance these values are of a periodical character, circular functions ought to be employed. But where no such indications exist it is customary to assume for the general expression of the values under consideration a rational and integral function of x, and to determine the coefficients by the given conditions. This assumption rests upon the supposition (a supposition however actually verified in the case of all tabulated func- tions) that the successive orders of differences rapidly dimi- nish. In the case of a rational and integral function of x of the nth degree it has been seen that differences of the n + 1th B. F. D. 3 34 ON INTERPOLATION, [CH. III. and of all succeeding orders vanish. Hence if in any other function such differences become very small, that function may, quite irrespectively of its form, be approximately repre- sented by a function which is rational and integral. Of course it is supposed that the value of x for which that of the function is required is not very remote from those, or from some of those, values for which the values of the func- tion are given. The same assumption as to the form of the unknown function and the same condition of limitation as to the use of that form flow in an equally obvious manner from the expansion in Taylor's theorem. 2. The problem of interpolation assumes different forms, according as the values given are equidistant, i.e. corre- spondent to equidifferent values of the independent variable, or not. But the solution of all its cases rests upon the same principle. The most obvious mode in which that principle can be applied is the following. If for n values a,b, ... of an independent variable x the corresponding value ua, ub) ... of an unknown function of x represented by ux, are given, then, assuming as the approximate general expression of ux, ux = A + Bx + Cx* ...+Exn-1 (1), a form which is rational and integral and involves n arbitrary coefficients, the data in succession give ua = A -f Ba + Co? . . . + Ean~\ a system of n linear equations which determine A, B...E. To avoid the solving of these equations other but equivalent modes of procedure are employed, all such being in effect reducible to the two following, viz. either to an application of that property of the rational and integral function in the second member of (1) which is expressed by the equation &nux = 0, or to the substitution of a different but equivalent form for the rational and integral function. These methods will be respectively illustrated in Prop. 1 and its deductions, and in Prop. 2, of the following sections. PKOP. 1. Given n consecutive equidistant values uof ult ... M,_t of a function ux) to find its approximate general expres- sion. ATIT. 2.] AND MECHANICAL QUADRATURE. 35 By Chap. II. Art. 10, m (m — 1) A» .o u*+m = % + m&ux + ^ 2 AX + &c. Hence, substituting 0 for x, and x for m, we have + &c. But on the assumption that the proposed expression is rational and integral and of the degree n — 1, we have A*^ = 0, and therefore A*\ = 0. Hence , X (x — 1) A2 the expression required. It will be observed that the second member is really a rational and integral function of x of the degree n— 1, while the coefficients are made determinate by the data. In applying this theorem the value of x may be con- ceived to express the distance of the term sought from the first term in the series, the common distance of the terms given being taken as unity. Ex. Given log 3-14 = '4969296, log 3*15 = -4983106, log 316 = '4996871, log 317 = '5010593; required an approxi- mate value of log 314159. Here, omitting the decimal point, we have the following table of numbers and differences : 4969296 4983106 4996871 5010593 A 13810 13765 13722 A3 -45 -43 A3 2 The first column gives the values of UQ and its differences up to A8w0. Now the common difference of 314, 315, &c. 3-2 36 ON INTERPOLATION, [CH. III. being taken as unity, the value of x which corresponds to 3-14159 will be 159. Hence we have u, = 4969296 + 159 x 13810 + O159) C1^ " *) x( - 45) (169) (-159-1) (-159 -2) 1.2.3 Effecting the calculations we find ux = '4971495, which is true to the last place of decimals. Had the first difference only been employed, which is equivalent to the ordinary rule of proportional parts, there would have been an error of 3 in the last decimal. 3. When the values given and that sought constitute a series of equidistant terms, whatever may be the position of the value sought in that series, it is better to proceed as follows. Let UQ, ul} u2, ...un be the series. Then since, according to the principle of the method, Anu0 = 0, we have by Chap. 11. Art. 10, ^-^w-1 + )^_2-...-f(-l)X=0 ...... (3), an equation from which any one of the quantities may be found in terms of the others. Thus, to interpolate a term midway between two others we have WI = -* ............ (4). Here the middle term is only the arithmetical mean. To supply the middle term in a series of five, we have u0 — 4wx + 6u2 — 4u3 + u^ = 0; -K + tQ^ ^ ART. 3.] AND MECHANICAL QUADRATURE. 37 ,00 Ex. Representing as is usual I e~e 6n~l d6 by F (?i), it is ° required to complete the following table by finding approxi- mately log T ( n log F (n), ft log F (ft) 9 7 4i 12 •74556, • 12 •18432, 3 8 Cr 12 •55938, 12 •13165, 12 •42796, 9 12 •08828, 5 12 •32788, 10 12 •05261. Let the series of values of log F (ft) be represented by ttt, w2, ... u9) the value sought being that of u5. Then pro- ceeding as before, we find 8.7 8.7.6 or, Wj + w9 - 8 (ut + wa) + 28 (MS + 1*7) - 56 (w4 whence 56 + t - 28 ^5 = 0 ; ,,, Substituting for Mlf ?/2, &c., their values from the table, we find = -24853, the true value being '24858. To shew the gradual closing of the approximation- as the number of the values given is increased, the following results are added : 38 ON INTERPOLATION, [CH. III. Data. Calculated value of uy u4 u6 ............... -25610, wa, wa, w4 t*fl, M7, u8, ........... „.. -24865, Mlf «*„ MB, w4 we, w7, «8, w9 ............... -24853. 4. By an extension of the same method, we may treat any case in which the terms given and sought are terms, but not consecutive terms, of a series. Thus, if ult w4, u5 were given and us sought, the equations A3^ = 0, A8^2 = 0 would give from which, eliminating w2, we have 3^-8^ + 6^-^ = 0 ..................... (7), and hence u3 can "rf found. But it is better to apply at once the general methdfi of the following Proposition. PROP. 2. Giyen^Ti values of a function which are not consecutive and equidistant, to find any other value whose place is given. Let ua) ub, ue, ...uk be the given values, corresponding to a, 6, c ... k respectively as values of x, and let it be required to determine an approximate general expression for ux. We shall assume this expression rational and integral, Art. 1. Now there being n conditions to be satisfied, viz. that for x = a, x = b . . . x — Jc, it shall assume the respective values ua, ub, ... uk, the expression must contain n constants, whose values those conditions determine. We might therefore assume u. = A + Bx+Ctf ... + EaT* .................. (8), and determine A, B, C by the linear system of equations formed by making x — a, 6 . . . k, in succession. The substitution of another but equivalent form for (8) enables us to dispense with the solution of the linear system, ART. 4.] AND MECHANICAL QUADRATURE. 39 Let ux — A (x — b) (x — c) ... (x — k) + B (x — a) (x — c) ... (x — ti) + C (x-a) (x-b) ... (x-k) + &c ........................................... (9) to n terms, each of the n terms in the right-hand member wanting one of the factors x — a, x — b, ... x — k, and each being affected with an arbitrary constant. The assumption is legitimate, for the expression thus formed is, like that in (8), rational and integral, and it contains n undetermined coefficients. Making x — a, we have ua = A (a — b) (a — c) ... (a — k); therefore A = In like manner making x = b, we have 7?== u> (b-a) (b-c) ... (b-k)' and so on. Hence, finally, (x — b) (x — c) . . . (x — k) t (x — a) (x — c) ... (x— k) ux = uc (a - 6) (a - cj .*. (a - * b (b^rijlb - c) . . . (b - A-, ' ' „ (x-a)(x-b)(x-c) ... + &C-~+U*(k-a)(k-b)(k-c)... the expression required. This is Lagrange's* theorem for interpolation. If we assume that the values are consecutive and equi- distant, i.e. that UQ) u^ ... un_^ are given, the formula be- comes x(x-l) ... Q-71+2) a?(a?-l) ... (as-n+l) U* ~ U-> 1.2. 3. .>-!) ' W-2 "T7l~ 2~(*r^2)~~ + &c. * Journal de VEcole Poly technique, n. 277. The real credit of the discovery must, however, be assigned to Euler ; who, in a tract entitled De eximio v*u iiH'thotli interpolationum in xcricnim doctrina, had, long before this, obtained a closely analogous expression. 40 ON INTERPOLATION, [CH. III. where This formula may be considered as conjugate to (2), and possesses the advantage of being at once written down from the observed values of ux without our having to compute the successive differences. But this is more than compensated for in practice, especially when the number of available obser- vations is large, by the fact that in forming the coefficients in (2) we are constantly made aware of the degree of closeness of the approximation by the smallness of the value of AKw0, and can thus judge when we may with safety stop. As the problem of interpolation, under the assumption that the function to be determined is rational and integral and of a degree not higher than the (n — l)th, is a determinate one, the different methods of solution above exemplified lead to consistent results. All these methods are implicitly contained iu that of Lagrange. The following are particular applications of Lagrange's theorem. 5. Given any number of values of a magnitude as ob- served at given times ; to determine approximately the values of the successive differential coefficients of that magnitude at another given time. Let a, 6, ... k be the times of observation, ua,ub,... u^ the observed values, x the time for which the value is required, and ux that value. Then the vaMe of ux is given by (10), and the differential coefficients cai thence be deduced in the usual way. But it is most convenient to assume the time represented above by x as the epoch, and to regard a, b, ... k as measured from that epoch, being negative if measured backwards. The values of -~ ^ ~T~£> ^a w^ ^en ^e ^e coefficients of x, x*, &c. in the development of the second member of (10) multiplied by 1, 1 . 2, 1 . 2 . 3, &c. successively. Their general expressions may thus at once be found. Thus ART. 6.] AND MECHANICAL QUADRATURE. 41 in particular we shall have fc ... du .. . A ,.. a^' (a -b) (a- c) ... (a — &) Laplace's computation of the orbit of a comet is founded upon this proposition (Mecanique Celeste). 6. The values of a quantity, e. g. the altitude of a star at given times, are found by observation. Required at what intermediate time the quantity had another given value. Though it is usual to consider the time as the independent variable, in the above problem it is most convenient to con- sider the observed magnitude as such, and the time as a function of that magnitude. Let then a, 6, c, . . . be the values given by observation, ua) ubt uc, ... the corresponding times, x the value for which the time is sought, and ux that time. Then the value of ux is given at once by Lagrange's theorem (10). The problem may however be solved by regarding the time as the independent variable. Representing then, as in the last example, the given times by a, b, ... k, the time sought by x, and the corresponding values of the observed magnitude by ua, ub, ... ut, and uxt we must by the solution of the same equation (10) determine x. The above forms of solution being derived from different hypotheses, will of course differ. We say derived from dif- ferent hypotheses, because whichsoever element is regarded as dependent is treated not simply as a function, but as a rational and integral function of the other element ; and thus the choice affects the nature of the connexion. Except for the avoidance of difficulties of solution, the hypothesis which assumes the time as the independent variable is to be pre- ferred. 42 ON INTERPOLATION, [CH. III. Ex. Three observations of a quantity near its time of maximum or minimum being taken, to find its time of maxi- mum or minimum. Let a, b) c, represent the times of observation, and ux the magnitude of the quantity at any time x. Then ua, ub and uc are given, and, by Lagrange's formula, (x — b) (x — c) (x — c)(x — a) (x — a) (x — b) ni - ni 1 _ ' ^ _ • I ni N _ ' \ _ '_ I nj \ _ / \ a (a-b) (a-cy b(b - C) (6 -a)* c (c - a) (c - b) ' and this function of x is to be a maximum or minimum. Hence equating to 0 its differential coefficient with respect to x, we find = This formula enables us to approximate to the meridian altitude of the sun or of a star when a true meridian observa- tion cannot be taken *. 7. As was stated in Art. 4, Lagrange's formula is usually the most convenient for calculating an approximate value of ux from given observed values of the same when these are not equidistant. But in cases where we have reason to believe that the function is periodic, we may with advantage substitute for it some expression, involving the right number of undetermined coefficients, in which x appears only in the arguments of periodic terms. Thus, if we have 2?i + 1 obser- vations, we may assume ux = A0 + A^ cos x + A9 cos 2x + . . . + An cos nx -f 5j sin x + B^ sin 2% + . . . + Bn sin nx. . . (15), and determine the coefficients by solving the resulting linear equations. Gauss •(• has proved that the formula sin ^ (x — b) sin -= (x — c) . . . sin -=(x — k) sin (a — b) sin x (a — c) ... sin -(a — k) * A special investigation of this problem will be found in Grunert, xxv. 237. i Werke, Vol. ni. p. 231. ART. 8.] AND MECHANICAL QUADRATURE. 43 is equivalent to (15), ua, ub, ...ut being assumed to be the 2ra 4- 1 given values of ux. It is evident that we obtain ux = ua when for x we substitute a in it, and also that when expanded it will only contain sines and cosines of integral multiples of x not greater than nx, and as the coefficients of (15) are fully determinable from the data, it follows that the two expressions are identically equal. 8. Cauchy* has shewn that if ra + n values of a function are known, we may find a fraction whose numerator is of the 7ith, and denominator of the (ra — I)01 degree, which will have the same m + n values for the same values of the variable. He gives the general formula for the above frac- tion, which is somewhat complicated, though obviously satis- fying the conditions. We subjoin it for the case when ra = 2, n — 1, _ MfcM.(6-c)(a?--g) + &c' /-, 7x ua(b-c)(x-a) + &c. ' When ra = 1 it reduces of course to Lagrange's formula. Application to Statistics. 9. When the results of statistical observations are pre- sented in a tabular form it is sometimes required to narrow the intervals to which they correspond, or to fill up some particular hiatus by the interpolation of intermediate values. In applying to this purpose the methods of the foregoing sections, it is not to be forgotten that the assumptions which they involve render our conclusions the less trustworthy in proportion as the matter of inquiry is less under the dominion of any known laws, and that this is still more the case in proportion as the field of observation is too narrow to exhibit fairly the operation of the unknown laws which do exist. The anomalies, for instance, which we meet with in the at- tempt to estimate the law of human mortality seem rather to * Analyse A Igebralque, p. 528, but it is better to read a paper by Brassine (Liouville, xi. 177), in which it is considered more fully and as a case of a more general theorem. This must not be confounded with Cauchy's Method of Interpolation, which is of a wholly different character and does not need notice here. He gives it in Liouville, n. 193, and a consideration of the advantages it possesses will be found in a paper by Bienayme, Comptes Rendus, xxxvu. or Liouville, xvin. 299. 44 ON INTERPOLATION, [CH. III. be due to the imperfection of our data than to want of conti- nuity in the law itself. The following is an example of the anomalies in question. Ex. The expectation of life at a particular age being defined as the average duration of life after that age, it is required from the following data, derived from the Carlisle tables of mortality, to estimate the probable expectation of life at 50 years, and in particular to shew how that estimate is affected by the number of the data taken into account. Age. Expectation. Age. Expectation. j 10 48-82 = w, 60 14-34 = M6 20 41-46 = ^2 70 918 = WT 30 34-34 = u3 80 5'51=^8 40 27-61 = u4 90 3-28 = u9 The expectation of life at 50 would, according to the above scheme, be represented by ua. Now if we take as our ooly data the expectation of life at 40 and 60, we find by the method of Art. 3, If we add to our data the expectation at 30 and 70, we find 2071 ......... (&). If we add the further data for 20 and 80, we find ^8) = 2075.. .(c). And if we add in the extreme data for the ages of 10 and 90, we have 8 4 W5= JQ K + *0 ~ 10 We notice that the second of the above results is consider- ably lower than the first, but that the second, third, and fourth exhibit a gradual approximation toward some value not very remote from 20'8. ART. 9.] AND MECHANICAL QUADRATURE. 45 Nevertheless the actual expectation at 50 as given in the Carlisle tables is 21'H, which is greater than even the first result or the average between the expectations at 40 and 60. We may almost certainly conclude from this that the Carlisle table errs in excess for the age of 50. And a comparison with some recent tables shews that this is so. From the tables of the Registrar-General, Mr Neison* deduced the following results. Age. Expectation. Age. Expectation. 10 477564 60 14-5854 20 40-6910 70 9'2176 30 34-0990 80 5-2160 40 27-4760 90 2-8930 50 20-8463 Here the calculated values of the expectation at 50, corre- sponding to those given in (a), (6), (c), (d), will be found to be 21-0307, 20-8215, 20-8464, 20'8454. We see here that the actual expectation at 50 is less than the mean between those at 40 and 60. We see also that the second result gives a close, and the third a very close, approxi- mation to its value. The deviation in the fourth result, which takes account of the extreme ages of 10 and 90, seems due to the attempt to comprehend under the same law the mortality of childhood and of extreme old age. When in an extended table of numerical results the differ- ences tend first to diminish and afterwards to increase, and some such disposition has been observed in tables of mor- tality, it may be concluded that the extreme portions of the tables are subject to different laws. And even should those laws admit, as perhaps they always do, of comprehension under some law higher and more general, it may be inferred that that law is incapable of approximate expression in the particular form (Art. 2) which our methods of interpolation presuppose. * Contributions to Vital Statistics, p. 8. 46 ON INTERPOLATION, [CH. III. Areas of Curves. 10. Formulae of interpolation may be applied to the ap- proximate evaluation of integrals between given limits, and therefore to the determination of the areas of curves, the con- tents of solids, &c. The application is convenient, as it does not require the form of the function under the sign of in- tegration to be known. The process is usually known by the name of Mechanical Quadrature. PROP. The area of a curve being divided into n portions bounded by n + 1 equidistant ordinates UQ} ut,...un, whose values, together with their common distance, are given, an approximate expression for the area is required. The general expression for an ordinate being ux) we have, if the common distance of the ordinates be assumed as the fn unit of measure, to seek an approximate value of I uxdx. J o Now, by (2), x (x — 1) 2 x (x — 1 ) (x — 2) A 3 ux = UQ + x&u0 + ^ 2 ; AX + - -y^y- - A'ti0 + &c. Hence cn r* rtt AV rn M«C&c~«J efo + AwJ ffdoj + r— °l #(#- J0 J o Jo x • ^Jo + 0^3 [ x (x ~ 1} ( AX = u2 — 2wx + w0; whence, substi- tuting and reducing, If the common distance of the ordinates be represented by h, the theorem obviously becomes , Qh ^, uxdx=— - ^ - -h ............. (19), 6 and is the foundation of a well-known rule in treatises on Mensuration. Sndly. If there are four ordinates whose common distance is unity, we find in like manner Srdly. If five equidistant ordinates are given, we have in like manner •ithly. The supposition that the area is divided into six portions bounded by 7 equidistant ordinates leads to a re- markable result, first given by the late Mr Weddle (Math. Journal, Vol. IX. p. 79), and deserves to be considered in detail. Supposing the common distance of the ordinates to be unity, we find, on making n = 6 in (18) and calculating the 48 ON INTERPOLATION, [CH. III. coefficients, 123 . uxdx = 6uQ -f- 18Awrt + 27AV0 4- 24A3w0 H — — JOAw°+i40A6^ (2 ,. differs from - .. ^ _ 140 140 10 41 42 3 Now the last coefficient — - differs from - ^ or ~ by the I Ail I 1 .1.1 I 111 " small fraction •— r , and as from the nature of the approxima- tion we must suppose sixth differences small, since all suc- ceeding differences are to be neglected, we shall commit but o a slight error if we change the last term into ^. A6^0- Doing this, and then replacing A^0 by wa — u0 and so on, we find, on reduction, [6 3 I™* a ~ 10^0 + W2 + W4 + ^6+ K + ^5)- "Wl which, supposing the common distance of the ordinates to be h, gives 3A, ^ao; = YA luo + WSA + UM + W6A + ° (w& + W5A/ + ot^j . . . (23), 10 the formula required. It is remarkable that, were the series in the second member of (22) continued, the coefficient of A7w0 would be found to vanish. Thus while the above formula gives the exact area when fifth differences are constant, it errs in excess by only — AV when seventh differences are constant. 140 The practical rule hence derived, and which ought to find a place in elementary treatises on mensuration, is the fol- lowing: The proposed area being divided into six portions by seven equidistant ordinates, add into one sum the even ordinates 5 times the odd ordinates and the middle ordinate, and mul- ART. 10.] AND MECHANICAL QUADRATURE. 49 g tiply the result by ^ of the common distance of the ordi- nates. Ex. 1. The two radii which form a diameter of a circle are bisected, and perpendicular ordinates are raised at the points of bisection. Required the area of that portion of the circle which is included between the two ordinates, the diameter, and the curve, the radius being, supposed equal to unity. The values of the seven equidistant ordinates are V8 V8 V35 x/35- V8 V3 2 ' 3 ' 6 ' ' 6 ' 3 ' 2 ' and the common distance of the ordinates is ~. The area 6 hence computed to five places of decimals is '95661, which, on comparison with the known value -~ + ™- , will be found to be correct to the last figure. The rule for equidistant ordinates commonly employed would give '95658. In all these applications it is desirable to avoid extreme differences among the ordinates. Applied to the quadrant of a circle Mr Weddle's rule, though much more accurate than the ordinary one, leads to a result which is correct only to two places of decimals. Should the function to be integrated become infinite at or within the limits, an appropriate transformation will be needed. IT log sin 6d0. „ The function log sin 6 becomes infinite at the lower limit. We have, on integrating by parts, flog sin 0dO = 0 log sin 0 - j 6 cot 0d0, B. F. D. 4 50 ON INTERPOLATION, [CH. III. hence, the integrated term vanishing at both limits, 7 { J o log sin 6de-=-e cot Ode. The values of the function 6 cot 0 being now calculated for the successive values 6 = 0, 0 = ^r, 0 = — > ...... ^ = ~<> 1 -j .1 iw iu the theorem being applied, we find IT -(20 cot 0 0 The true value of the definite integral is known to be |log , or - 1-08882. 11. Lagrange's formula enables us to avoid the interme- diate employment of differences, and to calculate directly the coefficient of um in the general expression for I uxdx. If we represent the equidistant ordinates, 2n -f 1 in number, by MO, Mt . . . w2n, and change the origin of the integrations by assuming x~n = y, we find ultimately 2n i, where generally A ,= /; ... (n-r) A similar formula may be established when the number of equidistant ordinates is even. 12. The above method of finding an approximate value for the area of a curve between given limits is due to Newton and Cotes. It consists in expressing this area in terms of observed values of equidistant ordinates in the form Area = J0w0 + ^4lul + &c., ART. 12.] AND MECHANICAL QUADRATURE. 51 where A0) Al &c. are coefficients depending solely on the number of ordinates observed, and thus calculable beforehand and the same for all forms of ux. It is however by no means necessary that the ordinates should be equidistant; Lagrange's formula enables us to express the area in terms of any n ordinates, and gives \ujiLx = Aju,u + AM + &C (25), Jq where Now it is evident that the closeness of the approximation depends, first, on the number of ordinates observed, and secondly, on the nature of the function ux. If, for instance, ux be a rational integral function of oc of a degree not higher than the (n — l)th, the function is fully determined wheii n ordinates are given, whether these be equidistant or not, and the above formula gives the area exactly. If this be not the case, it is evident that different sets of observed ordinates will give different values for the area, the difference between such values measuring the degree of the approximation. Some of these will be nearer to the actual value than others, but it would seem probable that a know- ledge of the form of ux would be required to enable us to choose the best system. But Gauss* has demonstrated that we can, without any such knowledge, render our approxi- mation accurate when ux is of a degree not higher than the (2?i — l)th if we choose rightly the position of the n observed ordinates. This amounts to doubling the degree of the approximation, so that we can find accurately the area of the curve y = ux between the ordinates to x = p, x = q, by observing n properly chosen ordinates, although ux be of the (2ft— l)th degree. The following proof of this most remarkable proposition is substantially the same as that given by Jacobi (Crelle. Vol. I. 301). * Werke, Vol. in. p. 203. 4—2 52 ON INTERPOLATION, [CH. III. ,'p Let I uxdx be the integral whose value is required, where •* q ux is a rational and integral function of the (2n — l)th degree. Let uay ub...\)e the n observed ordinates, andy(£c) the ex- pression which they give for ux by substitution in Lagrange's formula. Let A(x-a) (x-l) ...... = M, where A is a constant. r I J Since ux —f (x) vanishes when x = a, b, ... it must be equal to MN where N is rational, integral, and of the (n — l)th degree, and the error in the approximation is p MNdx, which we shall now shew can be made to vanish q by properly choosing M, i.e. by properly choosing the ordi- nates measured. Now dx - [ = &c. &c. - - denoting by MK the result of integrating M K times, and by N^ the result of differentiating .ZV" K times ; and remembering that N(n~l] is a constant. Taking the above integrals between the given limits, we see that the problem reduces to making Mr vanish at each limit for all values of r from r = 1 to r = n. This is at once accomplished by taking for it is thus a rational and integral function of x of the nth degree, such that all its first n integrals can be taken ART. 13.] AND MECHANICAL QUADRATURE. 53 to vanish at the given limits. That this is the case is seen at once when we consider that the parts independent of the arbitrary constants will contain some power of (x—p) (& — <£) as a factor, and will thus vanish at both limits. The coefficients Aa, A...m \ x dx will of course be functions of p and q of the form given in (26). In order to save the trouble of calculating them for all values of the limits, it is usual to transform the integral, previously to applying the above theorem, so as to make the limits 1 and — 1. We then have d"(x*-l?= gg _ ~~ af 1 1.2.2»(2»-l)(2»^2)(2n-3) and a, 6, c ... are the roots of M= 0, which are known to be real, since those of (a? — l)n = 0 are all real. 13. We shall now proceed to demonstrate a most im- portant formula for the mechanical quadrature of curves. It was first given by Laplace*, and will be seen to be closely allied to (18). Since d & d L A 1 1 ) =^t1+2-i2A+2iA &c-r"f; _ii Integrate between limits 1 and 0, remembering that [ * Mtcanique Ctleste, iv. 207. t The coefficients of the powers of t in j — -^ — -. may be calculated either directly, or by the method in Ex. 18 at the end of this Chapter. 54 ON INTERPOLATION, [CH. III. and we easily get, writing ux for Writing down similar expressions for I uxdx, &c., and J i adding, we obtain -&c ....(27), since AX + AX + &c. = A^1 (A^0 + A^ + &c.) = Ar-a (un - u0). This formula has the disadvantage of containing the dif- ferences of un, which cannot be calculated from the values UQ, u^ ... un. We may remedy this in the following way: log (1 + A) / _ _A_\ log (1 - 10 - 1 A£" - 1 A^- - 1 As^-' - &cj «-.. ART. 14.] AND MECHANICAL QUADRATURE. 55 Removing the first two terms from each side since they are obviously equal, and writing un for Awn, we get ~ 12 Aw" 4 24 AV» " &C* = ~ T2 Aw*-1 ~~ 24 A2"«-* ~ &C'' and the formula becomes — -&C, ................................. (28). In the above investigation we have in reality twice per- formed the operation— on botk sides of an equation. We shall see that Awa. = Ava. only enables us to say ux = vx + C and not ux — vx\ hence we should have added an arbitrary constant. But the slightest consideration is sufficient to shew that this constant will in each case be zero. 14. The problems of Interpolation and Mechanical Quadrature are of the greatest practical importance, the formulas deduced therefrom being used in all extended calculations in order to shorten the labour without affecting greatly the accuracy of the result. This they are well capable of doing; indeed Olivier maintains (Crelle, n. 252) that calculations proceeding by Differences will probably give a closer approximation to the exact result tkan corresponding ones that proceed by Differential Coefficients. In con- sequence of this practical value many Interpolation-formulae have been arrived at by mathematicians who have had to do with actual calculations, each being particularly suited to some particular calculation. All the most celebrated of these formulae will be found in the accompanying examples. Examples of calculations based upon them can usually be found through the references; the papers by Gruuert (Archiv, xiv. 225 and xx. 361), which contain a full inquiry into the subject, may also be consulted for this pur- pose. Numerical examples of the application of several Interpolation-for- mulae may also be found in a paper by Hansen (Relationenzwischen Summcnund Dijferenzen, Abhandlungen der Kon. Sachs. GeseUschaft, 1865), in which also he gives a very detailed inquiry into the various methods in use, with numerical calculation of coefficients, &c. We must warn the reader against the notation, which is unscientific and wholly in defiance of convention, e.g. Ay,H anl 56 ON INTERPOLATION, [CH. III. A"t/z are used to represent the Ayx and A?yx-i of the ordinary notation. A good paper on the subject by Encke (Berlin. Astron. Jahrbuch, 1830), from which Ex. 7 is taken, labours under the same disadvantage ; and Stirling's formula (Ex. 9) is seldom found stated in the c.orrect notation. In speaking of the developments which -the theory has received we must mention an important Memoire by Jacobi (Crelle, xxx. 127) on the Cauchy Interpolation-formula of Art. 8. In it the author points out the advantages that it possesses over others, and subjects it to a very full investigation, representing the numerator and denominator in various forms as determi- nants, and considering especially the case when two or more -of fhe values of the independent variable approach equality. A paper by Eosenhain which follows immediately after it treats also of the above formula in repre- senting the condition that two equations (x) = 0 and / (x) = 0 should have a common root, in terms of the values of the expression j~ for different values of x. But the most important researches in the theory of Interpolation have had reference to the Gauss-formula of Art. 12. Minding (Crelle, vi. 91) extends it to the approximate evaluation of double integrals between constant limits. Christoffel (Crelle, LV. 61) investigates the more general problem of deter- mining the ordinates we should choose for observation when certain ordinates are already given, so that the approximation may be as close as possible. Mehler (Crelle, LXIII. 152) shews that a closely analogous method enables us to calculate integrals of the form J-l with great accuracy, the position of the ordinates chosen being in -this case determined by the roots of the equation of the wth degree 0, dxn X and /j. being each > - 1. Jacobi had previously examined the case in which A=/i= -^-j in other words, he had shewn that in f -$=dx ar/' /Wl-a2 jfo the positions of the co-ordinates to be chosen after the analogy -of the Gauss- formula are given by the roots of which is equivalent to cos (n cos*1 x) = 0. Hence a;=cos ~ — ir. In this case the coefficients Aa, Ab, ... (see (26), page 51) are all equal each being — , and the formula becomes ART. 14.] AND MECHANICAL QUADRATURE. 57 In most of the above papers the magnitude of the error caused by using the approximate formula instead of the exact value of the function is investigated. The special importance of the method becomes evident when we con- sider the close relation between it and the celebrated Laplace's functions. This is seen by comparing the expression for the nth Laplace's coefficient of one variable, _1 1 dxn with the value of M in Art. 12; and the similarity of the corresponding expressions for two variables is equally great. In fact the Gauss-method may be represented as follows : — Let ux be a rational integral function of the (2n - l)th degree, and Yn be the nth Laplace's coefficient. Divide ux by Fn, and let N be the quotient and f(x) the remainder which is of the (n - l)th degree. Thus ux=f(x) + Yn . N. Integrate between the limits 1 and - 1. and since N is of a lower degree FI r than Yn, I YnNdx = 0, and we are left with I f(x)dx which is accurately found by the Lagrange-formula from the n observed values of ux. In consequence of this close connexion the method is of great import- ance in the investigation of Laplace's Functions and of the kindred subject of Hypergeometrical Series. Heine's Handbuch der Kugelfunctionen will supply the reader with materials for discovering the exact relation in which they stand to one another, or he may compare a paper by Bauer on Laplace's functions (Crelle, ivi. 101) with that by Christoffel given above. For in- stances of numerical calculation he may consult Bertrand (Int. Cal. 339), where, however, the limits 1 and 0 are taken. EXERCISES. t- 1. Required, an approximate value of log 212 from the following data: log 210 = 2-3222193, log 213 = 2<3283796, log 211 = 2-3242825, log 214 = 2-3304138. 2. Find a rational and integral function of x of as low a degree as possible that shall assume the values 3, 12, 15, and - 21, when x is equal to 3, 2, 1, and - 1 respectively. v 3. Express v2 and v3 approximately, in terms of VQ, vlt v4, and v^, both by Lagrange's formula and the method of (7), Art. 4. 58 EXERCISES. [CH. III. 4. The logarithms in Tables of n decimal places differ from the true values by ± ^.-T^at most. Hence shew that ths errors of logarithms of n places obtained from the Tables by interpolating to first and second differences cannot exceed 1 19 ± ^ + e an(l ± TTT« x ZT + e respectively, e and e being the errors due exclusively to interpolation. (Smith's Prize) 5. The values of a function of the time are alt a2, aa, #4, at epochs separated by the common interval h\ the first dif- ferences are dlt d\, d"l} the second differences are c£2, cT2, and the third difference d3. Hence obtain the following formulae of interpolation to third differences : t being reckoned in the first case from the epoch of «2, and in the second from that of as. 6. If P, Q, E, S, ... be the values of X, an unknown function of a?, corresponding to x—p^ q, r, s, ..., shew that (under the same hypothesis as in the case of Lagrange's formula), X= P + (x -#) {p, q} + (x -_p) (x - q) [p, q, r] + &c._ where generally 7. Shew that, in the notation of the last question, if —p = r — q = s — r = &c. = 1, A3P EX. 8.] EXERCISES. 59 and apply the theorem to demonstrate that (2) 8. Shew that the function a — 6 (a — &) (a — c) becomes unity when t = a, and zero when £ = &, c, ..., and deduce Ex. 6 therefrom. 9. Demonstrate Stirling's Interpolation-formula (Smith's Prwre, 1860.) 10. Deduce Newton's formula for Interpolation from Lagrange's when the values are equidistant. 11. If /A radii vectores (/t being an odd integer) be drawn from the pole dividing the four right angles into equal parts, shew that an approximate value of a radius vector (ue) which makes an angle 6 with the initial line is where a, 6, ... are the angles that the fj, radii vectores make with the initial line. CO EXERCISES. [CH. III. 12. Assuming the formula for resolving into Partial Fractions, deduce Lagrange's Interpolation- formula. 13. If (£(#) = 0 be a rational algebraical equation in x of any order, and zv z2...zk be taken to represent <£ (1), (2), ... (f) (k), find under what conditions '=* r *,*,...*, 2 —7— — r - j— — ; • may be taken as an approximate root of the equation. 14. Demonstrate Simpson's rule for finding an ap- proximate value for the area of a curve, when an odd number of equidistant ordinates are known, viz.: To four times the sum of the even ordinates add twice the sum of the odd ones ; subtract the sum of the extreme ordinates and multiply the result by one-third the common distance. 