CORNELL UNIVERSITY LIBRARY FROM Cornell University Library arV19572 A treatise on the calculus of finite dif 3 1924 031 240 934 olin,anx fl Cornell University & Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031240934 4 TREATISE ON THE CALCULUS OF FINITE DIFFERENCES. A TREATISE CALCULUS OE FINITE DIFFERENCES. GEORGE BOOLE, D.C.L. LATE HONORARY MEMBER OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY AND PROFESSOR OF MATHEMATICS IN THE QUEEN'S UNIVERSITY, IRELAND. EDITED BY jTft moult on, FELLOW AND ASSISTANT TUTOR OF CHRIST'S COLLEGE, CAMBRIDGE. THIRD EDITION. Hott&on : MACMILLAN AND CO. 1880 [The Right of Translation is reserved,] © PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PKESS. 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 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. GEORGE BOOLE. Queen's College, Coek, April 18, 1860. 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. MOULTON. Cheibt's College, Oct. 1872. CONTENTS. DIFFERENCE- AND SUM-CALCULUS. CHAPTER L PAGE NATURE OF THE CALCULUS OF FINITE DIFFERENCES . 1 CHAPTER II. DIRECT THEOREMS OF FINITE DIFFERENCES 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. Eefer- ences on Interpolation, &e. 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 SEEIES 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. Rational and integral Functions, 68. Integrable Fractions, 70. Functions of the form a x (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 u x , the suffix taking the place of the symbol which in the former mode of notation is enclosed in brackets. Thus, if u x = (x), 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. P. D J 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 Ax of the variable x. Then, representing as above the proposed function of x by u„ we have an d Au z ^ u x+Az -u x Ax Ax j Here then we might say that as -v- is the fundamental ope- ration of the Differential Calculus, so -r— is the fundamental 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 j- is not a true fraction, nor have du and dx any distinct meaning as symbols of quantity. The fractional form is adopted to express the limit to which a true fraction approaches. Hence -T- , and not d, there represents a real operation. But in the Calculus of Finite Differences -j- 5 is a true fraction. Its nu- Ax merator Au x stands for an actual magnitude. Hence A might itself be taken as the fundamental operation of this Calculus, always supposing the actual value of Ax to be given; and the Calculus of Finite Differences might, in its symbolical charac- ter, be denned either as the science of the laws of the operation A, the value of Ax 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 Ax 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 T, 2 2 , 3 2 , 4 2 , ... the general or x^- term is x\ It is an explicit function of x, but the values of x are the series of natural numbers, and Ax=\. 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 At = h ; then assuming t = hx we have At = JiAx, whence Ax = 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 develope-the theory of the operation denoted by A and defined 'by the equation Au x = u x+1 — u x . But we shall, where convenience suggests, consider the more general operation Au x= _ u^-u m Ax h ' where Ax = h. 1—2 ( 4 ) CHAPTER II. DIRECT THEOREMS OF FINITE DIFFERENCES. 1. 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 tbe 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 AAV 5 AX AA n "VsAX (1), the last member being termed the n th difference of the function u x . If we suppose u x — x 3 , the successive values of u x with their successive differences of the first, second, and third orders will be represented in the following scheme : Values of x 1 2 3 4 5 6... 1 u x 1 8 27 64 125 216... o A«, 7 19 37 61 91.. •7 AX 12 18 24 30.. f AX 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 Am, A 2 m, &c. by assign- ART. '2.] DIRECT THEOREMS OF FINITE DIFFERENCES. 5 ing to x the successive values 1, 2, 3, &c. Since u x = x s , we shall have Au x = (x + If - X s = 3a? + 3x + 1, AX = A (3a? + 3x + 1) = 6x + 6, AX = 6. It may also be noted that the third differences are here constant. And generally if u x be a rational and integral function of x of the n th - degree, its w th differences will be constant. For let u x = ax" + bx"' 1 + &c, then Au x =a(x+ 1)" + b (x + 1)"" 1 + &c. - ax" - bx"' 1 - &c. = anx"- 1 + b^ + b^x""* + &c, b lt & 2 , &C, being constant coefficients. Hence Aw^/is a rational and integral function of x of the degree n — 1. Kepeating the process, we have AX = an(n- 1) a;" -2 + c^"" 3 + c^"" 4 + &c, a rational and integral function of the degree n—2; and so on. Finally we shall have c - A\ = a«()i-l)(n-2)...l, a constant quantity. Hence also we have AV = 1.2...ra (2). 2. While the operation or series of operations denoted by A, A 2 , ... A" are always possible when the subject-function u x 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 u x = x (x — 1) (x — 2) ... (x — m + 1). Then by definition, Au x =(a)+l)x(x-l) ... (x-m+%)-x(x— l)(x-2)...(x-m+l) ' = mx(x-l)(x-2) ...(x-m + 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-\)... (iS-m + l) = aiM'..„ (1), doing which, we have A« (m, = ma! (m - 1, (2). Hence, as 1 " 1 " 11 being also a factorial term, AV^mfm-l)^, and generally % AV""=m (m -V)T.. "(ro-#+ 1) d m T> (3). 2ndly. Let u x = — — -^ -. — ■ T r. J x(x + l) ... (sc + ra— 1) Then by definition, * (x+l)(x + 2)...(x+m) x(x + l)...(x + m-l) = (_L_ -I) I \x + m x/(x + l) (oc + 2) ... (x + m-1) = x(x + l)...(x + m) ^' ■■at*, ART. 2.] DIRECT THEOREMS OF FINITE DIFFERENCES. 7 Hence, adopting the notation 1 jc(x + l) ... (x + rii—f)' we hare . . Aa;(- m »=-ma;l- m " 1 > (5). Hence by successive repetitions of the operation A, AV™> = - to (- m - 1) ... (- to - n + 1) a;'-""' 1 = (-1)" m (m+1) ... (m + n -1) x^^ (6), and this may be regarded as an extension of (3). 3rdly. Employing the most general form of factorials, we find Aty^ . . . tt^ = K +1 - tv^J X «A-! • • • Vtr ( 7 )' A 4 = ^"^ (8), and in particular if u x = oa; + 5, Av M 1 ■ ■ m^h.! = cmu^M^ . . . m„ h (9), A * = ~ am (10). U X U X+1 . . . M x+m _, U^M^ . . . U x+m In like manner we have A log w, = log u^ - log u x = log -f* . u x To this result we may give the form Alog Ws = log(l+ A j) (11). So also A log (ty^ ... u^J = log £**- (12). 4thly. To find the successive differences of a". 8 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. We have Ao" = a artl -cs» = (a-l)a x (13). Hence AV = (a-l) 2 a*, and generally, 8 AV=(a-l)"a* (14). v Hence also, since a mx = (a m ) x , we have • AV a! = (a ffl --l) B a , " ai (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) o •- a I 7 a\ = 2 sin 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. 9 To find the differences of tan u, and of tan -1 M.. A tan u x — tan u x+l — tan u x miK^ sin u x cos u x+1 cos u x . sin (««„ - u x ) Wi c sin Au. cos m„. +1 cos u x (!)■ wo M-ar + | ^vo wa; Next, cos m^, cos u x A tan -1 ^ = taxT 1 u xt , l — tan'V; = tan- 1 >'" M - vi - = tan- V A% (2). From the above, or independently, it is easily shewn that Atana# = -. rr- (3), cos ax cos a{x+l) w A tan -1 a* = tan -1 s — ■ — =-= (4). 1 + a'x + aV v ' 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 -r— merges, on supposing Ax to become 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 OP FINITE DIFFERENCES. [CH. II. Ex. We have A sin x _ sin (x + Ax) — sin x Ax Aa; _ . , A - . / , Aa! + w\ 2 sm ^ Aa; sm to; H = — j Ax And, repeating the operation n times, / Ax ~\~ 7r\ .„ . (2 sini Aa5)"sin (#+«. — ~ — ) A" sm x _ v 2 ' V 2 / ,j. (Ax) n ~ (Aa)" ~ W ' It is easy to see that the limiting form of this equation is ^-=sm[x + -^j (2), d" sin x dx' a known theorem of the Differential Calculus, Again, we have Aa* = a°** x - a* Ax Ax -C^)-- And hence, generally, .AV (Ax) ?-rsrj° (3) - Supposing Ax to become infinitesimal, this gives by the ordinary rule for vanishing fractions ££ = (lpg«)-a' (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* j;, ART. 5.] DIRECT THEOREMS OP 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 <£> (as), a given rational and integral function of x of the m th degree, in a series of factorials. Assume ${x)=a + bx + cx m + dx i *> ... + hx™ (1). The legitimacy of this assumption is evident, for the new form represents a rational and integral function of x of the m th degree, containing a number of arbitrary coefficients equal to the number of coefficients in (as). 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<£ (at)=b + 2cx + 3dx« ) ... + mhx {m - 1) (2), A*(x) = 2c + 3 . 2da> ... + m(m - 1) W H, ...(3), A m (x)=m(m-l)...lh (4). • And now making x = in the series of equations (1)...(4), and representing by A<£ (0), A 2 <£ (0) ? &c. what A (a:), A 2 # (as), &c. become when x = 0, we have 0(0) = a, A0(O)=6, A>(0) = 2c, A m (0) = 1.2...mh Whence determining a, b, c, ... h, we have *(*)=* (°) + w (°) * + ^x^ <** + ^jr «? + &c - ( 5 >- If with greater generality we assume (x) = a + bx + ex (x — h) + dx (x — h) (x — 2h) + &c, 12 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. we shall find by proceeding as before, (except in the employ- ing of — for A, where Ax = h,) j.^ r. ,m (A0(aO) , fA*A (*)!»(« a; -A) ^(A*) 8 } 1.2.3 + * - -W. 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 Aa; 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 Au x = u x+1 -u x 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, du x ld*u x 1 d*u x dx + 2 da? + 2 . 3 dx* 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 "x+ 1 — «•* — J77 t a -j j l o^ q j„8 T (t) is capable of being developed in a series of powers of t, the general term of the expansion being represented by u x f, then (t) is said to be the generating function of u x . And this relation is expressed in the form Thus we have '-«^ since = — ^ is the coefficient of f in the development of e'. In like manner Ug. since * ~1.2:..(a>+l)' 1 — 2 — / i i\ * s *^ e coefficient of f in the development of the first member. And generally, if Gu x = ^> (t), then «W-*£ *.-*& (2). Hence therefore Gu^-Gu x =(l-l}$(t), But the first member is obviously equal to GAu x , therefore tfA«„=(i-l)^) (3) . ART. 8.] DIRECT THEOREMS OF FINITE DIFFERENCES. 15 And generally GA°u x = (±-lf(t) W- To apply these theorems to the problem under considera- tion we have, supposing still Gu x = (t), = fin. + n GAu x + n ^ n ~ V > #A\ + &c. = £ {«. + nAu x + ^f^ A\ + &c.} . * Hence w,*: = «„ + , «A«, + — ^ — A s w + &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 G is, from its very nature, a symbol of inversion {Biff. 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 , w S*«=iW" • (!)• Laws and Relations of the symbols E, A and -*- . 1st. The symbol A is distributive in its operation. Thus A(m x + ^+&c.) = Am x + A^ + &c (2). For A (u x + v x + &c.)=u x+1 + v M ...-(u x + v. x ...) = Au x + Av x ... In like manner we have A ( u x~ v x + &c.) = A« aj - Av x + &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, Aau x = au M -au x = aAu x ......(4). And from this law in combination with the preceding one, we have, a, &,... being constants, A(au x + bv x +&c...)=aAu x + bAv x + &c (5). 1 * 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 AbeVare given in Grunert's Archii'. xviii. 381. ART. 9.] DIRECT THEOREMS OF FINITE DIFFERENCES. 17 3rdly. The symbol A obeys the index law expressed by the equation A?"AX = A m+n u x (6), m and n being positive indices. For, by the implied definition of the index m, A^A"^ = (AA-.-ra times) (AA...K times) u x = {A A. . . (m + n) times} u x = A m+ 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. + a)u denoting Aw + ait, we should have, in virtue of the first two of the above laws, (A+a)(A + 6)w={A 2 + (a + &)A + a&}w = A a M + (a + b) Am + abu (7), the developed result of the combination (A + a) (A + b) 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 E= 1 + A, * For instance, if denote an operation which, when performed on two quantities x, y, gives a single function X, it is an operation incapable of repe- tition in the sense of the text, since 2 (a;, y) = (X) is unmeaning. But ifit be taken to represent an operation which when performed on x, y, gives the two functions X, Y, it is capable of repetition since

\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 Jftt. = «. + A«.= (l + A)«. (8), and they are connected with -j- by the relation E=eK (9), founded on the symbolical form of Taylor's theorem. For „ du , ld*u x , 1 d*u d . 1 d* . 1 d 3 /.did", 1 eF , a \ = €"«„. It thus appears that E, A, and -r- , are connected by the two equations & E=l+A = e ex (10), and from the fact that E and A are thus both expressible by means of j- we might have inferred that the symbols E, A, and j- * combine each with itself, with constant quantities, 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 D. The equations will then he E=l + A=e D , 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 u x+n in a series consisting of u x and its successive differences (Ex. of Art. 7, resumed). By definition w^i = Eu„ u x ^ = Wu x , &c. Therefore « X+ „ = #X=(1+A)"«* (1), = | 1 + raA + T ^A' + w ^- 2 1 ^- 2) A-..j^ - K + n*u x + 1^ AX+ ^-lK"- 2 V +&c...(2). Ex. 2. To express AX in terms of w^ and its successive values. Since Au x = u x ^. l — u x = Eu x — u x , we have Au x =(JE-l)u x , and as, the operations being performed, each side remains a function of oc, A"u x = (E-iru x = {& - «E" + ^=^ E" - &c.j u x . Hence, interpreting the successive terms, AX = u x+H - nu xV _ 1 + n Y ~ ^ m^»_ 2 .- + (- l)X-.(3). Of particular applications of this theorem those are the most important which result from supposing u x = x m . We have iy f (*+«)"-(i (;?+»-l)" , +^fc 1) («+n-2) M -&c...(4). 2—2 20 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. Now let the notation A"(T be adopted to express what the first member of the above equation becomes when x = ; then A n 0r = n m -n(n-l) m t n(n-l)(n-8r n(n-l)(n-2)(n-8)- j &c _ (5)> The systems of numbers expressed by A"0 m are of frequent occurrence in the theory of series*. From (2) Art. 1, we have A"0" = 1.2...», and, equating this with the corresponding value given by (5), we have 1 . 2 ... n = n" -n (n - 1Y + w< ""^ (n - 2)" - &c....(6)f. Ex. 3. To obtain developed expressions for the^n", differ- ence of the product of two functions u x and v x . _ 1 Since Au x v x = u x+1 .v xyi -u x v x = Eu x .E'v x -u x v x , where 'E applies to u x alone, and E' to v x alone, we have Au x v x =(EE'-l)u x v x , and generally *" A"V.= (EE'-l)»u x v x (7). It now only remains to transform, if needful, and to de- velope the 1 operative function in the second member according to the nature of the expansion required. Thus if it be required to express A n u x v x in ascending differ- * A very simple method of calculating their values will be given in Ex. 8 of this chapter. ■J- This formula is of use in demonstrating Wilson's Theorem, that 1 + 1 7i - 1 is divisible by n when n is a prune number. ART. 10.] DIRECT THEOREMS OP FINITE DIFFERENCES. 21 ences of v x , we must change E' into A' + 1, regarding A' as operating only on v x . We then have A"m A = {#(1 + A')-1}X^ = (A + J BA')"«A = | A" + nA-^JEA' + W <>-1) a-^^A' 2 + &c.| u x v x . Remembering then that A and E operate only on u x and A' only on v x , and that the accent on' the latter symbol may be dropped when that symbol only precedes v x , we have AX«» = AX . v x + nA" _1 M m . Aw, + l^L) A -^ li .A^ + 4c (8), the expansion required. As a particular illustration, suppose u x = a". Then, since A*"n^. = A^a"* = a r £"eT = . + &c.} ...(9). Again, if the expansion is to be ordered according to suc- cessive values of v x , 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«- = (-!)" fa«. - ™WW + * 2 Wx * Vx ™ ~ &C?i " "^ Lastly, if the expansion is to involve only the differences of u x and v x , then, changing E into 1 + A, and E' into 1 + A', we have .■AX«.= (A + A' + AA')X«. (11). and the symbolic trinomial in the second member is now to be developed and the result interpreted. 22 DIRECT THEOEEMS OF FINITE DIFFERENCES. [CH. II. Ex. 4. To express A"u x in terms of the differential co- efficients of u x . By (10), Art. 9, A = e* - 1. Hence AX = (^-1)X (12). Now t being a symbol of quantity, we have (e*-ir=(^ T ^ + r |- 3 + &c.)" (13), on expansion, A lt A 2 , being numerical coefficients. Hence and therefore The coefficients A^, A t ,...&o. 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. •♦UK 2.3 where !-^A 0™ denotes what $ f -j- J x m becomes when x = 0. ART. 10.] DIRECT THEOREMS OF FINITE DIFFERENCES. 23 First, we shall shew that if (oo) and ty (x) are any two functions of x admitting of development in the form a + bx+cx i + &c, then *(£)+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 j- J x n , while in corresponding terms of the two members the order of the indices m and n will be reversed. f d\ m Now I -j- J x n is equal to if m is greater than n, to 1 . 2...w if m is equal to n, and again to if m is less than n and at the same time x equal to ; for in this case a;" - "' is a factor. Hence if x = 0, \dooJ \a *V'-®> and therefore under the same condition the equation (15) is true, or, adopting the notation above explained, *Gfc) + <0)- + (|j)*C0) '-( 16 >- Now by Maclaurin's theorem in its known form ■*W = *(0)+35*(0).*+J^(0). I ^ 1 +&c (17). Hence, applying the above theorem of reciprocity, *(*) = *(0) + *(^)o.* + *(^)o^ + ^...(18), the secondary form in question. The two forms of Mac- laurin's theorem (17), (18) may with propriety be termed conjugate. 24 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. A simpler proof of the above theorem (which may be more shortly written (t) = (D\ e" ■') is obtained by regarding it as a particular case of Herschel's theorem, viz. 4> (e«) = 4, (1) + (E) . * + <£ (E) 0* . j^ + &c. . . .(19), or, symbolically written, (e*) = (E) e -'.* The truth of the last theorem is at once rendered evident by assuming A n e nt to be any term in the expansion of (e') in powers of e'. Then since A n e nt = A n E n e -' the identity of the two series is evident. But 4> (t) = (log e') =

(t) = (e'-l)", then by means of either of the above theorems we easily deduce (e 8 - 1)" = A"0 . t + A"0 a . ^-= + A*0 3 . =-J-5 + &c. But A n m is equal to if m is less than n and to 1 . 2 . 3. . .n if m is equal to n, (Art. 1). Hence A n/\n+l A m n+1 Hence therefore since A"m = (e** — l)"w we have AM -^ s+ 1.2...(n+l)-cS iH:i + 1.2...( TO +2)d^ +&C -^ 1 )' the theorem sought. The reasoning employed in the above investigation pro- ceeds upon the assumption that n is a positive integer. The * Since hoth A and D performed on a constant produce as result zero, it is obvious that 0(Z>)C=0(O) 0=0 (A) O, and (E)C=^[l)0. It is of coarse assumed throughout that the coefficients in are'constants. ART. 11.] DIRECT THEOREMS OF FINITE DIFFERENCES. 25 very important case in which n = — 1 will be considered in another chapter of this work. Ex. 5. To express -^— n in terms of the successive differences ofu. £ ■ Since e** = 1 + A, we have jUogd + A), therefore (£f = {log (-1 + A)Jv (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 . A 2 w , A 3 m A*m , . .„„. 5 - = A^_ + _-_ + &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 x f d*u r , „ Perform <£(A) on both sides (A having reference to t alone), and subsequently put t = 0. This gives *(A)^^.*(0) + *(A)0.^ + 4^.gf + 4a...(M), of which (21) is a particular case. 28 DIRECT THEOREMS OF FINITE DIFFERENCES. [CH. II. By (2) we have u x+t = u x + t.Au x + j-^A\ + &c. Perform ( -5- J on each side, and subsequently put t = ; + KI)° (2, -r^ +&c --( 25 )- of which (23) is a particular case. 12. We have seen in Art. 9 that the symbols A, E and ■j- or D have, with certain restrictions, the same laws of com- 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 Memoirs by Cauchy (Journal Polytechnique, Vol. xvn.). Ex. 6. To shew that B (m + 1, n) = (- l) m A m - , where m is a positive integer. We have -=f e**dx; . • . a™ 1 = A ra f e^dx = / A V*cfo f e 1 "(€*-l)"*B • Jo =| B_1 (z - l) m dz (assuming z = e*) Jo = {-l) m B(m + l,n). ART. 12.] DIRECT THEOREMS OF FINITE DIFFERENCES. 27 Ex. 7. Evaluate u=\ A" „ a (x, n) dx expresses the integral 28 EXERCISES. [CH. II. of a quantity of m + 1 terms of the form A s j> (x, n +p), while f 00 A m | £ (x, n) dx expresses the sum of m + 1 separate inte- 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 ${E)O n = E<$>'(E)Q"- 1 (28). Let A r E r O n and ErA r E'~ 1 0' l ~ 1 be corresponding terms of the two expansions in (28). Then, since each of them equals A r r", the identity of the two series is manifest. Since E= 1 + A the theorem may also be written (A) 0" = E' (A) 0"- 1 , and under this form it affords the simplest mode of calcu- lating the successive values of A m 0". Putting # (A) = A™ we have A m 0" = E . mA^O" -1 = m (A m - 1 n - 1 + A m 0" -1 ), and the differences of 0" can be at once calculated from those ofO- ^ Other theorems about the properties of the remarkable set of numbers of the form A"'0" 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. VIII. and IX. EXEKCISES. 1. Find the first differences of the following functions ; 2*-sin ^ , tan ^ , cot (2* a). EX. 2.] EXEECISES, 29 2. Shew that v* VJ> t 3J w »fl 3. Prove the following theorems; ^»Q»+l _ n \ n + ■*■ ) A 1>Q7I ■ * 2 ;' - A0"*Aq» + &». = 0(1) . A*"0" + n A^O"- 1 + V ^ ; A m B - 2 + . . . + r-^- = , . 1.2 « — m m + 1 . 0(JB")O"=«n"0(iF) 2. Prove that {log J E , }"O m = 0, unless m = n when it is equal to |«. 11. Prove that _1_ = ,»<-« + (1 - n\u*+Q-n) (2 -*) »"*+ &c. a? + nx x 12. If a; = e 9 , prove that / rf\" _ A0" d A 2 0" , cf A°0 ft , d° ' \de) ~~T- a! Tx + i.2 ai d^ + T^73 ar d^ + '- 13. If &u XtS =u x+liV n-u x , v and if AX, Vl be expanded in a series of differential coefficients of u ti „, shew that the general term will be 14. Express AV 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 EX. 15.] EXERCISES. 31 where m is a positive integer, and reduce the result when m = 4. 15. Prove that F (JE) a'Qx = a*F (aE) fa, (*A) (K V = { x + n - l)""Aw„ / («A) (xE) m u x = {aE) m f (eA + m) u x , and find the analogous theorems in the Infinitesimal Calculus. 16. Find u x from the equations (1) Wf. = g« ' (2) Qu m =f{f). 17. Find a symbolical expression for the w* 11 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 P n be the number of ways in which a polygon of n sides can be divided into triangles by its diagonals, and f a-" (e-- l)"«r I dsB = TaA m n a , n and a being positive quantities. *20. Shew that •I sin 2nx sin"a! ^ irA m (2ra - to)" 2 m+1 r (a + 1) cos— 5— 7T if In > m > a all being positive. * In Questions 19 and 20 A acts on n alone. 32 EXERCISES. [EX. 21. f° sin m a; Hence, shew that I sin 2nx . m+1 . dx is constant for J all values of n between ^ an< i °° • 21. Shew that if p be a positive integer [V* sin 8 '* &- 1-8-S--8P (Bertrand, CaZ. 7w*. p. 185.) 22. Shew that A" I** 1 + A" -1 1* A n_1 2 ! ' = - n+1 23. Demonstrate the formula A" l 1 * 1 = (n + 1) A" 1» + wA" -1 1* and apply it to construct a table of the differences of the powers of unity up to the fifth power. ( 33 ) 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 magr 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 n th degree it has been seen that differences of the n + 1 th B. F. D. 3 3-4 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 u a , u t , ... of an unknown function of x represented by u x , are given, then, assuming as the approximate general expression of u x , u x = A+Bx+Ca?...+Ex n - 1 (1), a form which is rational and integral and involves n arbitrary coefficients, the data in succession give u a = A + Ba + Co* ... + EcF 1 , u b = A+Bb + CV... + Eb' 1 Ln-1 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 A"u x = 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. Prop. 1. Given n consecutive equidistant values u , w lt ... «,., of a function u x , to find its approximate general expres- sion. ART. 2.] AND MECHANICAL QUADRATURE. 35 By Chap. n. Art. 10, . , m (m — 1) A - , o ««.» = «* + ^Aw, + ^ 2 AX + &c Hence, substituting for x, and x for m, we have u x = u + xAu ^^~^' A\ + &c. But on the assumption that the proposed expression is ' rational and integral and of the degree n ^- 1, we have A n u x = 0, and therefore A"w„ = 0. Hence . , x (x — 1) . , , = u + xAu + ^ 2 'A\ . . . x(x-l)...(x-n + 2) + 1.2... (n-1) A U " {Z) ' 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 314 = -4969296, log 315 = -4983106, log 316 = -4996871, log 317 = "5010593 ; required an approxi- mate value of log 3-14159. Here, omitting the decimal point, we have the following table of numbers and differences : u M . % M 3 4969296 4983106 4996871 5010593 A 13810 13765 13722 A a -45 -43 A 3 2 The first column gives the values of w and its differences up to A s m„- 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. Ilence we have u, = 4969296 + -159 x 13810 + C 159 ^^ 9 - 1 ) x ( _ 45 ) (159) (-159-1) (-159-2) + 17273 " XA Effecting the calculations we find u x = "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 u , u lt u 2 , ... u n be the series. Then since, according to the principle of the method, A"m = 0, we have by Chap. II. Art. 10, ^ I i £ « n -^ n . 1 + ^^ ) «^-... + (-l)X=0 (3), an equation from which any one of the quantities u ,u v ...u n may be found in terms of the others. Thus, to interpolate a term midway between two others we have w 8 -2 Ml+Mi = 0; /. Ml = ^* (4). Here the middle term is only the arithmetical mean. To supply the middle term in a series of five, we have u - 4m, + 6m, — 4w 8 + % = 0; . ,. _ 4(m 1 + m s )-(m + m 4 ) • ■ u i § \°l- ART. 3.] AND MECHANICAL QUADRATURE. 37 -co Ex. Eepresenting as is usual / e~" fl"" 1 dd by T (n), it is required to complete the following table by finding approxi- mately logrQ: n logr(m), n logr(«), 12 - 74556 > U F2 ,55938 ' V2 T2 - 42796 > T* k - 32788 ' § •74556, ^ -18432, •55938, ^ -13165, 4 -42796, £ -08828, 5 10 ■32788, ~ -05261. Let the series of values of logr(«) be reprefcnted by «,,,«,,...«,, the value sought being that of u s . Then pro- ceeding as before, we find Q , 8.7 8.7.6 ■ . n «i - 8m s + i72 m s ~Y^73 m 4 + &c - = 0» or, u t + u a - 8 (a 4 + « 8 ) + 28 (u t + u,) - 56 (w 4 + m 6 ) + 70m 6 = ; whence _ 56 K + tQ - 28 K + m 7 ) + 8 (u, + u a ) - (u, + tQ M * 70 W- Substituting for «,, w 2 , &c, their values from the table, we find log r(J) = -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 « £ . m 4 « 6 -25610, u s ,u t u e ,u, -24820, m 2 , m 3 , m 4 u e , m 7 , u e , -24865, u lt w 2 , u a , u 4 u e , «„ u a , u a -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 u , w 4 , u 6 were given and w 3 sought, the equations A 3 u x = 0, A w 2 = would give w i — 3 u s + 3w 2 — «! = 0, w 6 -3m 4 + 3m s -w 2 = 0, from which, eliminating u s , we have 3m 6 -8m 4 + 6m 3 -m 1 = (7), and henfla u s can be found. But it is better to apply at once the general method of the following Proposition. Prop. 2. Given n values of a function which are not ^jgwrecutive and equidistant, to find any other value whose P U 1S gi ven - Hit u a , u b , u c , ...u k be the given values, corresponding to a^, c ...k respectively as values of x, and let it be required to determine an approximate general expression for u^ 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 — k, it shall assume the respective values u a , u b ,... %, the expression must contain n constants, whose values those conditions determine. We might therefore assume u x = A + Bx+ Co? ...+Ex n ~ 1 (8), and determine A, B, C by the linear system of equations formed by making x = a, b ... k, in succession. The substitution of another but equivalent form for (8) enables us to dispense with the solution of the linear system. AHT. 4.] AND MECHANICAL QUADRATURE. 