15*/\ f jw that Simpson's rule is tantamount to consider- ing theSrtrrve between two consecutive odd ordinates as pa- rabolic. Also, if we assume that the curve between each ordinate is parabolic, and that it also passes through the extremity of the next ordinate (the axes of the parabolae being in all cases parallel to the axis of y), the area will be given by Area = h gy - 15 (y0 + yn) - 4 (yx + yn_,) + 16f. Given ux and ux+l, and their even distances, shew that * On the comparative merits of these and similar methods see Dupain (Nouvelles Annales, xvn. 288). t The notation in this formula (due to Gauss) is that referred to on the top of page 56. EX. 17.] EXERCISES. Cl 17. Shew that x (x + 2r — 1) A In what cases would the above formulas be especially useful ? 18. Shew that the coefficient of Arwn in (27) is equal to \^ndx> and hence shew the exact relationship in which (27) and (18) stand to each other. 19*. If from the values uai ub... of a function corre- sponding to values a, b, c ... of the variable, we obtain an Interpolation-formula, ux = ua + B(x-a) + C(x-a)(x-b) + D(x-a)(x-b)(x-c) shew that B = u (a, b, . . .). Deduce (2), page 35, from the above formula. * Newton's Principia, Lemma v. Lib. in. This is the first attempt at finding a general Interpolation-formula, and gives a complete solution of the problem. The result is of course identically that obtained by Lagrange's formula, though in a very different form. 62 ) CHAPTER IV. FINITE INTEGRATION, AND THE SUMMATION OF SERIES. 1. THE term integration is here used to denote the process by which, from a given proposed function of #, we determine some other function of which the given function expresses the: difference. Thus to integrate ux is to find a function vx such that The operation of integration is therefore by definition the inverse of the operation denoted by the symbol A. As such, it may with perfect propriety be denoted by the inverse form A"1. It is usual however to employ for this purpose a distinct symbol, 2, the origin of which, as well as of the term inte- gration by which its office is denoted, it will be proper to explain. One of the most important applications of the Calculus of Finite Differences is to the finite summation of series. Now let uw u^ Up &c. represent successive terms of a series whose general term is ux> and let ...+ ux_l ..................... (1). Then, a being constant so that ua remains the initial term, we have vx+1 = ua+ ua+1 + ... +ux^ + ux .................. (2). Hence, subtracting (1) from (2), It appears from the last equation that A"1 applied to ux expresses the sum of that portion of a series whose general term is uz, which begins with a fixed term ua and ends with M^J. On this account A"1 has been usually replaced by the ART. 1.] FINITE INTEGRATION, &C. 63 symbol 2, considered as indicating a summation or integra- tion. At the same time the properties of the symbol 2, and the mode of performing the operation which it denotes, or, to speak with greater strictness, of answering that question of which it is virtually an expression, are best deduced, and are usually deduced, from its definition as the inverse of the symbol A. Now if we consider 2^ as defined by the equation 2tt, = w_1 + «^ + . ..+«.. ........... (3), it denotes a direct and always possible operation, but if we consider it as defined by the equation 2«. = A-X, ........................ (4), and as having for its object the discovery of some finite ex- pression vx, which satisfies the equation ^vx — ux) it is inter- rogative rather than directive (Diff. Equat. p. 376, 1st ed.), it sets before us an object of enquiry but does not prescribe any mode of arriving at that object ; nor does it give us the assurance that there is but one answer to the question it virtually propounds. A moment's consideration, indeed, will assure us that the number of expressions that can claim to be denoted by A"1^ is infinite, since it includes the quantity whatever value a may be supposed to have, provided only that it is one of the series of integral values which x is sup- posed to take. We cannot therefore consider the definitions of 2,ux contained in (3) and (4) as identical, and shall there- fore proceed to investigate the relation between them and the restrictions as to the use of each. It is obvious that the 2^ of (3) is one of the functions represented by the A~X. in (4), since it satisfies the equation &vx = ux- But this is of no value to us unless we can recog- nize to which of the functions represented by A~X. in (4) it is <• [iial, or obtain an expression for it in terms of any one of them. This last we shall now proceed to do. 64 FINITE INTEGRATION, [CH. IV. Let (f) (x) be a function such that A$ (as) = ux. /.<£(#)-<£ (a) = wa + wa+1 ...... +^_1 = ^in (3). Hence retaining for ^ux the definition of (4) we should write (3) thus : (5). Again suppose %ux to be defined by (3) and be equal to <£ (#), and let the %ux of (4) be given generally by $ (a?) + wx, then M,. = A {<£ (as) + w,} = A<£ (a?) + Aw,. = ux + AwK ; /. A^ = 0, or wx does not change when x is increased by unity ; hence it remains constant while x takes all the series of values which it is permitted to take in any problem in Finite Differences. Since then wx will remain unchanged, so far as we shall have to do with it, we shall denote it by C and regard it as a constant, and examine its true nature later on. (Art. 4, Ch. n.) Hence regarding ^ux as defined by (3) we should write (4) thus : * Were it not that in so fundamental a theorem it is advisable to use only such methods as are beyond all suspicion as to their rigour, we might have arrived more easily at the same result symbolically, thus: = ~_ ua = (E'~a - 1) A~lua = (E*~a - 1) Zwa, from (4) ... (7) , = Z«*-2«a (8), which agrees with (5). But the method in the text is preferable, since the steps in (7) and (8) presuppose a rigorous examination into the nature of the symbols A"1 and 2 before we can state the arithmetical equivalence of the quantities with which we are dealing, i.e. some such investigation as that in the text. ART. 2.] AND THE SUMMATION OF SERIES. 65 We shall not dwell farther on this point, since the differ- ence between the %ux of (3) and that of (4) is precisely analogous to that between the definite integral / $(x)dx, J a .j and the indefinite integral I (x) dx, and the precautions necessary to be taken in using them are identical with those to which we are accustomed in the Integral Calculus. In fact we adopt a notation for definite Finite Integrals stri- kingly similar to that for Definite Integrals in the Infi- nitesimal Calculus, writing the *%ux of (3) in the form Integrdble Forms. 2. As in Integral Calculus, we shall be able to obtain finite expressions for the integrals of but few forms, and must be content to express the integrals of others in the form of infinite series. Of such integrable forms the following are the most important, as being of frequent recurrence and re- ducible under general laws. 1st Form. Factorial expressions of the form in the notation of Ch. n. Art. 2. We have = (m-H)tf(w); ~("'+l) m + 1 or ^(a-l)...(x-m + i),*(*-V-^I™).+ C. ...... (1). Taking this between limits x=n and x = m, (n >m), we get _ n(n — B. F. D. 66 FINITE INTEGRATION, [CH. IV. Or we may retain G and determine it subsequently, thus ^ Pat n = m + 1 and the series on the left-hand side reduces to its first term, and we obtain Thus also if ^=003 + &, we have '~x-m< ri /9\ Ex. 1. Sum the series 3 . 5 . 7 + 5 . 7 . 9 + &c. to n terms. Here a = 2, 6 = 5, m=3, and since we have to find the sum of n terms we must change n into 71 + 1 in the last formula, and we obtain (2n + 7) (2n + 5) (2» + 3) (2n + 1) , 4x2 But n — 1 gives us o K 7 9 x7x5 x 3 ^ „ 105 ~TT~ '~8~; .'. 3 . 5 . 7 + 5 . 7 . 9 + &c. to w. terms _ (2n + 7) (2n + 5) (2n + 3) (2n + 1) 105 8 8 ' 2nd Form. Factorial expressions of the form nr «.(-») ART. 2.] AND THE SUMMATION OF SERIES. 67 We have by Ch. II. Art. 2, So also if «,,.= aa; 4- 6, we liave A «A« • • • «W* VWc ••**•*« = O — or, writing m — 1 for wi, •X- 2 - - - =C-— - — — - ....(5). It will be observed that there must be at least two factors in the denominator of the expression to be integrated. No finite expression exists for 2 — '—j-. Ex. 2. Find the sum of n terms of the series 1.4.7 4.7.10 ' We have here a=3, J = —2, m = 8. .'. Sum of (w. — 1) terms Put w = 2 and we obtain JL -o--i- • c-1 1.4.7" 6.4.7' 24' 68 FINITE INTEGRATION, [CH. IV. Hence (writing n for n — 1 and therefore n -f 1 for ri) Sum of n terms = ^7 — 24 6 (3a + 1) (3n + 4) ' As all that is known of the integration of rational functions is virtually contained in the two primary theorems of (2) and (5), it is desirable to express these in the simplest form*. Supposing then ux = ax + 6, let then whether m be positive or negative. The analogy of this result with the theorem is obvious. We shall now shew how to reduce other forms to one of the preceding. 3rd Form. Rational and integral functions. * As most of the summations of series whose nth term is a rational function of n will have to be effected by these methods, and as such sum- mations are of very frequent occurrence, it is still more important to have a readily applicable rule for effecting them. The following is perhaps the most convenient form for finding the sum of n terms of such series : — " Write down the wth term with its factors in ascending order of mag- ' (take away one factor at the beginning)' factors now remaining, and by the coefficient of x (in each factor), and j add to ) constant." (subtract from) It is scarcely necessary to add that the upper line in the brackets must be taken when the terms are of the form ux ux^... vx-m+i and the lower when of the form . ART. 2.] AND T£E SUMMATION OF SERIES. 69 . By Ch. II. Art, 5 Let <£ (a?) = 2vx and put G for 0 (0), — . and the number of terms will be finite if vx be rational and integral. ^ The series in (§) comes from the equivalence of the opera- tions denoted by the symbols Ex and (1 + A)*. In like manner we may obtain a cognate expression from the equivalence of E~x and (1 + A)~*. This gives us, when we perform them on <£ (x), Putting as before <£(#) = 2V* and C for 0(0), and trans- posing, we get — /„ . -i \ (8)*. In applying the above to the summation of series we may avoid the use of an undetermined constant and render the demonstration more direct by proceeding as follows : Ex-l (9). * That the constants in (7) and (8) are the same appears evident when we consider that (8) may be obtained from (7) by mere algebraical transforma- tion. The series-portions are in fact the results of performing the equivalent direct operations (1± i and L£+£? f> on tv 70 FINITE INTEGRATION, [CH. IV. Here all the operations performed on va are direct, and the result is given in differences of the first term. Ex. 3. To find the sum of x terms of the series 12+ 22+ . . . Applying* (7) we have (since Ava=l, Putting x — 2 we see that C is zero, and adding y? to both sides we obtain 6 Ex. 4. Find the sum of n terms of the series whose 71th term is n3 + 7n. We* shall here apply formula (9). The first terms are 8 22 48 92 ...... „ „ differences „ 14 26 44 ......... „ second „ „ 12 18 ............. „ third „ „ 6 ................ . \ sum of n terms = 8ra + 14 — n(n-l)(n-2) fin(n-l) (n-2) (n-3) 1.2.3 1.2.3.4 4th Form. Any rational fraction of the form * In practice it will be found better to resolve the 71th term into factorials and apply the rule given in the note to page 68. ART. 2.] AND THE SUMMATION OF SERIES. 71 ux being of the form ax + 6, and (x) a rational and integral function of x of a degree lower by at least two unities than the degree of the denominator. Expressing <£ (x) in the form 0 (x) = A + Bux + Cuxux+l + . . . + Euxux+l . . . ux+m_z, A, B ... being constants to be determined by equating coeffi- cients, or by an obvious extension of the theorem of Chap. II. Art. 5, we find ...+E2- and each term can now be integrated by (5). Again, supposing the numerator of a rational fraction to be of a degree less by at least two unities than the denominator, but intermediate factors alone to be wanting in the latter to give to it the factorial character above described, then, these factors being supplied to both numerator and denominator, the fraction may be integrated as in the last case. Ex. 5. Thus ux still representing ax + 6, we should have with the second member of which we must proceed as before. Ex. 6. Find the sum of n terms of the series 2 3 Here the nth term n + 1 n* + 2w + 1 n (n + 2) (n + 3) n (n + 1) (n + 2) (n + 3) 72 FINITE INTEGRATION, [CH. IV. n(n + I)+n + l _ 1 ~ n (n + 1) (n + 2) (n + 3) ~ (n + 2) (n + 3) i «V\~ "* / • t The sum of n terms therefore, by the rule on page 68, 11 1 2(n + 2)(rc m2 + 6(» + l)(* + 2)(n + 3)' 17 and C can easily be shewn to equal ^ . ou We thus can find the sum of n terms of any series whose n* term is <£> (n), provided that (n) enx = tf> (D) enx we may write 0 (a) -i- $ (a + 1) + ... 0 (a + n - 1) = [0 (D) (e0* + & and the series may therefore be summed by the methods of Differential Cal- culus or Differential Equations according as (n) is an integral function of n or not. That the result thus obtained is identical with that in the text follows from the identity demonstrated in (16) page 23, viz. For this gives = S •which agrees with the previous expression. ART. 2.] AND THE SUMMATION OF SERIES. 73 X From (13) page 8, we obtain at once %ax= — — y. For the integration of ax(x) we shall have recourse to sym- bolical methods. a* (ae° - 1)^(«) = a* {a (1 + A) - l}- to which of course an arbitrary constant must be added. It will be found that the direct application of this theoremf is the simplest method of summing such series as have their #* term of the form a*. $(x). * By means of the well-known formula f(D)emx (x) = emxf(D + m) (x). The proof of this formula is given in Boole's Diff. Eq. (First Ed., p. 385), and in many other books. t The demonstration of (10) can be still farther simplified by quoting the theorem, f(E)ax(x)=axf(aE)(x). This may be deduced from the formula above quoted, but is more readily demonstrated independently, since if AnEn be one term of the expansion off(E) in powers of E we have An En ax<}>(x) = Anax^ (x + n) = ax . An an En 0(a:) = a* . An summing all such terms we get f(E)a*(x)=axf(aE)(x), and the demonstration of (10) runs thus, A'1 a*0 (*) = (E - I)-1 ax (x) = ax(aE - I)"1 0 (x) 74 FINITE INTEGRATION, [CH. IV. Ex. 7. Find the sum of the series Sum to n terms The method just given may be generalized to apply to all functions of the form ux.(#) is rational and integral, and ux is a function such that we know the value of A~nux for all integral values of n. In this case we have (comp. Ex. 3, p. 20) 2w,£(aO - (EE' - irX(x}y (x) being a ra- tional fraction the constitution of which would be suggested by that of -»/r(#). Thus also, since A sin""1^ (a?), A tan"1^ (x\ &o.., are of the respective forms sin'1^^), tan~Sj^(#), &c,, ty(x) being an algebraic function when (x) is such, and, in the case of tan~J<£(#), rational if <£(#) be so, it is usually not difficult to conjecture what must be the forms, if finite forms exist, of 2 silftyfc), 2 tan^f (x\ &c., A/T(CC) being still supposed algebraic. The above observations may be generalized. The opera- tion denoted by A does not change or annul the functional 76 FINITE INTEGRATION, [CH. IV. characteristics of the subject to which it is applied. It does not convert transcendental into algebraic functions, or one species of transcendental functions into another. And thus, in the inverse procedure of integration, the limits of conjec- ture are narrowed. In the above respect the operation A is unlike that of differentiation, which involves essentially a procedure to the limit, and in the limit new forms arise. Instances of the above will be given in the Examples at the end of the chapter, but we subjoin the following by way of illustration* Ex. 9. To sum, when possible, the series 273 ^¥ ~^ The wth term, represented by nn> being we have n*x* n*x Now remembering that the summation has reference to n} assume -, nzx* an + b n (TI + 1) (n + 2) ** "n+l Then, taking the difference, we have a?V n ' = xn That these expressions may agree we must have a (x- 1) = 1, (2a + b) (x - 1) = 0, (a + b) x - 2& = 0. Whence we find 1 2 ART. 3.] AND THE SUMMATION OF SERIES. 77 The proposed series is therefore integrable if x = 4?*, and we have '' Substituting, determining the constant, and reducing, there results r.4 2*. 4* nV 4n+1 w-1 2 273 + 3 . 4 '" + (n+ 1) (7i + 2) ~ IT '» + 2 + 3 ' 3. 3 is of course, like A, .Z?, and D, an operation capable of repetition and therefore obeying the index-law ; SVe being defined as 2 (StiJ, Our symbolical methods will render it an easy matter to obtain expressions for Sn (or A~w) analogous to those already obtained for 2, but we shall have to add, as in Integral Calculus, a function of the form (where (70, Gv &c. are arbitrary or undetermined constants) in- stead of the single arbitrary constant which we added in the previous instance. We shall merely give the formula for 2n analogous to (10) and leave the others as an exercise for the ingenuity of the student. It is (12). * The explanation of this peculiarity is very easy : 4 1 (n + l) (n + 2) " ^T2 S+ir* and the summation of the above series would require a finite expression for xn yf+l 2 — if x had not such a value that the term - - which occurs in the n r+2 — (r + l)th term exactly cancelled the term - ~ *^a* occurs in ^Q rth term, i.e. unless x=4. 78 FINITE INTEGRATION, [CH. IV. It will be found that the 1st, 3rd, and 5th forms can have their nth Finite Integrals expressed in finite terms, but that the 2nd and 4th only permit of this if n be not too great. Conditions of extension of direct to inverse forms. Nature of the arbitrary constants. 4. From the symbolical expression of S in the forms (e^-r1), and more generally of 2n in the form (e^-l)"1, flow certain theorems which may be regarded as extensions of some of the results of Chap. 11. To comprehend the true nature of these extensions the peculiar interrogative character <3 of the expression (edx — !)"*»* must be borne in mind. Any legitimate transformation of this expression by the develop- ment of the symbolical factor must be considered, in so far as it consists of direct forms, to be an answer to the question which that expression proposes; in so far as it consists of inverse forms to be a replacing of that question by others. But the answers will not be of necessity sufficiently general, and the substituted questions if answered in a perfectly un- restricted manner may lead to results which are too general In the one case we must introduce arbitrary constants, in the other case we must determine the connecting relations among arbitrary constants ; in both cases falling back upon our prior knowledge of what the character of the true solution must be. Two examples will suffice for illustration. Ex. 1. Let us endeavour to deduce symbolically the ex- pression for 2%B, given in (3), Art. 1. Now 2u = E-l-*ux Now this is only a particular form of %ux corresponding to a = — oo in (3). To deduce the general form we must add an arbitrary constant, and if to that constant we assign the value we obtain the result in question. ART. 4.] AND THE SUMMATION OF SERIES. 79 Ex. 2. Let it be require oceeding according to 2^, We have by (11), page 74, Ex. 2. Let it be required to develope ^uxvx in a series proceeding according to 2^, 22vx, &c. 2V + A2M Z - &c, In applying this theorem, we are not permitted to introduce unconnected arbitrary constants into its successive terms. If we perform on both sides the operation A, we shall find that the equation will be identically satisfied provided A2n% in any term is equal to 2B~VX in the preceding term, and this imposes the condition that the constants in 2*~1wa. be retained without change in %nux. And as, if this be done, the equa- tion will be satisfied, it follows that however many those constants may be, they will effectively be reduced to one. Hence then we may infer that if we express the theorem in the form lujo. = C + u_i 2vx - A^.t 2X + AX_2 2X ..... (1), we shall be permitted to neglect the constants of integration, provided that we always deduce 2ntk by direct integration from the value of 2""1wa in the preceding term. If ux be rational and integral, the series will be finite, and the constant C will be the one which is due to the last inte- gration effected. We have seen that C is a constant as far as A is con- cerned, i.e. that A (7 = 0. It is therefore a periodical con- stant going through all its values during the time that x takes to increase by unity. The necessity of a periodical constant G to complete the value of "S,ux may also be esta- blished, and its analytical expression determined, by trans- forming the problem of summation into that of the solution of a differential equation. Let 2M* = yt then y is solely conditioned by the equation &y=ux) or, putting edx — 1 for A, by the linear differential equation 80 FINITE INTEGRATION, [CH. IV. Now, by the theory of linear differential equations, the complete value of y will be obtained by adding to any par- ticular value vx the complete value of what y would be, were ux equal to 0. Hence ............. (2), C,, (72, &c. being arbitrary constants, and mlt m2, &c. the different roots of the equation em-l = 0. Now all these roots are included in the form i being 0 or a positive integer. When i= 0 we have m— 0, and the corresponding term in (2) reduces to a constant. But when i is a positive integer, we have in the second member of (2) a pair of terms of the form which, on making C+C' = A., (G—C') ^— l = Bt, is re- ducible to At cos 2iV + B. sin 2iir. Hence, giving to i all possible integral values, 2wx = Vx + C + A^ COS 27T# + A 2 COS 4f7TX + A3 COS §TTX + &C. + Sl sin %TTX + Bz sin ^TTX + B3 sin GTTX + &c ....... (3). The portion of the right-hand member of this equation which follows vx is the general analytical expression of a periodical constant as above defined, viz. as ever resuming the same value for values of x, whether integral or fractional, which differ by unity. It must be observed that when we have to do, as indeed usually happens, with only a particular set of values of x progressing by unity, and not with all possible sets, the periodical constant merges into an ordinary, i.e. into an absolute constant. Thus, if x be exclusively integral, (3) becomes c being an absolute constant. ART. 5.] AND THE SUMMATION OF SERIES. 81 It is usual to express periodical constants of equations of differences in the form <£ (cos 2 THE, sin 2-7r#). But this nota- tion is not only inaccurate, but very likely to mislead. It seems better either to employ C, leaving the interpretation to the general knowledge of the student, or to adopt the correct form C + 2< (A. cos 2i7rj5 + Bi sin 2iVa?) ..... . ...... (4). We shall usually do the former. 5. The student will doubtless already have perceived how much the branch of mathematics that forms the subject of our present consideration suffers from its not possessing a clear and independent set of technical terms. It is true that by its borrowing terms from the Infinitesimal Calculus to supply this want, we are continually reminded of the strong analogies that exist between the two, but in scientific language accuracy is of more value than suggestiveness, and the closeness of the affinity of the analogous processes is by no means such that it is profitable to denote them by the same terms. The shortcomings of the nomenclature of the subject will be felt at once if one thinks of the phrases which describe the operations analogous to the three chief operations in the Infinitesimal Calculus, i.e. Differentiation, Integration, and Integration between limits. There is no reason why the present state of confusion should be permanent, so that we shall in future (in the notes at least) denote these by the unambiguous phrases, performing A, taking the Difference-Integral (or performing S), and summing, and shall name the two divisions of the calculus, the Difference- and the Sum-Calculus respectively, and consider them as together forming the Finite Calculus. The preceding chapters have been occupied with the Difference -Calculus exclusively — the present is the first in which we have approached problems analogous to those of the Integral Calculus; for it must be borne in mind that such problems as those on Quadratures are merely instances of use being made of the results of the Difference-Calculus, and have nothing to do with the Sum-Calculus, except perhaps in the case of the formula on page 55. Enough has been said about the analogy of the various parts of our earlier chapters with corresponding portions of the Differential Calculus, and we shall here speak only of the exact nature and relations of the Sum-Calculus. If the ?tth term of a series be known, and its sum be required, it is tanta- mount to seeking the difference-integral, and our power of finding the difference-integral is coextensive with our power of finding the sum of any number of terms. Hence the summation of all series, whose sum to n terms can be obtained, is the work of the Sum-Calculus. It is true that there are many series, that can be summed by an artifice, of which we have taken no notice, but that is not because they do not. belong to our subject, but because they are too isolated to be important. But it must be remembered that the difference-integral is only obtainable when we can find the sum of any number of consecutive terms we may wish. But there are many cases in which we seek the sum of n terms of a series which is such that each term of the series involves n, e.g. we might desire the sum of the series l.n + 2.(n-l) + 3. (n-2) + &c. to n terms. Now in a certain sense this is not a case of summation ; we do not seek the B. F. D. G 82 FINITE INTEGRATION, &C. [CH. IV. sum of any number of terms, but of a particular number of terms depending on the first term of the series itself. And, as might be expected, this opera- tion has not the close connexion that we previously found with that of finding the difference-integral of any term ; for though the knowledge of the latter would enable us to sum the series, yet the knowledge of the sum of the series will not enable us to find the difference-integral of any term. These must be called definite difference-integrals, and hold exactly the same posi- tion that Definite Integrals occupy in the Infinitesimal Calculus. No one would think of excluding from the domain of Integral Calculus the treatment of such functions as the definite integral j^f (a-x)mdx, because the know- •'o ledge of its value does not give us any clue to that of the indefinite integral Jxtfa-x^dx, and is obtained indirectly without its being made to depend on our first arriving at the knowledge of the latter. By similar considerations we shall arrive at a right view of the relation of infinite series to the Sum-Calculus. It is often supposed that it has nothing to do with such series— that the summation of finite series is its business, and that this is wholly distinct from the summation of infinite series. This is. by no means correct. The true statement is that such series are definite difference-integrals, whose upper limit is oo , and so far they as much belong to our subject as / e-x^dx does to the Infinitesimal Calculus. •'o How is it then that the whole subject of series is not referred to this Calculus, but is separated into innumerable portions, and treated of in all imaginable connexions ? It is that in the expression of such series as those we are speaking of, reference being only made to finite quantities, there is nothing to distinguish them from ordinary algebraical expressions, except that the symmetry is so great that only a few terms need be written down. Hence when it is summed by an artifice, and not by direct use of the laws of the Sum-Calculus, there is nothing to distinguish the process from an ordinary algebraical transformation or demonstration of the identity of two different expressions. Now in Definite Integrals that are similarly evaluated by an artifice, there is perhaps just as little claim for the evaluation to be classed as a process belonging to the Infinitesimal Calculus, but the expression of the subject of that process involving the notation and fundamental ideas of the Calculus, it is naturally classed along with processes that really belong to the Calculus. Thus the Infinitesimal Calculus has a wide field to which no recognized branch of the Finite Calculus corresponds, not because it does not exist, but because it is not reserved for treatment here. No doubt this has its disadvantages. Series would be more systematically treated, and the processes of summation more fully generalized, if they were dealt with collec- tively ; yet on the other hand it is a great advantage in the Finite Calculus to have to do only with such processes as really depend on its laws, and not with processes that are really foreign to it, and are only connected therewith by the fact that their subject-matter in these particular instances is expressed in the form of a series, i.e. in the notation of the Calculus. It is not usual to speak of such identities as Definite Difference-Integrals, but a certain class of them are considered in this light in a paper by Libri (Crelle, xn. 240). Before leaving the subject of Definite Difference-Integrals we must men- tion a paper by Leslie Ellis (Liouville, ix. 422), in which he demonstrates a EX. 1.] EXERCISES. 83 theorem analogous to the well-known one on the value of ...)dxdydZ,.., where x+y + z+ ...^1. The method is a very beautiful one, but we must not be supposed to endorse it as rigorous, since one part involves the 00 evaluation of S a^ cos ax. o The fundamental operations of the Finite Calculus are taken as A with its correlative S. In this view of the subject the sign of each term is supposed to be + , not that its algebraical value is supposed to be positive, but that its sign must be accounted for by its form. Thus if we take the series vt0 - u^ + u2 - &c., we must call the general term ( - l)xux. To avoid this com- plication in the treatment of series whose terms are alternately positive and negative, some have wished to have a second Calculus whose fundamental operation is f = 1 + .E, the correlative of which, f"1, would of course denote the operation of summing such a series. A series of papers by Oettinger, the inventor of it, will be found in Crelle, Vols. xi. — xvi. In these he developes the new Calculus in a manner strictly analogous to that in which he subse- quently treats the Difference-Calculus, connects them similarly with the Infinitesimal Calculus, demonstrates analogous formulae, and applies them at first to simple cases and then to more complex ones, especially to those series whose terms are products of the more simple functions and those most suitable to such treatment. The work is unsymbolical, and therefore clumsy and tedious compared with more recent work, and we should not have referred to the papers here (for we consider it highly unadvisable to invent a new Calculus for a comparatively unimportant class of questions that can very easily be dealt with by our present methods) were it not that his results are very copious and detailed. The student who desires practice in the symbolical methods cannot do better than take one of these papers and employ himself in demonstrating by such methods the results there given. Should he desire however a statement of the nature and advantages of this more elaborate treatment of series, he will find it in a review by Oettinger. (Grunert, Archiv. xm. 36.) This is not the only attempt to introduce a new Finite-Calculus. A certain class of series is treated in a paper by Werner (Grunert, Archiv. xxn. 264), by means of a calculus whose fundamental operation, A = JE-vx> is almost the most general form of linear fundamental operation that can be imagined. . EXERCISES. 1. Sum to n terms the following series : 1.3.5.7 + 3.5.7.9 + .. . , 1 .3. 5. 7^3. 5. 7. 9"*" C— 2 84 EXERCISES. [CH. IV. 1.3. 5. 10 + 3. 5. 7. 12 + 5. 7. 9. 14+... 10 12 14 1.3.5 + 3.5.7 5.7.9"1" 1 . 3 . 5 . cos 6 + 3 . 5 . 7 . cos 20 + 5 . 7 . 9 . cos 30 + . . . 1 + 2a cos 6 + 3a2 cos 20 + 4a3 cos 30 + ... 2. The successive orders of figurate numbers are defined by this ; — that the #th term of any order is equal to the sum of the first x terms of the order next preceding, while the terms of the first order are each equal to unity. Shew that the #th term of the nth order is 3. If 2' ua denote the sum of the first • n terms of the n series w0, u^ u^ &c. shew that 1 L, 1 A A» . and apply this to find the sum of the series 1.3.5+5.7.9 + 9.11.13 + &C. 4. Expand 2<£ (us) cos mx in a series of differences of 5. Find in what cases, when ux is one of the five forms given as integrable in the present Chapter, we can find the sum of n terms of the series and construct the suitable formulas in each case. 6. Sum the following series to n terms : JL l , l * /I * • d /I > A /I I* • • • 1 +.. cos 0 . cos 20 ^ cos 20 . cos 30 EX. 7.] EXERCISES. 85 7. Shew that cot'1 (p + qn + r?i2) is integrable in finite terms whenever being any con- stants and ux = ax+b. (Herschel's Examples of Finite Differ ences^ p. 47.) 9. Shew that cos 20 + u cos 40 + &c. = - - « . sin 0 + -^ cos 20 - >8 sin 30 - &c. 8sm30 16sm40 32sm50 10. If &ux = ux]^ — ux and X = — — =- , shew that ux + \&ux + X*AX + &c. + X" AX = a- {(a* - 1) SaX + Xw2az^Att+1wx} Find the sum of w terms of the series whose nth terms are (a+n- l)mxn~l and (a + n - l)(m)aTl. 11. Prove the theorem + n('t1) A2n+2 - &c. 12. If (x) = VQ -f v^ + v2#2 + &c. , shew that &c. = w> 86 EXERCISES. [CH. IV. and if (x) = VQ + v,x + vzx(*] 4- &c., then u$ (as) + #A$ (a?) . Av0 (Guderman, Crelle, vn. 306.) 13. Sum to infinity the series fV_i.T- *4.9»- ^(5~1) rQr * fe - 1) (* - 2) . 1 •;* vj^T)4 • •*(*-i)(«-2) + - 14. If ^> («r) = v0 4- ^a? + V2cc2 + &c., shew that nW^ + &c. where a is an 71th root of unity. (aa)] Au. . « + &c.}, 15. If ln-f 2n + ..<-f mn = #nand m(ra + l)=p, shew that 8n—p*f(p) or (2m 4- 1) pf(p), according as n is odd or even. (Nouvelles Annales, x. 199.) CHAPTER V. THE APPROXIMATE SUMMATION OF SERIES. 1. IT has been seen that the finite summation of series depends upon our ability to express in finite algebraical terms the result of the operation S performed upon the general term of the series. When such finite expression is beyond our powers, theorems of approximation must be employed. And the constitution of the symbol % as expressed by the equation 2 = (e" -!)-'...(!) renders the deduction and the application of such theorems easy. Speaking generally these theorems are dependent upon the development of the symbol S in ascending powers of D. But another method, also of great use, is one in which we expand in terms of the successive differences of some im- portant factor of the general term, i. e. in ascending powers of A, where A is considered as operating on one factor alone of the general term, and is no longer the inverse of the 2 we are trying to perform*. * Let us compare these methods of procedure with those adopted in the Integral Calculus. If /<£ (x) dx cannot be obtained in finite terms it is usual either (1) To expand 0 (x) in a series proceeding by powers of x and to integrate each term separately ; (2) To develope j. By means of this formula we can obtain developed expres- sions for S2, S3, &c. with great readiness in terms of the co- efficieats in the expansion of 2, i.e. in terms of Bernoulli's numbers. Ex. To develope S3 in terms of D. From (17), = £ + 3 Tt + 2) {l - 1 + ^'e + ^ + &c j suppose> D whore ^2r = 0 for all values of r and J.2r+1 = (- l)r B. F. D. 7 08 THE APPROXIMATE SUMMATION OF SERIES. [CH. Y. -J-jl+f Hence 6. PROP. III. To develope 2,nux in a series, proceeding by successive differential coefficients of u^_n . ' >y-l ...... (18). Suppose #" cosecn# = 1 • - <72#2 + (74x4 - &c., then 2X -D- fl + 02 g)2 + 04 (f)4 + &c.l ^_n ....(19)*. V. \^' V*1/ J 2 ,-. . Ijb. must be mentioned that the Summation-formula of Art. 2 (which is due to Maclaurinf) is quite as applicable in the form to the evaluation of integrals by reducing it to a summation, as it is, in its original form, to the summation of series by reducing it to an integration. It is thus a substitute for (27), page 54. * This remarkably symmetrical expression for 2" is due to Spitzer (Grunert, Archiv. xxiv. 97). t Tract on Fluxions, 672. Euler gives it also (Trans. St Petersburg, 1769), and it is often ascribed to him. ART. 7.] THE APPROXIMATE SUMMATION OF SERIES. 99 7. PROP. IV. To expand %ux and *%nux in a series pro- ceeding by successive differences of some factor ofux. It will be seen that the formula of (11) page 74 and Ex. 11 page 85, accomplish this object. We shall only treat here of the very important case when ux = a* (x) and more especially regard the form which the result takes when a = — 1, i.e. when the series is We have in general, = (E- l)~V£( = a'(aE - l)"1^^) (note, page 73) which may be now expanded. If a = — 1, we obtain - f ^ - &c.} ^ («). This enables us to transform many infinite series into others of a more convergent character ; for . ad inf. which is very rapidly convergent if the other is but slowly so. Ex. Transform the series f o ~" 13 + f more convergent form. Here 0(0) = (0 + 12)™, .'.we have by (21) 1(1 = 1 12 2(12^2.12. 13 4. 12. 13. 14 23 ' which converges rapidly. 7—2 100 THE APPROXIMATE SUMMATION OF SERIES. [CH. V. 8. It is very often advisable to find the sum of the first few terms of a series by ordinary addition and subtraction, and then to apply our formulas to the remaining terms, as in this way the convergence of the resulting series is usually greater. Thus, if we had applied the formula just obtained to the series we should have obtained 1 f 1 ^ 23 ' a much more slowly converging series. This remark is of great importance with reference to all the formulae of this Chapter. We shall see that the Mac- laurin Sum-formula of Art. (2) usually gives rise to series that first converge and then diverge, but that by keeping only the convergent part we obtain an approximate value of the function on the left-hand side of the identity ; and also that the closeness of the approximation depends on the smallness of the first of the terms in the rejected portion. From this it follows that by applying the formula in the manner just indicated we can greatly increase the closeness of the approximation. An example will make it clearer. Ex. Let ux = -5 , then the formula becomes 3D Taking this between limits and 1, we obtain Now, remembering tint we must only keep the convergent part of the series, we find that we must stop at J35, since ART. 9.] THE APPROXIMATE SUMMATION OF SERIES. 