39 Let u x = A (x, — b) (x — c) ... (x — k) + B (x — a) (x — c) ... (x — k) + (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 u a = A (a — b) (a — c) ... (a — k) ; therefore A = (a —b)(a — c) v . (a — k)' In like manner making x = b, we have R _ % (b-a)(b-c)...(b-k)' and so on. Hence, finally, (x — 6) (x — c) ... (x — k) (x — a) (x — c) ...(x-k ) U *~ Ua &=bj(a -c)...(a-k) + Ui (b^ajlb - c) ... (b-k) - „ (x — a) (x — b) (x — c) ... ,-. m + &c ---- + Sk-a)(k-b)(kJ)... s"M> the expression required. This is La grange's * theorem for interpolation. If we assume that the values are consecutive and equi- distant, i.e. that « , w a ...«„_! are given, the formula be- comes x(x-l) ...(x-n+2) _ x(x- 1) ...(x-n +V) %- M „-i 1.2.3... (n-1) M "- 2 1.1.2... (n-2) + &c. * Jownal de VEcole Polytechnique, n. 277. Thereal credit of the discovery must, however, be assigned to Euler ; who, in a tract entitled De eximio usu meiho&i interpolationum in serierum, doctrina, had, long before this, obtained a closely analogous expression. 40 ON INTERPOLATION, [CH. III. _ x(x-l)...(x-n + l) ( Un _ t c Un _, +&c l (n) |»-1 \x-n+l 1 x-n.+ ^ J ^ ' [w-1 where 0.= \ | r | ra ~l~ r ' This formula may be considered as conjugate to (2), and. possesses the advantage of being at once written down from the observed values of u x 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 A"w , 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 — 1)* is a determinate one, the different methods of solution above exemplified lead to consistent results. All these methods are implicitly contained in 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, b, ... k be the times of observation, u a , u b , ... u k the observed values, x the time for which the value is required, and u x that value. Then the value of u x is given by (10), and the differential coefficients can thence be deduced in the usual way. But it is most convenient to assume the time represented above by a; as the epoch, and to regard a, b, ... k as measured from that epoch, being negative if measured backwards. The values of -7-*, -=-£, &c. will then be the coefficients of x, a?, &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 , be ...k( T + - ,.. + T ) du„ \b c k J , o , 1S)N -7— =+ / t, / v / ,, «„+ &c (12), dx (a — b) (a — c) ... (a-ft) a ~ v ; js 6c... ft ( j- +T-7+— r + &c.) j-T = + 1 .2. , ,. , r 7 , M„+ &C....(13). da^ (a ^b) {a -c) ... (a-k) " x ' Laplace's computation of the orbit of a comet is founded upon this proposition (M&anique Celeste). 6. The values of a quantity, e. g. the altitude of a star at given times, are found by observation. Ee'quired 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, b, c, . . . be the values given by observation, u a , u b , u c , ... the corresponding times, x the value for which the time is sought, and u x that time. Then the value of u x 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, ... ft, the time sought by x, and the corresponding values of the observed magnitude by u a , u„, ... u k , and u x , 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 u x the magnitude of the quantity at any time x. Then u a , u„ and u c are given, and, by Lagrange's formula, (x — b)(x — c) (x — c)(x — a) (x — a) (x — b ) Ux ~ U " (a-b)(a-c) + U \b-c){b-a) + U ° (c - a) (o - 6) ' and this function of x is to be a maximum or minimum. Hence equating to its differential coefficient -with respect to x, we find (& a - o g ) u a + (c* - a°) u 6 + (a 2 - V) u c 2{(b-c)u a + (c-a) Ul ,+ (a-b)u c } v ; " 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 u x 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 2n + 1 obser- vations, we may assume u x = A + A 1 cos x + A 2 cos 2x + . . . + A n cos nx + B t sin x + _B 2 sin 2a? + . . . + B n sin nx. . . (15), and determine the coefficients by solving the resulting linear equations. Gauss-f - has proved that the formula sin n (x — b) sin ^ (x — c) . . . sin ^(x — k) «„= — j j j— m„ + &c....(16), sin g (a — b) sin « (a — c) ... sin ^ (a —h) * A special investigation of this problem will be found in Grunsri, xxv. 237. 1 Werke, Vol. in. p. 281. ART. 8.] AND MECHANICAL QUADRATURE. 43 is equivalent to (15), u a , u b , ...u h being assumed to be the 2n + 1 given values of u,. It is evident that we obtain M » = K 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 to + w values of a function are known, we may find a fraction whose numerator is of the n th , and denominator of the (to — l)" 1 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 to = 2, n—\, = u b u e (b -c){x-a) + &c. 7 u m (b-e)(x-a) + &o. K h When to = 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 Igeiraique, p. 528, but it is better to read a paper by Brassine (Liouville, zi. 177), in which it is considered more fully and as a case of a more general theorem. This must not be confounded with Cauohy'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', Corruptee Bendus, xxxvh. or Liouville, xviii. 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. 10 48-82 = 1*, 60 14-34 = it, 20 41-46 = « a 70 918 = « 7 30 34-34 = m„ 80 551 =u a 40 2761 =u t 90 3-28 = m 9 The expectation of life at 50 would, according to the above scheme, be represented by w 6 . Now if we take as our only data the expectation of life at 40 and 60, we find by the method of Art. 3, u _u ± +}h ==20 . 97 (a)> If we add to our data the expectation at 30 and 70, we find * «*=gK + tO-g fa + «?) =20-71 (b). If we add the further data for 20 and 80, we find 3 3 1 « 6 =4 K + « 6 ) - Jo («* + «») + 25 fa + u t>~ 20 '7S--(o)- And if we add in the extreme data for the ages of 10 and 90, we have 8 , v 4 . Ui= To ( M « + Mfl ' ~ 10 ^ a ^ + ^ («, + u a ) - Tjj fa + m 9 ) = 20-776 (d). "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 2111, 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 208463 Here the calculated values of the expectation at 50, corre- sponding to those given in (a), (b), (c), (d), will be found to be 21-0307, 20-8215, 20-8464, 208454. 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, wrTich 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. Ill, 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+l equidistant ordinates u , u 1 ,...u n , whose values, together with their common distance, are given, an approximate expression for the area is required. The general expression for an ordinate being u x , we have, if the common distance of the ordinates be assumed as the unit of measure, to seek an approximate value of I u x dx. Jo Now, by (2), x(x — 1) .. , x(x — 1) (x — 2) .. u x = u„ + xAu + ^ 2 ' A\ + v 1 g 8 A u o + &c - Hence ru x dx = u \ dx + Au a \ xdx+~ — ^1 x(x — l)dx Jo J J o ^ ' «i/ o and effecting the integrations + &c (18). ART. 10.] AND MECHANICAL QUADRATURE. 47 It will be observed that the data permit us to calculate the successive differences of u up to A'\. Hence, on the assumption that all succeeding differences may be neglected, the above theorem gives an approximate value of the integral sought. The following are particular deductions. 1st. Let n = 2.- Then, rejecting all terms after the one involving A 2 w , we have f J u x dos = 2u + 2Am + £A\. But Am = « 1 -m , A\ = u i — 2u 1 + u f) ; whence, substi- tuting and reducing, ■j , u + 4m + M Ujx = -2 g 1 * , If the common distance of the ordinates be represented by h, the theorem obviously becomes J Wfa = W ° + t* + M * A (19), o and is the foundation of a well-known rule in treatises on Mensuration. 2ndly. If there are four ordinates whose common distance is unity, we find in like manner \ ^■ '(H. + H + Ss + V m 3rdly. If five equidistant ordinates are given, we have in like manner / \^x = 14 K + *0 + 64K + tt 8 ) + 24 Ma (21) 40 4thly. 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, / u,dx = 6w + 18Am + 27A\ + 24A\ + -^- A\ +ro A ^ + m A ^--^ 2) - 41 42 3 Now the last coefficient ^ta differs from =-tf; or tk by the small fraction y^ > an( l 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 3 a slight error if we change the last term into j= A"m . Doing this, and then replacing Ait by w, — u and so on, we find, on reduction, / i 3 u x dx= j-: {u 6 + m 2 + u t + u e + 5 (it, + u B ) + 6k 8 }, f Jo 10 which, supposing the common distance of the ordinates to be h, gives r ^ = V2' ^ = 1' an< ^ the theorem being applied, we find -f*0 cot 0d0 = - 1-08873. The true value of the definite integral is known to be ;-log(J),or-: Jlog 11. Lagrange's formula, enables us to avoid the interme- diate employment of differences, and to calculate directly the coefficient of u m in the general expression for | u x d%. If we represent the equidistant ordinates, 2ra + 1 in number, by « , m, . . . u in , and change the origin of the integrations by assuming x — n = y, we find ultimately u x dx=A u n +A 1 (M n+1 +0 + A Kw+WnJ.-.+AO^+Wo). where generally a. <-a>: 1.2...(n + r)1.2...(n-r) «[ v-*v;%~<*-* i, w 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 = Aji Q + A t u k + &c, ART. 12.] AND MECHANICAL QUADRATURE. 5 J, where A , A t &c. are coefficients depending solely on the number of ordinates observed, and thus calculable beforehand and the same for all forms of u x . 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 / j i u x dx = A a u a + A„u b + &c (25), where A a = f ^-^-Ai^dx (26) f J q (a-b) (a-c)... v " 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 u x . If, for instance, u x be a rational integral function of a: of a degree not higher than the (n — l) 1 *, the function is fully determined when 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 u x 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 u x is of a degree not higher than the (2w — 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 = u x between the ordinates to on = p, x = q, by observing n properly chosen ordinates, although u x be of the (2m — l) th degree. The following proof of this most remarkable proposition is substantially the same as that given by Jacobi {Grelle, Vol. I. 301). • Werke, Vol. in. p. 203. 4—2 52 ON INTERPOLATION, [CH. III. Let I ujlx be the integral whose value is required, where J q u x is a rational and integral function of the (2ra — I)* degree. Let u a ,u b ...be the n observed ordinates, and f(x) the ex- pression which they give for u x by substitution in Lagrange's formula. Let A(x-a) (x-b) = if, where A is a constant. . Since u x —f(x) vanishes when x = a, b,... it must be ^equal to MJy where JV is rational, integral, and of the (n — l) 411 degree, and the error in the. approximation is MNdx, which we shall now shew can be made to vanish by properly choosing M, i.e. by properly choosing the ordi- nates measured. Jq Now JMFdx = M v Nr-JM 1 N' dx = M t N- MJT + \MJX"dx = &c. = MJST - MJV' + &c. - (- 1)" M n N^\ denoting by M K the result of integrating M k times, and by JVW the result of differentiating JV k times ; and remembering that W** is a constant. Taking the above integrals between the given limits, we see that the problem reduces to making M, r vanish at each limit for all values of r from r = 1 to r = n. This is at once accomplished by taking M _ d«{(x-p)(x -g)}" . dx" for it is thus a rational and integral function of . x of the w th 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) {x — q) as a factor, and will thus vanish at both limits. The coefficients A a , A t ...'m I / (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 M ~ dx n ~\n_\ 2n(2rc-l) a ' n*(n-iy(n-2)(n-S) •&c, }■ M.2.2ra(2ra-1) (2m-2)(2ra-3) and a,b, c ... are the roots of M= 0, which are known to be real, since those of (** — 1)" = 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 1 + A = 6 D , /.A =4-»{, log (1+ A) J ^E)} W * d f A ' dx (log (1 + ' Integrate between limits 1 and 0, remembering that [» =^|i-|a^-1a^-^a^-»-&c.} w .. AKT. 14.] AND MECHANICAL QUADRATURE. 55- Removing the- first two terms from each side since they are obviously equal, and writing w„ for Aw„ we get "12 AM " + 21 A ^- &c - = -T2 Aw ^-ii ^V^&c, and the formula becomes J a 7 U. U UjiiC = -^ + U 1 + U i +J -i2( Au «-*~ Aw °y -&c .(28). In the above investigation we have m reality twice per- formed the operation -^ on. both sides of an equation. We shall -see that Au x = Av x only enables us to say u x = v x + C- and not u x — v x ; 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 formulae 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 (GreUe, n. 252) that calculations proceeding by Differences will probably give a closer approximation to the exact result than corresponding ones that proceed by Differential Coefficients. In con- sequence of this practical value many Interpolation-formulas 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 Grunert (Anihiv, 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 {BelationenzwischenSummenund Differenzen, Abhandlungen der Kan. Sdchs.Gesellschaft, 1S65), in which also he gives a very detailed inquiry into the various methods in use, with numerical calculation of coefficients, &o. We must warn the reader against the notation, which is unscientific and wholly in defiance of convention, e.g. Ay r(i and 56 ON INTERPOLATION, [CH. III. A 5 jk are used to represent the Ay, and A l y x . 1 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 correct notation. In speaking of the developments which the theory has received we must mention an important MSmoire by Jacobi (Crelle, xxfcT 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 the 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 - 1. Jacobi had previously examined the case in which X=//= - = ; in other words, he had shewn that in dx or ff(eoB0)d$, •J> the positions of the co-ordinates to be chosen after the analogy of the Gauss- formula are given by the roots of 4 i which is equivalent to cos (n cos- 1 x) = 0. Hence x = cos " jr. 2n In this case the coefficients A., A (see (26), page 61) are all equal, ih being — , and the formula becomes n /"/(cos^^Ij^cos^./^sg,...,/^ 2 ^.)!. 