101 after that the numbers begin to -increase. This gives us 1.65714, the true value being ^- or 1.64493. Now let us find the sum thus l + l + l + &c.adinf. = l+l + l+±. + J^ 205 1 1 B. B, On examination it will be found that we may in this case keep the terms at least as far as B19*, while the convergence is so rapid at first that by only retaining as far as JSl we obtain 1.64494. The general advantage of using the formula may be gathered from this example. To obtain an equally close approximation by actual summation, some hundred thousand terms would have to be taken. 9. We can also expand ^ax(j>(x) in a series proceeding by successive differential coefficients of <£ (x). For 2a*^(a?) = (E- 1)~V0 (x) = ax (aE - l)~l(j> (x) (23). But by Herschel's Theorem i^(e') = ty(E) e0'*, . '. tjr (E) = ty (en) = tjr (E) e*'D as operating factors, where E' affects 0 only, .-. 2ax(x) = (x)=(-l)x(-eD-irl6(x), puttiDg a = -l in (23), * In reality we may keep all terms up to 1? , a quantity whose first significant figure is in the fourteenth decimal place. 102 THE APPEOXIMATE SUMMATION OF SERIES. [CH. V. 112 which determines the coefficients*. 10. Expansion in inverse factorials. The most general method of obtaining such expansions is by expressing the write r°° given function (x) in the form I e */($ dt. If we then J o = 1 - z, we get $ (x) = J" (1 - *)"/ {leg (j4-)j &. must now be expanded in some way in powers of z, and each term must be integrated separately by means of the formula ton By performing 2 on this we can expand in a similar way I"30 e~* — e~xt the more complicated form I . —=-t — *-f(t) dt. The most in- J o ^ •*• teresting cases are those in which $(x) = logx or = — (see page 115). The method is obviously very limited in its application. A paper on it by Schlomilch will be found in Zeitschrift fur * Compare (7), page 108. Ex. 12, page 85, is closely connected with the problem of this article. EX. 1.] EXERCISES. 103 Math, und Physik, IV. 390, and a review of this in Tortolini (Annali, 1859, 367) has sufficiently copious references to enable any one who desires it to follow out the subject. Stirling's formula — the earliest of the kind — is given in Ex. 11, page 30. The very close connection that Factorials in general have with the Finite' Calculus renders it worth while to give special attention to them, and to in- vestigate in detail the laws of their transformations. For this purpose .the student may consult a paper by Weierstrass (Crelle, LI. 1). Oettinger has also written on the subject (Crelle, xxxui. and xxxviir.), and Schlafli (Crelle, XLIII. and LXVII.). Ohm has an investigation into -the connection, between them and the Gamma-function (Crelle, xxxvi.), with a continuation on Fac- torials in general (Crelle, xxxix.). The papers on the subject of the Euler-Maclaurin Sum-formula are very numerous. Characteristic examples have been selected from them where it was possible, and placed, with references, in the accompanying Exercises. By far the most important application of the principle of approximation is to the evaluation of Tx, or rather of log Tx and its differential coefficients when x is very large* Eaabe has two papers on this (Crelle, xxv. 146 and xxvin. 10). See also Bauer (Crelle, LVII. 256) and Guderrnan (Crelle, xxix. 209). Reference will be made to these papers when we consider Exact Theorems. See also a paper by Jeffery (Quarterly Journal, vi. 82) on the Derivatives of the Gamma-function. The constant C of Ex. 3 is of great importance in this theory. For its value, which has been calculated to a great number of decimal places, see Crelle, LX. 375. Closely connected with the subject of differential coefficients of log Tx ia- that of the summation of harmonic series ( S r- — ; - TT^T. I • On this see (n-l)d}'J papers by Knar (Grunert, XLI. and XLIII.). EXERCISES. 1. Find an expression for fa + T¥ + 02 + &c-> to n terms, and obtain an approximate value for the sum ad infinitum. 2. Find an approximate expression for 2 -5 and also the value of 1 + gj + 35 + &c., ad inf., to 10 places of decimals. 104 , EXERCISES. [CH. V. 3. Find an approximate value of 3.5 ...... (2a?+l) 2.4 ...... 2# ' supposing x large but not infinite. 4. Find approximately S —$ - 7 and obtain an exact for- 27 •*- a mula when a is an integral multiple of . 5. Transform the series 1 1 1 ~ 2) (a; + 3) into series of a more convergent character, and find an approximate value of the sum of each when x — 5, that is, correct to 6 places of decimals. 6. If u0 4- utx + ujx? + &c. = / (x), shew that «W. + V*+1 + &c. =/(!) +/ (1) &vx +-OJ A2^ + &c. and apply this theorem to transform the series to one proceeding by factorials only. 7. Shew that 1.2 & EX. 8.] EXERCISES. 105 8. Find the sum to n terms of the series and shew that its sum ad inf. is z-x + 9. Shew by the method given in the note to page 72, that « + where Br_: = -^—r \ ^ ~v numerically. ax (L — e )x=0 [Schlomilch, Grunert X. 342.] 10. Shew that the sum of all the negative powers of all whole numbers (unity being in both cases excluded) is unity ; o if odd powers are excluded it is ^ . 11. Expand 2 -. j-- in terms of successive differences (ax + b)n of log (ax + b) and deduce 2 cot x = C + •! log sin x — -^ log sin x -f — log sin x — &c. !• . [Tortolini, V. 281.] 12*. If Sn = UQ + un + u2n + &c., ad inf., shew that 13. Find X — , in factorials, and determine to 3 places xi of decimals the value of the constant when the first term is 1 tw; If the Maclaurin Sum-formula had been used, to what degree of accuracy could we have obtained C ? * De Morgan (Diff. Gal. 554). Compare (27), page 54. 106 EXERCISES. [EX. 14. 14. Shew that - log* 1 '.(*,)• x ^4 22 4i and apply this to the summation of Lambert's* series, viz. x x2 1 r-— + 2 -f &c., when a; is - nearly. [Zeitschrift, VI. 407.] 15. Shew that _ where /c = J— 1, and deduce similar formulae for the sums of the series Find an analogous expression for the sum of the last mentioned to n terms. 16. Shew that sin x sin 2x sin So; p , . - H -- ^ + &c., ad inf., a+1 a + 2 a + 3 _ /• -J o if a; lie between TT and — TT. [Schlomilch, Crette XLII. 130.] * On the application of the Maclaurin Sum-formula to this important series see also Curtze (Annali Math. i. 285). ( 107 ) CHAPTER VI. BERNOULLI'S NUMBERS, AND FACTORIAL COEFFICIENTS. 1. THE celebrated series of numbers which we are about to notice were first discovered by James Bernoulli. They first presented themselves as connected with the coefficients of powers of x in the expression for the sum of the nih powers of the natural numbers, which we know is 1 4 v / \ -&c. (1), or rather as the coefficient of x in the successive expressions when n was an even integer, and De Moivre pointed out that by taking this between limits 1 and 0 we obtain the formula from which the numbers can be easily calculated in succes- sion by taking n — 2, 4, . . . ...... After the discovery of the Euler-Maclaurin formula [(6), page 90] the coefficients were shewn to be those of > — - from the application of it to 2e**, which gives 6 ~~ J. -&c ....... (3), 108 BERNOULLI'S NUMBERS, AND [CH. vi. which gives ~* — T ==£~~2~*~T91^~~uif^3"*~ ^c ^' 2. Many other important expansions can be obtained by consideration of this identity. Thus, for Ti write 20V — 1 ; then, since =^_i_1|= _icot,_i we at once obtain cot0 = i-|'2*0-|°2^-&c ................... (5). /I Again cosec 0 = cot » — cot 0, 2 c. ... (6). r li Similarly from cot 6 — 2 cot 20 = tan 0 we obtain tan 6 = ^0 + ^ + &c ............. (7). B I- 8. An expression for the values of the numbers of Bernoulli can be obtained from (5). For cot 6 = - (log sin 0) and - 2V2 2V ART. 3.] FACTORIAL COEFFICIENTS. 109 -&c Equating the coefficients of the same powers of 6 in (5) and (8), we obtain . ++ -- - 2 3 From this we see that the values of .Z?2n_, increase with T> very great rapidity, but those of -,-J^ ultimately approach to 1_ equality with those of a geometrical series whose common ratio is -r— « 4t7T * A variation of (9), due I believe to Raabe (Diff. und Int. Rechnung, i. 412), depends on the following ingenious transformation : *J~~JL ' Ote ' K"n O n O and all the terms of the form — — -^ are removed. Proceeding as before Thus we ultimately get S= •where 2, 3, 5 ... are the prime numbers taken in order. This formula "would ho of great use if we wished to obtain approximate values of Bn correspond- ing to largo values of n, as it is well adapted for logarithmic computation. 110 BERNOULLI'S NUMBERS, AND [CH. vi, 4. If m be a positive integer and p be positive **aT*dx = &c. = Hence we can write (9) thus An-i = 4» f "a2"-1 [e~27r* + e'4™ + &c. J 5. Euler was the first to call attention to a set of numbers closely analogous to those of Bernoulli. They appear in the coefficients of the powers of x when sec x is expanded. Thus (11). The identity sec # = -7- log tan f-j — 5) will giye> when Ctt£ \ TP —/ treated as before, r i i "* (12), while a consideration of the identity fcr^'f=-^ .(i3)t ^o -o« -;• ^ + 6* e +e will give (14), formulae analogous to (9) and (10), from which (12) may be deduced. * Due to Plana (Mem. de VAcod. de Turin, 1820). t SchlomUcli (Grunert, I. 361). ART, 6,] FACTORIAL COEFFICIENTS. Ill 6. Owing to the importance of Bernoulli's and Euler's numbers a great many different formula have been investigated to facilitate their calculation. Most of these require them to be calculated successively from Bl and E2 onwards, and of these the most common for Bernoulli's numbers is (2). Others of a like kind may easily be obtained from the various expansions which involve them. Thus from (5), multiplying both sides by sin 6, cos# = ( 6— T^ and equating coefficients of #2n we obtain / _ i \n 1 ...(15). f The simplest formulae of this nature both for Bernoulli's and Euler's numbers are obtained at once from the original assumptions by this method. 7. But direct expressions for the values of the numbers may be found. Thus -, — =- = -— y = ^— ^ eQ ' ' (by HerscheFs theorem) ==log(l+A)cf>.v Hence, equating coefficients, we find 112 BERNOULLI'S NUMBERS, AND [CH. vi. and in like manner we obtain "+>>0) .............. (17). 8. These formulae are capable of almost endless trans- formation. Thus, since An~J O*'1 = — - A^'1 (Ex. 8, n page 28), we can write (16) thus 3.-, - (- 1)™ {(A - 1' + §,'- fc.) o— = (-!)"« (A- |2 + |3-&c.)o<»+' ........... (18), since the other term is 9. A more general transformation by aid of the formula is as follows : -^O- ...... (19). Also |log (1 + yE)\ 0/(0) = yf(l) - £. 2/(2) +&c. /(O) ........................ (20), 1 + yE if/(0)=0. In (19) write E' for x and operate with, each side 6n/(0'). ART. 10.] FACTORIAL COEFFICIENTS. 113 Then {log (1 + A^')} 0*/(0') = I- = - {log (1 + A£')l O71'1 0'A'/(0') by (20), since OlrtA//(0/) = 0 = - {log (l + A^O"-1 /'(<>'), where /'(O') = 0'A'/(0'). Eepeating this n — 1 times we get {log (1 + A£')} 0«/(0') = (- irl {log (1 + A#)} Of"(tf) = E'fn-\V) = [(x + 1) A (x + 1) A.../(a; + l)]*=o. This transformation has been given because it leads to a remarkable expression due to Bauer (Grelle, LVIII. 292) for Bernoulli's numbers. Denote by A' the operating factor (x + 1) A, and write - for/(#) and 2n + 1 for n, and we obtain from (18) x Q2n+l Factorial Coefficients. 10. A series of numbers of great importance are those which form the coefficients of the powers of x when x(n] is expanded in powers of x. These usually go by the name of factorial coefficients. It is evident by Maclaurin's Theorem that the coefficient _D*0(n) of 3* in the expansion of x(n) is -- *. Although it is not * Comparing (22) page 25, and (25) page 26, we see that — r-— is the coefficient of An in the expansion of {log (1 +A)}*. That this is the case is B. F. D. 8 114 BERNOULLI'S NUMBERS, AND [CH. vi. easy to obtain an expanded expression for this, it is very easy to calculate its successive values in a manner analogous to that used in Ch. n. Art. 13. Let CZ = numerical value of the coefficient of of in the expansion of x(n\ Then since x(n+l} = (x — n) x(n}, we obtain C*""** <%* + *€,? (22), and we can thus calculate the values of Cn+1 from those of Cn\ and we know that the values of Cl are 1, 0, 0, ... 11. Let us denote by CKn the numerical value of the coeffi- cient of — in the expansion of x(~n) in negative powers of x, cc so that Then x™ = ^ j- A*"1 ^ = v ~' I A"-* - (where A now refers to p alone) »-« [n-l n-l also evident from the following consideration : ]n ^ | (D] = Dn~\ and , ' cc we get after division by (- I)""1 1 n — 1, Dn-'0(tt) , x 4-&c (27), in the notation of Art. 10*. * It will be seen that, as in the analogous case we coull expani {log (!+«)}" in terms of Cn, we can expand (e-1)* in terms of -Oc+l) CH . In fact ......... <26' where we have given C7n its numerical value, disregarding its sign. 8—2 116 BERNOULLI'S NUMBERS, &c. [CH. vi. 13. There is another class of properties of Bernoulli's numbers that has received some attention ; these relate to their connection with the Theory of Numbers. Staudt's theorem will serve to illustrate the nature of these properties. It is that \ - ff where m is a divisor of n such that 2m + 1 is a prime number. Thus, taking 7i = 8, we have (since the divisors of 8 are 1, 2, 4, 8) J515 = integer + f | + 1 + 1 + ~j = integer + 1/^. It will be found on reference to page 91 to be l-flfc. Staudt's paper will be found in Crelle (xxi. 374), but a simpler demonstration of the above property has been given by Schlafli (Quarterly Journal, vi. 75). On this subject see papers by Kummer (Crelle, XL. XLI. LVI.). Staudt's theorem has also been given by Clausen. 14. To Raabe is due the invention of what he names the Bernoulli- Function, i.e. a function F(x) given by when x is an integer, and which is given generally by AF(x)=xn. He has also given the name Euler-Function to the analogous one that gives the sum of 1»_ 2« + 3n - &c. + (2x - l)n when x is integral. See Brioschi (Tortolini, Series II. i. 260), in which there is a review of Raabe's paper (Crelle, XLII. 348) with copious references, and Kinkelin (Crelle, LVII. 122). See also a note by Cayley (Quarterly Journal, n.198). \ 15. The most important papers on the subject of this Chapter are a series by Blissard (Quarterly Journal, Vols. iv. — ix.) under various titles. The de- monstrations shew very strikingly the great power obtainable by the use of symbolical methods, which are here developed and applied to a much greater extent than in other papers on the subject. They include a most complete investigation into all the classes of numbers of which we have spoken in this Chapter; the results are too copious for any attempt to give them here, but Ex. 15 and 16 have been borrowed from them. The notation in the original differs from that here adopted. B2n there denotes what is usually denoted by -Ban-!- See also two PaPers on AnOm and its congeners by Horner (Quarterly Journal, iv.). 16. Attempts have been made to connect more closely Bernoulli's and Euler's Numbers, which we know already to have markedly similar properties. Scherk (Crelle, iv. 299) points out that, since tan ( j + | J = secx+tana;, the expansion of this function in powers of x will have its coefficients depending alternately on each set of numbers |see (7) and (11), of this Chapter f. This idea has been taken up by others. Schlb'milch (Crelle, xxxn. 360) has written a paper upon it. It enables us to represent both series by one expression, but there is no great advantage in doing so, as the expression referred to is very complicated. Another method is by finding the coefficient of xn in the ex- EX. 1.] EXERCISES. 117 pansion of — ^ — =-, from which both series of numbers can be deduced by taking a=± 1 (Genocchi, Tortolini, Series I. Vol. in. 395). 17. Schlomilch has connected Bernoulli's numbers and factorial coeffi- cients with the coefficients in the expansions of such quantities as Dnf (logo;), Dnl - *- . ), &c. (Grunert, vin. ix. xvi. xvin.). Most of his analysis could be rendered simpler by the use of symbolical methods. This is usually the case in papers on this part of the subject, and the plan mentioned in the last Chapter has therefore been adhered to, of giving characteristic examples out of the various papers with references, instead of referring to them in the text. We must mention, in conclusion, that the numbers of Bernoulli as far as J531 have been calculated by Eothe, and will be found in Crelle (xx. 11). EXERCISES. 1. Prove that 2. Prove that if n be an odd integer 1 n(n-l)(n-2) n(n-l)(n-2)(n-8)(n-4)n - IT — — *•', — &c., to n — 1 terms. 3. Obtain the formula of page 107, for determining suc- cessively Bernoulli's numbers, by differentiating the identity t = — u + ue* where u = e'-l* 4. Shew that [Catalan, Tortolini 1859, 239.] 118 EXERCISES. [CH. VI. 5. Shew that "-1 22*-l'2 + A 6. Apply Herschel's Theorem to find an expression for a Bernoulli's number. 7. Demonstrate the following relation between the even Bernoulli's numbers : [Knar, Grunert, xxvu. 455.] 8. Assuming the truth of the formula deduce a value of #,„_,. 9. Prove that the coefficient of ff* in the expansion of 8 Y. 22"(2»-l) D ^j is equal to —^--B^. 10. Express log sin x and log tan # in a series proceeding by powers of x by means of Bernoulli's numbers. [Catalan, Comptes Rendus, Liv.] 11. Shew that the coefficient of n r* T> - in I log (1 — e*) dt — z log z is — ^ numerically. J7l J Q W 1 12. Shew by Bernoulli's numbers or otherwise that r 22 32 27r EX. 13.] EXERCISES. 119 13. Prove that 14. Express the sums of the powers of numbers less than n and prime to it in series involving Bernoulli's numbers. [Thacker, Nouvelles Annales, x. 324.] 15. If -- = 1 + Pj + Pf + &c., shew that E*}Pt = 0, -H ^0=0, X 16. Shew, in the notation of the last question, that 17. Shew that since sin2# sin3# „ __ __ I ____ Xrn 2r+1 - K) ' in the notation of Art. 10. [Schlomilch, Grunert, xvm. 315.] 34. Shew that (with the notation of (21), page 113) and find the general formula for r = K. Shew that 35. If x -r- = Diy shew that A-in Dr/(») s *f(x) +~ *f (x) + *f"(tt) + &c. 36. Find expressions for Bernoulli's numbers and Fac- torial-coefficients in the form of determinants. [Tortoliniy Series II. VII. 19.] ( 123 ) CHAPTER VII. CONVERGENCY AND DIVERGENCY OF SERIES. 1. A SERIES is said to be convergent or divergent accord- ing as the sum of its first n terms approaches or does not approach to a finite limit when n is indefinitely increased. This definition leads us to distinguish between the con- vergency of a series and the convergency of the terms of a series. The successive terms of the series converge to the limit 0, but it will be shewn that the sum of n of those terms tends to become infinite with n. On the other hand, the geometrical series is convergent both as respects its terms and as respects the sum of its terms. 2. Three cases present themselves. 1st. That in which the terms of a series are all of the same or are ultimately all of the same sign. 2ndly. That in which they are, or ulti- mately become, alternately positive and negative. Srdly. That in which they are of variable sign (though not alter- nately positive and negative) owing to the presence of a periodic quantity as a factor in the general term. The first case we propose, on account of the greater difficulty of its theory, to consider last. 124 CONVERGENCY AND DIVERGENCY OF SERIES. [CH. VII. 3. PROP. 1. A series whose terms dimmish in absolute value, and are, or end with becoming, alternately positive and negative, is convergent. Let ul — u^ + us — ui + &c. be the proposed series or its terminal portion, the part which it follows being in the latter case supposed finite. Then, writing it in the successive forms ul-Ui+(ua-uJ + (ut-uJ + &e. ............ (1), ul-(u2-u3)-(Ut-u!.)-&c ................... (2), and observing that wt — u2 , U2 — us , &c. are by hypothesis positive, we see that the sum of the series is greater than w± — u2 and less than ut. The series is therefore convergent. Ex. Thus the series 1~~2~I~3~~4+5~~&C'C^ inf' tends to a limit which is less than 1 and greater than 3 *. 4. PROP. II. A series whose nih term is of the form un sin nO (where 0 is not zero or an integral multiple of 2?r) will converge if, for large values of n, un retains the same sign, continually diminishes as n increases, and ultimately vanishes. Suppose un to retain its sign and to diminish continually as n increases after the term ua. Let (3); * Although the above demonstration is quite rigorous, still such series pre- sent many analogies with divergent series and require careful treatment. For instance, in a convergent series where all the terms have the same sign, the order in which the terms are written does not affect the sum of the series. But in the given case, if we write the series thus, in which form it is equally convergent, we find that its value lies between ^ and - while that of the original series lies between l-5 and 1-^-f ^, i.e. 6 m . A 9 . . 1,6 between and . ART. 5.] CONVERGENCY AND DIVERGENCY OF SERIES. 125 .'. 2sm|tf=wa jcos (a- |) 0-c 4- ua+l jcos (a+^0- cos (a + |) 4 + &c- = WaCOS a - 0 + *<« -w«) cos ( a+ Now w^ — wa, ^+3 — ^0+!, &c, are all negative, hence f\ / 2 sin ^ 5- ua cos f a - ^ numerically,1 or < UM - wa ; .'. < - ua, since uw = 0. ^ Hence the series is convergent unless sin - be zero, i.e. un- less 6 be zero or an integral multiple of 2?r*. An exactly similar demonstration will prove the propo- sition for the case in which the nih term is un sin (n0 — fi). Ex. The series a sin 26 sin 30 sin 6 + — ^— + — g— +&c. is convergent unless 0 be zero or a multiple of 2?r. This is the case although, as we shall see, the series 1 + 5 + » + &c- is divergent. ^ o 5. The theory of the convergency and divergency of series whose terms are ultimately of one sign and at the same time converge to the limit 0, will occupy the remainder of this chapter and will be developed in the following order. 1st. A * Malmstdn (Grunert, vi. 38). A more general proposition is given by Chartier (Liduville, xvm. 21). 126 CONVERGENCY AND DIVERGENCY OF SERIES. [CH. VII. fundamental proposition, due to Cauchy, which makes the test of convergency to consist in a process of integration, will be established. 2ndly. Certain direct consequences of that pro- position relating to particular classes of series, including the geometrical, will be deduced. Srdly. Upon those conse- quences, and upon a certain extension of the algebraical theory of degree which has been developed in the writings of Professor De Morgan and of M. Bertrand, a system of criteria general in application will be founded. It may be added that the first and most important of the criteria in question, to which indeed the others are properly supplemental, being founded upon the known properties of geometrical series, might be proved without the aid of Cauchy's proposition ; but for the .sake of unity it has been thought proper to exhibit the different parts of the system in their natural relation. Fundamental Proposition. *v 6. PROP. III. If ike function (x) be positive in sign but diminishing in value as x varies continuously from a to oo , then the series <£ (a) + <£ (a + 1) + <£ (a + 2) + &c. ad inf. ......... (4) ,« will be convergent or divergent according as I $ (x) dx is * a finite or infinite. , For, since (x) diminishes from x = a to a? = a + 1, and again from # = a4-ltoo; = a + 2, &c., we have (f>(x) f-a+2 4>(x •'o and so on, ad inf. Adding these inequations together, we have f%<«) J a dx < <£(a) + £ (a + 1) + &c. ad inf. ...... (5). ART. 7.] CONVERGENCY AND DIVERGENCY OF SERIES. 127 Again, by the same reasoning, r<£ (x) dx> <£ (a + 1), ^ /•a+2 J^t and so on. Again adding, we have > (6). f °° Thus the integral I (#) dx, being intermediate in value between the two series (a) + (a +1) + £ (a + 2) + &c. which differ by (j> (a), will differ from the former series by a quantity less than

(a) + $ (a + 1) + <£ (a + 2) + &c. ad inf. has for its inferior and superior limits ! J a (7). 7. The application of the above proposition will be suffi- ciently explained in the two following examples relating to geometrical series and to the other classes of series involved in the demonstration of the final system of criteria referred to in Art. 5. Ex. 1. The geometrical series l + A + Aa + ^8 + &c. ad inf. is convergent if h < 1, divergent if h ~ 1. 128 CONVERGENCY AND DIVERGENCY OF SERIES. [CH. VII. The general term is h*, the value of x in the first term being 0, so that the test of convergency is simply whether hxdx is infinite or not. Now *j ixdx = . log h If h > 1 this expression becomes infinite with x and the series is divergent. If h < I the expression assumes the finite value ^ — T • The series is therefore convergent, If h = 1 the expression becomes indeterminate, but, pro- ceeding in the usual way, assumes the limiting form xhx which becomes infinite with x. Here then the series is divergent. Ex. 2. The successive series ^ / >/' ^ + (a + ir+(a+2r+&C' L 1 oQoga)- (a + 1) (log (a+1)} >(8)*, i : ; i <&v aloga(logloga)m T (a+l)log(a+l){loglog(a+l))w' a being positive, are convergent if m>l, and divergent if m< 1 The determining integrals are 100 I and divergent if m ^ 1. Perhaps there is no other mode so satisfactory for esta- blishing the convergency or divergency of a series as the direct application of Cauchy's proposition, when the inte- gration which it involves is possible. But, as this is not always the case, the construction of a system of derived rules not involving a process of integration becomes important. To this object we now proceed. First derived Criterion. 8. PROP. IV. The series u0 + ut + u2 + . . . ad inf., all ivhose terms are supposed positive, is convergent or divergent accord- ing as the ratio -x+1 tends, when x is indefinitely increased, to a limiting value less or greater than unity. Let h be that limiting value ; and first let h be less than 1, and let k be some positive quantity so small that h + k shall also be less than 1. Then as ^±1 tends to the limit h, it is ux possible to give to x some value n so large, yet finite, that for that value and for all superior values of x the ratio U*+i shall ux he within the limits h + k and h-k. Hence if, beginning with the particular value of x in question, we construct the B. F. D. 130 CONVERGENCY AND DIVERGENCY OF SERIES. [CH. VII. three series un + (h + k} wn -f (h - k) u each term after the first in the second series will be inter- mediate in value between the corresponding terms in the first and third series, and therefore the second series will be intermediate in value between and l-(h + k) which are the finite values of the first and third series. And therefore the given series is convergent. On the other hand, if h be greater than unity, then, giving to k some small positive value such that h — k shall also exceed unity, it will be possible to give to x some value n so large, yet finite, that for that and all superior values of x, l^±1 shall lie between h + k and h — k. Here then still each Ux term after the first in the second series will be intermediate between the corresponding terms of the first and third series. But h + k and h — k being both greater than unity, both the latter series are divergent (Ex. 1). Hence the second or given series is divergent also. £ f Ex. 3. The series l+t + =— 5+, 0 Q+&c-> derived 1 . £ 1 .. Zi . o from the expansion of e\ is convergent for all values of t. For if then 1.2. .. t Ux X + 1 ' and this tends to 0 as x tends to infinity. ART. 9.] CONVEKGENCY AND DIVERGENCY OF SERIES. 131 Ex. 4. The series s a(g + l)(a+2) + is convergent or divergent according as t is less or greater than unity. a (a + 1 ) (a + 2)... (a + x-I) "•=&(6 + l) (6 + 2) ... Therefore ^ = , wa 6 4- # and this tends, x being indefinitely increased, to the limit t. Accordingly therefore as t is less or greater than unity, the series is convergent or divergent. If t = 1 the rule fails. Nor would it be easy to apply directly Cauchy's test to this case, because of the indefinite number of factors involved in the expression of the general term of the series. We proceed, therefore, to establish the supplemental criteria referred to in Art. 5. Supplemental Criteria. 9. Let the series under consideration be «*« + ^i + N«« + tta«+ ... ad inf. (10), the general term ux being supposed positive and diminishing in value from x = a to x — infinity. The above form is adopted as before to represent the terminal, and by hypothesis positive, portion of series whose terms do not necessarily begin with being positive ; since it is upon the character of the terminal portion that the convergency or divergency of the series depends. It is evident that the series (10) will be convergent if its terms become ultimately less than the corresponding terms of a known convergent series, and that it will be divergent if its terms become ultimately greater than the corresponding terms of a known divergent series. 9—2 132 CONVERGENCY AND DIVERGENCY OF SERIES. [CH. VII. Compare then the above series whose general term is ux with the first series in (8), Ex. 2, whose general term is — . Then \K a condition of convergency is m being greater than unity, and re being indefinitely increased. Hence we find .'. m log x< log — ; -5 - , loga? and since m is greater than unity log — ~^>i. log x On the other hand, there is divergency if 1 *>j* x being indefinitely increased, and m being equal to or less than 1. But this gives and therefore log — logo? ART. 9.] CONVERGENCY AND DIVERGENCY OF SERIES. 133 It appears therefore that the series is convergent or divergent l°9~ according as, x being indefinitely increased, the function -7 L\jCj tJC approaches a limit greater or less than unity. But the limit being unity, and the above test failing, let the comparison be made with the second of the series in (8). For convergency, we then have as the limiting equation, m being greater than unity. Hence we find, by proceeding as before, log; And deducing in like manner the condition of divergency, we conclude that the series is convergent or divergent according as, log - x being indefinitely increased, the function-; — ^Z- tends to log log x a limit greater or less than unity. Should the limit be unity, we must have recourse to the third series of (8), the resulting test being that the proposed series is convergent or divergent according as, x being indefinitely log —j — increased, the function ; — f °9 xu* tends to a limit greater log log log x or less than unity. The forms of the functions involved in the succeeding tests, ad inf., are now obvious. Practically, we are directed to construct the successive functions, 134 CONVERGENT AND DIVERGENCY OF SERIES. [CH. VII. and the first of these which tends, as x is indefinitely increased to a limit greater or less than unity, determines the series to be convergent or divergent. The criteria may be presented in another form. For representing — by <£ (x), and applying to each of the functions ux in (J.), the rule for indeterminate functions of the form — , CO we have l (x) (x) * ;*(») ">'(*) 1) . 1 p (x) x) ' x log x and so on. Thus the system of functions (A) is replaced by the system It was virtually under this form that the system of functions was originally presented by Prof. De Morgan, (Differential Calculus, pp. 325 — 7). The law of formation is as follows. If Pn represent the ?ith function, then (11). 10. There exists yet another and equivalent system of de- termining functions which in particular cases possesses great advantages over the two above noted. It is obtained by sub- stituting in Prof. De Morgan's forms — - — 1 for . , The uw * (*) lawfulness of this substitution may be established as follows. ART. 10.] CONVERGENCY AND DIVERGENCY OF SERIES. 135 Since ux = -T-/-T , we have (x) tends to become infinite as x is indefinitely increased, and therefore \,^ assumes the form ^ ; therefore (x) $ (x) And thus the second member has for its limits VT\ *(») (ic + l) . - ,(x + I) . .. , \ , x , i.e. 1 and \ . . ; or in other words tends to <#> 0) * («) the limit 1. Thus (12) becomes Substituting therefore in (B), we obtain the system of functions the law of formation being still Ptt+1 = lnx (Pn - 1). 136 CONVERGENCY AND DIVERGENCY OF SERIES. [CH, VII. 11. The extension of the theory of degree referred to in Art. 5 is involved in the demonstration of the above criteria. When two functions of x are, in the ordinary sense of the term, of the same degree, i.e. when they respectively in- volve the same highest powers of x, they tend, x being indefinitely increased, to a ratio which is finite yet not equal to 0 ; viz. to the ratio of the respective coefficients of that highest power. Now let the converse of this proposition be assumed as the definition of equality of degree, i.e. let any two functions of x be said to be of the same degree when the ratio between them tends, x being indefinitely increased, to a finite limit which is not equal to 0. Then are the several functions x (lx}m, xlx (llx}m, &c., with which — or <£ (x) is successively compared in the de- monstrations of the successive criteria, so many interposi- tions of degree between x and a?1+a, however small a may be. For x being indefinitely increased, we have r x(lx)m Irm— -=00, , xlx(llx}m xlx(llx]n lim— —j -- = oo , lim — ,, ' = 0, xlx x so that, according to the definition, x (lx)m is intermediate in degree between x and x^a, xlx (llx)m between xlx and x (fo?)1+0, &c. And thus each failing case, arising from the sup- position of m = 1, is met by the introduction of a new function. It may be noted in conclusion that the first criterion of the system (A) was originally demonstrated by Cauchy, and the first of the system (C) by Raabe (Crelle, Vol. IX.). Bertrand*, to whom the comparison of the three systems is due, has de- monstrated that if one of the criteria should fail from the absence of a definite limit, the succeeding criteria will also fail in the same way. The possibility of their continued failure through the continued reproduction of the definite limit 1, is a question which has indeed been noticed but has scarcely been discussed. * Liouville's Journal, Tom. vn. p. 35. ART. 12.] CONVERGENCY AND DIVERGENCY OF SERIES. 137 12. The results of the above inquiry may be collected into the following rule. . RULE. Determine first the limiting value of the function -^ . According as this is less or greater than unity the series ux is convergent or divergent. But if that limiting value be unity, seek the limiting values of whichsoever is most convenient of the three systems of func- tions (A), (B), (C). According as, in the system choseny the first function whose limiting value is not unity, assumes a limiting value greater or less than unity, the series is conver- gent or divergent. Ex. 5. Let the given series be Here ux = —-^ , therefore, 1 xx x and x being indefinitely increased the limiting value is unity. Now applying the first criterion of the system (A), we have , 1 x + l? I— - Ix U,_ X X+l . and the limiting value is again unity. Applying the second criterion in (A), we have Ux 138 CONVERGENCY AND DIVERGENCY OF SERIES. [CH. VII. the limiting value of which found in the usual way is 0. Hence the series is divergent. Ex. 6. Resuming the hypergeometrical series of Ex. 4, viz. we have in the case of failure when t = l, Therefore ux b + x and applying the first criterion of (C), / ux \ fb + x \ as [ —2- — 1 ) = x( -- 1 1 \ux+l ) \a + x J which tends to the limit b — a. The series is therefore con- vergent or divergent according as b — a is greater or less than unity. If b — a is equal to unity, we have, by the second criterion of(C), — olx since b — a — 1. The limiting value is 0, so that the series is still divergent. It appears, therefore, 1st, that the series (14) is convergent or divergent according as t is less or greater than 1 ; 2ndly, that if t = 1 the series is convergent if 6 — a > 1, divergent ART. 13.] CONVERGENCY AND DIVERGENCY OF SERIES. 139 It is by no means necessary to resort to the criteria of system (C) in this case. From (13) page 94 we learn that Tx bears a finite ratio to Jx( - J , and by writing the 71th term in the form ^^? + ^ f , it will be found to be com- 1 a- 1 (o + n) tn parable with -^ , whence follows the result found above. 13. We will now examine the series given us by the methods of Chap. V. By (22) page 100 we have ,1 1 1 2 B 2 . 3 . 4 s - &c-> Here numerically = ultimately (see (9) page 109}, and thus the series ultimately diverges faster than any diverg- ing geometrical series however large x may be. As it stands then our results are utterly worthless since we have obtained divergent series as arithmetical equivalents of finite quantities and in order to enable us to approximate to the numerical values of the latter. We shall therefore recommence the investigations of Chap. V, finding expres- sions for the remainder after any term of the expansion obtained, so that there will always be arithmetical equality between the two sides of the identity, and we shall be able to learn the degree of approximation obtained by examining the magnitude of the remainder or complementary term. 14. The solution of the problem of the convergency or divergency of series that has been given is so complete that it is scarcely possible to imagine how a case of failure could arise. But we have not only obtained a test for con- 140 EXERCISES. [CH. VII. vergence, we have also classified it. Let us consider for a moment any in- finite series. Its nth term un must vanish, if the series is convergent, but it must not become a zero of too low an order ; otherwise the series will be divergent in spite of un becoming ultimately zero. Thus the zero - is of n too low an order, since un ~ - gives a divergent series ; — 2 is of a sufficiently high order, since un = — 2 represents a convergent series. Now the series on page 128 give us a classified list of forms of zero. The zeros of any one form are separated by the value m=l into those that are of too low an order for convergency and those that are not. But between any zero value that gives convergency and that corresponding to m=l (which gives divergency) come all the subsequent forms of zero. Series comparable with the series produced by giving m any value >1 in the rth class converge infinitely more slowly than those with a greater value of m, but infinitely faster than any similarly related to the (r+l)th or subsequent classes, whatever value be given to m in the second case. Thus we may refer the convergency of any series to a definite standard by naming the class and the value of m of a series with which it is ultimately comparable. 15. Tchebechef in a remarkable paper (Liouville, xvn. 366) has shewn that if we take the prime numbers 2, 3, 5... only, the series will be convergent if the series .F(4) log 2 ^logS"*" log 4 is convergent. Compare Ex. 10 at the end of the Chapter. A method of testing convergence is given by Kummer (Crelle, xin.), in- ferior, of course, to those of Bertrand, &c., but worthy of notice, as it is closely analogous to his method of approximating to the value of very slowly converging series (Bertrand, Diff. Gal. 261). It is by finding a function vn such that vnun=Q ultimately, but V-^ - vn+l >0 when n is oo . His further paper is in Crelle, xvi. 208. We shall not touch the question of the meaning of divergent series; De Morgan has considered it in his Differential Calculus, or an article by Prehn (Crelle, XLI. 1) may be referred to. EXERCISES. 1. Find by an application of the fundamental proposition two limits of the value of the series a2 + 9 EX. 2.] EXERCISES. 141 In particular shew that if a = 1 the numerical value of the series will lie between the limits - and T . 2 4 2. The sum of the series (where 8 is positive) lies between & l 2 , 2s 3. Examine the convergency of the following series 1 1 sin s a; sin s a; sin as 2 in o 4. Are the following series convergent ? xn 4- . . . where x is real, + x cos a 4- #2 cos 2«+ where cc is real or imaginary. 142 EXERCISES. [CH. VII. 5. The hypergeometrical series ab " x ( . -n j u t i\ - c(c+I) a (a + 1) is convergent if x < 1 , divergent if x> 1. If x — 1 it is convergent only when c + d — a — 5 > 1. 6. For what values of x is the following series convergent ? 7. In what cases is #2 + x x* 4- x x* + x „ . ~ — . — i — - . ~6 — - finite j 8. Shew that 1 , 1 , 1 , * — I 1 h &c. is convergent if wft+2 — 2wn+1 + un be constant or increase with n. 9. If un n ri* shew that the series converges only when a, < 1, or when o = l, and /3 > 1. 10. A series of numbers pi}p2... are formed by the formula nss shew that the series F (p^) + F(p^) + &c., will be convergent rgent. [Bonnet, Liouville, VIII. 73.] F '%\ if "— ^ + r ^ o + &c- is convergent, log 2 log 3 EX. 11.] EXERCISES. 143 11. Shew that the series + &c., and -1 H -- ? — I -- 3- -- h ......... converge and diverge a together. Hence shew that there can be no test-function <£ (ri) such that a series converges or diverges according as $ (ri) + un does not or does vanish when n is infinite. [Abel, Crelle, in. 79.] 12. Shew that if /(a?) be such that */'(*), 7W" when ie=0,the series ut + uz+ ...... and/(wj +f(uj+ ...... converge and diverge together. 13. Prove from the fundamental proposition Art. 6 that the two series A (1) + 6 (2) + 6 (3) + ............. } , . i/\Tii/fvi f w being positive are con- $ (1) + wz («n) + ra2c/> (m2) + ...... j vergent or divergent together. 14. Deduce Bertrand's criteria for convergence from the theorem in the last example. [Paucker, Crelle, XLIII. 138.] 15. If «0 + CLjX + a,£? + &c. be a series in which a0 at &c., do not contain x and it is convergent for x = 8 shew that it is convergent for x < S even when all the coefficients are taken with the positive sign. 16. The differential coefficient of a convergent series remains finite within the limits of its convergency. Examine the case of un = (ri) cos nd. Ex. (ri) = - , when the sum of the original series is - log (2 — 2 cos x\ 144 EXERCISES. [CH. VII. 17. Find the condition that the product U1u2u3 ...... should be finite. Ex. 2* . 3^ . 4* 18. If the series uQ + ul + u2+ has all its terms of the same sign and converges, shew that the product - (1 +u0) (1 +HJ) is finite. Shew that this is also the case when the terms have not all the same sign provided the series and that formed by squaring each term both converge. [Arndt, Grunert, XXL 78.] CHAPTER VIII. EXACT THEOREMS. 1. IN the preceding chapters and more especially in Chapter n. we have obtained theorems by expanding func- tions of A, E and D by well-known methods such as the Bi- nomial and Exponential Theorem, the validity of which in the case of algebraical quantities has been demonstrated else- where. But this proceeding is open to two objections. In the first place the series is only equivalent to the unexpanded function when it is taken in its entirety, and that is only pos- sible when the series is convergent ; so that there can in this case alone be any arithmetical equality between the two sides of the identity given by the theorem. It is true that the laws of convergency for such series when containing algebra- ical quantities have been investigated, but it is manifestly impossible to assume that the results will hold when the sym- bols contained therein represent operations, as in the present case. And secondly, we shall very often need to use the method of Finite Differences for the purpose of shortening numerical calculation, and here the mere knowledge that the series obtained are convergent will not suffice ; we must also, know the degree of approximation. To render our results trustworthy and useful we must find the limits of the error produced by taking a given number of the terms of the expansion instead of calculating the exact value of the function that gave rise thereto. This we shall do pre- cisely as it is done in Differential Calculus. We shall find the remainder after n terms have been taken, and then seek for limits between which that remainder must lie. We shall con- sider two cases only — that of the series on page 13 (usually called the Generalized form of Taylors Theorem) and that on page 90. The first will serve for a type of most of the theo- rems of Chapter II. and deserves notice on account of the B. F. D. 10 146 EXACT THEOREMS. [CH. VIII. relation in which it stands to the fundamental theorem of the Differential Calculus; the close analogy between them will be rendered still more striking by the result of the investiga- tion into the value of the remainder. But it is in the second of the two theorems chosen that we see best the importance of such investigations as these. Constantly used to obtain numerical approximations, and generally leading to divergent series, its results would be wholly valueless were it not for the information that the known form of the remainder gives us of the size of the error caused by taking a portion of the series for the whole. Remainder in the Generalized form of Taylor s Theorem. 2. Let vx be a function defined by the identity (a-a)vx==ux-ua (1). By repeated use of the formula Awwvx = wx+l &vx + vx kwx (2) we obtain (x — a + 1) &vx + vx = Aux, (a- - a + 2) A\ + 2Avx = AX, Substituting successively for vxt &vx> A^.-.we obtain after slight re-arrangement + (.-«)...(a-,--. + l)AX (3), -here J> = (*-*) fr-*-!) -. («-*-n) A^_ 4) |n vx representing -*— - , as is seen from (1). ~~~ a ~~~ cc AET. 4.] EXACT THEOREMS. 147 3. This remainder can be put into many different forms closely analogous, as has been said, to those in the ordinary form of Taylor's Theorem. For instance, if ux =/(#) we have where 0 is some proper fraction. If we write x + h for a, this last may be written An/' (x + h&) where A# is now supposed to be 1 — 6 instead of unity, and R appears under the form from which we can at once deduce Cauchy's form of the re- mainder in Taylor's Theorem, i.e. after the easy generalization exemplified at the bottom of page 11. 4. Another method of obtaining the remainder is so strik- ingly analogous to one well known in the Infinitesimal Cal- culus that we shall give it here. (Compare Todhunter's Diif. Cal 5th Ed. p. 83.) Let fy— /rV2> *(*)-* (x) -(,-*) A* (*) - 2-L A2^, (x) - &c. be called F(x) ; where (z-x)(r]=(z-x)(z-x-\]...(z-x-r+l). 10—2 148 EXACT THEOREMS. [CH. VIII. Then, since from (2) we obtain . AF(*0 = -(£~^V' *) suPP°se- We shall now shew that (2n, z) does not change sign be- tween the limits of the integral, remains positive or negative as m is even or odd, and has but one maximum (or minimum) value in each case. We see from (11) that (r} z) vanishes when 3 = 1, as it also does when z — 0. 7. Assume the above to hold good for some value of ??, say an even one, so that (2n • + 1, z). Now this vanishes at both limits, and there- fore its differential coefficient (Zn, z) 4- AZn must vanish at some point between them. Now this last is negative at each limit and has but one maximum, thus it must vanish twice, — in passing from negative to positive and from positive to negative, — so that (2n + 1, z) has only one. minimum fol- lowed by a maximum between 0 and 1, and thus can vanish but once. Adding A2n+l (which is zero) to it, for the sake of symmetry, and integrating again we obtain (2n + 2,z). This vanishes also at both limits, and its differential coeffi- cient is, as we have seen, at first negative and then positive, changing sign but once. Thus <£(2ri + 2, z) has but one maximum and remains positive, which was what we sought to prove. Continuing thus, the theorem is proved for all subsequent values of n> if it be true for any particular one ; sfi — Z and as it is true for (2, z) or — — , it is generally true. ART. 10.] EXACT THEOREMS. 151 8. Since (2», z) retains its sign between the limits £»— f *(^ *)£?-.<** =-ur+?f* (^ *)<**• 0o 2n+l in virtue of (11). x+e Now perform 2 on both sides of (9) and write i for M x+6 J*M Let 3/ be the greatest value irrespective of sign that — — —- (J/*JU has between the limits of summation, x and x + m suppose. Then 2u must lie between the limits + mM. x+9 9. Other conclusions may be drawn relative to the size of the error when other facts are known about the behaviour of ux and its differential coefficients between the limits. For in- stance, if u*n keeps its sign throughout, we may take 0 in- stead of — mM as one of the limits. The sign of the error will therefore be that of ( - l)n M, and, should u?** keep the same sign as u** between the limits, the error made by taking one term more of the series will have the same sign as (—1}**1M, i.e. the true value will lie between them. This is obviously the case in the series at the top of page 101, hence that series (without any remainder-term) is alternately greater and less than the true value of the function. 10. If u**1 retain its sign between the limits in (10) we have E2w = - T $ (2n, z) u?"dz = - <£ (2», 0) Aw.8", 6 < 1. J a 152 EXACT THEOREMS. [CH. VIII. ISTow it can be shewn that (j>(Zn, 0) is never greater nu- merically than — 2 A 2n; hence the correction is never so much as twice the next term of the series were it continued instead of being closed by the remainder-term. Thus, wher- ever we stop, the error is less than the last term, provided tfyat the differential coefficient that appears therein either constantly increases or constantly decreases between the limits taken. This condition is satisfied in all the important series of the form 2 —^ . The series to which they lead on cc application of the Maclaurin sum-formula all converge for a time and then diverge very rapidly. In spite of this diverg- ence we see that they are admirably adapted to give us approximate values of the sums in question, for we have but to keep the convergent portion and then know that our error is less than the last term we have kept; and by artifices such as that exemplified on page 100, this can be made as small as we like. 11. Several solutions have been given of the problem of finding the re- mainder after any number of terms of the Maclaurin sum-formula. The one in the text is by Malmsten, and the proof given was suggested by that in a paper by him in Crelle (xxxv. 55). It has been chosen because the limits of the error thus obtained are perfectly general and depend on no property of nx or the differential coefficients thereof, save that such as appear must vary continuously between the limits. The idea of the method used in this very valuable paper was taken from Jacobi, who used it in a paper on the same subject (Crelle, xn. 263), entitled De usu legitimo formula summatoria Maclauriarue. Malmste'n's paper contains many other noteworthy results, and in various cases gives narrower limits to the error than those obtained by other processes, while at the same time they are not too complicated. But the whole paper is full of misprints, so that it is better to read an article of Schlomilch (Zeitschrift, i. 192), in which he embodies the important part of Malmsten's article, greatly adding to its value by shewing the connection between the remainder and Bernoulli's Function of which we have spoken in Art. 14, page 116. The paper is written with even more than his usual ability, and is to be highly recommended to those who wish further informa- tion on the subject. 12. The chief credit of putting the Maclaurin sum -formula on a proper fooling, and saving the results it gives from the suspicion under which they must lie as being derived from diverging series, is due to Poisson. In a paper on the numerical calculation of Definite Integrals (Memoires de V Academic, 1823, page 571) he starts from an expansion by Fourier's Theorem, and obtains for the remainder an expression of the form ART. 15.] EXACT THEOREMS. 153 and he then investigates the limits between which this will lie. The investi- gation is continued by Raa.be (Crelle, xvm. 75), and the practical use of the results in the calculation of Definite Integrals examined and estimated, and modifications suitable for the purpose obtained. A method of obtaining the supplementary term which possesses many advantages is based on the formula -,Z7T3 -| 6 — J. where KZ^/-!. On this see a paper by Genocchi (T&rtolim, Ann. Series, i. Vol. in.), which also contains plentiful references to earlier papers on the subject. Tortolini in the next volume of the same Journal extends it to S". See also Schlomilch (Grunert Archiv, xn. 130). 13. The investigation which appeared in the first edition of this book is subjoined here (Art. 16). The editor thinks that the fundamental assump- tion, viz. that the remainder may be considered as being equal to cannot be held to be legitimate, since the series which the latter represents may be and often is divergent. For the conditions under which the series 'itself would be convergent, see a paper by Genocchi (Tortolini, Ann. Series, i. Vol. vi.) containing references to some results from Cauchy on the same subject. There is a very ingenious proof of the formula itself by integration by parts, in the Cambridge Mathematical Journal, by J. W. L. Glaisher, wherein the remainder is found as well as the series, and Schlomilch (Zeitschrift^ IT. 289) has obtained them by a method of great generality, of which he takes this and the Generalized Taylor's Theorem as examples. 14. By far the most important case of summation is that which occurs in the calculation of log Tn and its differential coefficients. For special examina- tions of the approximations in this case we may refer to papers by Lipschitz (Crelle, LVI. 11), Bauer (Crelle, LVII. 256), Raabe (Cr«He,xxv. 146, and xxvm. 10). It must be remembered that there is nothing to prevent there being two semi-convergent expansions of the same function of totally different forms, so that the discrepancy noticed by Guderman (Crelle, xxix. 209) in two expansions for log Tn, one of which contains a term in , and the other does n not, does not justify the conclusion that one must be false. 15. The investigation into the complete form of the Generalized Taylor's Theorem is derived from a paper by Crelle in the twenty-second volume of his Journal. Other papers may be found in Liouville, 1845, page 379, (or Gruncrt Arc hiv, vm. 166), Grunert, xiv/337, and Zeitschrift, n. 269. The convergence and supplementary term of the expansion in inverse factorials (Stirling's Theorem) have also been investigated by Dietrich (Crelle, LIX. 163). The degree of approximation given by transformations of slowly converg- ing series has been arrived at by very elementary work by Poncelet (Crelle, xin. 1), but the results scarcely belong to this chapter. 154 EXACT THEOREMS. [CH. VIII. Limits of the Remainder of tJie Series for ~2ux. (BOOLE.) 16. Representing, for simplicity, ux by w, we have The second line of this expression we shall represent by .R, and endeavour to determine the limits of its value. Now by (9), page 109, 1.2...2r Therefore substituting, Assume - •=»+i (2mir)2r dx2--1 And then, making - — = e0, we are led by the general theorem for the summation of series (Diff. Equations, p. 431) to the differential equation d't (- 1) the complete integral of which is (Diff. Equations, p. 383) t-— ,J — ri— r !sin 2mirx I cos Zrnirx ^T— -; dx ; I si or, since we have to do only with integer values of x for which sin (2??i3ne) =0, cos Hence ART. 16.] EXACT THEOREMS. 155 the lower limit of integration being such a value of x as makes to vanish, the upper limit x. Hence if within the limits of integration -j-3--+~ retain a constant sign, the value of R will be numerically less than that of the function d?n+lu . sJ ~t . *1 (47r)2"+1 therefore, than that of the function A therefore, by (9), page 109, than that of the function An-i d*nu When n is large this expression tends to a strict interpolation of form between the last term of the series given and the first term of its remainder, viz., omitting signs, between B^ d^-^u JBW d*+Hi l.2...2ndxin-i 1.2...(2w + 2)dx2«+1"< it being remembered that by (9), page 109, the coefficient of -r-^ in (1) is, in ^ the limit, a mean proportional between the coefficients of -^-^.^ and +1 in (2). And this interpolation of form is usually accompanied by interpo- lation of value, though without specifying the form of the function u we can never affirm that such will be the case. The practical conclusion is that the summation of the convergent terms of the series for 2w affords a sufficient approximation, except when the first differential coefficient in the remainder changes sign within the limits of integration. DIFFERENCE- AND FUNCTIONAL EQUATIONS. CHAPTER IX. DIFFERENCE-EQUATIONS OF THE FIRST ORDER, 1. AN ordinary difference-equation is an expressed "rela- tion between an independent variable #, a dependent variable ux, and any successive differences of uxt as AMZ, AX,...A*wc. The order of the equation is determined by the order of its highest difference ; its degree by the index of the power in which that highest difference is involved, supposing the equa- tion rational and integral in form. Difference-equations may also be presented in a form involving successive values, in- stead of successive differences, of the dependent variable ; for Anwx can be expressed in terms of ux, ux+1. . .ux+n. Difference-equations are said to be linear when they are of the first degree with respect to ux, Awx, AX,, &c.; or, sup- posing successive values of the independent variable to be employed instead of successive differences, when they are of the first degree with respect to ux, ur+l, ux+z, &c. The equi- valence of the two statements is obvious. 158 DIFFERENCE-EQUATIONS OF THE FIRST ORDER. [CH. IX. Genesis of Difference-Equations. 2. The genesis of difference-equations is analogous to that of differential equations. From a complete primitive F(x,ux,c) = 0 ........................ (1), connecting a dependent variable ux with an independent variable x and an arbitrary constant c, and from the derived equation AF(*,n,,c)-0 ..................... (2), we obtain, by eliminating c, an equation of the form <£(#, ux, AtO=0 ..................... (3). Or, if successive values are employed in the place of dif- ferences, an equation of the form (4). of diffe In like manner if, from a complete primitive Either of these may be considered as a type of difference- equations of the first order. (5), and from n successive equations derived from it by successive performances of the operation denoted by A or E we elimi- nate cv C2,...cn, we obtain an equation which will assume the form $(x,ux> A^,...AW*0 = 0 ............... (6), or the form ux,ux+i,...ux+J = 0 ............... (7), according as successive differences or successive values are employed. Either ,of these forms is typical of difference- equations of the nih order. ART. 3.] DIFFERENCE-EQUATIONS OF THE FIRST ORDER. 159 Ex. 1. Assuming as complete primitive ux = cx + c*, we have, on performing A, Aw« = c, by which, eliminating c, there results ux = ac&u, + (Aw,)8, the corresponding difference-equation of the first order. Thus too any complete primitive of the form ux = ex +f (c) will lead to a difference-equation of the form .................. (8). Ex. 2. Assuming as complete primitive ux = ca* + cbx, we have u^ccf^ + c'b*". Hence ux+i — au* = c' (b — a) b*, Therefore Ux+Z "" aUx+l ~~ & (Wx+l ~ aWx) = ^J or = 0 (9). Here two arbitrary constants being contained in the com- plete primitive, the difference-equation is of the second order. 3. The arbitrary constants in the complete primitive of a difference-equation need not be absolute constants but only periodical functions of x of the kind whose nature has been explained, and whose analytical expression has been deter- mined in Chap. IV. Art. 4. They are constant with reference only to the operation A, and as such, are subject only to the condition of resuming the same value for values of x differing by unity ; a condition which however reduces them to abso- lute constants when x admits only of such systems of values, as for instance in cases when it must be integral. 160 DIFFERENCE-EQUATIONS OF THE FIRST ORDER. [CH. IX. Existence of a Complete Primitive. 4. We shall now prove the converse of the theorem in Art. 2, viz. that a difference-equation of the n* order implies the existence of a relation between the dependent and inde- pendent variables involving n arbitrary constants. We shall do so by obtaining it in the form of a series. Let us take (6) as the more convenient form of the equa- tion, and suppose that on solving for A*X we obtain AX=/(#» w^Aw,... A"-1^) (10). Performing A we get A*"1"1^ = some function of x, ux, &ux ... AM", and on substituting for Anw* from (10) this will reduce to an equation of the form A"+X=/,(*,«,,A«,...A"-X) (11). Continuing this process we shall obtain A"+X=/, (*, ux, Aw, ... A-V) (12). But -u^+Cn + rX+^-c. + Ac.-, A -~ r(n) + &c7+fr(-n, u_n,Cl> ... O ............... (13), where clt ca . . . cn_j are the values of Aw_w . . . An~lu_n, and with the value of u_n form ?i arbitrary constants in terms of which and r the general value of ur is expressed. Thus (13) con- stitutes the general primitive sought. It is evident that it satisfies the equation for &pux for all values of p, since it is derived from those equations. 5. Though this is theoretically the solution of (6) it is practically of but little use. On comparing a with the cor- responding theorem in Differential Equations, we see that both labour under the disadvantage of giving the solution ART. 6.] DIFFERENCE-EQUATIONS OF THE FIRST ORDER. 161 in the form of a series the coefficients of which have to be calculated successively, no law being in general discovered which will give them all. And in one point the series in Differences has the advantage, for it consists of a finite number of terms only, while the other is in general an infinite series. On the other hand, the latter is usually convergent (at all events for small values of r, since the (m + l)th term f* contains -. — as a factor), so that the first portion of the series m suffices. But in our case the last part of the series is as important as the preceding part, since there is no reason to think that the differences will get very small and the (r 4. %)(m> factor - — - is never less than unity. m Having shewn that we may always expect a complete primitive with n arbitrary constants as the solution of a difference-equation of the nih order*, we shall take the case of equations of the first order, beginning with those that are also of the first degree. Linear Equations of the First Order. 6. The typical form of this class of equations is where Ax and Bx are given functions of x. We shall first consider the case in which the second member is 0. To integrate the equation ^+1-^* = ° ..................... (15), we have ««« = 4cM«» whence, the equation being true for all values of xt * An important qualification of this statement will be given in the next chapter. B. F. D. 11 162 DIFFERENCE-EQUATIONS OF THE FIRST ORDER. [CH. IX. Hence, by successive substitutions, v.-AA-A-,...^ (16), r being an assumed initial value of x. Let C be the arbitrary value of ux corresponding to x = r (arbitrary because, it being fixed, the succeeding values of ux, corresponding to x = r + 1, x = r + 2, &c., are determined in succession by (15), while ur is itself left undetermined), then (16) gives ^i^^A-i — A-, whence «.= C!A,A*-A, (17), and this is the general integral sought*. 7. While, for any particular system of values of as differ- ing by successive unities, C is an arbitrary constant, for the aggregate of all possible systems it is a periodical function of x, whose cycle of change is completed, while x varies con- tinuously through unity. Thus, suppose the initial value of x to be 0, then, whatever arbitrary value we assign to u0, the values of uv u2, u3, &c. are rigorously determined by the equation (15). Here then C, which represents the value of u^ is an arbitrary constant, and we have ^+1 = oiA-i-'.A- Suppose however the initial value of x to be J, and let E be the corresponding value of ux. Then, whatever arbitrary * There is another mode of deducing this result, which it may be well to notice. Let ux=ef. Then MiM.1=ef+Af, and (15) becomes e<+A<-^e' = 0; .•.>«-4, = 0, whence At = log -4 * , t=2logAf+C =log- .-. ux = x(x -!)...! x (x+C), where C is an arbitrary constant. * The simplest method of solving the equation ux+l- Axux=B, is derived from its analogy with the equation In this latter we sought for a factor u which should make the first side a perfect differential, and found that it was given by solution of the equation In the present case suppose C'x to be the factor which makes the left-hand side a perfect difference, i.e. of the form vx+l iix+l - vxiix. Then -v ,+1 = C, and vx = AXCX. Thus as above, putting the arbitrary constant equal to unity, since we only want one integrating factor, not the general expression for such. Multiplying by vx+l we get n U.} ART. 9.] DIFFERENCE-EQUATIONS OF THE FIRST ORDER. 165 Ex. 2. Given ux+l — aux = b, where a and b are constant. Here Ax=a, and II (Ax) = ax, therefore ix=arl\Z-x + c\ = arl CL 1| -^ - W * — a where Cl is an arbitrary constant. We may observe, before dismissing the above exam pie, that when Ax = a the complete value of II (Ax) is ax multiplied by an indeterminate constant. For — ci.a.a...(x — r •}• 1) times, But were this value employed, the indeterminate constant cT*1 would in one term of the general solution (20) disappear by division, and in the other merge into the arbitrary con- stant C. Actually we made use of the particular value corre- sponding to r = l, and this is what in most cases it will be convenient to do. 9. We must here make a remark about the solution of linear equations of the first degree, which will be easily appre- hended by those who are acquainted with the analogous pro- perty of linear differential equations. The solution of consists of two parts, one of which contains the arbitrary con- stant and is the solution of ux+l-Axux = 0 (22), and the other is a particular solution of the given equation (21). It is evident that these parts maybe found separately; the general solution of (22) being taken, any quantity that satisfies (21) may be added for the second part and the result 166 DIFFERENCE-EQUATIONS OF THE FIRST OKDER. [CH. IX. will be the general solution of (21). It will be often found advisable to use this method in solving such equations, and to guess a particular integral instead of formally solving the equation in its more general form (21). Ex. 3. Given Awx + Zux = - x — 1. Replacing Awa by ux+1 — ux) we have Here Ax = — 1, Bx — — (x + 1), whence Ex. 4. We find When, as in the above example, the summation denoted by 2 cannot be effected in finite terms, it is convenient to employ as above an indeterminate series. In so doing we have sup- posed the solution to have reference to positive and integral values of x. The more general form would be r being the initial value of x. Difference- Equations of the first order, but not of the first degree. 10. The theory of difference-equations of the first order but of a degree higher than the first differs much from that of the corresponding class of differential equations, but it throws upon the latter so remarkable a light, that for this end alone ART, 10.] DIFFERENCE-EQUATIONS OF THE FIRST ORDER. 167 it would be deserving of attentive study. Before however proceeding to the general theory, we shall notice one or two great classes of such equations that admit of solution by other ways. The analogy between these and well-known forms of differential equations is too evident to need special notice. A. Clairault's Form. ux = x& A solution of this is evidently which gives A^ = c. Ex.5. u gives u = ex -f c2. B. One variable absent. Writing ux+1 — ux for Aux and solving we obtain u**i = ^ K) suppose ; denoting by ^2 (#) the result of performing i|r on ^ (a?). Continuing we shall have This may fairly be called a solution of the equation, but its interpretation and expansion may offer greater difficulties than the original equation presented. This subject will be considered under the head of Functional Equations. Ex. 6. uw = 2u.' ; .'. «„ = 2 (2«,')' = 2V and continuing we obtain « =i(2«r. »fn g ^ *' " 168 DIFFERENCE-EQUATIONS OF THE FIRST ORDER. [CH. IX. C. Equations homogeneous in u. The type of such equations is *?/ Solve for -^ and we obtain an equation of the form Hi -*+1 = AxJ which leads to a linear equation in ux. u* Ex.7. ^'-3^. + 21^ = 0 (23). Solving ux+1 = 2ux or ux, hence uf = 2Z<7 or (7. We shall examine furthe-r on whether these are the only solutions of (23). Many other difference-equations may be solved by means of relations which connect the successive values of well-known functions, especially of the circular functions. Ex. 8. ux+1ux - ax (ux+l - uj +1 = 0. Here we have Now the form of the second member suggests the trans- formation ux = tan V0 which gives 1 tan vx+1 — tan vx tan vx = tan whence 2tair'cT- EX. I.] EXERCISES. 160 Ex. 9. Given ux+lux +y{(l - ux+l2) (l - OJ = <**• Let ux = cos VK, and we have ax = cos vx+1 cos vx + sin vx+l sin va = cos (v — 1}%} = cos At^, whence finally MS = cos ((7 + S cos"1 aj. But such cases are not numerous enough to warrant special notice, and their solution must be left to the ingenuity of the student. We subjoin examples requiring these and similar devices for their solution. EXERCISES. 1. Find the difference-equations to which the following complete primitives belong. 1st. u = ex* + c\ 2nd. u = ]c (- 1)«- ^ - ~ . ( *' 3rd. u—cx + cax. 4th. u = cax + c2. 5th. u + «/ (1 + «)* Solve the equations 2. wa.+1~X^=gaa:2. 3. ^^J.+1 — aux — cos na;. 4. W^M,. + (a? + 2) ux+1 + xux = -2-2x- x\ 5. ux+l — ux cos ao; = cos a cos 2a ... cos (x — 2) a. 6. w^+1 + aux + b = 0. 7. w^+1-a^ + 6= 0. 8. w^-e^M^e-'. 170 EXERCISES. [EX. 9. 9. ux+1$mcc0 — uxsm (a?+l) 0 = cos (as — 1)9- cos 10. ux+l - aux = (2x + 1) a*. 11. 12. 13. 14. 15. 16. 17. 3 _ 3aWuxux 2 + 2aVi3 = 0. 18. If PK be the number of permutations of n letters taken K together, repetition being allowed, but no three con- secutive letters being the same, shew that where a, /? are the roots of the equation tf = (n - 1) (x + l). [Smith's Prize.] ( 171 ) CHAPTER X. GENERAL THEORY OF THE SOLUTIONS OF DIFFERENCE- AND DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 1. WE shall in this Chapter examine into the nature and relations of the various solutions of a Difference-equation of the first order, but not necessarily of the first degree, and then proceed to the solutions of the analogous Differential Equations in the hope of obtaining by this means a clearer insight into the nature and relations of the latter. Expressing a difference-equation of the first order and nih degree in the form -j-g • • - -E*n being functions of the variables x and u, and then by algebraic solution reducing it to the form (&u-Pl) (AM-jpt)...(Aw-pJ=0 ......... (2), it is evident that the complete primitive of any one of the component equations, A^-^ = 0, Att-^2 = 0... Aw-^w = 0 ......... (3), will be a complete primitive of the given equation (1) i. e. a solution involving an arbitrary constant. And thus far there is complete analogy with differential equations (Diff. Equa- tions, Chap. vil. Art. 1). But here a first point of difference arises. The complete primitives of a differential equation of the first order, obtained by resolution of the equation with du respect to ~ and solution of the component equations, may without loss of generality be replaced by a single complete primitive. (Ib. Art. 3.) Referring to the demonstration of 172 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. this, the reader will see that it depends mainly upon the fact that the differential coefficient with respect to x of any func- tion of Vv F2>. .. Ftt, variables supposed dependent on x, will be linear with respect to the differential coefficients of these de- pendent variables [16. (16), (17)]. But this property does not remain if the operation A is substituted for that of -,- ; and therefore the different complete primitives of a difference- equation cannot be replaced by a single complete primitive *. On the contrary, it may be shewn that out of the complete primitives corresponding to the component equations into which the given difference-equation is supposed to be re- solvable, an infinite number of other complete primitives may be evolved corresponding, not to particular component equations, but to a system of such components succeediDg each other according to a determinate law of alternation as the independent variable x passes through its successive values. Ex. Thus suppose the given equation to be (&ux¥-(a + x) &uf + ax = 0 (4), which is resolvable into the two equations Att,-a = 0, &ux-x = 0 (5), and suppose it required to obtain a complete primitive which shall satisfy the given equation (4) by satisfying the first of the component equations (5) when x is an even integer, and the second when x is an odd integer. * This statement must be taken with some qualification. The reason why the primitives in question Vl- Cl — 0, F2-C2 = 0, &c., can be replaced by the single primitive ( Ft - C) ( F2 - C) . . . = 0 is merely that the last equation exactly expresses the facts stated by all the others (viz. that some one of the quantities Vlt Fg,... is constant) and. expresses no more than that. In a precisely similar way the primitives of a difference-equation of the same kind, being represented by /t (x, ux, G^} = 0, /2 (x, UK, <72) =0, &c., may be equally well re- presented by /! (x, ux, C) x/2(#, ux, C) x &c. = 0. But we shall see that the latter equation must be resolved into its component equations before any conclusion is drawn as to the values of Aux. It is not Zoss of generality that is to be feared when we combine the separate primitives into a single one, but gain. The new equation is the primitive of an equation of a far higher degree (though still of the first order), and though including the original difference-equation is by no means equivalent to it. We shall return to this point (page 184). ART. 2.] OF THE FIRST ORDER. 173 The condition that AM^ shall be equal to a when x is even, and to x when x is odd, is satisfied if we assume _ a + x , 1Ve a-x 2- H-i; -3- the solution of which is and it will be found that this value of ux satisfies the given equation in the manner prescribed. Moreover, it is a com- plete primitive*. 2. It will be observed that the same values of Aw^ may recur in any order. Further illustration than is afforded by Ex. 1 is not needed. Indeed, what is of chief importance to be noted is not the method of solution, which might be varied, but the nature of the connexion of the derived complete pri- mitives with the complete primitives of the component equa- * To extend this method of solution to any proposed equation and to any proposed case, it is only necessary to express AM,. as a linear function of the particular values which it is intended that it should receive, each such value being multiplied by a coefficient which has the property of becoming equal to unity for the values of x for which that term becomes the equivalent of Aw,, and to 0 for all other values. The forms of the coeffi- cients may be determined by the following well-known proposition in the Theory of Equations. PROP. If a, /?, 7, ... be the several nth roots of unity, then, x being an integer, the function ^ — — — is equal to unity if x be equal to n or a multiple of n, and is equal to 0 if a; be not a multiple of n. Hence, if it be required to form such an expression for Aux as shall assume the particular values plt p.2,...pnin succession for the values 35=1, x = 2,...x = n, and again, for the values a5 = n + l, x = n + 2,...x = 2n, and so on, ad inf., it suffices to assume where a, |3, 7,... being as above the different ?tth roots of unity. The equation (6) must then be integrated. 