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 n' h Laplace's coefficient of one variable, p 1_ tPfo'-l)" " _ 2»|V dx« ' 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 u z be a rational integral function of the (2n - 1)" 1 degree, and Y a be the n ,h Laplace's coefficient. Divide u, by Y n , and let N be the quotient and / (x) the remainder which is of the (n- l) ,h degree. Thus ««=/(x) + Y n . N. Integrate between the limits 1 and - 1, and since N is of a lower degree f 1 ( l than Y„, I Y n Ndx=0, and we are left with I f(x) dx which is accurately found by the Lagrange-formula from the n observed values of «,. 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 (Grelle, lvi. 101) with that by .Christqffel 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. 1. Kequired, 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. z. 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. 3. Express v 2 and v s approximately, in terms of v , v 1 , v t , and v s , both by Lagrange's formula and the method of (7), Art. 4. 58 EXERCISES. > [OH. Ill: 4. The logarithms' in Tables of n decimal places differ from the true values by + 1 n „+i at most. Hence shew that- the errors of logarithms of n places obtained from the Tables by interpolating to first and second differences cannot exceed, 1 19, - fn?* ~*~ e an ^ - Tn» x x "*" e ' res P ec ti v€ !ly> e an( i e ' being the errors due exclusively to interpolation. (Smith's Prize) 5. The values of a function of the time are a lt a 2 , a s , a 4 , at epochs separated by. the common interval h; the first dif- ferences are d lt d\, d'\, the second differences are d a , d\, and the third difference d a . Hence obtain the following formulae of interpolation to third differences: fffi-a+fd'-^-^t+dj t + k t. J{t)-a, + \a, 2 6JA + 5i-A s+ 6-A s ' t being reckoned in the first case from the epoch of a 2 , and in the second from that of a a . 6. If P, Q, R, 8, ... be the values of X, an unknown function of x, 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) {p, q] + {x -p) (x - q) {p, q, r) + &c, where generally P Q {p, q, r...} =-. r-. r — + -. ~ h &c. *l. Shew that, in the notation of the last question, if q —p = r — q = s — r = &c. = 1, . , A'P ^ 2,r ' s,= rx3 ; EX. 8.] EXERCISES. 59, and apply the theorem to demonstrate that (!) «.» = », + AV, + 1.2.3 A »~+ 1.2.3.4 A V + ^ '(2) ^^, + ^+^-Mav, ^-1M £ +1) ^(^-1)^ + 2) 4&c- + 1.2.3 aw »-* + 1.2.3.4 "*~* T 8. Shew that the function i+iz^ + («-gf-*) +te . a — o (a — o)(a — e) becomes unity when £ = a, and zero when t = b, c, ..., and deduce Ex. 6 therefrom. 9. Demonstrate Stirling's Interpolation-formula u t = m + 1 A (w + « J + j^2 A'w., + 2 i,~2.3 A3 ^- + U ^ + nM A ^ +&c - (Smith's Prize, I860.)' " 10. Deduce Newton's formula for Interpolation from Lagrange's when the values are equidistant. ■11. If ft radii vectores (ft. 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 (-u e ) which makes an angle with the initial line is 1 sin|(0-^a)- " sin±(0-a) where a, b, ... are the angles that the /* radii vectores make with the initial line. 60 EXERCISES. [CH. III. 12. Assuming the formula for resolving (x — a) (x — b)...(x — k) into Partial Fractions, deduce Lagrange's Interpolation- formula. 13. If (j> (x) = be a rational algebraical' equation in x of any order, and z v z 2 ...z H be taken to represent (1), (2),.... (Jc), find under what conditions 2 r= * r *&■•'** r=lZr {z 1 -z r )...{z h -z r ) 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*. Shew that Simpson's rule is tantamount to consider- ing the curve 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 parabola being in all cases parallel to the axis of y\ the area will be given by Area = h]%y- 1 jl5 (y„ + yj - 4 {y x + y^) + y 2 +y^}j . 16-J-. Given u x and u^ v and their even distances, shew that *W = Ml_J:A 8 + 1S A 4 - 13 ' 5 A 6 + &c. 2j 8 + 8.16 8.16.24 T % + «*. * On the comparative merits of these and similar methods see Dupain (Nouvelles Annates, xvii. 288). t The notation in this formula (due to Gauss) is that referred to on the top of page 56. EX. 17.] BXEECISES. Gl 17. Shew that , . , x (x + 2r — 1) . , ««. = «. + ^v + ^ 1 2 A V» *(a ; + 3r-l)( a; + 3r-2) ~ j 2 3 »-»• T ""-• AX = A"^ + "A-X*. + !L T^ ) An+2M — + &C " In what cases would the above formulae be especially useful ? 18. Shew that the coefficient of A r u a in (27) is equal to ) r+jdx, i and hence shew the exact relationship in which (27) and (18) stand to each other. 19*. If from the values u a , u b ... of a function corre- sponding to values a, b, c ... of the variable, we obtain an Interpolation-formula, u x = u a + B (x — a) + C (jc - a) (x — b) + B (x — a) (x — b) (x — c) , + &c, shew that D Au a n AB „ AG . B = - 7 — — ,- G= , D = j , &c. b—a c—a a— a where" A<£ (a, b, . . .) = (b, c, . . .) - {a, b, . . .). Deduce (2), page 35, from the above formula. * Newton's Principia, Lemma v. Lib. hi. 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 x, we . determine some other function of which the given function expresses the difference. Thus to integrate u x is to find a function v x such that Av x = u x . 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 m , u v w 2 , &c. represent successive terms of a series whose general term is u x} . and let »« = ".+\ 1 +V"+»m' (!)• Then, a being constant so that u„ remains the initial term, we have »«H=M.+ ««.l+—+«*4+«. (2). Hence, subtracting (1) from (2), Av x = u x , :. v x = A~ 1 u x . It appears from the last equation that A -1 applied to u x expresses the sum of that portion of a series whose general term is u x , which begins with a fixed term u a and ends with u z _ v On this account A -1 has been usually replaced by the AET. 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 2m,. as denned by the equation 2«, = ««_! + «„, + ...+«„. (3), it denotes a direct and always possible operation, but if we consider it as defined by the equation ; tu x = A'\ (4), and as having for its object the discovery of some finite ex- pression v x , which satisfies the equation Av x = u x , it is inter- rogative rather than directive {Biff. Equqt. 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 tbat 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 u a + u M +... +u x _ v whatever value a may be supposed to have, provided only that it is one of the series of integral values which a; is sup- posed to take. We cannot therefore consider the definitions of 2m x contained in (3) and (4) as identical, and shall therer fore proceed to investigate the relation between them and the restrictions as to the use of each. It is obvious that the %u x of (3) is one of the functions represented by the A~ i u x in (4), since it satisfies the equation Av x = u x . But this is of no value to us unless we can recog- nize to which of the functions represented by A"^ in (4) it is equal, 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 j> (x) be a function such that A0 [x) = u x . .'. <]>(a + l)-(a + 2)-(a+l)=u w , (x)-(x-l) = u x _ 1 , .: (x) - (a)=-u a + u M +« M = 2u x in (3). •• .A'^ - A"'«-, - •u-.UJU+i + 'M-x-i ^t J^-w -1», Hence retaining for 2%, the definition of (4) we should write (3) thus : *S«,-S«„ = M a +v +%_! (5). Again suppose %u x to be defined by (3) and be equal to $ (#), and let the tu x of (4) be given generally by <£> (a?) +w x , then «,. = A { (x) + w x ] = A<£ (x) + Aw x = u x + Aw x ; .*. Aw x = 0, or w x 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 w x will remain unchanged, so far as we shall have to do with it, we shall denote it by and regard it as a constant, and examine its true nature later on. (Art. 4, Ch. n.) Hence regarding %u x as defined by (3) we should write (4) thus : A-Hi„ = Su.+ CJ (6)- * 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: «.+««+!+... + u^_ 1 =\l+E + E i -i- ...+E'~^)u. = E eL~x' u ' = &~- 1 ) A- j «,= (E— -1)2«„ from (4)...(7), =Su x -Su. (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 Btate 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 £w x of (3) and that of (4) is precisely analogous to that between the definite integral I (x) dx, J a ., 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 %u x of (3) in the form r=x~l Integrctble 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 x(x-l)...(x-m + l) or x {M] in the notation of Ch. n. Art. 2. We have A* (ra+1> = (m + 1) * lm) ; .-. £*<"" = ^— T +C, m+l or 2,x(x-l)...(x-m + l) = — — ~ -+C. (1). \ / \ / m + l Taking this between limits x — n and x = m, (n > to), we get 1.2...TO + 2.3...(TO + l)+... + (n-TO)...(n-2)(n-l) n(n — l)...(?i— to) TO+1 B. F. D. 5 66 FINITE INTEGRATION, [OH. IV. Or we may retain C and determine it subsequently, thus 1.2...m + 2.2...(m + l) + ...+(n-m)...(n-2)(n-l) ^ rc(w-l) ...(n-m) p m + 1 Put n = m + 1 and the series on the left-hand side reduces to its first term, and we obtain l,2...m J m + 1) T-" 1 +Oi .'.0=0. m + 1 Thus also if - ( +"K-x-i.+ - Here a = 2, 6 = 5, m=3, and since we have to find the sum of n terms we must change n into n+1 in the last formula, and we obtain S(2w + 7)(2/i + 5)(2» + 3) (2n + 7) (2n + 5) (2» + 3) (2n ■ + ,1) ^ 4x2 But w = 1 gives us ,, 9x7x5x3 , „. . „ 105. 3 - 5 ' 7 4~2 + G ' ' ,C7 T' .-. 3 . 5 . 7 + 5 . 7 . 9 + &c, to n terms _ (2n + 7V(2n + 5) (2n + 3) (2n + 1) _ 105 8 8 ' 2nd Form. Factorial expressions of the form #(#+1) ... (x + m — 1) ART. 2.] AND THE SUMMATION OF SERIES. 67< We have by Ck n. Art. 2> A we Have 1 A = M„M I+I ...M I+m _ 1 M A+i •■•«**». ,-.2 l =C- u x u x+i • • • u x+m 1 or, writing m — 1 for m, 1 -1) «**"*« •■■ M »tm-!l (4); (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 r . 1 ax + b Ex, 2, Find the sum of n terms of the series 1 + , „* +&c 1.4.7 4.7.10 We have here q= 3, b = — 2, m = 3. .". Sum of (n — 1) terms -S 1 =G- ' Bx2xu n .u H+l CV- 6(3ri-2)(3n,+ l)' Put « = 2 and we obtain 1 ; -(l-i;,(j4. 1.4.7 6.4.7' " ' 24 68 FINITE INTEGRATION, [CH. IV. Hence (writing n for n - 1 and therefore n + 1 for w) 1 1 Sum of n terms = ^ - 6(8 „ + 1) (3b + 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 u x =ax + b, let «*M*-1 • • • «*-*,-= («» + &)'""> = («w + &) 1 -', then s ^ + ^ ! =^w +a - «>' whether m be positive ox negative. The analogy of this result with the theorem JX ' a (w» + 1) is obvious. We shall now shew how to reduoe other forms to one of the preceding. 3rd Form. Kational and integral functions. * As most of the summations of series whose nfi 1 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 moat convenient form for finding the sum of n terms of such series : — " Write down the 7fi> term with its factors in ascending order of mag- nitude, \ff one faotor ** ^ e end tl , . . • I, divide by the number of 1 (take away one faotor at the beginning^' l """ D ' factors now remaining, and by the coefficient of x (in each factor), and Subtract fromj a constant.- It is scarcely necessary to add that the upper line in the brackets must be taken when the terms are of the form u x u x _ 1 ... v x _ mi . 1 and the lower when of the form r-. . . ART. 2.} AND THE SUMMATION OF SERIES. 69 By Ch. II, Art. 5 $( x ) = (0) + A(0)x + ^p-*w+&c. Let (%) = 2«„ and put G for (0), a; 121 »-. 2^=0 + *.^ + j— -.Avo + Scc* ,...(7>, and the number of terms will be finite if v* be rational and integral. The series in (6) comes from the equivalence of the opera- tions denoted by the symbols E" and (1+A)*. In like manner we may obtain a cognate expression from the equivalence of ja x and (1 + A)"*. This gives us, when we perform them on <£ (x), <£ (0) = («») - a; . A<£ (x) + ^+11 ' tf(x) - && Putting as before $>($))=.%% and G for <£(()), and trans- posing, we get S». = G + xv, - 1^+1) At>, + 4a . . . ... .,.,(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 : v a + v a+1 + ... + v we _ 1 = {l + E + E° + ...E x - 1 }v a E*-l (1 + A)"-1 E-\ " A = {x + ^> A + *.}., (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 *■ (1+Af-l ,1-(1 + A)-* direct operations -~ and — *-^ — '— Er on «„. 70 FINITE INTEGRATION, [CH. IV. ( Here all the operations performed on i/„ 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 T+ 2 s + . . . Applying* (7) we have (since A« = l, A\=2) i> +y+ ... +( ,^iy,^, g+ ^j) + "<»-^(;-») .l Putting x = 2 we see that G is zero, and adding 'a? to hoth sides we obtain c " v, _x (os+-l)(2x+l) ~. 6 ' Ex. 4. Find the sum of n terms of the series whose n & term is n" + 7m. 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 = 8» + 14 — - — —' n(«-l)(n-2) , a »(«i-l)(n-g)(»-8) + 1.2.3 " t "° 1.2.3.4 ■ 4th Form. Any rational fraction of the form »(g) «»««! — %♦*, ' * In practice it -will be found better to resolve the n th term into factorials and apply the rule given in the note to page 68. ART. 2.] AND THE SUMMATION OF SEEIES. 71 u x being of the form ax + b, and (x) a rational and integral function of a; of a degree lower by at least two unities than the degree of the denominator. Expressing ^> (x) in the form 4>(x)=A+ Bu x + Gu x u x+1 + ... + Eu x u x+1 ... u^^, A, B ... being constants to be determined by equating coeffi- cients, or by an obvious extension of the theorem of Chap. n. Art. 5, we find v ._!(•) = ^ S JL + B %- 1 + ... + EX- 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 u x still representing ax + b, 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 , , 1.S.4 ' 2.4.5 Here the to* term n + 1 n* + 2n + l n{n + 2)(n + 3) n(n + 1) (re+ 2) (n + 3) 72 . . FINITE INTEGRATION, [CK IV. n(n + l)+n+l _ 1 _ m(w + l)(n + 2)(w + 3)^(« + 2)(n + 3) '(ra + l)(n + 2)(n + 3)^»(n + l) (w + 2) (ft + 3) " The sum of ra terms therefore, by the rule on page 68, 11 1 = C- = 0- w + 3 2(n + 2)(ft + 3) 8 <(n + 1) (» + 2) (n + $j 6ra' + 21w + 17 , 6 (n + 1) (n + 2) (ft + 3) ' 17 and (7 can easily be shewn to equal -^ . We thus can find the sum of n terms of any series whose n th term is $ (w), provided that $ (n) be either (1) a rational integral function of n, or (2) a fraction whose denominator is the product of terms of an arithmetical series, $hat, re- main a constant distance from the re 01 tern^ and whose numerator is of a degree lower by at least two than its denominator*. 