174 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. tions into which the given difference-equation is resolvable. It is seen that any one of those derived primitives would geometrically form a sort of connecting envelope of the loci of what may be termed its component primitives, i.e. the complete primitives of the component equations of the given difference-equation. If x be the abscissa, ux the corresponding ordinate of a point on a plane referred to rectangular axes, then any particular primitive of a difference-equation represents a system of such points, with abscissse chosen from a definite system dif- fering by units, and a complete primitive represents an infi- nite number of such systems, the system of abscissae being the same for all. Now let two consecutive points in any system be said to constitute an element of that system, then it is seen that the successive elements of a derived primitive (according to the definitions implied above) will be taken in a determinate cyclical order from the elements of sys- tems corresponding to what we have termed its component primitives. 3. It is possible also to deduce new complete primitives from a single complete primitive, provided that in the latter the expression for ux be of a higher degree than the first with respect to the arbitrary constant. The method, which con- sists in treating the constant as a variable parameter, and which leads to results of great interest from their connexion with the theory of Differential Equations, will be exemplified in the following section. Solutions derived from the Variation of a Constant. A given complete primitive of a difference-equation of the first order being expressed in the form u=f(x, c) (7), let c vary, but under the condition that AM shall admit of the same expression in terms of x and c as if c were a constant. It is evident that if the value of c determined by this condition as a function of x be substituted in the given primitive (7) we shall obtain a new solution of the given equation of dif- ferences. The process is analogous to that by which from ART. 3.] OF THE FIKST ORDER. 175 the complete primitive of a differential equation we deduce the singular solution, but it differs as to the character of the result. The solutions at which we arrive are not singular solutions, but new complete primitives, the condition to which c is made subject leading us not, as in the case of differential equations, to an algebraic equation for its discovery, but to a difference-equation, the solution of which introduces a new arbitrary constant. The new complete primitive is usually termed an indirect integral*. Ex. The equation u = x&u + (Aw)2 has for a complete primitive . .......... (8), an indirect integral is required. Taking the difference on the hypothesis that c is constant, we have and taking the difference of (8) on the hypothesis that c is an .unknown function of x, we have Aw = c + (x + 1) Ac + 2cAc + (Ac)2. Whence, equating these values of Aw, we have Ac(a; + l + 2c + Ac) = 0 ................... (9). Of the two component equations here implied, viz. Ac = 0, Ac + 2c + x + 1 = 0, the first determines c as an arbitrary constant, and leads back to the given primitive (8) ; the second gives, on integra- tion, (10), * We shall see reason to doubt the propriety of giving to it any special name that would seem to imply that it stood in a special relation to the original difference-equation. 17G NATURE OF SOLUTIONS OF EQUATIONS [CH. X. C being an arbitrary constant, and this value of c substituted in the complete primitive (8) gives on reduction Now this is an indirect integral. "We see that the prin- ciple on which its determination rests is that upon which rests the deduction of the singular solutions of differential equations from their complete primitives. But in form the result is itself a complete primitive; and the reader will easily verify that it satisfies the given equation of differences without any particular determination of the constant C. Again, as by the method of Art. 1 we can deduce from (9) an infinite number of complete primitives determining c, we can, by the substitution of their values in (8), deduce an infinite number of indirect integrals of the equation of differ- ences given. 4. The process by which from a given complete primi- tive we deduce an indirect integral admits of geometrical in- terpretation. For each value of c the complete primitive u=f(x, c) may be understood to represent a system of points situated in a plane and referred to rectangular co-ordinates ; the changing of c into c + Ac then represents a transition from one such system to another. If such change leave unchanged the values of u and of Aw corresponding to a particular value of x, it indicates that there are two consecutive points, i.e. an element (Art. 2) of the system represented by u—f(xt c), the position of which the transition does not affect. And the successive change of c, as a function of x ever satisfying this condition, indicates that each system of points formed in suc- cession has one element common with the system by which it was preceded, and the next element common with the sys- tem by which it is followed. The system of points formed of these consecutive common elements is the so-called indi- rect integral, which is thus seen to be a connecting envelope of the different systems of points represented by the given complete primitive. ART. 6.] OF THE FIRST ORDER. 177 5. It is proper to observe that indirect integrals may be deduced from the difference-equation (provided that we can effect the requisite integrations) without the prior know- ledge of a complete primitive. Ex. Thus, assuming the difference-equation, u = xkux + (kuxy (12), and taking the difference of both sides, we have Aw, = Aux + # AX + AX + 2A^AX + (AX)8 ; .'. AX (AX + 2A^ + x + 1) = 0, which is resolvable into AX=0 (13), AX + SAw. + a+l =0 (14). The former gives, on integrating once, kux = c, and leads, on substitution in the given equation, to the com- plete primitive (8). The second equation (14) gives, after one integration, Au,= C(-l)"-|-i, (15), and substituting this in (12) we have on reduction (-l)--*}'-£, (16), which agrees with (11). 6. A most important remark must here be made. The method of the preceding article is in no respect analogous to the derivation of the singular solution from the differential equation. It is precisely analogous to Lagrange's method of ''••'ing differential equations by differentiation (Boole, Diff. K ......... (18). This appears to be a third complete integral, "but it is only another form of (11), which may be written thus =C*(-ir-ic(-ir + i; ART. 9.] OF THE FIRST ORDER. since C (— I)2* is constant as far as the operation A is con- cerned. Substituting in (11) we obtain a result equivalent to (18)*. General Theory of Difference- Equations of the first order and their solutions. 8. We shall now examine the meaning and relationship of difference-equations, their complete primitives and indirect integrals ; and to render our ideas clearer shall notice first the analogous cases in differential equations. If we have a differential equation of the first order and first degree ~- has but one value at each point, and the dx solution consists of a series of curves one of which passes through every point and no tiuo cut ; for if two members of the family of curves coincided in one point they would co- incide during the remainder of their course. But if — be dx given by an equation of a higher (suppose the ?ith) degree this is not the case. Writing the equation in the form dx we see that at every point -*- may have any one of the values plt p^ . . . pnt but must have one of them. 9. This and only this is told us by (19); the statement at the end of the last paragraph is identically the same as the statement contained in (19). Hence anything further that we can extract from (19) must come from laws independent of * It may be shewn independently, that if one integral of (14) gives a complete primitive, the other must give, the same. For if (17) hold, the solution must come under the complete primitive of (14), involving two arbitrary constants. But for all such solutions, (15) must also hold. Hence all solutions derived from (17) and (12) must come among those derived from (15) and (12), and as the converse proposition is also true, the results of the two methods must be identical. This can only be asserted when (11) is of the iirst degree in S-ux; in all other cases we shall see that there is no single complete primitive. 12—2 180 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. this special equation, which impose conditions on the systems of values that -/- can take. The law that effects this is the ax law of continuity, which requires that -?- should vary continu- ously, or that there should not be a finite change in -^- corresponding to indefinitely small changes in x and y. Thus if we would trace out a continuous curve that shall be a solution of the equation, and commence moving in the direc- tion given by -j- =p1, we shall be compelled to continue moving in the direction given by ~ =p^ at each point, and (LOG shall not be able to change to the direction ~- = p2 at any point* even though motion in that direction is equally contem- plated in equation (19). Thus the law of continuity renders equation (19) the same as the system of equations (20), and permits us to solve them separately and take the com- bined results as forming the solution of (19). 10. Now take the case of difference-equations. As before, if kux or Ay be given uniquely by the given equation, there exist definite point-systems beginning with any point arbi- trarily chosen, but entirely fixed by the choice of it. But when the equation is of the form (*y-p1)(*y-pa)...(*y-pj = 0 ......... (21), A?/ may have any of the n values plt p9, ... pn at each point. And, as before, this and this only is told us by (21), and any further information must be gained by consideration of the general laws that govern AT/ and not from the special case before us. * This is purposely overstated. A case of excsption will be noticed later. Art. 20. ART. 12.] OF THE FIRST ORDER. 181 11. But here no law of continuity comes to our aid. The changes in x and y are finite and so will therefore that in A?/ generally be. Thus there is no reason why A?/ should continue to be equal to p^ because it is so at the particular point which may be under consideration. In fact, if you will trace out a series of points forming a solution, starting from an arbitrarily chosen point, you have at each point the choice of n different values of A?/, that is, of n different directions in which to go to the next point, and your past choice in no way binds your present*. At most it can be demanded that A?/ should be analytically expressible, and that the values should not be arbitrarily chosen at each point, but, as we saw in Art. 1, this merely implies that the succession of values of A?/ should obey some law, and places no restriction on what that law shall be. The number of point-systems satis- fying the equation is therefore infinite, and must defy all attempts at expression, and the equation (21) reduces to the system of equations Ay -^ = 0, Ay-pa = O...Ay-pB = 0 ...... (22), but we are not permitted to solve these separately and take the combined results as the full solution of (21). 12. But in spite of all this, if we integrate separately the various equations contained in (22), the resulting series of n families of point-systems (any one point in the plane form- ing a part of one member of each system and of only one) has great claims to be called a complete solution of (21). Let it be denoted by /, 0, y, C'J = 0, /, („, y, 02) = 0 . . /. (x, y, C.) = 0. . .(23). In the first place, they together impose exactly the same * The consideration that the equation means simply that Ay is at every point equal to one of the quantities Pi> Ps,---Pni gives us the important limitations under which the proof on page 1GO of the existence of a complete primitive must be taken. Unless the equation is of the first degree there will at every fresh step be a choice of values for A«,t4_r, which will of course affect &n+ru, and thus the number of distinct expansions will be infinite. When however we have adopted a law as to the recurrence of the values of Ay, the expansion at once becomes definite. 182 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. restraints on the values of Ay that (21) does, since the first member of the series permits it to equal pl , the second per- mits it to equal p2, and so on, and thus if taken as alternative equations they lead to the original equation for Ay. And in the second place, if you stand at any point, the n permissible changes of y will be those of such members of these n point- systems as actually pass through this point. Hence all per- missible elements are elements of members of (23), and thus all possible solutions of the equation are made up of elements of the point-systems included in (23). 13. That the statements in the last paragraph may be true of any series similar to (23), it is necessary and sufficient that it should at every point give all the admissible values of Ay and no more. But this is attainable in many ways other than by taking the integrals of (22). For instance, if equation (21) be &) = 0 .................. (24), it is equivalent to the alternative equations where r is some fixed value of x. If then these be integrated, they have exactly the same claim to be considered as con- stituting a complete solution of (24) as have the solutions of Ay-a = 0, Ay-Z> = 0 ............... (26). Thus, following the nomenclature of Art. 2, we see that we shall have sets of n associated derived primitives, forming as complete a solution of the equation as the set of n com- ponent primitives. And in no respect do these solutions yield * It must not be supposed that the presence of a constant r renders these more or less general than (26). Any expression in finite differences implies that some system of values of x (differing by units) has been chosen, fixing the ordinates on which all our points lie, so that r may be said to define the space about ivliich we are talking, and is wholly distinct from a constant that determines y, i.e. the position of the point on some one of, those ordinates which form our working-ground. ART. 15.] OF THE FIRST ORDER. 183 to the others in closeness of connection with the original equation. Had (24) been given in the form as it might equally well have been, the above solutions would have changed places, and the last found would have played the part of component primitives to those obtained from the solution of the factors of (24). 14. But in differential equations the solutions of the dif- ferential equations dy n dy A dy -^'-p=0, -f -0=0... -j--P«=0 das •ri dx ra dx being supposed to be Fi-C^O, F2-C, = 0... Fn-(7w = 0 ...... (27), where C19 (72,...(7B are arbitrary constants, the single solution (F,-C')(F2-C)...(^-0) = 0 ......... (28) can be substituted for them, since the latter signifies that the solution consists of all the curves obtained by giving C all possible values in it. This is obviously tantamount to giving Gl , O9 . . . Cn all possible values in the alternative equa- tions (27) from which (28) is formed, and taking all the curves so given. And this being the case, the differential equation obtained from (28) must be the original differential equation, since (28) comprehends exactly all solutions of it and no more. 15. And the reasoning which permits us to write (28) in- stead of the system of alternative equations (27), holds when they are solutions of a difference- instead of a differential equation. But it no longer follows that we may use (28) to derive our difference-equation from. This may be seen ana- lytically from the following consideration. Suppose, for sim- plicity's sake, that Ft, F2, &c. Fn are all linear. The equation obtained by performing A on (28) will generally be of the (n - l)th degree in C and of the ?ith in A#. On eliminating C between it and (28), we shall in general obtain an equation of 184 NATUKE OF SOLUTIONS OF EQUATIONS [CH. X. the ft2 degree in Ay instead of the equation of the nih degree from which we obtained (28). But it may also be seen geometrically thus. Suppose we stand at a point and choose C so that (28) contains the point in virtue of Ft — C' = 0 containing it. Then if we put x + 1 for x in (28) we shall obtain for y + Ay all the values of y corresponding to x + 1 on the curves 7,-C^O, F,- 0=0. ..F.- (7=0, no one of which except the first contains the point at which we start. Take now the value of C which causes Vz— 0=0 to contain the point, we have a similar set of values of Ay, and so on for the rest. All these values will of course be given by the equation for A?/ derived from (28) in the ordi- nary way. Thus we see that in general such an equation as (28) will lead to a difference-equation of a much higher order than the one of which it is a solution, and which per- mits values of Ay wholly incompatible with that difference- equation. And hence we must in general be content with a system of alternative solutions like (23), or if we com- bine them as in (28) we must understand that the equation in C must be solved before we can deduce the equation in question. It is by no means necessarily the case that a single equation exists that will lead to the given difference- equation, and even if such a solution exists it does not follow that it is the full solution of the difference-equation. 16. But though it is not necessarily so, it may be so. For instance, the equation y = <%c + c2 leads to a difference-equa- tion of the second order, i.e. there are two permissible values of Ay. But substituting in the original equation the co-ordinates of any point, c is found to have two values, so that there are two possible values of Ay corresponding to these two values of c. Hence here the single equation can be taken as a complete substitute for the system of alterna- tive equations with which we are usually obliged to content ourselves. This may fairly be called a complete primitive, but it is by no means the case, as we have seen, that every difference-equation has a complete primitive in this sense of the word. Suppose now two such primitives can be dis- covered— primitives that it leads to and that lead to it — ART. 18.] OF THE FIRST ORDER. 185 then the second one will be what has been named an indirect integral. The name is very unfortunate, for regarded as an integral it stands exactly on the same footing as the other complete primitive*. 17. It is obvious that if such integrals exist they must be discoverable by the process of rendering C variable, but assum- ing that the variation of C will not affect Ay. It must be noticed that any integral of the resulting equation will lead to a new and complete integral of the original equation. We need not wait to get a complete primitive (in the stricter sense of the word) of this equation, a component or derived integral will serve. Nor does the method of deriving them from the difference-equation demand special n'otice here. We shall see better its meaning and scope by working out fully an example. 18. We have seen that the equation ux = cx + c* (29) leads to the difference-equation ^ = tfA^+(Aw,)2 (30). Representing, as before, by ux the ordinate of a point whose abscissa is x, we see that (30) represents a family of point- systems such that at any point there are two values of Aux1 or, in other words, two points with abscissa x + 1 that form with the chosen point an element of the point-systems (see Art. 2). Now (29) represents also a family of point-sys- tems such that two contain each point, these two having for their distinguishing constants the roots of the equation in c formed from (29), by substituting therein the co-ordinates of the chosen point. Thus (29) and (30) are co-extensive, the elements that satisfy (30) are elements of the point-systems included in (29). * In the first edition of this work an analytical proof was given that, if indirect integrals existed, any one might be taken as the complete primitive, and the others as well as the former .complete primitive would appear as indirect integrals. This seems to be unnecessary. Any indirect integral conducts to the difference-equation, i.e. it gives precisely the same liberty of choice for Ay that the complete primitive did. Considering it as the complete primitive, any solutions that satisfy these conditions for Ay are therefore, in relation to i't, derived or indirect integrals, according as they do not or do leave to A?/ the full liberty that the equation does. From this the proposition is evident. 186 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. On solving (30) we obtain ^- ...... (31), / 2 where \f ux + — is taken to represent the numerical value* * As students are so constantly told that the square root of a quantity has necessarily a double sign, and that it is impossible algebraically to distinguish between them or to exclude one without excluding the other, it is necessary to caution them here that, whatever be the truth of the state- ment as far as analysis is concerned, it is certainly not true when the functions are represented geometrically, or perhaps we should rather say graphically. Nothing is easier than to distinguish between curves satisfying the equations y= + Jc* - a2 and y= - Jc2 - x2. It is true that they will not be what we are accustomed to call complete curves, but they will be perfectly definite. And with this understanding it will be evident that the / a^ x equation Aux~ + \/ ux+— — - gives a unique value of Aux at every point just as much as if the right-hand side were rational, and it is just as im- possible for two members of the family it represents to include the same point without wholly coinciding. But not only does a stipulation such as the one we have made about the sign to be taken with /v/w + ^- remove all indefiniteness geometrically, it also (as must necessarily be the case) removes it arithmetically. As an instance take the theorem in italics. The next value of (x + 1)2 x + 1 IJT- • — X+l X* X+l / & 1 X+l / tf X = + V U* + T + 2 ~ ~= + V W* + T ~ 2 =its former value. If at any step the wrong sign had been taken to the square root we should have failed to bring the right result, but by adhering to the stipulation, not only do we obtain the right result, but it forms a rigidly accurate proof of the theorem. It is the neglect of the above principle of the uniqueness of such / x2 x expressions as + A/ u + -r- - -= that causes much of the obscurity that sur- rounds singular solutions in differential equations. ART. 18.] OF THE FIRST ORDER. 187 of the square root of u + -7- , Equation (29) gives us the same values for c. And the result of performing A on (29) tells us that Awaj = c, in other words The point-system ob- tained by taking at each step x* x will keep the latter function wholly unaltered, and thus the solution of this equation is In a similar way the solution of Aw = — is We have divided then our point-systems into two totally distinct families, and elements of members of these families are alone permitted by (30). Now suppose we first choose to take the element given by the first equation of (31), and then we change and take that given by the second. We shall then have -•/- x+l or =-(»+!)- AM,, ..' since our first element belonged to the family (32) Aw, 188 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. or its equivalent Let us for the next element return to the element belong iog to the first family. As before, (33), since the last element was taken from the system x2 x 4"~2 (32) and (33) give the same equation, viz. Aux+1 = - (x + 1) - Aux (34), which is identical with (14), page 177. This on being integrated leads to the equation (35). The undetermined constant enables us to make it give the right value c for &ux at the point chosen, and then &ux as given by (35) will continue at each point to have a value permitted by (30), but belonging alternately to each of the two systems of values into which we have divided it. Thus (30) and (35) are both true along the whole of our new solution, and we ought to represent this new solution by them as a system of simultaneous equations. But we know from Algebra that we can take as an equivalent system either of them together with the equation produced, by eli- minating &ux between them. This last does not involve A% at all and is a complete primitive. ART. 20.] OF THE FIRST ORDER. 189 19. It is so obvious that all solutions of a difference- equation must be included in those of the equation obtained by performing A on it, that it is natural that we should try to obtain new solutions of (*ux-Pl) (A% -jpa) ... (Ati, -pn) = 0 (36), by this method. The important thing to bear in mind is that which has been illustrated in the foregoing investigation, viz. that all that the method leads to is that Aux must either always continue equal to a particular one of the roots p1} p2, •"Pn> or it must change so that it jumps from the value of one at one point to the value of another at the next, i.e. AX. = Apr or ( pr)x+l — ( PJ)X. Arid it is the alternatives of the latter class that make the sole difference between this method and the method of Lagrange of solving differential equations. In the latter \i ~=pr at a point d.-¥ can in general only equal dpr since -g- cannot jump from being equal to pr to being equal to pt. 20. We say that it can in general only equal dpr. It is only prevented from taking the specified jump by that jump being finite, and hence when we get to a point where pk = pr the change is possible. If at the next point pk is still equal to pr, ~ can change back again to pk, and so on. This will hap- pen if it should chance that at the point where pk is equal to pr the curve ~ — pk is going in the direction of the curve Pk — Pr- ID this case then there will be a solution analogous to our indirect solutions to difference- equations — its equation will be pk=pr) and it will only exist when the curves given by ~d7 ~ Pr touch the curve pk=pr at the point where they meet it, or, in other words, if the value of j- derived from dx pk =pr is pk. Such a solution is termed a singular solution*. * Few people seem aware of what might be called the rarity of singular solutions. The chances are infinity to one that a differential equation of the first order, but not of the first degree, has no singular solution. As far 190 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. 21. The question at once suggests itself — are there such singular solutions to difference- equations ? But the answer is obvious. If there be any such they are included in the indi- rect integrals. It is true that they will have a peculiarity. If pk=pr gave Ay =pk, it is evident that the point-system pk = pr might be called a solution of the equation (Ay-K) (Ay-A) ...(Ay-A) = 0 ......... (37), in virtue of it satisfying Ay —pr = 0 at every point, or of satisfying Ay —pk = 0 at every point, or of satisfying them alternately in any cycle. Hence it might with propriety be called a multiple solution, since it would appear many times over in the list of solutions. But it can never fail to be in- cluded in the complete primitive or its indirect integrals or associated integrals. Poisson (Journal de VEcole Poly technique) Tom. vi. p. 60) has written a paper on such solutions. An instance of them is given by the equation y_4"(Ay)' Ay i -- 9~ f ' of which a complete primitive is and for which he obtains the singular solution If two of the values of Ay given by (38) be equal we must have 0\ / = - — as analysis is concerned it is a mere accident that in certain cases pk=pr gives pr as the value of --. In any equation given for examination, or even in one met with in actual investigations, the chance of the existence of a singular solution is much greater, for it has probably not been written down at random, but has been derived from a complete primitive which represents a family of curves having an envelope. ART. 22.] OF THE FIRST ORDER. 191 On substitution in (38) we obtain according as we take the upper or lower sign within the bracket. O / T \ 3X Thus y—± o ( —• o ] gives us a singular solution or, as it J \ 2t) might better be called, a multiple solution*. 22. Leaving these and returning to the solutions of differ- ential equations, we must remark that not only may the change from -^-^ = 0 to -r—pz = ® be made at a point d'3C CLX where pt=p2 without obtaining a discontinuous curve, but as a rule it actually is made in every complete curve that satisfies the equation, provided that a singular solution exists. Take, for instance, the equation y = ex + c2, this leads to the alternative differential equations 4 ! , dy x x - and the singular solution is of course This represents a parabola touching the axis of x at the origin and having its axis in the negative direction of y. The two equations in (41) denote the tangents to it through the chosen point, the first representing the one that makes the algebraically greater angle with the axis of x, since -j- is ctx greater along it. Now take a tangent and beginning from x = — GO move along it. At every point it is the solution of * As in differential equations the results of this method need not be solutions, but if they are solutions, they are singular solutions. Compare Art, 0. 192 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. the second of equations (41), since the other tangent through the point has its -—- algebraically greater, as will be seen at once from a figure. But as soon as it has touched the en- velope it takes at once the r6le of being the solution, at every point of its length, of the first of equations (41). So that if we take the complete curve, i.e. the whole of the tangent line, as a solution of the equation, we shall have changed from satisfying the second of the alternative equations to satisfying the first ; the change taking place at the point of contact with the envelope. 23. This enables us to see very clearly that the envelope is in reality an indirect integral. For let us start from a point on a tangent just before it meets the envelope and proceed along it — of course in the positive direction of x — to a point on it just after it meets the envelope. Our path at first satisfied the second and now satisfies the first of equations (41). Let us now change and take the path through the point at which we now are that satisfies the second of those equations. It will be the tangent through the point which is just going to touch the envelope. On continuing this process we see that we have a circumscribing polygon, the limit of which when the sides are indefinitely diminished is the curve. And this was generated by pursuing exactly the same method that we observe in obtaining derived or indirect integrals from com- ponent integrals or complete primitives, viz. by alternating between different solutions*. 24. It will not be necessary to dwell upon the derivation of * The Singular Solution (or rather Multiple Integral) of Art. 21 partakes, as we have seen, of the nature of the singular solution of a Differential Equation, since it is derived from the difference-equation in the same way, viz. by taking the condition that two of the alternative solutions should coincide. And hence it is not to be wondered at that the singular solution of a differential equation should have somewhat in it of a multiple integral. In point of fact, portions of it form part of all solutions of the original equation. For instance, in the case we are considering the solution of, y + -. - | is obtained by always choosing the one of the two permissible paths that lie most to the right, supposing that we start from a point in the third quadrant. This takes you in a straight line as far as the curve and then takes you round during the rest of your motion, since any departure must be along a tangent, i. e. more to the left than along the curve. ART. 24] OF THE FIEST ORDER. 193 indirect integrals or singular solutions from the complete primitive. What has been said will be guide sufficient. But before leaving this part of the subject we will examine how far these views enable us to explain the anomalies connected with Singular Solutions in Differential Equations. Boole (Diff. Eq. Ch. vm.) gives the following four Properties of Singular Solutions : I. An exact differential equation does not admit of a singular solution. II. The singular solution of a differential equation of the first order and degree renders its integrating factors infinite. III. A differential equation may be prepared (even with- out the knowledge of its integrating factors) so as no longer to admit of a given singular solution of the envelope species. IY. A singular solution will generally make the value of ~? as deduced from the differential equation as- (US sume the ambiguous form - . The first of these seems self-contradictory. An envelope ft ?/ has the same value of -~ as the enveloped curve at the point of contact. Hence it must satisfy the differential equation of the latter, i. e. the equation that gives — . Now the dif- ferential equation to any family of curves whatever, say F '(as, y, c) = 0, can be given in the form of an exact equation. All that is necessary is to solve for c and to differentiate the resulting equation c =^r (x, y). Thus (I.) seems tantamount to saying that no family of curves can have an envelope. (II.) stands or falls with (I.), but is at least remarkable that an integrating factor should have any essential connection with that which is represented by the equation. The inte- grating factor is simply the reciprocal of the factor by which the equation, when in its exact form, was multiplied to bring it into its present form. It is therefore a purely arbitrary B. F. D. 13 194 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. thing, and has nought whatever to do with the nature of the equation or with that which it represents. And (III.) is not less puzzling. For since the geometrical envelope has two consecutive points in common with each member of the family, it would seem probable that it would continue to have that property after any transformation of x and y. But were this the case it would continue to touch them all, and thus to be a singular solution according to our previous remark. 25. It cannot be doubted that these anomalies demand explanation, and if our theory of the nature of a singular solu- tion be the right one it must render them intelligible. And from our theory we see no reason why exact differential equations should be more or less likely to have singular solu- tions than others. It is true that they are of the first degree, and of course no differential equation that gives a single value of -p at every point can have a singular solution (Art. 8). CiuC But there is no reason to expect that an exact equation will give one value and one only of ~ at every point; it will usually give the value in terms of quantities such as roots of algebraical functions of the co-ordinates, which will have more than one value, and no attempt is made in such equa- tions to limit the interpretation of these to one of their many values. Yet, although our theory declines to take special notice of exact equations, it still gives us a clue to the inter- pretation of their peculiarity by pointing out a class of equa- tions which possess the property in question, viz. those that give but one value to -^ at each point, and which may be for shortness' sake termed unique equations. It must be that by our treatment of exact equations we make them to all intents and purposes unique equations. 26. Let us take the instance given by Boole, On dividing by V^a 4- y* — a? to render it an exact equation, we obtain ART. 27.] OF THE FIRST ORDER. 195 dy - j dy_^ Now it is not fair yet to say that this is not satisfied by the singular solution X* + y* = a2, for that causes the first term to assume the indeterminate form ^ ; but as soon as we write it in the form -y- V#2 + y2 — a2 — ~ = 0, we see that the singular solution has ceased to satisfy it, and hence it must be in this step that we have converted the equation into a unique equation. Writing r for Vic2 + y* — a2, it becomes •T- - -~ = 0, the integral of which is y — r = c, representing a series of parabolas touching the circle r = 0. As y is made to increase from its greatest negative value (c being taken posi- tive) r, which at first would generally be negative, gets smaller numerically, vanishes, and then becomes positive. This confirms our remark that the complete curves which are solutions of the equation require #* + y* — a? to be taken partly with a plus and partly with a minus sign, and thus are partly solutions of -f- dr — dy = 0, and partly of — dr — dy = 0, the change occurring at the point of contact with the enve- lope*. Of course this is allowable in consideration that the sign of r is arbitrary at each point, but it will be seen that this stipulation renders the equation a unique equation just as much as the stipulation that r shall always be taken positive. 27. But a difficulty arises here. Since the stipulation, which, as we see, renders the equation unique, enables us to trace out the whole of each curve, it will enable us to trace out all the solutions of the equation, and thus is it not a complete form of the equation ? It is true that at any point when two of the curves intersect we shall pass along one or the other accord- ing as we reckon that we have or have not passed the point of contact with the envelope, and thus when we make the * Should this contact not be real, then, so far as real space is concerned, there will be no change in the equation satisfied at every point, and ac- cordingly there will be at 110 point nn alternative path, and therefore no real portion of the singular solution corresponding thereto. 