5th Form. Functions of the form o* or d°$>(%) where (x) is rational and integral. * Since tp (n) e"*=0 (D) e nx we may write 4> (a) + (o + l) +-... 0(a + n-l)=[0(D){e« + e(» +1 '»+... e^- 1 *}],^ =[(a + m) - ^(a)> = S0(a + m) - S^(ct), which agrees with the previous expression. ART. 2.] AND THE SUMMATION OF SERIES. 73 From (13) page 8, we obtain at once 2o" = — ^-'. For a— 1 the integration of a x (%) we shall have recourse to sym- bolical methods. Za*{x)=A- 1 a x (x) = a x (ae° - l)^(x) = a x [a (1 + A) - l}-ty(*) + ( _^A s (x)=^"f(D+m) $ (»). The proof of this formula is given in Boole's Diff. Eq. (First Ed., p. 385), and in many other books. f The demonstration of (10) can be still farther simplified by quoting the theorem, f(E) d"(x)-a i f(aE) (x + H)=cc c . A n a" E" {x), summing all such terms we get f(E)a x ${x)=a x f(aE) (x) = {E- 1)" 1 a% (as) = a x (aE - l)" 1 {x) =a x {a(l + A)-l) #c)=&c. 74 FINITE INTEGRATION, Ex. 7. Find the sum of the series l , .2 , + 2f.2 , + 3\2 , + ^ Sum to n terms = n\2* + $n\2" 2" [ch. IV. = n\2 n + 2-1 2 A,° + * 2-1 (2-1) iAV + G = 2*{2w , -4w + 6} + C. The method just given may be generalized to apply to all functions of the form u x . #(#), where (&) is rational and integral, and u x is a function such that we know the value of A^Mj, for all integral values of n. In this case we have (comp. Ex. 3, p. 20) tu x (x) = (EE' - irn^(«) = (AE' + ±T«M*) (E' and A' being supposed to operate on and E and A on u x alone) 1 f A' A' s ) = AF J 1 - AW + W ! _ &c j u *^ x) = A _ \, . <£ (x - 1) - A'X .A(x~ 2) + A _ X.A a ^(a;-3)-&c (11), dropping the accents as no longer necessary, Ex. 8. A good example of the use of the above formula is got by taking u x = sin (ax + b). From (17), page 8, we get easily A~"sin.(aa;+&) , f , n(a + ir)\ sin \aac + o ^ — '-> + *>' ■' /, . ay ■ Let us take then the series whose « th term is (n — 7) sin (an + b) ; ART, 2.] AND THE SUMMATION OF SERIES. 75 the sum of n terms' will bes (n- 7) sin 0n+ 6) +2 (n-7) sin (an + 6) in I an + b ^ — \ sin 2sm 2 ^$m[an + b- (a + v)\ _^ ^ (»*D' 6th. Miscellaneous Forms. When a function proposed for integration cannot be referred to any of the preceding forms, it will be proper to divine if possible the form of its integral from general knowledge pf the effect of the operation A, and to determinethe constants by comparing the difference of the conjectured integral with the function proposed. Thus since Aa x (x) = a*f (»), where ^(x) = a (x + 1) — tf>(x), it is evident that if ^(a>) be a rational fraction yfr-(x) will also be such. Hence if we had to integrate a function of the form a x ty(x), ifr(«) being a ra- tional fraction, it would be proper to try first the hypothesis that the integral was of the form a x tj>(x), (x) being a ra- tional fraction the constitution of which would be suggested by that of ■$■(«). ' Thus also, since A sin _1 <£ (a;), A tan _1 <£(a;), &c, are of the respective forms sin -1 ^r(a;), tan" 1 ^(a;), Sue, ^-(oo) being an algebraic function when 0(a?) is such, and, in the case of ta,n~ 1 tf>(x), rational if (x) be so, it is usually not difficult to conjecture what must be the forms, if finite forms exist, of 2 sin rl ^r(a;), 2 tan"^r(a;), &c, •^r(i») being still supposed algebraic. The above observations may be generalized. The opera- tion denoted by A does not change of 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 + YA + 475 + &a *° n terma ' «* . «" The n th term, represented by u n , being -. =-^-. — , we have (»+l)(n + 2) .' ~{b + 1)(» + 2)' -Now remembering that the summation has reference to n, assume Then, taking the difference, we have a;V _ ( a (n + 1) + b an + b ) (n+l)(«+2)' -flr f n + 2, IT+l) = - n ci(a;-l)w !! +(2a+5)(a;-l)w+(ffi+Zi)a;-26 (n + l)(» + 2) That these expressions may agree We must have a(x-l) = l, (2a + ft)(«-l) = 0, (« + 6)a;-2& = 0. Whence we find a 1,2 ART. 3.] AND THE SUMMATION OF SEKIES. 77 The proposed series is therefore integrable if x = 4*, and we have 4T.rf 1 n-2 r+a (w+l)(w+2) B'n + 1 Substituting, determining the constant, and reducing, there results r.4 ff.ff n'4' _ 4" +1 w-1 2 2.3 + 3. 4 "• + (w + l)(M + 2) _ X"m + 2 + 3' 3. 2 is of course, like A, E, and D, an operation capable of repetition and therefore obeying the index-law ; 2V,. being defined as 2 (2m,.). Our symbolical methods will render it an easy matter to obtain expressions for 2" (or A"") analogous to those already obtained for 2, but we shall have to add, as in Integral Calculus, a function of the form (where G ,G V &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 2" analogous to (10) and leave the others as an exercise for the ingenuity of the student. It is S*«'* (»> - jjjziji {*<•> -"£=! 4 «*> + C -t-C l x + ... + C t _ l ar 1 (12). * The explanation of this peculiarity is very easy : _ n'x" _ ( 1 4 1 ) "» = («+l)(n + 2)"i S3 + i+I| ' and the summation of the above series would require a finite expression for 2 — if x had not such a value that the term — = which occurs in the n T + i — 4af (r+1)" 1 term exactly cancelled the term — -= that occurs in the r" 1 term, ' '" J* -J- a i.e. unless x = i. 78 FINITE INTEGRATION, [CH. IV. It will be found that the l rt , 8* and 5 th forms can have their n th Finite Integrals expressed in finite terms, but that the 2 nd and 4 th 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 £ in the forms ^-l" 1 ), and more generally of 2" in the form (e°-l)"", flow certain theorems which may be regarded as extensions of some of the results of Chap. II. To comprehend the true nature of these extensions the peculiar interrogative character d of the expression (e^ — l)~"tt* 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« x , given in (3), Art. 1. Now Zus-iE-iy 1 ^ = {E~ l + E^ + &c)u x 5= M^j + U x _ 2 + U^ , . . + &C.--U. _ ,* Now this is only a particular form of %u x corresponding to a = — in (3). To deduce the general form we must add an arbitrary constant, and if to that constant we assign the value -K-i +«..,... +&a), we obtain the result in question. AST. 4.] AND THE SUMMATION OF SERIES, 79 Ex. 2. Let it be required to develope Xw a v s in a series proceeding according to %v x , %*v x , &c, We have by (11), page 74, 2,u x v x = u x _ x Sf„ - Ait M %\ + AX_ 3 Vv. - &c. In applying this theorem, we are not permitted tp introdnce 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 &£ n u x in any term is equal to % n ~ l u x in the preceding term, and this imposes the condition that the constants in S" -1 ^ be retained without change in X"u x . 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 tujo. =C + u x _ t Xv x - Au^, %\ + A ! m^ Vv x (1), we shall be permitted to neglect the constants of integration, provided that we always deduce %"% by direct integration from the value of "Z n ~ 1 v x in the preceding term. If u x be rational and integral, the series will be finite, and the constant G 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 AC = 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 2m„ 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 2w x = y, then y is solely conditioned by the equation Ay=u x , or, putting e** — 1 for A, by the linear differential equation (e"-l)y.= u x . 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 v the complete value of what y would be, were u x equal to 0. Hence tu x = v x + G^ x + Cj™** + &c (2), 0,, C 2 , &c. being arbitraiy constants, and m lt m 2 , &c. the different roots of the equation e"*-l = 0. Now all these roots are included in the form m=± 2iir V— 1, i being or a positive integer. When i = 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 C e 2W-i + C" € -2*V-i, which, on making 0+ C = A { , (C—C) ^—l = B t , is re- ducible to A, cos 2itr + B i sin 2iir. Hence, giving to i all possible integral values, %u x = v x + G + A, cos 27nB + A t cos 4nrx + A a cos 6ttoi + &c. + B 1 sin 27rx + B s sin birx + B a sin Qirx + &c (3). The portion of the right-hand member of this equation which follows v x 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 a? be exclusively integral, (3) becomes %u x = v x + C + A, + A 2 + A a + Sec = v x + c, 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 2w?, sin Ittx). But this nota- tion is not only inaccurate, but very likely to mislead. It seems better either to employ G, leaving the interpretation to the general knowledge of the student, or to adopt the correct form + Sj (A ( cos 2ivx + J9 4 sin 2iir%) (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 2), and summing, and shall name the two divisions of the calculus, the Difference- and the Sum-Galculus 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 71 th 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 1 . n + 2 . (re - 1) + 3 . (n-2) + &a. 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, bat 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 / V (a—x) m dx, because the know- ''0 ledge of its value does not give us any clue to that of the indefinite integral /* xf (a - x) m 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 /.go much belong to our subject as / e - x> dx does to the Infinitesimal Calculus. 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 UBe 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 Beries, 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 (IAouville, ix. 422), in which he demonstrates a EX. 1.] EXERCISES. 83 theorem analogous to the well-known one on the value of ffff...f{x+y+...)dxdydz..., wherea;+j+2+...^-l. The method is a very beautiful one, but we must not be supposed to endorse it as rigorous, since one part involves the CO evaluation of 2 rf>) cos ax. o The fundamental operations of the Finite Calculus are taken as A with its correlative 2. 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 «o _ "i + u i ~ * c -, we must call the general term ( - l)*^. 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 sl+.E, the correlative of which, J" 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 s imil arly -with the Infinitesimal Calculus, demonstrates analogous formula, 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 unsymbolieal, 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. xin. 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. xxii. 264), by means of a calculus whose fundamental operation, A = E — v„ 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 + 315. 7.9 + ... 1 1 1.3.5.7 + 3.5.7.9 6—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 . 3 . 5 . cos + 3 . 5 . 7 . cos 20 + 5 . 7 . 9 . cos 30 + . . . 1 + 2a cos + 3a ! cos 20 + 4a s cos 30 + ... 2. The successive orders of figurate numbers are define4 by this ; — that the x* 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 X th term of the »i th order is x(x + l). (x+n-2) n-1 3. If 2' u x denote the sum of the first n terms of the series u , w 2 , u t , &c. shew that and apply this to find the sum of the series 1.3.5 + 5.7.9 + 9.11.13 + &C 4. Expand 2(#) cosma; in a series of differences of 5. Find in what cases, when u x is one of the five forms given as integrable in the present Chapter, we can find the sum of n terms of the series m -m 1 + m 2 -m 8 + &c, and construct the suitable formulae in each case. 6. Sum the following series to n terms : 1 1 1 + z^ir a + ^nrn + sin0 sin 20 sin 40 1 ,+_!_. . cos 6 . cos 20 cos 20 . cos 30 EX. 7.] EXERCISES* 85 7. Shew that cot" 1 (p + qn + to 2 ) is integrate in finite terms whenever q*-r> = k{pr-l): Obtain -. ., x • , ..logtan^ , _,2"(»-l) Stan 1 — — - — -r- andS ° on — — , and S ,,-./ . l+n(n— l)af 2 n(n + l) 8. It is always possible to assign such values to s, real or ' imaginary, that the function (a + $x + yd* + ... + vx n ) s* shall be integrable in finite terms ; a, /3 ... v being any con- stants and u x = ax + b. (Herschel's Examples of Finite Differences, p. 47.) 9. Shew that u, A 2 w„ . - , A\ aa AX . „-. ,. + o-=-fa • sm + nr^Ta cos 2e ~ -*sr^sa sm 3 ^ _ &c - 8 sm 6 16 sin 32 sin a h 10. If Au x = u x+h — u x and X = — — =- , shew that u x + \&u x + X 2 AX + &c. + X" AX = a x {(a" - 1) SaX + X B Sa* + *A n+ X}- Find the sum of w terms of the series whose w'" terms are (o+n-iraT 1 and (o + »-l) w «r*. 11. Prove the theorem Vuje. = M ,XX - "Au^X. + !L ^4p AXX""^ - &c. 12. , If (f) (x) = v„ 4- t^a; + vg? + &c, shew that u v + u^x + ujjji? + &c. = u$ (x) + x'(x) . Au 9 + ^"(x).A\ + &c; 86 EXERCISES. [CH. IV. and if (x) + xA (x) . Av + ^^(x).A\+&o. (Guderman, Crelle, vn. 306.) 13. Sum to infinity the series +1 -x + Z '~x!^\) + A ■x(a 1 -l)(x-2) + - 14. If (x) = i>„ + v x x + lye* + &c, shew that «r«/ + a^u^aT" + a^u^^x^ + &c. = 1 {t [a-Wra,)] «o + 1 [V"^' M] A** . x + &<;.}, where a is an 71 th root of unity. 15. If 1" + 2" + . .. + m" = S B and »»(»*,+ l).=p, shewthat S n =p*f(p) or (2m + l)pf(p), according as n is odd or even. (Nouvelles Armales, x. 199.) ( 87 ) 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 2 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 2 as expressed by the equation S = (e*-ir...(l) renders the deduction and the application of such theorems easy. Speaking generally these theorems are dependent upon the development of the symbol 21 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/actor 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 X we are trying to perform*. * Let us compare these methods of procedure 'with those adopted in the Integral Calculus. If f(f> (cc) dx cannot be obtained in finite terms it is usual either (1) To expand (x) in a series proceeding by powers of x and to integrate each term separately ; (2) To develops ftp (x) dx by Bernoulli's Theorem (i.e. by repeated inte- gration by parts) in a series proceeding by successive differential coefficients of some factor of the general term; or 88 THE APPROXIMATE SUMMATION OF SERIES. [CH. V. As our results are no longer exact it becomes a matter of the greatest importance to determine how far they differ from the exact results, or, in other words, the degree of approxima- tion attained. But this is usually a difficult task, and in order to lessen the difficulty of the subject to the student, we shall separate such investigations from those which first give us the expansions. The order in which we shall treat the subject will therefore be as follows : I. We shall obtain symbolical expansions for %, ^ 9 , &c. (Chapters V. and vi.) II. We shall examine the general question of Convergency and Divergency of Series, to ascertain if we may assume the arithmetical equivalence of the results of performing on u x the operations that we have just found to be symbolically equivalent. (Ch. vn.) III. Finding that many of our results do not stand the test we shall proceed to find the exact theorems corresponding to them, i. e. to find expressions for the remainder after n terms, and thus we shall reestablish the approximateness of these results. (Ch. vin.) (3) To develops /^(a;) dx in a series proceeding by successive differences of -l) , ~ l ""a> 2a;' 6a; 3+ 30;e s ffiC ' Let aj = oo", then the first member is equal to -^ by a known theorem, while the second member reduces to G. Hence l , + 2»" , + (»-l)* 6 a %x* 6z 8 + 30^ ' and if x be large a few terms of the series in the second member will suffice. 4. When the sum of the series ad inf. is unknown, or is known to be infinite, we may approximately determine C by giving to x some value which will enable us to compare the expression for %u x , in which the constant is involved, with the actual value of ~%u x obtained from the given series by addition of its terms. Ex. 3. Let the given series be 1 + s + h ■ • • + - . 2 O # Representing this series by u a , we have ART. 4.] THE APPROXIMATE SUMMATION OF SERIES. 93 1 j.^ J .1 ! 1 j 1 To determine (7, assume a; = 10, then Hence, -writing for log, 10 its value 2-302585, we have approximately G= '577215. Therefore u x = -577215+ log, + 2 V 1 -^ + I 4?-&c Ex. 4. Required an approximate value for 1 . 2 . 3 . . . x. If u x = 1 . 2 . 3 ... x, we have log u x = log 1 + log 2 + log 3 . . . + log x =log» + 21oga;. But 2' log x = C+ I log ot&c — o log a; •' •■■ "^1.2 -2)V(2a>) = /(w\ 3.5.7... {2x- 1) Vw* whence by division .'. C=logV(27r). And- now, substituting this value in (9) and determining u x , we find u x = V(2tt) x ar"* s x e -a!+ i25-86e +&0 - _i i_ = V(27rar).af.e"* + i^ 3«o# +Sc - (13). If we develope the factor ei2«~S6o5 +&c - in descending powers of x, we find i.M..^VH.^(i + i +i ^-^ ?+ fa) (14). ART. 4.] THE APPROXIMATE SUMMATION OF SERIES. 95 Hence for very large values of x we may assume 1.2.3... - By means of this formula we can obtain developed expres- sions for S 2 , % 3 , &c. with great readiness in terms of the co- efficients in the expansion of 2, i.e. in terms of Bernoulli's numbers. Ex. To develope S 3 in terms of J). From (17), L2 {6 -ir=g + 2)(| +1 ){e-ir =(l +3 i +2 ){i-J + ^ + ^ 2+&c -} suppose ' where, A„ = Q for all values of r and 4 2r+1 h (- l) r - ; jr+i- \ -i |2r + 2 B. F. D. ? f 98 THE APPROXIMATE SUMMATION OF SERIES. [CH. V. -p-p + | + (24.+ 3^,-1); + .2 L(r + 2) (r + 1) ^ + S (r + 1) ^ r+1 + 2J r ] f . r = l Hence SX -/JIM- "1 fjujx + juj* -§* +^£ X - &a - 6. Prop. III. To develope % n u x in a series, proceeding by successive differential coefficients of u^* . _,, * e°-l ■ e D -l e io -e- io . : Dt = £"» cosec (| D V- 1) x (| 2) V - 1) .-. ■D"S» = J S~*coBetf , (g.DV-l) x (|z>V-l)" (18). Suppose aj" cosec"a; = 1 - C^a? + C 4 a: 4 - &c, then fu. =d- |i + o a gy + c t gy + &c.j v r -a»)*- It must be mentioned that the Summation-formula of Art. 2 (which is due to Maclaurin-f-) 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. xxrv. 97). t Tract ore 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 ~Zu x and % n u x in a series pro- ceeding by successive differences of some factor ofu x . 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 u x = a x

(O)-0(l) + £(2)-&c. We have in general, ta x (x) = {E- l) _ Vc£(a;) = a(aE - l)~ty(a;) (note, page 73) which may be now expanded. If a = — 1, we obtain % (- TMx) = ( -^ {i - f + t " &c ] * (">• This enables us to transform many infinite series into others of a more convergent character ; for ^ (0) - <£ (1) + &c. ad inf. which is very rapidly convergent if the other is but slowly so. Ex. Transform the series f2 ~ 13 + 14 _ & °' into a more convergent form. Here *(0)= (0 + 12) 1 " 1 ', .'.we have by (21) 1 1 _,, 1(1, 1 , 2 12"r3 + &C - = 2ir2 + 2Tl27T3 + 4. 12. 13.14 23 + &c, T 8. 12. 13. 14. 15 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 formulae 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 I., ,4-5+ , A „ + o A 3 » , +&4 2.1.2 ' 4.1.2.3 8.1.2.3,4 a much more slowly converging series. This remark is of great importance with reference to all the formulas 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 jn 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 u x = -5 , then the formula becomes CD Taking this between limits oo and 1, we obtain 1 + l + l + & °. = l + l + -B l -B t + B v &c. 'Now, remembering that we must only keep the convergent part of the series, we find that we must stop at B B , since ART.: 9.] THE APPROXIMATE SUMMATION OB* 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 ii 111 *=°° 1 4 9 J 4 9 16 ,= 5 r _205 1 1 R B. x ~i44 + 5 + 275" 2+ 5 s ~5 6+,KC - On examination it will be found that we may in this case keep the terms at least as far as B 19 *, while the convergence is so rapid at first that by only retaining as far as B r 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 %cf(x) in a series proceeding by successive differential coefficients of (x). For 'taty (as) = {E- 1)-V0 (x) = a' (aE - 1)"^ (x) (23). But by Herschel's Theorem ^(e') = f(E)e°'', .-. i/r (E) — ■yjr (e°) = ty (E) e°° as operating factors, where E' affects only, . : %a?4>(x) = a x (aE - 1)" 1 |l + . D + ^ 2> 2 + &c.J 0(a>) -j&H^+iWW < 21 >> , In the case of a = — 1 an expression for A a in terms of Bernoulli's numbers can be obtained. For S(-l)"^(*)=(-l)"(-«"-l)"'*(«). putting a = -l in (23), =(-ir'(«*+ir*G»o- * In reality we may keep all terms up to - -^ L , a quantity whose first significant figure is in the fourteenth decimal place. 102 THE APPROXIMATE SUMMATION OF SERIES. [CH. V. 112 Now D 2 + |2 -2-^ i-i+|^- & 4 = i-j|<2'-l).D+jJ(2»-l)2*-to...<25), ■which determines the coefficients*. 10. Expansion in inverse factorials. The most general method of obtaining such expansions is by expressing the given function (x) in the form I e~ zt f(t) dt. If we then get + (a,) =j* (1 - dTf {% (-^)} dt. f hog (- J > must now be expanded in some way in powers ated \m of 2, and each term must be integrated separately by means of. the formula *-Wfe=- x{x + l)...{x + m)' By performing 2 on this we can expand in a similar way the more complicated form I _, f(t) dt. The most in- J $ e — -l teresting cases are those in which (as) = log x or =-; (see page 115). The method is obviously very limited in its application. A paper on it by Schlomilch will be found in Zeitsohrift Jur - 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, xxxiii. and xxxviii.), 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 Yx and its differential coefficients when x is very large* Eaabe has two papers on this (CreUe, xxv. 146 and xxviii. 10). See also Bauer (Crelle, lvii. 256) and Guderman(Crei?e, xxix. 209). Beference 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 Yx is that of the summation of harmonic series ( 2 ;-" — r — ' ,>■,.. I ■♦ On this, see . \ \a + (n^l)dYJ . : papers by Knar (Grunert, xli. and xliii.). EXEECISES. 1. Find an expression for ' -y+gj+gi + &c., toreterms, and obtain an approximate value for the sum ad infinitum. 2. Find an approximate expression for 2 ^ and also, the value of 1 1 l+2> + 2i+&c.,adinf., to 10 places of decimals. 104 EXERCISES. [CH. V. 3. Find ah approximate value of 3.5 (ftc+1) 2.4 2x ' supposing x large but not infinite. 4. Find approximately t -j , and obtain an exact for- SC •■" OL mula when a is an integral multiple of 5 . 5. Transform the series 1 1,1 ,2 _L 1 ,v.S I A. T" „2 I O . + V^ + &c. =/(l) +/' (1) Av x +tt) A 2 ^ + &c. and apply this theorem to transform the series to one proceeding by factorials only. ■ 7. Shew that 11 1 _1 , 1 , 1-2 ■s 2.*(*+l)~ 1 \3.s(* + l)(s+2) + 13. .Find 2 — A in factorials, and determine to 3 places CD of decimals the value of the constant when the first term is 1 If the Maclaurin Sum-formula, had been used, to what degree of accuracy could we have obtained C ? * De Morgan (Diff. Cat. 554). . Compare (27), page 54. 106 . EXERCISES. [EX. 14. 14. Shew that HIT + ^Ti + &c -> ad in f-' x 4 2 [2 4 |_4 and apply this to the summation of Lambert's* series, viz. a x is - nearly. [Zeitschrifi, vi. 407.] t; h ■= = + &c, when x is - nearly. \ — x 1 — ar e 15. Shew-that /(0) +/(!) + «««/ where k = N /— 1, and deduce similar formulae for the sums of the series /(0)-/(l)+/(2)-&a, /(l)+/(3)+/(5) + &c. , ;;. Find an. analogous expression for the sum of the last mentioned to n terms. 16. Shew that sin x sin 2x sin 3a; „ , . » 4 ^ H s- + &c, ad %nf., a+l a + 2 a + 3 ' J ' -f 'eC'-aO'-e- («-*)< atdt if a; lie between 7r and — it. [Schlomilch, Crelle xlii. 130.] * On the application of the Maolaurin 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 71 th powers of the natural numbers, which we know is l n +2 n ...+x" = x n + tx n -&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 we obtain the formula 1 , n „ w(w-l) (w-2) • + 5 + » + T2 T |2 ' [4 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 ~i — =• from the application of it to Se 5 *, which gives ^ = 2e^=Je^-i e -+|pe te -&c (3), 108 BERNOULLI'S NUMBERS, AND [CH. VI. which gives ^i-i-i + t*-|* +fc (4) - 2. Many other important expansions can he obtained by consideration of this identity. Thus, for h write 20 J — 1 ; then, since l( e MV^I + l ) 1 1 e 26V^l 1 lfe MVZT +l J 1 +/) 1 we at once obtain cot0 = |-|2*0-|w-&c (5). ; a Again cosec = cot ^ — cot 0, .•.cosec0 = J + 2(2-l)|*0 + 2(2 8 -l)^0 ,, + &c. ... (6). Similarly from cot — 2 cot 20 = tan we obtain . tang = 2a(2 '" 1) ^ + 2,( ^~ 1) ^ 8 + &c (7). , " 3. An expression for the values of the numbers of Bernoulli can be obtained from (5). For cot = -^ 0°g sin 0) and log S in0 = lag{0(l-5)(l-2^)...} = log 5 + log (l- J) + &c; ia 1 20 a, 0V 1 2 ^ fi ^V 1 x ■'■ coi6= -e-^-^) -2V^-¥?) - &c - 1 20.L-, 1 , I' . V ART.. 3.] -FACTORIAL COEFFICIENTS. , 109 -&c :..... ...(8). Equating the coefficients of the same powers of 6 in (5) and (8), we obtain ^^•"•f 2a »-i- 3a »i--.-y- ^^ . n 2|2w f. 1,1 } ■'■ B ^ = j^r{ 1 + 2- + w» + --'\ w- From this we see that the values of B 2n _ t increase with very great rapidity, but those of -r^ 1 ultimately approach to' equality "with those of a geometrical series whose common ratio is -r—. , . 47T 2 * A variation of (9), due I believe to Raabe [Biff, und Int. Eechnung, i. 412), depends on the following ingenious transformation : o-i ! ! 1 .. ' 1 c, 1 1 A. _1_ (2p) 2 ' and all the terms of the form /t> are removed. Proceeding as before ( x -p;) ( 1 - 3 ^) s=1+ i. + i +&c - Thus we' ultimately get i 8= "( 1 -A)( 1 -i)('-i) ■where 2, 3, 5 ... are the prime numbers taken in order. This formula would be of great use if we wished to obtain approximate values of B n correspond- ing to large 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 ["e^ardx = - ['far+ax = &c. = J^ • Jo Wo p m+l Hence we can write (9) thus An-i = 4mj as 2 " -1 {e -2 ™ + e" 4 ™ + &c. } dx J /•» S»-l = * n ] o ^r-l dx ( 10 )* 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 00 when sec x is expanded. Thus sec* = l + §«•+ £}<*+&c (11). The identity sec x — -j- log tan [-7 — 5) will give, when treated as before, ^ = 2 A{ 1 "3^ + 5^- &C -} (12) ' XV while a consideration of the identity e +e will give ^ = 2f^^ (14), formulae analogous to (9) and (10), from which (12) may be deduced. * Due io Plana (MSm. de VAcad. de Turin, 1820). + Sohlomilcli (Grvmert, 1. 361). ART. 6.] FACTORIAL COEFFICIENTS. Ill 6. Owing to the importance of Bernoulli's and Euler's numbers a great many different formulae have been investigated to facilitate their calculation. Most of these require them to be calculated successively from i? t and E 2 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 0, and equating coefficients of 6*" we obtain The simplest formulae of this nature both for Bernoulli's and Euler's numbers are obtained at once from the original assumptions -r^r = 1 -|-2(-ir^r and J_ = l+2§-» f e — 1 2 x ' Yin cos t vzn by this method. 7. But direct expressions for the values of the numbers may be found. Thus t loge 8 log 2? „ , .. _ , „ , - t — r = , _ .. = !?_-, e ' (by Herschel s theorem) _ Iog(l+A) ^A e ■ Hence, equating coefficients, we find , ir+1 x„_ 1= iog_(i+A) o^. \ / I9«. A ' \9m. ' ••■^-^(-ir X -2 2n A " |2ra ' A A 2 A 2n ) -4- &c - + 2^nr - (16) ' 112 BERNOULLI'S NUMBERS, AND [CH. VI. and in like manner we obtain 0=|l-| + | B -&c.Jo 2 » +1 (n>0) (17). 8. These formulae are capable of almost endless trans- formation. Thus, since A"' 1 (T 1 = — - A"*) 1 " 1 (Ex; 8, n : page 28),. we can write (16) thus . (4 4'4-. fc)0 »} = (-l) Wl (A-^ + | 3 -&c.)o s " +I (18), since the other term is log(l+A)0 s "=.D0 i!, *=0. ' ; 9. A m °re general transformation by aid of the formula is as follows : * ^u+^io^^o^^o- ^ Also {log (1 + yE)) 0/(0) = y/(l) ~ y j. 2/(2) + &c. = 2//(l)-yy(2) + &c. = I^/(°) M> if/(0)=0. _- -. In (19) write ■JE' for x and operate with each side oh/(0'). ART. 10.} FACTORIAL COEFFICIENTS. 113 Then {log (1 + AE')} 0"/(0') = T ^ KWf 0^/(0') = - {log (1 + AE')} 0"- 1 0'A'/(0') by (20), since O^A'/CO') = = -{log(l + A J E?')}0 n " , /'(0') J where /'(0') s 0'A'/(0'). Eepeating this n — 1 times we get {log (1 + AE;')} 0"/(0') = (- 1)** {log (1 + AE')} 0/"" 1 (0') = E'f"-\0') = [(x + 1) A (x+ 1) A.../0* + 1)]^ . This transformation has been given because it leads to a remarkable expression due to Bauer (Crelle, lviii. 