13—2 196 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. double supposition we shall, by the aid of the stipulation mentioned in the last paragraph, describe the curves without destroying the uniqueness of the equation. But this is equivalent to taking r of double sign at each point, and it is not to be expected that phenomena of intersection (such as singular solutions essentially are) will be discoverable by analysis which calls a point indifferently r, y, and — r, y. Whatever stipulation we make as to the sign of r to render dr — dy — 0, a unique equation renders it impossible that two such curves should intersect, i.e. should be satisfied by the same values of r and y, but if we consider it an intersection when the one is satisfied by r, y, and the other by — r, y, it is not to be expected that our analysis will be equally lax. 28. Assuming then that the true form of the exact differen- tial equation is dy ± dr = 0, we still have to explain how it is that r = 0 fails to satisfy the equation. The equation is no longer unique, but the alternative solutions do not seem to assist us, the change from the one to the other implies a sud- den change from -=- = 1 to -,- = — 1. This difficulty, which & \j is merely a particular case of the one arising from (III.), is of a wholly different nature to the last one. We have now at every point precisely the same liberty of path that we had in the original equation — the same number of alternative direc- tions. But we seem unable to change from one set to the other and thus to have no singular solution. Now the sole restrictions on change arise, as we see, from the law of conti- nuity, so that it is in connection with this that the solution of this difficulty must be found. We shall shew how it is that we have no longer the opportunity of choosing, at the points on the singular solution, along which of two paths we shall go. 29. For simplicity's sake, suppose that the appearance of uniqueness in the exact equation is produced, as in the instance that we have taken, by the presence of a quantity of the form ^u, where u is a rational integral function of x and y, so that u = 0 is the singular solution, since it renders equal the two values of -~ . This is a very common case, and the dx treatment -will apply to other more complicated cases. Let ART. 29.] OF THE FIRST ORDER. 197 x, y be the point of contact of a particular primitive with the singular solution, and x + dx, y + dy, a neighbouring point on the same primitive. Then since there is tangency with u = 0 at x, y} the value of u at x + dx, y + dy must be of the second order (arid hence *Ju is of the first order) in dx and dy. Now take *Ju and x as new variables, 77, %, expressing y in terms of them, and draw the curves represented by the primi- tives when x and 77 are considered as Cartesian co-ordinates. The axis of 77 is now the singular solution, and as we proceed along any primitive we find that in its neighbourhood — is dx finite, since 77 was of the first order along a primitive in the neighbourhood of 77 = 0. Thus the primitives seem to cut 77 = 0 at an angle. In fact near u = 0, du was of the order *Jdx excepting for small displacements in the direction of u = 0 at the point. Thus -^ is generally infinite for 77 = 0, or the distortion produced by the new representation is so great that all curves cutting 77 = 0 in the original will cut it at right angles now. Only those touching it will cut it at a smaller angle, and those that had a yet closer contact will appear to touch it. And, returning to the original, when we dr 1 remember that -j- is of the order — = for all directions of dis- dx placement but one coinciding with r — 0, we shall see that a solution of the equation ---^ = 0 dx dx must have the direction given by r — 0. So considered, the apparent absurdity of saying that ~ — -j- = 0 is satisfied by r = 0, -~- ^= 0, passes away. And -the preparation which Pois- son gives for getting rid of envelopes can be explained on exactly similar principles ; it differs chiefly in this, that he has made a rather more general supposition as to the origin of the alternative values of ~~ . dx 198 NATUKE OF SOLUTIONS OF EQUATIONS [CH. X. 30. We might have expected (IV.). The equation for -~ , CLtJC obtained by differentiating the differential equation after solving for ~ , must give the value of v| alike for the par- ticular primitive at the point and for the singular solution. And we should not expect these two values to be obtained by giving alternative values to the functions in -f- whose values are not unique, since such functions will naturally have unique values on the singular solution. Thus we should d*y expect that the equation for ~ would give an indeterminate result. We may remark in conclusion that we ought to expect no such anomalies in the solution of difference-equations, as they all arise from change of independent variable, a thing which cannot occur in Finite Differences excepting in the simple form of change of origin. The Principle of Continuity. 31. We have seen that the great distinction between the subject-matter of Difference- and Differential Equations is, that the law of Continuity rules in the latter and not in the former case. Hence we cannot expect that the results of the former will always be represented in the latter, and we have already dwelt upon cases in which they are not. It will not do to look on the Differential Calculus as a case of the Difference- Calculus, subject merely to the stipulation that the differences are infinitesimally small — while the latter deals with the ratios of simultaneous increments of the dependent and inde- pendent variables, the latter deals with the limits which these ratios approach when the increments are indefinitely small — and unless they approach definite limits the case can never be in the province of the Infinitesimal Calculus, how- ever small the differences be taken. We shall now examine ART. 33.] OF THE FIRST ORDER. 199 the conditions under which a point-system will merge into a curve, and apply our results to the case of solutions of a difference-equation. 32. It is a familiar but a partial illustration which presents a curve as the limit to which a polygon tends as its sides are indefinitely increased in number and diminished in length. Let us suppose the differences of the value of the abscissa x for the successive points of the polygon to be constant, the law connecting the ordinates of these points to be expressed by a difference-equation, and the corresponding law of the ordinates of the limiting curve to be expressed by a differ- ential equation. Now there is a more complete and there is a less com- plete sense in which a curve may be said to be the limit of a polygon. In the more complete sense not only does every angular point in the perimeter of the polygon approach in the trans- ition to the limit indefinitely near to the curve, but every side of the polygon tends also indefinitely to coincidence with the curve. In virtue of this latter condition the value of ~ in the polygon tends as A# is diminished to that of ~~- in the curve. It is evident that this condition will be ax realized if the angles of the polygon in its state of transition are all salient, and tend to TT as their limit. But suppose the angles to be alternately salient and re- entrant, and, while the sides of the polygon are indefinitely diminished, to continue to be such without tending to any limit in which that character of alternation would cease. Here it is evident that while every point in the circumference of the polygon approaches indefinitely to the curve, its linear elements do not tend to coincidence of direction with the curve. Here then the limit to which ,— approaches in the fl ?/ polygon is not the same as the value of •/• in the curve. CM? 200 NATUEE OF SOLUTIONS OF EQUATIONS [CH. X. 33. If then the solutions of a difference-equation of the first order be represented by geometrical loci, and if, as A# approaches to 0, these loci tend, some after the first, some after the second, of the above modes to continuous curves ; then such of those curves as have resulted from the former process and are limits of their generating polygons in re- spect of the ultimate direction of the linear elements as well as position of their extreme points, will alone represent the solutions of the differential equations into which the differ- ence-equation will have merged. This is the geometrical expression of the principle of continuity. 84 The principle admits also of analytical expression. Assuming h as the indeterminate increment of x, let yz, yx+7l, yx+tfr be the ordinates of three consecutive points of the polygon, let +(%>}• (44). A# VW This is the difference-equation sought. Taking the difference of (41), A# being still indeterminate but c a variable parameter, we have as in Ex. Art. 3, Ac + 2c = - (x + A 4j 4 It results then that (44) has for complete primitives (42) and (45), h being equal to A#. 2ndly. To determine tan 6 for the primitive (42), we have Ay = cA#, A2y = 0, whence, substituting in (A), we find tan 6 = 0. Thus the complete primitive (42) merges without limitation into a com- plete primitive of the differential equation. But employing the complete primitive (45), we have + 7*2 4 ' Hence 204 EQUATIONS OF THE FIRST ORDER. [CH. X. Now this value does not tend to 0 as h tends to 0, unless c = 0. Making therefore c = 0, h = 0, in (45), we have as the limiting value of y and this agrees with (43). Thus, while the complete primitive of the differential equation conies without any limitation of the arbitrary con- stant from the first complete primitive of the difference- equation, the singular solution of the differential equation is only the limiting form of a particular primitive included under the second of the complete primitives (45) of the difference-equation. Geometrically, that complete primitive represents a system of waving or zigzag lines, each of which perpetually crosses and recrosses some one of the system of parabolas represented by the equation W x* As h tends to 0, those lines deviate to less and less distances on either side from the curves ; but only one of these tends to ultimate coincidence with its limiting parabola. 38. As the nomenclature of this chapter is not very simple it may be useful to recapitulate the various kinds of solution that a difference- equation of the first order and wth degree may have : solutions involving an arbitrary constant from which the equation can be derived, and which can be derived from it. The two classes of solution are the same in their relation to the equation ; any one may be chosen as complete primitive, and the next become indirect integrals. Arts 15, 16. II. Complete primitive (in the less strict sense of the word)] Component primitive > solutions Derived primitive J which do not give to Aux all the freedom it may have, but which still allow it such values only as the difference-equation also permits. All these classes of solutions have the same relation to the equation, they are derived or component in relation to one another. Sets of n such equations granting to AWj. all the alternative values permitted by the equation form the only complete solution that most equations have, and if the members of any EX. 1.] EXEKCISES. 205 such set be called component primitives, all other solutions can be considered as derived primitives. Arts. 11 — 13. HI. Singular Solution ) q . , 2i Multiple Integral > EXERCISES. 1. Find a complete primitive of the equation which shall satisfy the equation At^ — a — 0 only when x is a multiple of 3. 2. The equation is satisfied by the complete primitive y — cx* + c2. Shew that another complete primitive may thence be deduced. 3. Shew that a linear difference-equation with rational and integral coefficients admits of only one complete primitive. 4. The equation has y — cax + c2 for a complete primitive. Deduce another complete primitive. 5. If uxux^— -- -, shew that CC ~p JL 2.4 ...... ( deduce thence a cycle of three complete primitives. 9. Form the difference-equation whose solution is the system of alternate equations y — ex 4- o? — 0) cy - x + a? = 0) ' and also form a difference-equation of the first order whose complete solution is one of the derived integrals of this equation. 10. Shew that if instead of putting equal arbitrary constants in (Vl — c:) (F2 — c2) =0 we put them alter- nately positive and negative, but of equal numerical value, the resulting differential will be the same, but the resulting difference-equation will be different. 11. Shew that the solution y = 0 of the equation %> (dy\- \M + 8y* = 0 (Boole, Diff. Eg., Oh. vill.) is analogous to the singular solutions of difference-equations spoken of in Art. 21. ( 208 ) CHAPTER XL LINEAR DIFFERENCE-EQUATIONS WITH CONSTANT COEFFICIENTS. 1. THE type of the equations of which we shall speak in the present chapter is -4,«.-* '.-.(I), where Alt A2, ...... A^ are constants and X is a function of the independent variable only. This form will manifestly include the form &c., +AHu, = X. ........... (2), and may be symbolically written , = X ..... .............. .....(3), where f(E) is a rational and integral function of E of the nih degree, with unity as the coefficient of the highest term, and with all its coefficients constant. 2. Now we know from (10) page 18 that E — eD, so that we might write (3) in the form /(e»X = Z ........................... (4), and consider it a linear differential equation of an infinite degree and solve it by the well-known rules for such equa- tions. The complementary function would then have an infinity of terms of the form Cemx where m would be deter- mined by the equation /(em) = 0 ; and to this we should ha~ve to add a particular integral obtained either by guess or ART. 3.] LINEAR DIFFERENCE-EQUATIONS. 200 by special rules depending on the form of X. But we shall not adopt this mode of procedure, and that for two reasons. In the first place we have to face the difficulty of an equation of an infinite degree, or rather of an equation that combines the difficulties of transcendental and algebraical equations ; and though we know from experience of Ex. 2, page 79, that these difficulties are more apparent than real, and that the infinitude of roots merely signify that the constants obtained are periodic and not absolute constants, the method still is open to the objection of being unnecessarily complex and intricate. But there is a more important reason for not adopting this method. The problems of Finite Differences are really phenomena of discontinuous change, the variables do not vary continuously but by jumps. And a method is open to grave objection that treats the change as a con- tinuous one the results of which are inspected only at certain intervals. At all events such a method should not be resorted to when the direct consideration of the operations properly belonging to the Difference-Calculus suffices to solve our problems. 3. We have seen in Chapter II. that E and A like D will combine with constant quantities and with one another as though they were symbols of quantity. And thus/(£") when performed on the sum of two quantities gives the same result as if it were performed on each and the results added. Hence if we take any two solutions of the linear difference-equation (5) the sum of these solutions will also be a solution. Also any multiple of a solution is obviously a solution. So that if we can obtain n particular solutions Vv F"2,...FM, connected together by no linear identical relation, then will C.V. ............ (6) be a solution, and in virtue of containing n arbitrary constants B. F. D. 14 210 LINEAR DIFFERENCE-EQUATIONS [CH. XI. it will be the most general solution*. We shall now proceed to find these particular integrals and shall then have solved equation (5), which is the form which (1) assumes when Z=0. 4. Let f(E) = 0 have as roots mlt w2, ...... mn; E being treated as a symbol of quantity. Then we know that f(E) = (E-mJ(E-m)...(E-mJ ........... (7), whether E be a symbol of quantity or of operation, so that we may write (5) thus, (E-ml)(E-ms)...(E-ma-)u!, = 0 ....... (8), where E—mn&c. denote successive operations the order of which is indifferent. But if we solve the equation (E — raj ux = 0 we obtain a particular solution of (8), since the operation (E—m^) (E — m2)...(E — mn_L) performed on a constant of value zero must of course produce zero. Putting in turn each of the other operational factors last, we obtain other particular integrals, and thus when the roots are all different we shall obtain the n particular integrals Vv F2,...FW (which give us by (6) the general solution) by solving n separate equations of the form (E-m)ux = 0 .... .................... (9). 5. But if one of the roots is repeated — say r times — this method fails ; for r of the solutions would be in point of fact identical or merely multiples of one another. But if the said root be mK and we take the full solution of the equation (10), (involving, as it will, r arbitrary constants), instead of taking the solution of the corresponding case of (9), we shall have as before the right number of arbitrary constants and there- fore the most general solution. * It must be noticed that in linear equations with constant or rational coefficients, there are no difficulties arising from alternative values of the increments of the dependent variables as in the cases which formed the subject of the last chapter. The value given for all successive differences is strictly unique, so that but one complete primitive exists. See note on page 181. ART. 8.] WITH CONSTANT COEFFICIENTS. 211 6. We have thus reduced the problem of solving (5) in all cases to that of solving a number of separate equations of the form (N-mYux = 0 ................ . ....... (11). But (see note page 73) f(E)a'ux = a"f(aE)u,....: ................. (12); hence (E - m}rux = mx (mE - m)rm~xux = m^^r(m~xux} = 0 by (11) ; .-. A" (m-xux) = 0, /. m~*ux = C0 + C,x + Cj? + . . . C^aT1 since the rih difference of such a function vanishes ; and thus ^ = (ao+C1*+... + CLX-')m* ............... (13). Thus the general solution of (5) is ux = 2(C0+Clx+... Cr_^)m* ............ (14), where r is the number of times the root m is repeated in the equation/^) = 0. 7. We will illustrate the foregoing by an example. Let the equation be ux+a-3ux+l-2ux= 0, .................. (15), or (E3- 3^-2)^ = 0. This is the same as 2 and thus the solution of (15) is ux=(C0+C^(-lY+C^ ............... (16). 8. A slight difficulty presents itself here — not in the theory of the solution, but in the interpretation of the result. It would seem as if we must content ourselves with results impossible in form whenever the roots of the equation for E 14—2 212 LINEAR DIFFERENCE-EQUATIONS [CH. XI. are impossible. This may be avoided thus. Impossible roots occur in pairs so that with any term Gxr (a + ft *J — 1)* in the solution, corresponding to a root (a + /3>J — 1) repeated at least (r + 1) times, there will be a term C'xr (a — J3j — l)x. Assuming a + ft N/^l = p (cos 6 + /^l sin 0), which gives the terms become #y { G (cos #0 + J^I sin a?0) + 0' (cos xO - /^l sin 050)}, or of/)* { Jf cos %0 + N sin #0}, where If and JV are still arbitrary constants. Thus the part of the solution of / (E) ux — 0 that corresponds to the pair of impossible roots a ± ft J — 1 repeated r times in/ (E) = 0 is ( Jf0 + MjX + . . . + M^aT1) />* cos xO + (N0 + JV> + . . . + ^aT1) p* sin a;0, which has, as we see, the right number of constants. Ex. 1. Let the equation be «XH! + 2wa48+tta = 0 .................. (17), or (#8+l)X = 0. The roots of f(E) =0 are 1, and L±N_IL?, each repeated twice, the solution is therefore since p = 1 and tan 0 9. We have thus obtained a solution of the most general form possible of the equation f(E) ux = 0. We shall now ART. 10.] WITH CONSTANT COEFFICIENTS. 213 proceed to the more general form of equation which we chose as the subject of this chapter, viz. f(E)u^X. .......................... (19). But our past work stands us here in good stead. For if to any solution of this equation we add a solution of f (E) ux = 0, the result of performing f(E) upon their sum will be X + 0 or X (see Art. 3). If then to a particular solution of (19) we add the general solution of (5), we shall get a solution of (19) involving n arbitrary constants, and which must therefore be the most general solution of (19) possible. Our task has therefore reduced itself to finding a particular integral of (19). And our first thought is to try if we cannot obtain it by a device similar to that which gave us the solu- tion of (5) — in other words, deduce it from the solutions of simpler equations. At first sight the method seems wholly to fail. For if we solve (E — mn) ux = X and obtain the solu- tion Xm, it is no longer a solution to the full equation. On performing f(E) upon it, we obtain (E-mi}(E-m,}...(E-m^X. ........ (20), which involves X and its next n — 1 consecutive values. Similarly if we find Xr the solution of (E—mr) ux — X, we should obtain, on performing f(E) upon it, 10. But a modification of our former method will still give us an integral. Instead of taking merely the solution of one of the simpler equations, take those of all and com- bine them by multiplying each by a constant and adding the results. If we perform / (E) on plXl + fj,zX2 + . . . + f^nXn— the roots of / (E) = 0 being for the present supposed all dif- ferent — we shall obtain the quantity E-ml)...(E-mn_l)}X ............ (22). And if by choosing pv /x2, . . . //,ft aright we are able to make the coefficients of all the powers of ^in (22) vanish and the 214 LINEAR DIFFERENCE-EQUATIONS [CH. XI. term independent of E become unity, we shall have a solu- tion of (19) in ux = ^X,+^X2+...^Xn (23). To do this we must have + &C.551 (24), when E is treated as a symbol of quantity. This proviso enables us to divide with confidence loy f(E), and we see that /^l t ^2 t , I ^» _ /9^"\ E-m+ttp + 3-mn=fW" or in other words JJLV /fca,... are the numerators of the partial fractions into which ^rfn can be resolved. 11. Nor will this method fail when a root is repeated. Let a root mK be repeated r times, then if we use for XK, XK+l,...XK+r_v the solutions of the equations (E-m.yu.~Z we shall have for the corresponding values of ^ the nume- rators of the partial fractions forming ^ , whose deno- minators are (E-mK\ (E-mKy,...(E-mKy. Thus we have reduced the solution of (19) to that of the equation (E-m)uxr = X (26), which we can write by (12) ra^A*" (m~*ux) = X ;' ART. 12.] WITH CONSTANT COEFFICIENTS. 215 or ux — mx 2r m~x~r X, and (19) is fully solved. And a little further consideration shews that this last investigation renders unnecessary that in Arts. 2 — 5, which suggested it. For in each of the quantities Xv X2,...Xn there is a term involving an arbitrary constant, and of the form Cm*, Cm*, &c. If we include these in the values of X, &c. which we substitute in (23) we get the general solution at once*. 12. Let us examine the results at which we have arrived. From the equation f(E) u = X we have deduced n ..................... (27), where X19 X^... are the solutions of (E — im^ux — X and kindred equations, and /JL^ yu-2. . . are the coefficients of the partial fractions into which irr™ is resolved when E is con- sidered a symbol of quantity. But it is natural to ask, — Could we not have obtained this at once by symbolical methods, thus : — ......... (28)" But, since X^ is a solution of (E — mj ux = X, ...... finXn ............ (30), agreeing with (27). * It might seem that we shall get more than sufficient constants by this method when roots are repeated. For (E-m)rux=x will give r constants, and (E -m)r-luz=x will give r-1 additional ones, while there should only be r hi all. But since all the solutions of the equation (E - m)r~1vf=Q are solutions of the equation (E-m)rug = fy and all the terms which we are considering come from these last equations, we neither gain nor lose in generality whatever solution of (E -m)r~1«ie = 0 we take, provided we take the full solution of (E - m)ruf = Q which gives r arbitrary constants. 216 LINEAR DIFFERENCE-EQUATIONS [CH. XI. 13. At first sight this method seems justified by the properties of E proved in Art. 9, Ch. II. And there is no doubt that, as far as suggestiveness is concerned, such an application of symbolical methods is all that could be desired. But as it stands it is not rigorous. So long as our operations are direct we may place absolute reliance on symbolical methods, for the results of the operations are unique, and hence equality in any sense must mean alge- braical equality. But so soon as any of the operations are indirect, further investigation is needed. The results of the indirect operations are not, in an algebraical point of view, definite, and we must carefully examine each case in order to discover the conditions of interpretation of the results that there may be algebraical equality. For instance, (E-a)(E-b)ux=(E-b)(E-a)ux (31), but (E — a) j-j ux does not equal ^ (E — a) ux. . .(32), jG/ — a Jit — o, since the left-hand side is definite and the right-hand side has an arbitrary constant. And, while the first may be taken as an equivalent of ux , the latter is only so when we stipu- late that the constant in the term Cax, resulting from the performance of ^ , shall be taken zero. One difficulty & — a of this kind we met with at the beginning of Chapter IV., and we shall content ourselves with investigating the present one, leaving all future cases to the student's own examination. 14. Take then (28). Since ux is not considered a definite quantity, but as a representative of all the quantities that satisfy (19), there is no absurdity in representing it as equal to the quantity on the right-hand side of (28) which has n undetermined constants. All we have to ask is, whether on performing/ (E) on the right-hand side of (28) we shall obtain Xj and, this last being a perfectly definite quantity, while the right-hand side of (28) is indefinite, we might expect that some conditions of interpretation would be necessary in (28) to render the equivalence algebraical. But it is not so. For on performing f(E) on the first term, viz. //_ , the opera- ART. 15.] WITH CONSTANT COEFFICIENTS. 217 tion (E—a), which is one of those composing f(E)*, is absorbed in rendering this indefinite term strictly definite, so that the whole result of performing f(E) on it is strictly definite. Thus the result of performing / (E) on the right- hand side of (28) is a strictly definite quantity, and as under some circumstances it must equal X (which we know from the laws of the symbol E)t it must be actually equal to it*f. Ex. 2. ux+2 - 5ux+1 + Gux = 5* ; or (#-3) (E-2)ux= 5*; 5* _f 1 1 ] U*~(E-Z)(E-2) (E-3 j£-2j = 5* + C3x-ox + C'2x = 5*+ (73* + (72*. Zoo 15. The above is a general solution of linear difference- equations with constant coefficients. But, as we have seen that the part involving arbitrary constants is readily written down after the algebraical solution of the equation / (E) = 0, and that any particular integral will serve to complete the * It must be remembered that these operations being direct it is wholly unimportant in what order we perform them. t While it is true that/(E) } *** +&c. j X =X whatever X may, it is by (£j -m^ ) no means true that J ^ + &c. j / (E) X*=X. The importance of care in this respect if we would avoid loose reasoning may be exemplified by an example. In Linear Differential Equations such a quantity as -^ --- is often evaluated thus : cosmac _ (D - a) cos mx _ -mammx-aco&mx -mainmx-acoamx ~ The first step with the interpretation afforded by the second is wholly inadmissible : It should be thus : —mammx-acosmx - 218 LISTEAK DIFFEKENCE-EQUATIONS [CH. XI. solution, it is usually better to guess a particular integral, or at all events to obtain it by some special method. The forms of X for which this can readily be done are three, viz. (I.) When X is of the form ax. Since f(&)ax=f (a) . a* we obviously have TTT^T ax = -^-r ax. (II.) When X is a rational and integral function of x. Here we have only to expand f(E) in a series of ascending powers of A, and perform it in this shape on X. The result* will of course terminate, since X is rational and integral. Should /(J?) when expressed in terms of A assume the form V" Ar (A + B A + &c.), we must evaluate -^ or SrJT before apply- ing this method, or may omit the factor A"**, apply the method, and then perform 2r on the result. (III.) When X is of the form ax(f) (x\ where (j> (x) is a rational and integral function of x. Here the formula / (E) off (x) = axf(aE) $ (x) gives us which comes under our second rule. Sin mx and cos mx are really instances of (I.), though the results will be given in an impossible form. 16. Special cases of failure of these rules will occur, as in the analogous cases in differential equations. We shall con- clude the Chapter with two examples of this. Ex. 3. (E -a)(E- 1} ux = ax. Here /(a)=0; .v^L=oo. * Its determinateness will serve as our warrant for its truth. ART. 17.] WITH CONSTANT COEFFICIENTS. 219 But we may in this case proceed thus : (E -a)(E-b)~ (aE - a) (aE - b) 1 =o" - which comes under (II.). Ex. 4. (E-2f(E- 1) ux = aj»2a. This will be done in a precisely similar way : 23 A3 2^-1) "1 + 2A 17. In a short note in Tortolini's Annali (Series i. vol. v.) Maonardi gives a solution of the linear difference-equation with constant coefficients that does not require the preliminary solution of the algebraical equation for JB, but the results do not seem of much value. EXERCISES. Solve the equations : I- ux+z-3ux+l-4 ux^+2ux^ + ux = x(x-V)(x- 4. ux^ — 2mux+i + (m* + ^2) ux = ^x- 5. &ux + AX = a; + sin a;. 6. w^- 61*^ + 81*^ -3^ = ^ + (- 7. AX - SA?^ + 4^ = 2X (1 -f cos x). 8. A6M-2A=« 3". 220 EXERCISES. [CH. XI. 9. ^+2 i W*M* — cos mx' 10. u ± 2?i + n'u = 0. 11. A person finds his professional income, which for the first year was £a, increase in A.P., the common difference being £b. He saves every year — of his income from all sources, laying it out at the end of each year at r per cent. per annum. What will be his income when he has been x years in practice ? 12. A seed is planted — when one year old it produces ten-fold, and when two years old and upwards eighteen-fold. Every seed is planted as soon as produced. Find the number of grains at the end of the x* year. CHAPTER XII. MISCELLANEOUS PROPOSITIONS AND EQUATIONS. SIMUL- TANEOUS EQUATIONS. 1. SINCE no class of equations of an order higher than the first have been solved with the completeness which marks the solution of linear difference-equations with con- stant coefficients, it becomes very important to find what forms of equations can be reduced to this class. The most general case of this reduction is with regard to equations of the form $ 0) «Wi + A£ (a?) <£ (x - 1) Z ...... (1), where A^ Az ... An are constant, and <$> (x) a known function. These may be reduced to equations with constant coefficients by assuming ............ (2). For this substitution gives ^M = (x — 1) ... (1), we get, &o. = ......... (3), an equation with constant coefficients. 222 MISCELLANEOUS PROPOSITIONS AND EQUATIONS. [CH. XII. In effecting the above transformation we have supposed x to admit of a system of positive integral values. The general transformation would obviously be ux — $ (x — n) (x — n — 1) . . . (r), r being any particular value of x assumed as initial. Equations of the form are virtually included in the above class. For, assuming (x) = ax, they may be presented in the form * 0*0 ^ ......... (4). Hence, to integrate them it is only necessary to assume (S-M) (s-w+1) », ..................... (5). 2. By means of the proposition in the last article we can solve all linear binomial equations. Let the equation be .=-B...... ......... . ..... ....(6). Assume •Ax = vxvx_1...vx_n+l ........ . .......... (7). Take logarithms of both sides and let log vx_H+l = wx, then we have wx = logAx ............ (8), a linear difference-equation with constant coefficients. Solving this we obtain wa and thence vx) which enables us to put (6) into the form U^+U^X ............ . ...... ..... ...(9) by Art. 1, and thus the equation is solved. Such equations are however substantially equations of the first degree, and should be treated as such. They state a ART. 4.] SIMULTANEOUS EQUATIONS. 223 connection between consecutive members of the series ur, ur+n> ur+2n &c., and leave these last wholly unconnected with intermediate values of u. We should therefore assume x — ny and the equation would become a linear difference-equation of the first order, the independent variable now proceeding by unit increments. 3. Equations of the form ux^ux + axux^+lxux = cx ...... .............. .(10) can be reduced to linear equations of the second order, and, under certain conditions, to linear equations with constant coefficients*. Assume «.=%-«.• Then for the first two terms of the proposed equation, we have v, ™ vx • Whence substituting and reducing, we find v*n + & - O v*n ~ (*J>x + cx) vx = 0 ............. (11), a linear equation whose coefficients will be constant if the functions bx — ax+l and axbx + cx are constant, and which again by the previous section may be reduced to an equation with constant coefficients if those functions are of the respective forms A* (a), B &c. = 0, Eliminating the quantities p^ p^ &c. we obtain ,(37), an equation which must be satisfied by every solution of (36). Now the solution of (36) is ux = AOL* + B$x + &c. to n terms (38), where A, B, ... are arbitrary and a, /3, ... depend on pl} pz, ... and these last do not appear in equation (37) which we are now considering. Hence (38) will be the solution of (37), a, /3, ... being also considered arbitrary, thus making the full number of 2n arbitrary constants. By a slight variation in the method of elimination we can obtain the solution of a yet more general equation. Taking the last term of each of the equations to the other side and eliminating pvp^..pn_v we obtain ^•••WK+H "x+i> • U .u. "x+Zn-2" ' (39), 230 MISCELLANEOUS PROPOSITIONS AND EQUATIONS. [CH. XII. or calling the last determinant Px P^ = (-^)nPnPx (40), the solution of which may be written Thus the solution of the equation (writing n 4- 1 for n) .(42) .-i» ux+n is um = AQ.* + Bfix + &c. to n + 1 terms (43), where A,B, &c. and a, & ... are arbitrary constants limited by the two equations of condition and C — the determinant P for some value of x. If we take this last-named value to be zero, it is evident that 1, 1, 1, Ac? a, /3, 7, o= Aai Bp c A T> f*1 A, ±>, O, 1,1,1, -ABC... ^f;.7;;;;;;; a71 = AB C . . . product of squares of differences of oc, /3, 7. . . taken with the proper sign. Ex. The equation uu -u*=C... ..,(44) X X+Z X \ / may be supposed to be derived from the equation which gives also ART. 8.] SIMULTANEOUS EQUATIONS. 231 Whence eliminating p we have ux+? — uxux+2 = ux2 — ux_^ux+l and .'. = constant, since it is equal to its consecutive value. Hence ux = A£ + B/3X, where a/3 - 1, and (AoT1 + B&*) (Aoi + B&) -(A + B)* = C; + AB& - ZABap = Ckfr or C= Simultaneous Equations. 8. Instead of a single equation involving one function we may find that we have a system of n equations involving n unknown functions of the independent variable. The method by which we reduce this to the former case is so obvious that we shall not dwell upon it. We must by the performance of A or E obtain a system of derived equations sufficient to enable us by elimination to deduce a final equa- tion involving only one of the variables with its differences and successive values. The integrations of this will give the general value of that variable, and the equations employed in the process of elimination will enable us to express each other dependent variable by means of it. If the coefficients are constant we may simply separate the symbols and effect the eliminations as if those symbols were algebraic. ux+l - a*xvt = 0) Vi- «*.-OJ" Ex.1. From the first we have Hence eliminating vx+l by the second ux+z - c?x (x + 1) ux = 0, the solution of which is u=x-\ 232 MISCELLANEOUS PROPOSITIONS AND EQUATIONS. [CH. XII. and by the first equation Ex.2. ^+ w*fi- This may be written ... { (E- I)2 + 4>E] ux = - 2Ea* ; or This gives and from the first equation 9 z+i f _ -i \ 1)' + (a + !)* 9. On the subject of linear equations with variable coefficients the student should see a remarkable paper by Christoffel (Crelle, LV. 281), in which he dwells on the anomalies produced by the passage through a value which causes the coefficient of the first or last term to vanish. On the con- dition that an expression in differences should be capable of immediate summation, i.e. should be analogous to an exact differential, see Minich, (Tortolini, Series i. vol. i. 321). EXERCISES. Integrate the equations !• ux+2 ~~ xux+i + (x ~ 1) ux = sm x> ODe portion of the com- plementary function being a constant. 3. ux = x 4. v = EX. 5.] EXERCISES. 233 5. w^- 2(0-1) ^+1+(*-l)0*-2K = |^ 6. <=«- . 8. Integrate the simultaneous equations ) J ' 9. 10. «U -»•«*•*(* "9* 12. When the solution of a non-linear equation of the first order is made to depend upon that of a linear equation of the second order whose second member is 0 by assuming (Art. 3), shew that the two constants which appear in the value of vx effectively produce only one in that of ux. 13. The equation may be resolved into two equations of differences of the first order. 14. Given that a particular solution of the equation u*+2 ~~ a (a* + 1) u*+i + ^ ux = 0 is ux = ca 2 , deduce the general solution, and also shew that the above equation may be solved without the previous knowledge of a particular integral. 234 EXERCISES. [CH. XII. 15. The equation **«Viw*« = a (ux + ux+1 + ux+2) may be integrated by assuming ux = */a tan vx. 16. Shew also that the general integral of the above equa- tion is included in that of the equation i^+3 - ux = 0, and hence deduce the former. 17. Shew how to integrate the equation *WV2 + « 18. Solve the equations and shew that if m be the integral part of Jn, — converges as x increases to the decimal part of Jn. 19. If at be a fourth proportional to a, b, c, bl a fourth proportional to b, c, a, and ct to c, a, b, and «2, 62, c2 depend in the same manner on alt b1} c±, find the linear equation of differences on which an depends and solve it. 20. Solve the equation AV, + &(!-#) Awx + kux = 0. 21. Solve the equation ux+5) considering specially the case when (7 is zero. 22. If v0, v,, va, &c. be a series of quantities the succes- sive terms of which are connected by the general relation VU = OT-^-I» and if vQ) vt be any given quantities, find the value of vn. [S.P.] 23. If n integers are taken at random and multiplied together in the denary scale, find the chance that the figure in the unit's place will be 2. EX. 24.] EXERCISES. 