292) for Bernoulli's numbers. Denote by A' the operating factor (a; + 1) A, and write - for/ (%) and 2n+ 1 for n, and we obtain from (18) B» - (- ir {log (1 + AE')} °^= [A- (^ I )]^...(21.) 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 of x* in the expansion of x w is — j *. Although it is not * Comparing (22) page 23, and (25) page 26, we see that — j — • is the coefficient of A" in the expansion of {log (1+ A) J", 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. II. Art. 13. Let C* = numerical value of the coefficient of of in the expansion of cc {n \ Then since a5 (n+1) = Qc — n) x w , we obtain C^-Ck + nCp :. (22); and we can thus calculate the values of C n+1 from those of C; and we know that the values of C 1 are 1, 0, 0, ... , ... , 11, Let us denote by CT" the numerical value of the coeffi- cient of — in the expansion of aj l_B> in negative powers of x,\ x so that a? x" Then eT* = \- ^ A"" 1 - = t-^J Fa"- 1 — 1 \n-l x |,n-l L x +!P\p** (where A now refers to p alone) \ n ~ 1 L l* *■■ » s ]Jp=o = (-irr_A^o &rv_ ) | H -i 1 of + x 3 &e y An-lrw-i also evident from the following consideration : \n ~ \n ~]n\ Z dz« lo B e \ t , putting x= log z 1 f d« k) 5=1 to 1 + &c J. , A following the notation of Art. 11. 1 Again in (25), page 26, put u x = - and <£ (D) s D"" 1 , and we get after division by (- 1)" -1 1 n — 1 , I jy»-lQ{n-l) pn-lQW X (-")_ n-\ a ,(-»-i) +&C- a n | w-l = Cf »-i ( -»,_ Cf « a .(-«-D + & c (27), in the notation of Art. 10*. * It will be seen that, as in the analogous case we couH expand {log (1 + a)}* in terms of c£, we can expand (e-l)" in terms of -(k+1) C?n .In fact where we have given C„ . its numerical value, disregarding its sign. 8—2 316 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 B^-tategar+M)- (l+ s l^Tl) where to is a divisor of n such that 2m + 1 is a prime number. Thus, taking «=8, we have (since the divisors of 8 are 1, 2, 4, 8) B 16 =integer+ Q + 3 + 5 + 17) = integer + 1^. It will be found on reference to page 91 to be 7**&. 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 papera 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 F(x)=l n +2»+ ... +(»-l) B when x is an integer, and which is given generally by AF(x)=x n . He has also given the name EuUr-Function to the analogous one that gives the sum of 1«- 2"+ 3»- &C. + (2ac- 1)" when x is integral. See Briosohi (Tortolini, Series II. 1. 260), in which there is a review of Raabe's paper (Crelle, sxn. 348) with copious references, and Kinkelin (Crelle, lvii. 122). See also a note by Cayley (Quarterly Journal, 11. 198). 15. The most important papers on the subject of this Chapter are a series ,hy 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; th& 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. JS 2n there denotes what is usually denoted by B^^. See also two papers on A"0™ 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 + ;r] = seca;+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 j. This idea has been taken up by others. Schlomilch (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 rinding the coefficient of x" in the ex- EX. 1.] EXERCISES. 117 pansion of — — T , from which both series of numbers can be deduced by ae x -l taking a=±l (Genocchi, TortoUni, Series I. Vol. in. 395). 17. . Schlb'milch has connected Bernoulli's numbers and factorial coeffi- cients with the coefficients in the expansions of such quantities as D n / (log x), &o. (Grunert, vm. ix. xvi. xviii.). Most of his analysis could be *(jh): 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 m 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 B 31 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 n(n -1) (h -2) (w-3) (n - 4) 1 5 *» — &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 = '-1' 4. Shew that [Catalan, TortoUni 1859, 239.] 118 EXERCISES. [CH. VI. 5. Shew that _ 2x (-1)' ^ s-- 1 2 aa, -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 : -^ + &c. + -h — hi = - |4w [2 |4ra-4 |6 |4w + 2 [Knar, Grvmrt, sxvn. 455.] 8. Assuming the truth of the formula + 1 2 , r smart , x f + 1 deduce a value of B^.^ 9. Prove that the coemcient of 0*" in the expansion of / V. ., 2 a "(2w-l) n UTn^J 1Se< l Ualt0 — ^~ ^-* 10. Express log sin x and log tan a; in a series proceeding by powers of x by means of Bernoulli's numbers. [Catalan, Comptes Rendus, LIV.] 11. Shew that the coemcient of log (l — e 7 'era — z log ^ is ' z" C" B -j— in I log(l— e"*)cZf — zlog^ is ™ numerically. 12. Shew by Bernoulli's numbers or otherwise that. I 8 2 a 3 s , . . Itt EX. ljj.] 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 Annates, X. 324.] 15. If ~j = 1 + Pf + Pf + &c, shew that {(l + JE) n -E n }P = Q, (E + l) P o =0, ( l + a)*P o J2£(i±£). OS 16. Shew, in the notation of the last question, that ^«P.- ( _ 1) .(!_A- +te )„.«. 17. Shew that sin x sin 2x sin 3x -I2P+X Q2r+1 *• 02T+1 " •&c. fl -j^ 1 )+^ a JrW + £^/"(*) + &c 36. Find expressions for Bernoulli's numbers and Fac- torial-coefficients in the form of determinants. [Tortolini, Series II. vn. 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. 3rdly. 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 CONVERGENCE AND DIVERGENCY OF SERIES. [CH. VII. 3. ProrI. A meri&s whose terms dimmish in absolute vali^WnV^are^ ol^nWwW^icommg, alternately positive and negative, is convergent. Let m, — w 2 + m s — m 4 + &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 u 1 -u i +(u 3 -u i ) + (u 6 -u^) + &c (1), m 1 -(m s -m s )-(m 1 -m s )-&c (2), and observing that w, — w 2 , u 2 — u B , &c. are by hypothesis positive, we see that the sum of the series is greater than Mj — u t and less than u x . The series is therefore convergent. Ex. Thus the series tends to a limit which is less than 1 and greater than ■=*. 4. Prop. II. A series whose n tu term is of the form u n sinnd (where is not zero or an integral multiple oj2tt) will converge if , for large values ofn, u n retains the same sign, continually diminishes as n increases, and ultimately vanislies. Suppose u n to retain its sign and to diminish continually as n increases after the term u a . Let S=u a sma0 + u m . l sin (a + 1) + &c (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* ('♦S)-i*G + ?H~ in which form it is equally convergent, we find that its value lies between - 6 4. Ill and - while that of the original series lies between 1 - ^ and 1 - s + - , i.e. ART. 5.] CONVERGENCY AND DIVERGENCY OP SERIES. 125 . '. 2 sin g S= u a |cos (a - 5 J - cos ( a + ^J 8 \ + u M jcos (a + 5J - cos fa + _-) 0* + &c. = M„ COS U - ^ J + (««+! ~ M a ) COS f <* + 2) ^ + ("a*, - m^) cos (a + |) ^ + &0 - Now m^, — m„, Ma+jj — Ma+i, &c. are all negative, hence 6 f 1\ 2 sin ^ 5- m„ cos [a - ^ J < (w^ - «„) + (« M - w aM ) + &c. numerically,' or < u m - u a ; .-. < - u a , Bince w M = 0. a Hence the series is convergent unless sin 5 be zero, i.e. un- less 6 be zero or an integral multiple of 27r*. An exactly similar demonstration will prove the propo- sition for the case in which the n th term is u n sin (nd - JS). Ex. The series . - sin20 sin 30 - sm + H s h&c, is convergent unless be zero or a multiple of 2tt. This is the case although, as we shall see, the series 1 + 5 + = + &c. is divergent. 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 * Malms ten (Grunert, yi. 38). A more general proposition is given by Chartier (Liouville, xvm. 21), 126 GONVERGENCY 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. ' 3rdly. 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. 6. Prop. III. If the function (x). be positive in sign, but diminishing in value as x varies continuously from a to , then the series ff>(a) + (a + 1) + tf>(a + 2) + &c. ad inf. (4) will be convergent or divergent according as I (x) dx<(d),. > J a I (x)dx<(f>(a + l), ■'o+i and so on, ad inf. Adding these inequations together, we. have J (x)dx<(a) + ^(d+l) + &c adinf. (5). ART. 7.] CONVERGENCE AND DIVERGENCY OP SERIES. 127 ' Again, by the same reasoning, 1 ra+i I (x) dx>(f> (ct+1), J a I ' .(tf>{a + 2), and so on. Again adding, we haye [ (a;)dx>(a + I) + 4>(a+2) + &c (6). Thus the integral I <£ (%) dx, being intermediate in value between the two series £(a)+0(a + l) + (a), therefore by a finite quantity. Thus the series and the integral are finite or infinite together. Cor. If in the inequation (6) we change a into a — 1, and compare ike result with (5), it will appear that the series (a) + <£ (a + 1) + (a + 2) + &c. ad inf. has for its inferior and superior limits I (x)dx...\ ....(7). J a J a-i 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 + h + h*+h s + &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 x , the value of x in the first term being 0, so that the test of convergency is simply whether / h x dx is infinite or not. Now h'-l I h x dx = •=— log h If h > 1 this expression becomes infinite with x and the series is divergent. If h < 1 the expression assumes the finite value ^ — ?-. The series is therefore convergent, log A If h = 1 the expression becomes indeterminate, but, pro- ceeding in the usual way, assumes the limiting form xh" which becomes infinite with x. Here then the series is divergent. Ex. 2. The successive series 1 1 _1_ . 1 i + &C a (log a) m T (a + 1) {log (a + 1)]" \ I \ . Xrn ologa(logloga) m "*" (a+l)log(a+l){loglog(a+l)}*" T ' > (8)*, a being positive, are convergent if m>l, and divergent if m < 1. The determining integrals are p dx r dx f M . dx Ja* m ' J a x{logx) m ' J„ x log x (log loga)"" * The convergency of these series can he investigated without the use of the Integral Calculus. See Todhunter's Algebra (Miscellaneous Theorems), or Malmsten (Grunert, vm. 419). ART. 8.] CONVERGENCY AND DIVERGENCY OP SERIES.- 129 and their values, except when m is equal to 1, are a,'-"- a'-" (toga) 1 -"- (logo) 1- * (log logic) 1 -'"- (log log a)"" 1 1 — m ' . 1 — m 1 — m in which x = oo . All these expressions are infinite if m be less than 1, and finite if m be greater than 1. If m = 1 the integrals assume the forms loga;— log a, log log x— log log a, log log log a?— log log log a &c. and still become infinite with x. Thus the series are con- vergent if m > 1 and divergent if m *Z 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 u +u 1 + u 2 + ...ad inf., all whose terms are supposed positive, is convergent or divergent accord- ing as the ratio - B±1 tends, when x is indefinitely increased, to a limiting value less or greaUr 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 -S* tends to the limit A, it is possible to give to * some value n so large, yet finite, that for that value and for all superior values of x the ratio - £tl shall lie 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. 9 130 CONVERGENCY AND DIVERGENCY OF SERIES. [CH. VII. three series Mh + «„ + l + w «« +&c. I (9), u n + (h - k) u n + (h-ky u n + &c. J 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 M - and U ~ l-{h + k) 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\ - £a shall lie between h + k and h — k. Here then still each 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 f Ex. 3. The series 1 + 1 + =—5 + , 2 o + &c-> derived from the expansion of e', is convergent for all values of t. For if f f* 1 then 1.2.. .a;' * +1 1.2...(*+1)' u^, t u x x+V and this tends to as a; tends to infinity. AET. 9.] CONVEEGENCY AND DIVEEGENCY OF SEEIES. 131 Ex. 4. The series 1+ b t + b(b + l) t+ b(b + l)(b + 2f + &C - is convergent or divergent according as t is less or greater than unity. Here M _ «(« + !) (« + 2) .-(« + «-!) ^ 66 Mit_ 6(6 + l)(6 + 2)...(6 + a; -l) r - Therefore ^*n = pL? t) 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 u a + u w + u a+2 + u ai . a + ... ad inf. (10), the g^agj^te^owbeing supposed positive and diminishing in valueTrrom x = Grt-, 3, log a; and therefore 1 l ° g u x , logsc ART. 9.] CONVERGENCY AND DIVERGENCY OF SERIES. 133 It appears therefore that the series is convergent or divergent l °9u according as, x being indefinitely increased, the function . * tog 3D approaches a limit greater or less than wnity. 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, «„ < (log*)" m being greater than unity. Hence we find, by proceeding as before, kg; XU. r _ j log log x 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 } x tends to a limit greater or less than wnity. 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 oc 1/0(1 csu increased, the function ? — ; — 4 — ~ tends to a limit greater J log log tog x a 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, 111 "• u x xu xlxu^ xlxllxu x . . . ix' ~M' liHT' • mix >&c ( A)) 134 CONVERGENCE AND DIVERGENCY OP 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 he convergent or divergent. The criteria may be presented in another form. For representing— by (j> (as), and applying to each of the functions u x in (A), the rule for indeterminate functions of the form — , we have fo(a») _ 0'(g) . l__ afl(x) ha (x) ' x {x) ' I x _ U' (x) 1J 1 Ux \ tf> (x) x) ' x log X and so on. Thus the system of functions (A) is replaced by the system x$> (x) Ix \4>{x) l }' It was virtually under this form that the system of functions was originally presented by Prof. De Morgan, {Differential Calculus, pp. 325 — 7). The law of formation is as follows. If P n represent the n* function, then P M = l n x(P n -l) (11). 10. There exists yet another and equivalent system of de- termining functions which in particular cases possesses great advantages over the two above noted. It is obtained by sub- stituting in Prof. De Morgan's forms — — 1 for ^ ) x { . The Vi (a?) lawfulness of this substitution may be established as follows. ART. 10.] CONVERGENCY AND DIVERGENCY OF SERIES. 