235 24. Shew that a solution of the equation u^Uz+n-t ...ux = a (ux+n + ux^_l + ...u) is included in that of and is consequently where a. is one of the imaginary (n + l)th roots of unity, the n -f 1 constants being subject to an equation of condition. 25. Solve the equation and shew that it is equivalent to _4n-6 p ^n+1 ~ -7- ^n' [Catalan, Liouville, in. 508.] 26. Shew that can be satisfied by M2JJ = u^x+1 or Wg^j, and that thus its solu tion is 3. 5. 7. ..(2a?-l) r, 2.4.6...2a? M" ''4 '' 2a?-2) '1.3.5 ... (2aj-3)' and deduce therefrom the solution of uX4l=ux+(xz-x)ux_l. [Sylvester, Phil. Mag.] ( 236 ) CHAPTER XIII. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS. SYMBOLICAL AND GENERAL METHODS. 1. THE symbolical methods for the solution of differential equations whether in finite terms or in series (Diff. Equations, Chap, xvil.) are equally applicable to the solution of differ- ence-equations. Both classes of equations admit of the same symbolical form, the elementary symbols combining according to the same ultimate laws. And thus the only remaining difference is one of interpretation, and of processes founded upon interpretation. It is that kind of difference which exists between the symbols f^-J and 2. It has been shewn that if in a linear differential equation we assume x = ee, the equation may be reduced to the form a), ction of 6. Moreover, the symbols -^ obey the laws, U being a function of 6. Moreover, the symbols -^ and And hence it has been shewn to be possible, 1st, to express the solution of (1) in series, 2ndly, to effect by general theorems the most important transformations upon which finite integration depends. ART. 1.] LINEAR EQUATIONS, &C. 237 Now -n and ee are the equivalents of x -7- and x, and it is dd dx proposed to develope in this chapter the corresponding theory of difference-equations founded upon the analogous employ- ment of the symbols x — and xE, supposing A# arbitrary, and therefore =»£(#+ Aa?) - <£ (a?), PROP. 1. If the symbols IT and p be defined by the equations they will obey the laws f (TT) pmu = pmf (7r + m)u\ . . ffj\ m.JffL^m f (*), £/ie subject of operation in the second theorem being unity. 1st. Let Ace = r, and first let us consider the interpretation of pmux. Now pux = xEux = a^+r ; whence generally pX = a? (x + r) . . . {x + (m - 1) r} ux+mr, an equation to which we may also give the form pX = a (x + r) ... {x + (m — 1) r} Emux (5). If ux = 1, then, since ux+mr = 1, we have pml = x (x + r) ... {x + (m — 1) r}, to which we shall give the form pm = x (x + r) ... {x + (m - 1) r}, the subject 1 being understood. 238 LINEAR EQUATIONS [CH. XIII. 2ndly. Consider now the series of expressions irpmux) Tr*pmux,...Trnpmux. Now irpmux = x -r- x (x + r) . . . {x + (m — 1) r] u x = pm (TT + m) W-c. Hence and generally ir^pmux — pm (TT + m)tlux. Therefore supposing /(TT) a function expressible in ascend- ing powers of TT, we have •f I \ m m£ I I \ IG\ j (7T) p u — p J (IT + m) u ^o;, which is the first of the theorems in question. Again, supposing u=l, we have ,/"«_ ART. 1.] WITH VARIABLE COEFFICIENTS. 239 But TT! = x A 1 = 0, 7T51 = 0, &c. Therefore /(TT) pml=pmf(m)l. Or, omitting but leaving understood the subject unity, /W =/(»*)/>"' ..................... (7). PROP. 2. Adopting the previous definitions of IT and p, every linear difference-equation admits of symbolical expres- sion in the form /.(*)». t/,(T)/»».+/,WA.-+/.W/>X=^ ...... (8). The above proposition is true irrespectively of the parti- cular value of AOJ, but the only cases which it is of any im- portance to consider are those in which Ax = 1 and — 1. First suppose the given difference-equation to be XQux+n + -XX^ ... + -XX = (*) ......... (9). Here it is most convenient to assume Aa? = 1 in the expres- sions of TT and p. Now multiplying each side of (9) by flf(fl?+l) ... (OJ + W-1), and observing that by (5) xux+l = pux, x (x + 1) ux+2 = pX, &c., we shall have a result of the form & (*K + A 0*)/™, .» + <#>„ (*) A. = & (®) ••• (10). But since A# = 1, 7T = icA, /D = «^ = #A + a;. Hence # = - TT + p, and therefore , ft (a?) = ft (- w + p), &c. These must be expressed in ascending powers of p, regard baing paid to the law expressed by the first equation of (4). 240 LINEAR EQUATIONS [CH. XIII. The general theorem for this purpose, though its applica- tion can seldom be needed, is F0 (TT-p) = Fa (TT) - F, (*) p + Ft («•) ^ where F^ (TT), Fz (TT), &c., are formed by the law jf. Equations, p. 439.) The equation (10) then assumes after reduction the form (8). Secondly, suppose the given difference-equation presented in the form Xpm + XpM... + Xnv_=X. .............. (12). Here it is most convenient to assume A# = — 1 in the ex- pression of TT and p. Now multiplying (12) by x(x — 1) ... (^r — n + 1), and ob- serving that by (5) xu^ = pux, x(x- 1) ux_z = p*ux , &c., the equation becomes 0 («) W, + & W PUx'~+n (®) P1U>* = -5 but in this case as is easily seen we have # = TT + p} whence, developing the coefficients, if necessary, by the theo- rem where as before ^(^=^,-,(^-^-,(^-1). we have again on reduction an equation of the form (8). ART. 2.] WITH VARIABLE COEFFICIENTS. 241 2. It is not always necessary in applying the above methods of reduction to multiply the given equation by a factor of the form x (x + 1) ... (x + n — 1), or x (x— 1) ... (x— n + 1), to prepare it for the introduction of p. It may be that the constitution of the original coefficients X0, X^ ... Xn is such as to render this multiplication unnecessary; or the requisite factors may be introduced in another way. Thus resuming the general equation X0ux + Xlux_l... + ZX^O ............ (14), assume We find X^ + Xjcv^ ... Hence assuming 7r = ^^.J f>=xEt where A# = — 1, we have Xj,. + XjH>f...+X.p'v. = 0 ........... (16), and it only remains to substitute TT + p for x and develope the coefficients by (13). 3. A preliminary transformation which is often useful consists in assuming ux = fj?vx. This converts the equation X0ux + X1ux_1... + Xnux_n = Q ............ (17) into frXf.+p+Xf^ ...z.^-o ......... (is), putting us in possession of a disposable constant p. 4. When the given difference-equation is expressed di- rectly in the form = Q ............ (19), it may be convenient to apply the following theorem. B. F. D. 16 242 LINEAR EQUATIONS [CH. XIII. Theorem. If TT = x — , p = xE, then (20). To prove this we observe that since F (TT) pnu = pnF (TT + n) u, therefore F (TT + n) u = p~nF (TT) p*u, whence F(ir^ — n}u — pnF (TT) p~nu. Now reversing the order of the factors TT, TT — 1,. . .TT — n + 1 in the first member of (20), and applying the above theorem to each factor separately, we have ... TTU But p-v = (xE)-*x A# Aa; But p"u = x (x + r) . . . {# 4- (ft — 1) ^ } Enuy whence / A \n (7r-n+l) (TT-n+2) ... TTU = X (x+r) ... [x + (w-l)r) which, since r = A#, agrees with (20). When Aa; = 1, the above gives Hence, resuming (19), multiplying both sides by x (x + 1) . . . (x + n — 1), ART. 5.] WITH VARIABLE COEFFICIENTS. 243 and transforming, we have a result of the form Q(x)7r(7r-l)...('jr-n + l)u + fa (x) TT (TT - 1) . . . (IT - n + 2) u + &c. = 0. It only remains then to substitute x = — IT + p, develope the coefficients, and effect the proper reductions. Solution of Linear Difference-Equations in series. 5. Supposing the second member 0, let the given equation be reduced to the form /. « » +/, W /» +/, ("•) A - +/. W />"« = 0 ...... (22), and assume u — %ampm. Then substituting, we have 2 {/„ (*•) ampm +/, (*) «W>"" • - +/. W arfw+"} = 0. whence, by the second equation of (4), 2 {/, (m) a.p- +/, (m + 1) a^"1" ...+/.(« + n) a^m+"J = 0, ^ in which the aggregate coefficient of pm equated to 0 gives This, then, is the relation connecting the successive values of am. The lowest value of m, corresponding to which am is arbitrary, will be determined by the equation and there will thus be as many values of u expressed in series as the equation has roots. If in the expression of TT and p we assume A#= 1, then since pw=»a;(a? + l)...(» + w-l) ............... (24), 16—2 244 LINEAR EQUATIONS [CH. XIII. the series %ampm will be expressed in ascending factorials of the above form. But if in expressing TT and p we assume AOJ = — 1, then since pm = x(x-l) ... (x-m + 1) (25), the series will be expressed in factorials of the latter form. Ex. 1. Given required the value of ux in descending factorials. Multiplying by x, and assuming TT = x — , p = xE, where Ace = — 1, we have x(x — a)ux — (2x — a — 1) pux + (1 - (x)=x(as+I) ... (x + a- p *( = the complex operation 2 - , denoting division of the subject sc by x and subsequent integration, being repeated i times. Should X however be rational and integral it suffices to express it in factorials of the forms a?, x (x + 1), x (x + 1) (x + 2), &c. ART. 6.] WITH VARIABLE COEFFICIENTS. 247 to replace these by p, p2, p3, &c. and then interpret (27) at once by the theorem As to the complementary function it is apparent from (28) that we have Hence in particular if i = 1, we find (TT - a)'1 0 = pV1 0 -l) ...... (30). This method enables us to solve any equation of the form x(x+l) ... (x+n-1) A"w + Ajc (x + 1) ... -2)kn-1u...+Anu = X ...... (31). For symbolically expressed any such equation leads to the monomial form = Z ............ (32). Ex. 2. Given a? (x + 1) A2w - 2#Aw + 2^ = ^ (x + 1) (# + 2). The symbolical form of this equation is 7r(7r-l) M- Or (7T2 Hence u = (vr2 - 3?r + 2) 248 LINEAE EQUATIONS [CH. XIII. since the factors of 7r2 — STT + 2 are IT — 2 and TT — 1. Thus we have ) + g (6). Binomial Equations. 7. Let us next suppose the given equation binomial and therefore susceptible of reduction to the form u + (7r)pnu=U .................. (33), in which U is a known, u the unknown and sought function of x. The possibility of finite solution will depend upon the form of the function 0 (TT), and its theory will consist of two parts, the first relating to the conditions under which the equation is directly resolvable into equations of the first order, the second to the laws of the transformations by which equations not obeying those conditions may when possible be reduced to equations obeying those conditions. As to the first point it may be observed that if the equa- tion be it will, on reduction to the ordinary form, be integrable as an equation of the first order. Again, if in (33) we have 0 (T) = ^ W ^ (T - 1) - - - ^ (IT - n + 1 ), in which ty (TT) = — '—=, the equation will be resolvable into a system of equations of the first order. This depends upon the general theorem that the equation $ (TT) pu + a$ (TT) (j> (TT — 1) p*u . . . 4- an(j) (TT) 0 (TT - 1) ... (IT - n + 1) pnu = U ART. 7.] WITH VARIABLE COEFFICIENTS. 249 may be resolved into a system of equations, of the form U — qcf) (TT) pu = U, q being a root of the equation (Differential Equations, p. 405.) Upon the same principle of formal analogy the propositions upon which the transformation of differential equations de- pends (76. pp. 408—9) might be adopted here with the mere substitution of TT and p for E and ee. But we prefer to in- vestigate what may perhaps be considered as the most general forms of the theorems upon which these propositions rest. From the binomial equation (33), expressed in the form [l+(^p"}u=U, we have u = {l+ (TT) and -v/r (TT), is it possible to determine an operation % (TT) such that we shall have generally p")Z ...... (36), irrespectively of the form of X ? Supposing F(t) = t, we have to satisfy tWp'xWX^xW + Wp'X ......... (87)- Hence by the first equation of (4), $ (TT) x (w - n) p'X = + (TT) x W p'X, to satisfy which, independently of the form of X, we must have •f W % W = 0 M % (^ - n) ; Therefore solving the above difference-equation, (7r) = Substituting in (37), there results, or, replacing n. and therefore X by Til, , ART. 7.] WITH VARIABLE COEFFICIENTS. 251 ft / \\ If for brevity we represent 11^ j y4— |r by P> and drop the suffix from Xl since the function is arbitrary, we have Hence therefore TT) pn}*X = P^r (TT) -p*p-lPty (rr) p and continuing the process, Supposing therefore .F (^) to denote any function develop- able by Maclaurin's theorem, we have F { (TT) pn] X=PF{^ (TT) pn} P~1X. We thus arrive at the following theorem. THEOREM. The symbols TT and p combining in subjection to the law the members of the following equation are symbolically equi- valent, viz. A. From this theorem it follows, in particular, that we can always convert the equation u +

(TT) pn] X = pmF {0 (TT + m) pn] p^X, and deduce hence the consequence that the equation u + <£ (TT) pnu — U may be converted into by assuming u = pmv. 8. These theorems are in the following sections applied to the solution, or rather to the discovery of the conditions of finite solution, of certain classes of equations of considerable generality. In the first example the second member of the given equation is supposed to be any function of x. In the two others it is supposed to be 0. But the conditions of finite solution, if by this be meant the reduction of the dis- covery of the unknown quantity to the performance of a finite ART. 8.] WITH VARIABLE COEFFICIENTS. 253 number of operations of the kind denoted by 5, will be the same in the one case as in the other. It is however to be observed, that when the second member is 0, a finite integral may be frequently obtained by the process for solutions in series developed in Art. 5, while if the second member be X, it is almost always necessary to have recourse to the trans- formations of Art. 7. Discussion of the equation (ax + 1} ux+(cx + e) ux_^ + (fa + g) ux_^ = X ...... (a). Consider first the equation (ax + b)ux + (ex + e) u^ +f(x — 1) u^ — X. ..... (b). Let ux = fjL°vx, then, substituting, we have I* (ax + b)vx + fj, (ex + e) vx_^ +f(x - 1) vx_z = pT^X. Multiply by x and assume TT = x — , p = xE} in which Ao; = — 1, then fj? (ax* + bx) vx + p (ex + e) pvx whence, substituting TT + p for x and developing the coeffi- cients, we find (a?r2 + &TT) vx + p {(2a/* + c) TT + (b — a) fi + e} pvx =:xiJrx+2X. ........... (c), and we shall now seek to determine fj, so as to reduce this equation to a binomial form. 1st. Let p be determined by the condition a//,2 + cfi +/= 0, then makin = w4, (Z> — a) p + e^B, 254 LINEAR EQUATIONS [CH. XIII. we have (TT + - J vx + A (TT + -) pvx = or A or, supposing V to be any particular value of the second member obtained by Art. 6, for it is not necessary at this stage to introduce an arbitrary constant, A 7r~*~li TT \ a This equation can be integrated when either of the func- tions, B B b A' A~a> is an integer. In the former case we should assume w i A 1 w = W (e) Wx ^7r+bpWx~ a whence we should have by (A)-, In the latter case we should assume as the transformed equation ART. 8.] WITH VARIABLE COEFFICIENTS. 255 and should find The value of Wx obtained from (/) or (h) is to be sub- stituted in (e) or (g), wx then found by integration, and vx determined by (/) or Qi). One arbitrary constant will be introduced in the integration for wx> and the other will be due either to the previous process for determining Wx, or to the subsequent one for determining vx. Tt Thus in the particular case in which -r is a positive inte- ger, we should have a particular value of which, derived from the interpretation / S\~: of TT + -j ) 0 and involving an arbitrary constant, will be \ -A-J rt found to be =— - — . Substituting in (e) and reducing the J. ~T~ X equation to the ordinary unsymbolical form, we have fi (ax + b)wx + (A -pa) xw^ = — ^- , i. -(- JC and wx being hence found, we have for the complete integral. 2ndly. Let fju be determined so. as if possible to cause the second term of (c) to vanish. This requires that we have 2a/4 + c = 0, ( b — a) p + e = 0, and therefore imposes the condition 2ae + (b-- a) c = 0. 256 LINEAR EQUATIONS [CH. XIII. Supposing this satisfied, we obtain, on making yu, = — , or, representing any particular value of the second member by F, h* where an equation which is integrable if - be an odd number whe- ther positive or negative. We must in such case assume and determine first Wx and lastly va by h. To found upon these results the conditions of solution of the general equation (a), viz. (cue + 1) ux + (ex + e) u^ + (fa +g) ux_z = X, assume Then comparing which with (b) we see that it is only necessary in the expression of the conditions already deduced to change a(l + g) b into b -- ^-H^ , e into e - AKT. 9.] WITH VARIABLE COEFFICIENTS. 257 Solution of the above equation when JT=0 l>y definite integrals*. 9. If representing ux by u we express (a) in the form -A 9A (ax + b)u+(cx + e)e dxu + (fa +g)e~ dxu = 0, or -*. _9<* -^ -2^ x (a + ce dx +/e <**) u + (6 + ee dic + #e dx) u = 0, its solution in definite integrals may be obtained by Laplace's method for differential equations of the form each particular integral of which is of the form ..-tf-S- , «= Hn I ^ external to F, as if IT were algebraic. And this enables us to transform some classes of equations which are not binomial. Thus the solution of the equation /. W « +/, O) <£ W p« +/, (") <£ (*•) (IT - 1) A = u will be made to depend upon that of the equation by the assumption TT "=IT 13. While those transformations and reductions which depend upon the fundamental laws connecting TT and p, and are expressed by (4), are common in their application to differen- tial equations and to difference- equations, a marked difference exists between the two classes of equations as respects the conditions of finite solution. In differential equations where TT- = — ? p = e9, there appear to be three primary integrable du forms for binomial equations, viz. CL7T + b , Tr u + — - p u = U, ^ AKT. 14.] WITH VARIABLE COEFFICIENTS. 263 a(7T-n}Z + b _.._ ^ ( /l> 7T 7T — ^ primary in the sense implied by the fact that every binomial equation, whatsoever its order, which admits of finite solution, is reducible to some one of the above forms by the trans- formations of Art. 7, founded upon the formal laws connecting TT and p. In difference-equations but one primary integrable form for binomial equations is at present known, viz. 1 u H i pu = u, a-TT + b^ and this is but a particular case of the first of the above forms for differential equations. General considerations like these may serve to indicate the path of future inquiry. 14. Many attempts have been made to accomplish the general solution of linear difference-equations with variable coefficients, but the results are in all cases so complicated as to be practically useless. It will be sufficient if we mention Spitzer (Grunert, xxxn. and xxxm.) on the class specially consi- dered in this chapter, viz. when the coefficients are rational integral functions of the independent variable, Libri (Crelle, xn. 234), Binet (Memoires de VAcademie des Sciences, xix.). There is also a brief solution by Zehfuss (Zeitschrift, in. 177). EXERCISES. 1. Of what theorem in the Differential Calculus does (20), Art. 4, constitute a generalization ? 2. Solve the equation x (x + 1) A2M + x&u — tfu = 0. 3. Solve by the methods of Art. 7 the difference-equation of Ex. 1, Art. 5, supposing a to be a positive odd number. 4. Solve by the same methods the same equation, sup- posing a to be a negative odd number. ( 264 ) CHAPTER XIY. MIXED AND PARTIAL DIFFERENCE-EQUATIONS. 1. IF ux>y be any function of x and y, then A." ^-v > (1)" kyUx>y~ Ay These are, properly speaking, the coefficients of partial dif- ferences of the first order of uxy. But on the assumption that A# and Ay are each equal to unity, an assumption which we can always legitimate, Chap. I. Art. 2, the above are the partial differences of the first order of ux>y. On the same assumption the general form of a partial dif- ference of uXiy is (Ao^Ay)^2" °r When the form of w^ is given, this expression is to be inter- preted by performing the successive operations indicated, each elementary operation being of the kind indicated in (1). Thus we shall find It is evident that the operations -r— and -r— in combination A^ Ay are commutative. ART. 1.] MIXED AND PARTIAL DIFFERENCE-EQUATIONS. 2G5 Again, the symbolical expression of -^— in terms of -j- being A 6*4-1 l\X i\OC in which A# is an absolute constant, it follows that 1.2 - &c. c/ (A/?)" and therefore AY1 f A^J %>" " p*"^* ~~ WM* / A \m / A \n So, also, to express f -r— j f-r— ) ux,y it would be necessary to substitute for — — , — their symbolical expressions, to effect their symbolical expansions by the binomial theorem, and then to perform the final operations on the subject func- tion ux y. Though in what follows each increment of an independent variable will be supposed equal to unity, it will still be necessary to retain the notation -r- , — for the sake of dis- A# Ay tinction, or to substitute some notation equivalent by defi- nition, e.g. A,, Ay. These things premised, we may define a partial difference- equation as an equation expressing an algebraic relation between any partial differences of a function uXiJfte^, the func- tion itself, and the independent variables x, yy 'z ... Or in- stead of the partial differences of the dependent function, its successive values corresponding to successive states of incre- ment of the independent variables may be involved. 266 MIXED AND PARTIAL [CH. XIV. A A Thus x — ux + y — ux = 0, and xux^y + yuXty+l -(x+y) ux>y = 0, are, on the hypothesis of A# and AT/ being each equal to unity, different but equivalent forms of the same partial difference-equation. Mixed difference-equations are those in which the subject function is presented as modified both by operations of the form — , — , and by operations of the form -7- , -y- , singly t\x l\y ax ay or in succession. Thus x&_u + d_u =0 X &xU*'y dyUx'y is a mixed difference-equation. Upon the obvious subordi- nate distinction of ordinary mixed difference-equations and partial mixed difference-equations it is unnecessary to enter. Partial Difference-equations. 2. When there are two independent variables x and y, while the coefficients are constant and the second member is 0, the proposed equation may be presented, according to con- venience, in any of the forms Now the symbol of operation relating to x, viz. A,, or Exy combines with that relating to y, viz. Ay or Ey, as a constant with a constant. Hence a symbolical solution will be ob- tained by replacing one of the symbols by a constant quan- tity a, integrating the ordinary difference-equation which results, replacing a by the symbol in whose place it stands, and the arbitrary constant by an arbitrary function of the independent variable to which that symbol has reference. This arbitrary function must follow the expression which contains the symbol corresponding to a. ART. 2.] DIFFERENCE-EQUATIONS. 267 The condition last mentioned is founded upon the inter- pretation of (E — aj^X, upon which the solution of ordi- nary difference-equations with constant coefficients is ulti- mately dependent. For (Chap. XI. Art. 11) whence (#- a)"1 0 = 0^5*0 = a*-<(co + c^...4-cn_1O, the constants following the factor involving a. The difficulty of the solution is thus reduced to the diffi- culty of interpreting the symbolical result. Ex. 1. Thus the solution of the equation ux+1 — aux = 0, of which the symbolical form is Exux — aux = 0, being um=0a*t the solution of the equation ux+ltV — uXty+l — 0, of which the symbolic form is Exux>y-EyuXty=Q, will be To interpret this we observe that since Ey = edy we have K«,~/*> (y) = <£ (y + x). Ex. 2. Given i^ltm - u,t m - uXi, = 0. This equation, on putting u for ux>y) may be presented in the form E^u-u^O ........................ (1). Now replacing E9 by a, the solution of the equation a&xu — u=0 is u = l + a^'C 268 MIXED AND PARTIAL [CH. XIV. therefore the solution of (1) is <£(2/) ..................... (2), where <£ (y) is an arbitrary function of y. Now, developing the binomial, and applying the theorem we find « = ^(2/)+^(2/-l)+^^iV(y-2)+&o ...... (3), which is finite when x is an integer. Or, expressing (2) in the form developing the binomial in ascending powers of Eyt and in- terpreting, we have (4). Or, treating the given equation as an ordinary difference- equation in which y is the independent variable, we find as the solution u = (Vr}(x).... .................... (5). Any of these three forms may be used according to the requirements of the problem. Thus if it were required that when x = 0, u should assume the form emy, it would be best to employ (3) or to revert to (2) which gives (f> (y} = em2/, whence u (6). 3. There is another method of integrating this class of equations with constant coefficients which deserves attention. We shall illustrate it by the last example. AET. 3.] DIFFERENCE-EQUATIONS. 269 Assume uxy= 'ZCcfb*, then substituting in the given equa- tion we find as the sole condition Hence 1+6 a = -r and substituting, As the summation denoted by 2 has reference to all pos- sible values of b, and C may vary in a perfectly arbitrary manner for different values of b, we shall best express the character of the solution by making C an arbitrary function of b and changing the summation into an integration ex- tended from — oo to oo . Thus we have As (b) may be discontinuous, we may practically make the limits of integration what we please by supposing (b) to vanish when these limits are exceeded. If we develope the binomial in ascending powers of Z>, we have bv~x $ (b) db -oo ^, (6) db + &c Now f ° V - ty (6) being arbitrary if <£ (b) is ; 'hence which agrees with (4), 270 MIXED AND PARTIAL [CH. XIV. Although it is usually much the more convenient course to employ the symbolical method of Art. 2, yet cases may arise in which the expression of the solution by means of a definite integral will be attended with advantage; and the connexion of the methods is at least interesting. Ex. 3. Given &*xux_liy = A2^^. Replacing uXtV by u, we have or (A/^-A;,E;>=O. But A^^-l, A, = ^-l; therefore (E*Ey + Ey- Ey2Ex - Ex) u = 0, or (ExEy-l)(Ex-Ey}u=0. This is resolvable into the two equations (#.#,- 1)« = 0, (Ex-Ey)u The first gives Exu-Ey-lu = 0, of which the solution is The second gives, by Ex. 1, Hence the complete integral is u = $ (y - x) + A/T (y + x} . 4. Upon the result of this example an argument has been founded for the discontinuity of the arbitrary func- tions which occur in the solution of the partial differential equation ART. 5.] DIFFERENCE-EQUATIONS. 271 and thence, by obvious transformation, in that of the equation d*u *d*u_ dx*~ ' dt*~ It is perhaps needless for me, after what has been said in Chap. X., to add that I regard the argument as unsound. Analytically such questions depend upon the following, viz. whether in the proper sense of the term limit, we can regard sin x and cos x as tending to the limit 0, when x tends to become infinite. 5. When together with A,,, and A^, one only of the inde- pendent variables, e.g. x, is involved, or when the equation contains both the independent variables, but only one of the operative 'symbols A^, Ay, the same principle of solution is applicable. A symbolic solution of the equation will be found by substituting Ay for a and converting the arbitrary constant into an arbitrary function of y in the solu- tion of the ordinary equation And a solution of the equation will be obtained by integrating as if y were a constant, and replacing the arbitrary constant, as before, by an arbitrary function of y. But if x, y, Ax and Ay are involved together, this principle is no longer applicable. For although y and Av are constant relatively to x and A^, they are not so with respect to each other. In such cases we must endeavour by a change of variables, or by some tentative hypothesis as to the form of the solution, to reduce the problem to easier conditions. The extension of the method to the case in which the second member is not equal to 0 involves no difficulty. Ex! 4. Given u-xu^Q. 272 MIXED AND PARTIAL [CH. XIV, Writing u for uXty the equation may be expressed in the form u-asE^E^u^Q ..................... (1). Now replacing E~l by a, the solution of u — axE~l u = 0 or ux — axux_^ — 0 is Cx (x -!)...!. ax. Wherefore, changing a into E~l, the solution of (1) is = x (x — 1) . . . 1 . (y — x). 6. Laplace has shewn how to solve any linear equation in the successive terms of which the progression of differences is the same with respect to one independent variable as with respect to the other. The given equation being Ax y) Bxy) &c., being functions of x and y, let y = x — k\ then substituting and representing uXiV by vx, the equation assumes the form X.vx + Xyx_, + X2vx_2 + &c. = X, JT0, Xl ... X being functions of x. This being integrated, k is replaced by x — y, and the arbitrary constants by arbitrary functions of x — y. The ground of this method is that the progression of dif- ferences in the given equation is such as to leave x — y un- affected, for when x and y change by equal differences x — y is unchanged. Hence if x — y is represented by k and we take x and k for the new variables, the differences now having reference to x only, we can integrate as if k were constant. Applying this method to the last example, we have v* - »V = 0, ART. 7.] DIFFERENCE-EQUATIONS. 273 VX=CX (#-1) ...1, ^".^ (•-•!) ^•i.-.#(*^y)i which agrees with the previous result. The method may be generalized. Should any linear func- tion of x and y, e.g. x + y, be invariable, we may by assum- ing it as one of the independent variables, so to speak reduce the equation to an ordinary difference-equation; but arbitrary functions of the element in question must take the place of arbitrary constants. Ex. 5. Given u^-pu^^ -(l-p) M_Lm = 0. Here x + y is invariable. Now the integral of Hence, that of the given equation is 7. Partial difference-equations are of frequent occurrence in the theory of games of chance. The following is an ex- ample of the kind of problems in which they present them- selves. Ex. 6. A and B engage in a game, each step of which consists in one of them winning a counter from the other. At the commencement, A has x counters and E has y counters, and in each successive step the probability of A's winning a counter from B is p, and therefore of B's winning a counter from A, 1 — p. The game is to terminate when either of the two has n counters. What is the probability of .A's win- ning it ? Let u be the probability that A will win it, any positive values being assigned to x and y. B. F. D. 18 274 MIXED AND PAETIAL [CH. XIV. Now A's winning the game may be resolved into two alternatives, viz. 1st, His winning the first step, and after- wards winning the game. 2ndly, His losing the first step, and afterwards winning the game. The probability of the first alternative is pux^y_^ for after A'u winning the first step, the probability of which is p, he will have x + 1 counters, B, y — 1 counters, therefore the probability that A will then win is w^.1(2/_r Hence the pro- bability of the combination is pux^>y_^ The probability of the second alternative is in like manner P;-j0V»i. Hence, the probability of any event being the sum of the probabilities of the alternatives of which it is composed, we have as the equation of the problem the solution of which is, by the last example, «*, = * (a + 30 + £4r)V (x + ti- lt remains to determine the arbitrary functions. The number of counters sc+y is invariable through the game. Represent it by my then Now A's success is certain if he should ever be in possession of n counters. Hence, if x = n, ux = 1. Therefore Again, A loses the game if ever he have only m — n counters, since then B will have n, counters. Hence ART. 8.] DIFFERENCE-EQUATIONS. 275 1—40 The last two equations give, on putting P — — — , _ pm-n whence which is the probability that A will win the game. Symmetry therefore shews that the probability that B will win the game is and the sum of these values will be found to be unity. The problem of the ' duration of play ' in which it is pro- posed to find the probability that the game conditioned as above will terminate at a particular step, suppose the rth, depends on the same partial difference-equation, but it in- volves great difficulty. A very complete solution, rich in its analytical consequences, will be found in a memoir by the late Mr Leslie Ellis (Cambridge Mathematical Journal, Vol. IV. p. 182). Method of Generating Functions. 8. Laplace usually solves problems of the above class by the method of generating functions, the most complete statement of which is contained in the following theorem. Let u be the generating function of um n which the difference-equation does not determine, then, correspond- ing to such initial values, terms will arise in the second member of (1) so that the differential equation will assume the form ' ar ••) *pe+gf"u=F(™. »•••) ......... (4)- If the difference-equation have constant coefficients the differential equation merges into an algebraic one, and the generating function will be a rational fraction. This is the case in most, if not all, of Laplace's examples. It must be borne in mind that the discovery of the gene- rating function is but a step toward the solution of the dif- ference-equation, and that the next step, viz. the discovery of the general term of its development by some independent process, is usually far more difficult than the direct solution of the original difference-equation would be. As I think that in the present state of analysis the interest which belongs to this application of generating functions is chiefly historical, I refrain from adding examples. ART. 9.] DIFFERENCE-EQUATIONS. 277 Mixed Difference-equations. 9. When a mixed difference-equation admits of resolution into a simple difference-equation and a differential equation, the process of solution is obvious. Ex. 7. Thus the equation A du , du , A -j — a&u — o -T- + abu = 0 doc doc being presented in the form the complete value of u will evidently be the sum of the values given by the resolved equations f\ \ r -j — au = 0, DM — ou=0. dx Hence where ct is an absolute, C2 a periodical constant. Ex. 8. Again, the equation d being resolvable into the two equations, dz dz\* we have, on integration, z — ex + c2, where c is an absolute, and C a periodical constant. 278 MIXED AND PAETIAL [CH. XIV. Mixed difference-equations are reducible to differential equations of an exponential form by substituting for Ex or ct d kx their differential expressions edx, edx— 1. Ex. 9. Thus the equation Aw — -r- = 0 becomes and its solution will therefore be the values of m being the different roots of the equation 10. Laplace's method for the solution of a class of partial differential equations (Diff. Equations, p. 440) has been ex- tended by Poisson to the solution of mixed difference-equa- tions of the form (1), where L, M, N, V are functions of as. Writing u for uat and expressing the above equation in the form ax ax it is easily shewn that it is reducible to the form u + (N-LH- L'} u=V, where L' = -7:- . Hence if we have ax (2), ART. 10.] DIFFERENCE-EQUATIONS. 279 the equation becomes which is resolvable by the last section into a mixed difference- equation and a differential equation. But if the above condition be not satisfied, then, assuming (E+L}u = v ........................ (3), we have + (N-LM-L'}u=V, whence -£*»),.tr N-LM-L' which is expressible in the form u = Ax~+Bxv + Ca, Substituting this value in (3) we have which, on division by AtH, is of the form The original form of the equation is thus reproduced with altered coefficients, and the equation is resolvable as before into a mixed difference-equation and a differential equation, if the condition £/ = () .................. (5) is satisfied. If not, the operation is to be repeated. 280 MIXED AND PARTIAL [CH. XIV. An inversion of the order in which the symbols -5- and dx E are employed in the above process leads to another reduc- tion similar in its general character. Presenting the equation in the form u = V where M_^ = E~1M, its direct resolution into a mixed differ- ence-equation and a differential equation is seen to involve the condition = 0 (6). If this equation be not satisfied, assume d and proceeding as before a new equation similar in form to the original one will be obtained to which a similar test, or, that test failing, a similar reduction may again be applied. Ex. 10. Given —^ — a -^ + (x ± n) ux+l — axux = 0. This is the most general of Poisson's examples. Taking first the lower sign we have i= — a, M=x—ny N= — ax. Hence the condition (2) is not satisfied. But (3) and (4) give (E — a) u = v, dv . N _ + (*-«)« u = — ~' whence AKT. 11.] DIFFERENCE-EQUATIONS. 281 or, on reducing, Comparing this with the given equation, we see that n reductions similar to the above will result in an equation of the form dwx, t dwx which, being presented in the form +• (*—)«.--<* is resolvable into two equations of the unmixed character. Poisson's second reduction applies when the upper sign is taken in the equation given : and thus the equation is seen to be integrable whenever n is an integer positive or nega- tive. Its actual solution deduced by another method will be given in the following section. 11. Mixed difference-equations in whose coefficients x is involved only in the first degree admit of a symbolical solution founded upon the theorem (Differential Equations, p. 445.) The following is the simplest proof of the above theorem. Since . fd\ . fd d'\ Y -7- #w = >Jr -j- + j~ xut T \dxj r \dx dxj if in the second member -7- operate on x only, and -r- on «, Ct;3? " Ctd? we have, on developing and effecting the differentiations which have reference to xt 282 MIXED AND PARTIAL [CH. XIV. — ) u = v, then , fd\ f . fd\rl ty hj- # 1^1 jr.lr v== r \dx) { r \dx)) or if i/r f-T-J be replaced by ev Vto/ , «*<£)«-*<£) Inverting the operations on both sides, which involves the inverting of the order as well as of the character of successive operations, we have the theorem in question. Let us resume Ex. 10, which we shall express in the form n being either positive or negative. Now putting u for ux A u + x (edx -a)u = Q. £ Let (eda: - a) M = ^, then we have j_ \dx ART. 11.] DIFFERENCE-EQUATIONS. 283 Or, d Hence, d and therefore by (1), ,= ......... (6). It is desirable to transform a part of this expression. By (1), we have and by another known theorem, The right-hand members of these equations being sym bolically equivalent, we may therefore give to (6) the form (6£_aro ......... (c). Now ^=(6^ — a)'1 zt therefore substituting, and replacing by #, ^ = (^-a)n-1€-?^) 'e^^-arO ......... (A). Two cases here present themselves. 284 MIXED AND PARTIAL [CH. XIV. First, let n be a positive integer ; then since / 77T st\~n (\ nX ( f> I l_ n \MJ ~~~ Ct) == (A ~\~ 1 ~~- d) we have u = (& + l-ay-1e-?{C+f£a*(cQ + clx... + cnixn-1} '(d), as the solution required. This solution involves superfluous constants. For inte- grating by parts, we have «2 «2 «2 «2 Je2 axxrdx = e2 axaTl + log a Je¥ axxr~ldx +(r — l)/e¥ afx^dx, and in particular when r = 1, ^ «2 *2 ^ Je2 CLxxdx = e2 a37 + log &/e2 ttxd^. These theorems enable us, r being a positive integer, to reduce the above general integral to a linear function of the elementary integrals Je2 axdx, and of certain algebraic X* terms of the form e2 ax xm, where ra is an integer less than r. Now if we thus reduce the integrals involved in (d), it will be found that the algebraic terms vanish. For (A + 1 - a)71'1 6~T (e? axxm) = (A + 1 - a = 0, since m is less than r, and the greatest value of r is n — 1 . It results therefore that (d) assumes the simpler form, u = (A + 1 - a)"'1 e^ (C0 + CJ? axdx) ; ART. 11.] DIFFERENCE-EQUATIONS. 285 and here 00 introduced by ordinary integration is an absolute constant, while C^ introduced by the performance of the operation X is a periodical constant. A superfluity among the arbitrary constants, but a super- fluity which does not affect their arbitrariness, is always to be presumed when the inverse operations by which they are introduced are at a subsequent stage of the process of solu- tion followed by the corresponding direct operations. The particular observations of Chap. xvn. Art. 4 (Differential Equations) on this subject admit of a wider application. Secondly, let n be 0 or a negative integer. It is here desirable to change the sign of n so as to express the given equation in the form du. du , while its symbolical solution (A) becomes .-<*-.y~."(|)V(*-«)-a And in both n is 0 or a positive integer. / d N"1 Now since (E - a)n 0 = 0, and [-5- 1 0 = C, we have \dxj u = (E-a)-n~lCe^ = C(E- ap-1 i^ + (E - a)""'1 0 >+1 0 = a^S^a^e 8 + ax (c0 + ctx . . . + cxn}. But here, while the absolute constant Cl is arbitrary, the n + l periodical constants c0, cr..cn are connected by n rela- tions which must be determined by substitution of the above unreduced value of u in the given equation. 286 MIXED AND PARTIAL [CH. XIV. The general expression of these relations is somewhat com- plex; but in any particular case they may be determined without difficulty. Thus if a — 1, n = 1, it will be found that If a = 1, n = 2, we shall have and so on. The two general solutions may be verified, though not easily, by substitution in the original equation. 12. The same principles of solution are applicable to mixed partial difference-equations as to partial difference- equations. If A,. and ^- are the symbols of pure operation cty involved, and if, replacing one of these by a constant m, the equation becomes either a pure differential equation or a pure difference-equation with respect to the other, then it is only necessary to replace in the solution of that equation m by the symbol for which it stands, to effect the corresponding change in the arbitrary constant, and then to interpret the result. Ex.11. A^-a^ = 0. dy Replacing -^- by m} and integrating, we have o u = c (1 + am)x. Hence the symbolic solution of the given equation is ART. 12.] DIFFERENCE-EQUATIONS. 287 (y) being an arbitrary function of y. Ex. 12. Given ux+li y- — ux>y = Vxt from the equation where A affects x only ; and, assuming as initial conditions d Ux'"~ ' ' dQUx'° shew that where A, \ and /JL are constants (Cambridge Problems). B. F. D. 19 290 EXERCISES. [CH. XIV. 11. Given with the conditions ^._x = 0, W0i0 = 0, and %>x+1 = 0, find ^>3,. [Cayley, Tortolini, Series II. Vol. n. p. 219.] 12. uXtV = M^§1 + u^ + &c. ... + 1^,. [De Morgan, Oam6. Math. Jour. Vol. IV. p. 87.] ( 291 CHAPTER XV. OF THE CALCULUS OF FUNCTIONS. 1. THE calculus of functions in its purest form is dis- tinguished by this, viz. that it recognizes no other operations than those termed functional. In the state to which it has been brought more especially by the labours of Mr Babbage, it is much too extensive a branch of analysis to permit of our attempting here to give more than a general view of its objects and its methods. But it is proper that it should be noticed, 1st, because the Calculus of Finite Differences is but a particular form of the Calculus of Functions ; 2ndly, because the methods of the more general Calculus are in part an application, in part an extension of those of the particular one. In the notation of the Calculus of Functions, <£ {^r (#)} is usually expressed in the form T/T#, brackets being omitted except when their use is indispensable. The expressions <£#, (jxjxfrx are, by the adoption of indices, abbreviated into 3x, &c. As a consequence of this notation we have $x = x independently of the form of <£. The inverse form ~1 is, it must be remembered, defined by the equation w*=* ........................... a). Hence ~1 may have different forms corresponding to the same form of . Thus if # = a? -f ax, we have, putting x = t, .-!, _ a ± V(<*2 + 4t) and (j>~* has two forms, 19—2 292 OF THE CALCULUS OF FUNCTIONS. [CH. XV. The problems of the Calculus of Functions are of two kinds, viz. 1st. Those in which it is required to determine a func- tional form equivalent to some known combination of known forms; e.g. from the form of tyx to determine that of ^x. This is exemplified in B, page 167. 2ndly. Those which involve the solution of functional equations, i.e. the determination of an unknown function from the conditions to which it is subject, not as in the pre- vious case from the known mode of its composition. We may properly distinguish these problems as direct and inverse. Problems will of course present themselves in which the two characters meet. Direct Problems. 2. Given the form of tyx, required that o There are cases in which this problem can be solved by successive substitution. Ex. 1. Thus, if tyx = x", we have •Wx = (of)*=af, and generally ^nx — of". Again, if on determining tf*x, ty*x as far as convenient it should appear that some one of these assumes the particular form x, all succeeding forms will be determined. Ex. 2. Thus if -fyx = 1 — x, we have -fy?x = 1 —(1 —x) =x. Hence -nx — 1 — x or x accordin as n is odd or even. Ex. 3. If tyx = — '— , we find X — X - - , ijr9x = x. X ART. 2.] OF THE CALCULUS OF FUNCTIONS. 293 Hence ^rnx = x, ^ — - or - : according as on dividing n by 3 the remainder is 0, 1 or 2. Functions of the above class are called periodic, and are distinguished in order according to the number of distinct forms to which ^nx gives rise for integer values of n. The function in Ex. 2 is of the second, that in Ex. 3 of the third, order. Theoretically the solution of the general problem may be made to depend upon that of a difference- equation of the first order by the converse of the process on page 167. For assume *>* = *„, -f1*-*.* .................. (2). , since ^r""1"1 x = ty^x, we have (3)- The arbitrary constant in the solution of this equation may be determined by the condition ^ = tyx, or by the still prior condition 0 2. The trigono- metrical solution therefore applies when the expression repre- sented by z/2 is positive, the exponential one when it is negative. In the case of v = 0 the difference-equation (12) becomes the integral of which is X y~ *~ P Determining the constant as before we ultimately get - (17) /u, x — fi — act a result which may also be deduced from the trigonometrical solution by the method proper to indeterminate functions. 298 OF THE CALCULUS OF FUNCTIONS. [CH. XV. Periodical Functions. 3. It is thus seen, and it is indeed evident a priori, that in the above cases the form of tyxt is similar to that of i/r^, but with altered constants. The only functions which are known to possess this property are a + bt , — — - and at . c + et On this account they are of great importance in connexion with the general problem of the determination of the possible forms of periodical functions, particular examples of which will now be given. Ex. 7. Under what conditions is a + bt a periodical func- tion of the #* order ? By Ex. 4 we have and this, for the particular value of x in question, must reduce to t Hence equations which require that b should be any Xth root of unity except 1 when a is not equal to 0, and any xih root of unity when a is equal to 0. Hence if we confine ourselves to real forms the only pe- riodic forms of a+bt are t and a — t, the former being of every order, the latter of every even order. Ex. 8. Required the conditions under which is a periodical function of the xih order. In the following investigation we exclude the supposition of e — 0, which merely leads to the case last considered. ART. 3.] OF THE CALCULUS OF FUNCTIONS. 299 Making then in (16) tyxt = t, we have tan"1 t-^-x tan"1-) (18), V fJL/ - u! ( . t - a ,v\ or - = tan tan"1 x tan - 1 , v \ v IJL/ an equation which, with the exception of a particular case to be noted presently, is satisfied by the assumption x tan""1 — = iir, i being an integer. Hence we have V ITT ,., m - = tan — (19), or, substituting for v and //, their values from (13), (20). x \ / whence we find 2^7r . » 4:0, COS — £P The case of exception above referred to is that in which v = 0, and in which therefore, as is seen from (19), i is a mul- tiple of x. For the assumption v = 0 makes the expression for t given in (18) indeterminate, the last term assuming the form 0 x oo . If the true limiting value of that term be found in the usual way, we shall find for t the same expression as was obtained in (17) by direct integration. But that expression would lead merely to # = 0 as the condition of periodicity, a condition which however is satisfied by all functions what- ever, in virtue of the equation fit — t. The solution (9) expressed in exponential forms does not lead to any condition of periodicity when a, 6, c, e are real quantities, 300 OF THE CALCULUS OF FUNCTIONS. [CH. XV. We conclude that the conditions under which - , when c + et not of the form A + Bt, is a periodical function of the Xth order, are expressed by (20), i being any integer which is not a multiple ofx*. 4. From any given periodical function an infinite number of others may be deduced by means of the following theorem. THEOREM. If ft be a periodical function, then f~lt is also a periodical function of the same order For let then And continuing the process of substitution tw* = <£/V*. Now, if /ft be periodic of the wth order, fnt = t, and Hence -^"t = f^t = t. Therefore tyt is periodic of the 71th order. Thus, it being given that 1 — t is a periodic function of t of the second order, other such functions are required. Represent 1 — t by/£. Then if t = t\ If xf then x=:f(xy c), and changing x into any given function tyx, Eliminating c between these two equations we have a result of the form F(x, fa <£tjr#) = 0 (1). This is a functional equation, the object of the solution of which would be the discovery of the form , those of F and -fy being given. It is evident that neither the above process nor its result would be affected if c instead of being a constant were a func- tion of x which did not change its form when x was changed into -fyx. Thus if we assume as a primitive equation (a), and change x into — x, we have £(-«) = -(» + - c Eliminating c we have, on reduction, 302 .OF THE CALCULUS OF FUNCTIONS. [CH. XV. a functional equation of which (a) constitutes the complete primitive. In that primitive we may however interpret c as an arbitrary even function of x, the only condition to which it is subject being that it shall not change on chang- ing x into — as. Thus we should have as particular solu- tions these being obtained by assuming c — cos x and a? respectively. Difference-equations are a particular species of functional equations, the elementary functional change being that of x into x + 1. And the most general method of solving func- tional equations of all species, consists in reducing them to difference-equations. Laplace has given such a method, which we shall exemplify upon the equation 0 ..................... (2), the forms of i|r and % being known and that of sought. But though we shall consider the above equation under its general form, we may remark that it is reducible to the simpler form (1). For, the form of ^r being known, that of ^/r"1 may be presumed to be known also. Hence if we put tyx — z and — ir we have and this, since ty'1 and ^ are known, is reducible to the general form (1). Now resuming (2) let (3) Hence vt and ut being connected by the relation the form of <£ will be determined if we can express vt as a function of ut. AKT. 5.] OF THE CALCULUS OF FUNCTIONS. 303 Now the first two equations of the system give on elimi- nating x a difference-equation of the form the solution of which will determine ut) therefore tyx, there- fore, by inversion, a? as a function of t. This result, together with the last two equations of the system (3), will convert the given equation (2) into a difference-equation of the first order between t and vt, the solution of which will determine vt as a function of t, therefore as a function of ut since the form of ut has already been determined. But this deter- mination of vt as a function of ut is equivalent, as has been seen, to the determination of the form of (/>. Ex. 9. Let the given equation be (f> (mx) — a(j> (x) = 0. Then assuming x = ut, mx^u^] $(x)=vt) (m*) = vj' we have from the first two u^ - mut = 0, the solution of which is Again, by the last two equations of (a) the given equation becomes whence v4 = CV .......................... (c). Eliminating t between (6) and (c), we have logut-logC v = C'a lo*m . Hence replacing ut by x, vt by x, and C'a~]°sm by Clt we have log*- x=Clalo«m ........................... (d). 304 OF THE CALCULUS OF FUNCTIONS. [CH. XV. And here C^ must be interpreted as any function of x which does not change on changing x into mx. If we attend strictly to the analytical origin of Cl in the above solution we should obtain for it the expression /_, log x \ ( . log x \ 0 cos STT + a, cos ^ + &c. sin + 6 sin ^ + &c. lora a0, a1? 6j, &c. being absolute constants. But it suffices to adopt the simpler definition given above, and such a course we shall follow in the remaining examples. Ex.10. Given <£ fr - «* 0») Assuming 1-f a? we have or w (ar), and eliminating „ -tan-1* ART. 6.] OF THE CALCULUS OF FUNCTIONS. 305 (7j being any function of x which does not change on chang- . . 1+ic ing x into - - . JL ~~ X 6. Linear functional equations of the form x ...+an(f>(x)=X ...... (6), where ^ (as) is a known function of x, may be reduced to the preceding form. For let TT be a symbol which operating on any function (x) has the effect of converting it into (fr-fr (x). Then the above equation becomes > (x) + ay-^ (x)... + an (x) = Xy or (7). It is obvious that TT possesses the distributive property expressed by the equation 7T (U + V) — 7TU + TTV> and that it is commutative with constants so that Trau = airu. Hence we are permitted to reduce (7) in the following manner, viz. j, ra2 ... being the roots of Q .................. (9), and NI , N2 . . . having the same values as in the analogous resolution of rational fractions. Now if (TT — m)~* X = $ (x), we have (TT — m) 0 (x} — X, B. F. D, 20 306 OF THE CALCULUS OF FUNCTIONS. [CH. XV. or <£i/r (as) — m (as) = X, to which Laplace's method may be applied. Ex. 11. Given (m*x) + acj> (mx) + b(f> (as) = xn. Representing by a and ft the roots of x* + ax + b = 0, the solution is „*, C and G' being functions of x unaffected by the change of x into mx. Here we may notice that just as in linear differential equations and in linear difference-equations, and for the same reason, viz. the distributive character of the symbol TT, the complete value of <£ (x) consists of two portions, viz. of any particular value of cj> (x) together with what would be its complete value where X = 0. This is seen in the above example. 7. There are some cases in which particular solutions of functional equations, more especially if the known functions involved in the equations are periodical, may be obtained with great ease. The principle of their solution is as follows. Supposing the given equation to be F(x, x, # as a function of x and tyx, and therefore since tyx is supposed known, as a function of x. If tyx is a periodical function of the third order, it would be necessary to effect the substitution twice in succession, and then to eliminate (ftifras, and $^x\ and so on according to the order of periodicity o ART. 7.] OF THE CALCULUS OF FUNCTIONS. 307 1 —X Ex. 12. Given -- -- L -J- (C 1 —x The function T - is periodic of the second order, Change _L ~\~ x 1 — x then x into ^ - , and we have , 2 # and <£i/r#. In such cases we must either, with Mr Babbage, treat the given equation as a particular case of some more general equation which is unsymmetrical, or we must endeavour to solve it by some more general method like that of Laplace. Ex. 13. Given This is a particular case of the more general equation m and n being constants which must be made equal to 1 and 0 respectively, and %# being an arbitrary function of x. 10—2 808 OF THE CALCULUS OF FUNCTIONS. [CH. XV. Changing & into ^ — x} we have 7T Eliminating 1-^ —x from the above equations we find Therefore (A i \i2 1 n ( fir \) \6 (x)Y = i— - 4- 1 -- o -\yx — my hv — ^ r • ir v /J 1+m l-m2]^ ^ \2 /j Now if m become 1 and n become 0, independently, the fraction = — : —z becomes indeterminate, and may be replaced by an arbitrary constant c. Thus we have {*(*)}'= I + <%(*) -< whence, merging c in the arbitrary function, The above is in effect Mr Babbage's solution, excepting that, making m and n dependent, he finds a particular value for the fraction which in the above solution becomes an arbi- trary constant. Let us now solve the equation by Laplace's method. Let [(j) (x)}* ~ ^j and we have Hence assuming ART. 7.] OF THE CALCULUS OF FUNCTIONS. 309 we have The solutions of which are Hence Therefore or Therefore in which C must be interpreted as a function of x which does not change when x is changed into ^ — x. It is in fact an 7T arbitrary symmetrical function of x and — — a?, The previous solution (12) is included in this. For, equating the two values of <£ (x) with a view to determine C, we find •310 OF THE CALCULUS OF FUNCTIONS. [CH. XV. n _ /7T _ • /,_%W %U IT IT 7T which is seen to be symmetrical with respect to x and -~—x. 8. There are certain equations, and those of no incon- siderable importance, which involve at once two independent variables in such functional connexion that by differentiation and elimination of one or more of the functional terms, the solution will be made ultimately to depend upon that of a differential equation. Ex. 14. Representing by P<£ (#) the unknown magnitude of the resultant of two forces, each equal to P, acting in one plane and inclined to each other at an angle 2x, it is shewn by Poisson (Mecanique^ Tom. i. p. 47) that on certain assumed principles, viz. the principle that the order in which forces are combined into resultants is indifferent — the principle of (so-called) sufficient reason, &c., the following functional equation will exist independently of the particular values of x and y, viz. Now, differentiating twice with respect to x, we have And differentiating the same equation twice with respect $'(X + y} + $' (*-y) = $(«)$" (y). Hence ART. 8.] OF THE CALCULUS OF FUNCTIONS. 311 • n / \ Thus the value of , ) is quite independent of that of x. $(x) We may therefore write m being an arbitrary constant. The solution of this equa- tion is <£ (x) = J-e7"* + -Be"77135, or <}>(x) = A cos m^ + B sin war. Substituting in the given equation to determine A and B, we find (f) (V) = €™ + (rm*i or 2 cos ra#. Now assuming, on the afore-named principle of sufficient reason, that three equal forces, each of which is inclined to the two others at angles of 120°, produce equilibrium, it fol- lows that (x) be taken, and that m be made equal to 1. Thus <£(#) = 2 cos x. And hence the known law of compo- sition of forces follows. Ex. 15. A ball is dropped upon a plane with the intention that it shall fall upon a given point, through which two per- pendicular axes x and y are drawn. Let (x) dx be the probability that the ball will fall at a distance between x and x + dx from the axis y, and (y) dy the probability that it will fall at a distance between y and y + dy from the axis x. Assuming that the tendencies to deviate from the respective axes are independent, what must be the form of the function (x) in order that the probability of falling upon any par- ticular point of the plane may be independent of the position of the rectangular axes ? (Herschel's Essays.) The functional equation is easily found to be 312 EXERCISES. [CH. XV. Differentiating with respect to x and with respect to y, we have r™ « <> <> Therefore r ' \ ( = -r~^. x (x) y$ (y) Hence we may write a differential equation which gives The condition that <^> (x) must diminish as the absolute value of x increases shews that m must be negative. Thus we have (x) = Ce'h'x\ EXERCISES. 2jj 1. If (x] = 1 -- „ , determine <£n (x). i. ~~ x 2. If $ (x) = 2#2 - 1, determine <^n (x). 3. If -^ (Q = - - and ty* (t) = -^ — „ , shew, by means of C ~i 6t *~ the necessary equation ^r^v (t} = ^^ (t), that A = E =°~B a ~ e c—b' EX. 4.] EXERCISES. 313 4. Shew hence that tyx (<) may be expressed in the form the equation for determining bx being - 66* - ae = 0, and that results equivalent to those of Ex. 5, Art. 2, may hence be deduced. Solve the equations 5- /(*)+/ (x, y) + *J— 1 ty (x, y) may be of the form F(x+y J-l). [Dienger, Grunert, x. 422.] 20. Shew that _dnu dn~lu n dxn ' dxn~l satisfies the equation dz ^=z^"' u being any function of x. If a regular polygon, which is inscribed in a fixed circle, be moveable, and if x denote the variable arc between one of its angles and a fixed point in the circumference, and zn the ratio, multiplied by a certain constant, of the distances from the centre of the feet of perpendiculars drawn from the nih and (n — l)th angles, counting from A, on the diameter through the fixed point, prove that zn is a function which satisfies the equation. 21. If (z) = (x) (j> (y), where z is a function of x and y determined by the equation /(z) =/(#)/(«/), find the form of (x). ( 316 ) CHAPTER XVI, GEOMETEICAL APPLICATIONS. 1. THE determination of a curve from some property con- necting points separated by finite intervals usually involves the solution of a difference-equation, pure or mixed, or more generally of a functional equation. The particular species of this equation will depend upon the law of succession of the points under consideration, and upon the nature of the elements involved in the expression of the given connecting property. Thus if the abscissse of the given points increase by a constant difference, and if the connecting property consist merely in some relation between the successive ordinates, the determination of the curve will depend on the integration of a pure difference-equation. But if, the abscissse still increas- ing by a constant difference, the connecting property consist in a relation involving such elements as the tangent, the normal, the radius of curvature, &c., the determining equa- tion will be one of mixed differences. If, instead of the abscissa, some other element of the curve is supposed to increase by a constant difference, it is necessary to assume that element as the independent variable. But when no obvious element of the curve increases by a constant difference, it becomes necessary to assume as in- dependent variable the index of that operation by which we pass from point to point of the curve, i.e. some number which is supposed to measure the frequency of the operation, and which increases by unity as we pass from any point to the succeeding point. Then we must endeavour to form two difference-equations, pure or mixed, one from the law of succession of the points, the other from their connecting pro- perty ; and from the integrals eliminate the new variable. ART. 2.] GEOMETRICAL APPLICATIONS. 317 There are problems in the expression of which we are led to what may be termed functional differential equations, i.e. equations fin which the operation of differentiation and an unknown functional operation seem inseparably involved. In some such cases a procedure similar to that employed in the solution of Clairaut's differential equation enables us to effect the solution. 2. The subject can scarcely be said to be an important one, and a single example in illustration of each of the dif- ferent kinds of proble'ms, as classified above, may suffice. Ex. 1. To find a curve such that, if a system of n right lines, originating in a fixed point and termina'ting in the curve, revolve about that point making always equal angles with each other, their sum shall be invariable. (Herschel's Examples, p. 115.) The angles made by these lines with some fixed line may be represented by n n Hence, if r = (0) be the polar equation of the curve, the given point being pole, we have a being some given quantity. Let 0 = -- , and let 6 ( -- ) = u, , then we have n \ n J the complete integral of which is n +(7acos^-...4-On.1cos 318 GEOMETRICAL APPLICATIONS. [CH. XVI. Hence we find r = a + GI cos 6 + C2 cos 20 . . . + Gn_^ cos (n-l}6, the analytical form of any coefficient C. being C. = A + Bl cos n6 + Bz cos 2nd + &c.} + Et sin nd + Ez sin 2nd + &c., A, BI} JE19 &c., being absolute constants. The particular solution r = a + b cos 6 gives, on passing to rectangular co-ordinates, and the curve is seen to possess the property that "if a system of any number of radii terminating in the curve and making equal angles with each other be made to revolve round the origin of co-ordinates their sum will be invariable." Ex. 2. Required the curve in which, the abscissae in- creasing by a constant value unity, the subnormals increase in a constant ratio 1 : a. Representing by yx the ordinate corresponding to the ab- scissa x, we shall have the mixed difference-equation mdm **+ dx Let y*~(jj; = ux> tnen ux - au^ = 0 .;. uf=Ca", whence - 0 ' Hence integrating we find ..... ..... (3), Cl being a periodical constant which does not vary when x changes to x + 1, and c an absolute constant. ART. 2.] GEOMETRICAL APPLICATIONS. 319 Ex. 3. Required a curve such that a ray of light pro- ceeding from a given point in its plane shall after two reflec- tions by the curve return to the given point. The above problem has been discussed by Biot, whose solution as given by Lacroix (Diff. and Int. Calc. Tom. in. p. 588) is substantially as follows : Assume the given radiant point as origin ; let x, y be the co-ordinates of the first point of incidence on the curve, and x, y those of the second. Also let ~r=P> -jr< =P- It is easily shewn that twice the angle which the normal at any point of the curve makes with the axis of # is equal to the sum of the angles which the incident and the cor- responding reflected ray at that point make with the same axis. Now the tangent of the angle which the incident ray at the point x, y makes with the axis of x is - . The tangent 00 of the angle which the normal makes with the axis of x is -- , and the tangent of twice that angle is P %P Hence the tangent of the angle which the ray reflected from x, y makes with the axis of x is - x Again, by the conditions of the problem a ray incident from the origin upon the point x, y would be reflected in the same 320 GEOMETRICAL APPLICATIONS. [CH. XVI. straight line, only in an opposite direction. But the two expressions for the tangent of inclination of the reflected ray being equal, 2afr' - y' (1 -ff 2) toy - y (1 - /) = x'(l-p*) + 2yp' x(l-/)+2yp while for the equation of that ray, we have Now, regarding # and y as functions of an independent variable z which changes to z + 1 in passing from the first point of incidence to the second, the above equations become The first of these equations gives - y 1 - a whence by substitution Ay Therefore Here C and C" are primarily periodic functions of z which do not change when z becomes z + 1. Biot observes that, if C be such a function, <£ (0), in which the form of 0 is arbi- trary, will also be such, and that we may therefore assume C' = <£ (C), whence and, restoring to (7 its value in terms of cc, y, and ^} given in (4), we shall have ART. 2.] GEOMETRICAL APPLICATIONS. 321 x . -- + 6 ,« * + * * (1 - p') + 2j

-a the integral of which is denoting a circle. * It is only while writing this Chapter that a general interpretation of this equation has occurred to me. Its complete primitive denotes a family of curves defined by the following property, viz. that the caustic into which each of these curves would reflect rays issuing from the origin would be identical with the envelope of the system of straight lines defined by the equation y = cx + (c), c being a variable parameter. This interpretation, which is quite irrespective of the form of the function , confirms the ob- servation in the text as to the necessity of restricting the form of that function in the problem there discussed. I regret that I have not leisure to pursue the inquiry. I have also ascertained that the differential equation always admits of the following particular solution, viz. (y-^)»+(a-B)a = 0, A and B being given by the equation 0 ( ^/Tf) =A -B*J~^l. (1st edition.) B. F. D. 21 322 GEOMETRICAL APPLICATIONS. [CH. XVI. If we make the arbitrary function a constant and equal to 2a, we find on reduction the complete primitive of which (Diff. Equations, p. 135) is flM / \2 22 Vb v (y - a)2 + cV = -= — -j , JL ~~ C the equation of an ellipse about the focus. 3. The following once famous problem engaged in suc- cession the attention of Euler, Biot, and Poisson. But the subjoined solution, which alone is characterized by unity and completeness, is due to the late Mr Ellis, Cambridge Journal, Vol. in. p. 131. It will be seen that the problem leads to a functional differential equation. Ex. 4. Determine the class of curves in which the square of any normal exceeds the square of the ordinate erected at its foot by a constant quantity a. If y* = ty (x) be the equation of the curve, the subnormal wiU be ^-|^ , and the normal squared ty (x) + I^J^f . The equation of the problem will therefore be Differentiating, we have which is resolvable into the two equations, ART. 3.] GEOMETRICAL APPLICATIONS. 323 The first of these gives on integration (4). Substituting the value of ty (x), hence deduced; in (1), we find as an equation of condition a=0, and, supposing this satisfied, (4) gives the equation of a circle whose centre is on the axis of x. It is evident that this is a solution of the problem, supposing To solve the second equation (3), assume and there results 0 ........... . ...... (5). To integrate this let x = ut , x (x) = ut_lt and we have ttt«-2wm + w( = 0, whence C and C' being functions which do not change on changing t into t + 1. If we represent them by P (£) and Pl (t), we have whence, since ut — x and wm = % (x) = x + J^' (x), we have 21—2 324 GEOMETRICAL APPLICATIONS. [CH. XVI. Hence +' (x) dx = PJ® [P1 (t) + P,(«) + tP,' (t)} dt, + p,(o + tp; (t) } dt. Keplacing therefore ty (x) by y"2, the solution is expressed by the two equations, *=p(<) +«>,«) i . tf = fpjf) [F (t) + Ptf) + tp; (t)} at] •• from which, when the forms of P(t) and P$) are assigned, t must be eliminated. If we make P (t) = ot, P^tf) = /3, thus making them constant, we have Therefore eliminating t and substituting e for c — aft Substituting this in (1), we find :£-* Thus, in order that the solution should be real, a must be negative. Let a = — h2, then 0 = ± 2A, and tf=±2hx + e , (7), the solution required. This indicates two parabolas. If a = 0, the solution represents two straight lines parallel to the axis of x. EXERCISES. 325 EXERCISES. 1. Find the general equation of curves in which the diameter through the origin is constant in value. 2. Find the general equation of the curve in which the product of two segments of a straight line drawn through a fixed point in its plane to meet the curve shall be in- variable. 3. If in Ex. 4 of the above Chapter the radiant point be supposed infinitely distant, shew that the equation of the reflecting curve will be of the form <£> being restricted as in the Example referred to. 4. If a curve be such that a straight line cutting it perpendicularly at one point shall also cut it perpendicularly at another, prove that the differential equation of the curve will be being restricted as in Ex. 4 of this Chapter. 5. Shew that the integral of the above differential equa- tion, when the form of (f> is unrestricted, may be interpreted by the system of involutes to the curve which is the envelope of the system of straight lines defined by the equation y — mx + <£ (m), m being a variable parameter. ( 326 ) ANSWEES TO THE EXAMPLES. CHAPTER II. 6. Obtained from the identity An (0 - 1) (0 - 2) 9. 14 (x — 6x8) cos # — (To;2 — a;4) sin x. 16. (2) i* =^ 0*. |# CHAPTER III. 1. 2-3263859 which is correct to the last figure. 2. a3- _ ^~ 10 '3 10 13. It will be so if (x) = 0 have one root, and (x) = 0 have no root between 1 and Jc. CHAPTER IV. 21 10 2 ' 90 6( (2w-l) (272 + 1) (27i +3) (27i + 5) (8/1+43) , 129 ANSWERS TO THE EXAMPLES. 327 <*> s- 12 4 (2ii + 1) (2n + 3) * (5) Apply the method of Ex. 8. (6) Write 2 cos 6 = x + - and use (10) page 73. 3. sin (2o-l) cos (2*) ( sin (2 sin g (2 sin ?V + &c. 6. (1) cot5-cot2n-l0. (2) cos (» + 1) # sin 26 ' Io£2sin2n0 7. tan ^n -!)#+£, (7 -- ^^ -- 8. Assume for the form of the interal , and then seek to determine the constants. lf * +-i-^ + CHAPTER Y. r VI^ W + T 2 I M+ T V 4/ V 4y where C= 1*0787 approximately and is the sum ad inf. 328 ANSWERS TO THE EXAMPLES. 2 sJL-r— 1 -I JL JL 5~ 4 5 "t" The sum ad m/". differs from that of the first nine terms by •0000304167. 3. 4. See page 71. 5. (1) Apply Prop. IV. page 99. If — a* be written for #2 in the first series it can be divided into two series similar to the Example there given. 2 (x(x+ 1) (#+2) 2 x (x + 1) (a? -f 2) (cc + 3) , 1 34 H s r-/ 1- &C. * ^T^- 13. See Ex. 7. Also page 115. CHAPTER VII. 1. - tan"1 a and — . a . 2a 3. (1) Divergent. (2) Convergent. (3) The successive tests corresponding to (C) are fit obtained by writing — Aux+n for ' - - — 1 therein. The set corresponding to (B) are obtained by writing (4) Convergent if x be positive, divergent if it be negative. (5) Divergent. (6) Divergent. ANSWERS TO THE EXAMPLES. 329 (7) Divergent unless a be greater than unity. (8) Divergent unless a be greater than unity. 4. (1) Divergent unless x be less than unity. (2) Convergent unless x or its modulus be numeri- cally greater than unity. 6. Divergent unless x < e'1. 7. x must not be less than unity numerically. 17. See Ex. 18. CHAPTER IX. °- <5) Thesame- 2. wa = C _ r O. U>x — \JUj 1 - - rr - - x - . 1 — 2a cos 7i + a 5. ux — {(7+coseca tan (35 — 1) a} cos a cos 2a... cos (as— 1) a. 6. Assume ux = vx + m where m is a root of m2 + a??i + 5 = 0, and there results a linear equation in — . 330 ANSWERS TO THE EXAMPLES. 2 sin xQ sin ( x — •= J 6 9. (7 Sm2 10. waj=a^1{^2+ (7). 11. ux = 12. uu = a*' 1 Q 1 f mM — S25) -Lo. U == — I d *~~ Cb f. 15. By writing ux+-^ = vx the equation may be reduced to vx+l = vl+C. When (7 = - 2 this gives vx = 2 cos 2* ft Jg ^,|^ __ 02^ i -J 17. -^ = ax or — 2o#. Hence two associated solutions ux (see Ch. x.) are u, = Co? Tx and ux= C(-2a)*Tx. CHAPTER X. 8. The two others are given by where z is a root of /i,2 + /^ + 1 = 0. , ANSWERS TO THE EXAMPLES. 331 9. CHAPTER XL «~,aj 3. 4. u, = (m2 + n2)"2 j<7 cos (x tan"1 ^ 2sini 6. «x= (-3)" ^ + +Cf' 7. The particular integral is obtained by (II) and (III) page 218. It is any value of + 0-2 332 ANSWEES TO THE EXAMPLES. + w2 cos mx + cos (x — 2) m 9. ux — — 4 , 0 2 - + complementary func- n ± 2ft,2 cos 2m + I tion, which is {~ 7TCC ~, . TTX] x (j cos h U sin — > n , / or Cnx+C' (—ri)x, according as the upper or lower sign is taken. 10. ri~xux = {C. ' + C x} cos -^r- + [C + C^x] sin -^- , L A 11. (a + bti) \ — j — [ — l)k where k = — ( "J J 7*. 12. _1((11 +^17^11-3^17 CHAPTER XII. 2. wa: = ^si to ^~ terms (supposing that n is odd) where X = — — . 5. ^^"2)] ^^ 2 g ANSWERS TO THE EXAMPLES. 333 6. loga. = (- n)« (7+ 0'* 7. For (a; + 2)" read (o; + 2)3. The equation is then re- /"V» duced into a very simple form by substituting - -- — 2 vx for ux. (x + 1) 8. •u= C + C-\ a 10. and ^ + &c., wa + &c. are obtained by writing a; + 2 and a? + 4 in the quantity on the right-hand side. .-2H-2 * -»-l 11. and vx+1 (and therefore vj is given at once by the first equa- tion. 13. It may be written (E-cT] (E- a") ux = 0. 14. t«,= a 2 /-, f ^ ZTTX , ^ . STTO?) lo. ux — va tan ^ C* cos — ^- + C/2 sin — ^- [• . I o 3 J 17. Compare with (15) after dividing by w^+^ 19. If log an = wn we have and the solution is !-(-£)* tool 334 ANSWERS TO THE EXAMPLES. 20. See page 228. Perform A on the equation and a linear equation in A2% results. 21. ux = POL* + Qfi* + R ( __ j. where 0 is a periodic and C' an absolute constant. CHAPTER XY. 2. ^•(*)*| 5. f(x)=Cx. _ ,, , 7- /^ 8- /(*) =/(*)-/ (2- 2^-2(-ir • 336 ANSWERS TO THE EXAMPLES. 12. y = ce*(x~a\ (j> (x) denoting an odd function of x. 13. Develope/(7r) in ascending powers of TT, and apply the conditions of periodicity. -i a J. f sm mx 16. <> sin (mx + c) * sine sin (mx + c) ' 22. Ate): (x) CHAPTEE XVI. CQ \ ffi __ ... \ o- ) -/ [ - - ] where/ (x} satisfies the equa- ZTT/ \ ZTT / tion A/(#) = 0. 2. Write log r for r in the answer to the previous ques- tion. CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. M <" i 9 ?/> / UNIVERSITY OF TORONTO ;: |f LIBRARY % CD $> ® > ^ \ Do not \ \ remove \ ' * :;l?f 1 « » 2 01 Tb CO CD CD O the card CD O / / from this o y 1 CD Acme Library Card Pocket Under Pat. "Ref. Index File." I Made by LIBRARY BUREAU, Boston ?L_ /