135 Since u, = -r-,-r , we have <$>{x) _ 4>(tB + l)-4>(x) (5 being some quantity between and 1) = *» *'(* + «). na x *(*) f'(*) : l ; ' Now y \ , . . has unity for its limiting value ; for, <£ (a;) tends to become infinite as x is indefinitely increased, and therefore . , > assumes the form |g ; therefore $ (a?) °° 0(g + fl) = f (g + fl) (») <£' (») A (a) And thus the second member has for its limits , , [ and ™ . „ . , i.e. 1 and , , . ; or in other words tends to (*) . <£ (*) the limit 1. Thus (12) becomes Jk _ i = ^'^ V (*) ' Substituting therefore in (B), we obtain the system of functions Ux[l*{a(£e--l\-l\-l\&tt (C), the law of formation being still P B+1 = Vx (P n — 1). 136 CONVERGENCY AND DIVERGENCY OP 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(lxy, 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+ °, however small a may be. For x being indefinitely increased, we have ,. x(lx) m .. x(lx) m hm— - — £-=oo, hm s ' = 0, x x .. xlx\llx) m ,. xlx(llx) n . so that, according to the definition, x (lx) m is intermediate in degree between x and x x+a , xlx(llx) m between xlx and x (lx) i+a , &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 (Grelle, 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. * Idouville's Journal, Tom. vii. p. 35. ART. 12;] CONVERGENCY AND DIVERGENCY OP 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 - JS1 . According as this is less or greater than unity the series u x 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), (G). According as, in the system chosen, the first function whose limiting value is not unity, assumes a limiting value greater or less than v/nity, the series is conver- gent or divergent. Ex. 5. Let the given series be 1 +-. + -. + -. + &o (13). 2* 3* 4* Here ' u x = —^ , therefore, 5+1 X % x+2 ■* (x+l) x+1 (x + 1) x+1 and x being indefinitely increased the limiting value is unity. Now applying the first criterion of the system (A), we have , 1 x + 1 1 I— Ix , . U x _ X _X+ 1 Ix Ix X ' and the limiting value is again unity. Applying the second criterion in (A), we have xu x _ lx° _ Ix Ux Ux xllx ' 138 . CONVERGENCE AND DIVERGENCY OP SERIES. [Cfi. Vll. 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. q a(a + l) fi a(a+lj(a + 2) f . ,^ 1+ b t + bJb+1) f+ b (b + 1) (6 + 2) f + &C --W> we have in the case of failure when t = l, _ a(a + l) ...(a + x — 1) U '~ b(b + l)...{b + x-l)' Therefore u^ 1 = a + x Ux b + x and applying the first criterion of (C), (b + x \u x .. J \a + x J x _(b — a)x a + x ' ;•_ 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(0), l x \x(^-l)-l\ = lx\ (b - a)x -l\ I Wh. / ) 1 a + x ) — alx ■ 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 b — a > 1, divergent if b — a ~ 1. -j* 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 *jM-) , and by writing the n* term in the form ^ -, ,i [ t", it will be found to be com- TaT (b + n) t* parable with -^ a , 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 t , 2.3.4 B s . i ^~ *~2^~^-[2 + — ^ _ -[4~ 1 in the r" 1 class converge infinitely more slowly than those with a greater value of m, but infinitely faster than any similarly related to the (r+l)" 1 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. Tohebechef in a remarkable paper (LiouviUe, xvii. 366) has shewn that if we take the prime numbers 2, 3, 5... only, the series F(2)+F(3) + F{5) + ... will be convergent if the series FJ2) FJS) F(4) log 2 "*" log 3 log 4 " is convergent. Compare Ex. 10 at the end of the Chapter. A method of testing convergence is given by Kummer {Orelle, xm.), 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, Biff. Cat. 261). It is by finding a function v„ such that v„u„=0 ultimately, but - & - s -v tii . 1 >0 when n is co . 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 + &c. o a + l ' a 3 + 4> ' a 2 + 9 EX. 2.] EXERCISES. 141 In particular shew that if a = l the numerical value of the series will lie between the limits „ and j- . 2. The sum of the series (where S is positive) lies between 2,2* and , 3. Examine the convergency of the following series l + 6 -i + e -( 1+ ^ + e -( 1+ H) + &c., l + e' + 2« l + 3 e + &c -i .1 .1 sin -xx sm s J! sin a; , 2 3 , „ l + 2^+3 _s + &c., l+2-*+3-S + &c., 1 +2^+3^ +&C " ^ + (l2|J) a +& c. 4. Are the following series convergent ? l a+ l tf+ Io a?+ A a;4+ - + 5+I ^ + ■" where * " rea ?' 1 + a cos a •+ « 2 cos 2« + where x is real or imaginary. 142 EXERCISES* [ch. vn; 5. The hypergeometrical series ah a(a + l)b(b + l) l+ cd X + c(c + l)d(d + l) ar + &C - is convergent if x < 1, divergent if x > 1, If x = 1 it is convergent only when c + d— a — ft > 1. 6. For what values of x is the following series convergent ? ; finite ? 7. In what cases is a? + x x* + x x* + x ^Ti'^+i'Sr+l' 8. Shew that 1 11 - + - + - + &C U U t M 2 is convergent if u n+2 — 2w M+1 + u a be constant or increase with n. 9. If ^ = a -^ + l s + &c, u n n n shew that the series converges only when a < 1, or when a = 1, and /3 > 1. 10. A series of numbers p v p 2 ... are formed by the formula Pn m = Alogp n + B' shew that the series F(pJ + F( Pi ) + &c, will be convergent. if ^?l + E^S +' &a is conTCr g ent - [Bonnet, Liouville, yili. 73..] , EX.11.] EXEECISES. 143 11, Shew that the series a + a l + a s + &c, and — H p — I ■ — 8 + converge and diverge a„ a^a t d t + a % + a t together, Hence shew that there can be no test-function (n) such that a series converges or diverges according as (n) -5- u„ does not or does vanish when n is infinite. [Abel, Crelle, in. 79.] 12, Shew that if/(#) be such that /(*) * wheu 33=0, the series u 1 + u i + and/^wj +/(m 8 )+ converge and diverge together. 13, Prove from the fundamental proposition Art. 6 that the two series ■m being positive are con- $ (1) + 4> (2) + (3) + <£ (1) -I- mj> (m) + m s (m 2 ) + . vergent or divergent together. 14. Deduce Bertrand's criteria for convergence from the theorem in the last example. [Paucker, Crelle, xliii. 138.] 15. If a + a t x + a 2 03 a + &c. be a series in which a a t &c, do not contain x and it is convergent for x = 8 shew that it is convergent for x<8 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 u n = (n) cos n6. Ex. (w) = - , when the sum of the original series is — 5 log (2—2 cos x). 144 EXERCISES. [CH. VII. 17. Find the condition that the product u t u a u a should be finite. Ex. 2*. 3*. 4* 18. If the series u + u 1 + u 1 + ...<.. has all its terms of the same sign and converges, shew that the product (1 +m ) (1 +M X ) 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, XXI. 78.] ( 145 ) CHAPTER VIII. EXACT THEOREMS. 1. In the preceding chapters and more especially in Chapter IL 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 qiiaiitities 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. VIH. 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 cf 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. Bemainder in the Generalized form of Taylor's Theorem. 2. Let v x be a function defined by the identity {sc-a)v x = u x -u a (1). By repeated use of the formula £w v v x = w x+1 Av x + v x Aw x , (2) we obtain (x — a + 1) Av x +v x — Au x , (x-a + 2) A\ + 2Av x = A%, (x-a + ri) A n v x + nA n ~ l v x = A"u x . Substituting successively for v x , Av x , A\...yre obtain after slight re-arrangement u a =v, x +(a-w)Au x + ± '-j-= '- A*u x + &c. + ( p-«Q...(q--± ^ i £__$ A«v x ....(4), v x representing - ■ , as is seen from (1). ART. 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 u x =f(x) we have ■'o , .«. AX = f A"/' {» + (a - x) z] d J a « A"/ {* + («-*)*}. where is some proper fraction. If we write x + h for a, this last may be written A"/' (a; + hd) where Ax is now supposed to be 1 — 6 instead of unity, and B n appears under the form X^ (1 0) — (A»)« (5) ' from which we can at once deduce Cauchy's form of the re- mainder in Taylor's Theorem, i.e. *^ (1 -*)•/■« (*+0A), 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 Biff. Gal. 5th Ed. p. 83.) Let $(z)-j> (x) ~{z-x) A (x) - { ' / A 2 <£ (x) - &o. \n T v ' be called F(x) ; where (z - x) m = (s - x) (z - x - 1) . . . (z - x - r + 1 ). 10—2 148 EXACT THEOREMS. [CH. VIII. Then, since from (2) we obtain Ai^ = - ( *~*~ 1) ''V ^(*) (6). Now if £ — a; be an integer F(z) - F (x) = AF(x) + AF(x + 1) +...+ AF (z-1) (7), and hence is equal to the product of (z — x) and some quantity intermediate between the greatest and least of these quantities, and as AF (x) is supposed to change continuously through the space under consideration, it will at some point between x and z (we might say between x and z — 1), take the value in question, and we may thus write (7), F(z) -F(x) = {z- x) AF{z + 6 (x-z)}. But F{z) = 0, :. (6) becomes F(x) =-{z-x) AF{z + (x-z)} . (*-«) {g(p«)-l E A „ { , + g( ,_, )}> or, if z — x = h, _ (6hy n+i) A n+1 (x + h-6h) (8). 5. A more useful form of the result would be derived at once by summing both sides of (6), remembering that F(z) is zero. Since (z — x — l) (n) is positive for all values of a; less than z, we see that F(x) lies between the products of the sum (z — x — lV r) of the coefficients of the form^ : '— by the greatest and least values of A n+1 <£ (x). But the sum in question is (z — x) {n+l) ^rj — , so that the form thus obtained is very convenient. ART. 6.] EXACT THEOREMS. 149 This last investigation only applies when z — x is an integer, or in other words when the series would terminate. It is evident that if it were not so we could not draw conclusions as to the magnitude of F (z) — F (x) from the successive differ- ences as we do above. The form of the periodical constant would affect F (z) — F (x) without affecting the other side of the equation. Remainder in the Maclaurin Sum-formula. 6. In finding the remainder in the Maclaurin sum-formula we shall take it in the slightly modified form obtained by writing u x for \u x dx and performing A on both sides. It then becomes §L* = A i A ^ + A A <^_ &c (9)) dx 2 dx 1.2 do? v " but for convenience we shall write it in the more symmetrical form (using accents to denote differentiation) uj = Au x + A^uJ+A^u." +...+ A, n _J\ur\ + R 2n . . . (10), where A=-\, A=A S = &c.= 0, and A 2r = (- 1)- %j* . By Taylor's Theorem we have (Todhunter's Int. Gal. Ch. iv.) 1 1 f l z* 1 Au„ = uJ + — 2 u: + &c. +] ^ur+]^Pdz, AuJ = u: + &c. + r^rj <» + fj^ Pdz, Au* n ~ l = uf+TePdz, Jo J2m+1 where P = u%£_ = ^^ u x+ ^. 150 EXACT THEOREMS. [CH. VIII. Substitute in (10) and the coefficient of u r x is ]r \r— 1 \r — 2 This must vanish through the identity expressed in (10). Our symbolical ■work is the demonstration of this. The coefficient of Pdx under the integral sign is Wn^ A * 12^1 + &C- + A *»- 1 * S * (2w ' Z) su PP° 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 z = 1, as it also does when z = 0. 7. Assume the above to hold good for some value of n, say an even one, so that $ (2??, z) is positive between and 1, has but one maximum and vanishes at the limits. Add thereto A 2n (which is negative) and integrate and we obtain 4> {2n + 1, z). Now this vanishes at both limits, and there- fore its differential coefficient (2w, z) + A^ 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 (2m + 1, z) has only one minimum fol- lowed by a maximum between and 1, and thus can vanish but once. Adding -4 2n+1 (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 (2ra + 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 ; z* — z and as it is true for (2, z) or — g— , it is generally true. ART. 10.] EXACT THEOREMS. 151 8. Since (2n, z) retains its sign between the limits K=-\\(2n,z)u^dz = -u^ U{2n,z)dz, 6<1>0 J a •'o = A„u „ in virtue of (11). Now perform t on both sides of (9) and write iu/x for ■ v _ f 1 B l du x . , (-l) n i? 2 „ , d in -°u x , (~ l)"' H -g i ,„-i 2» Let M be the greatest value irrespective of sign that . has between the limits of summation, x and x + m suppose. Then 2m must lie between the limits + mm. 9. Other conclusions may be drawn relative to the size of the error when other, facts are known about the behaviour of u x and its differential coefficients between the limits. For in- stance, if u x n keeps its sign throughout, we may take in- stead ot — mM as one of the limits. The sign of the error will therefore be that of ( - l) n M, and, should u? n ™ keep the same sign as u x n between the limits, the error made by taking one term more of the series will have the same sign as (— l) n+ W, 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 x K+1 retain its sign between the limits in (10) we have An = ~ l l ( 2n > Z ) U *" +ld Z = - ^ ( 2n > 6 ) A < n > & < 1. 152 EXACT THEOREMS. [CH. VIII. Now it can be shewn that $(2n, 6) is never greater nu- merically than — 2A ln ; 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 that 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 £ — . The series to which they lead on SO 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 n x 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 legitime formula summatoria Maclaurianiz. Malmsten'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, l. 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 footing, 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 (Mimoires de VAcademie, 1823, page 571) he starts from an expansion by Fourier's Theorem, and obtains for the remainder an expression of the form Xto= - 2 ( j^j J i «, ! " ' 2° r^ cos 2irzdz ART. 15J EXACT THEOREMS. 153 and he then investigates the limits between which this will lie. The investi- gation is continued by Raabe (Grelle, 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 where k = J-1. On this see a paper by Genocchi (Tortolini, 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 2*. 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 Genoeehi (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, n. 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 ease 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 (Grelle, lvi. 11), Bauer (CreHe, lvii. 256), Raabe (Crelle, xxv. 146, andxxvm. 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 (Grelle, xxix. 209) in two expansions for log Tn, one of which contains a term in - , and the other does 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 Grunert Archiv, vm. 166), Grtmert, 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 (Grelle, xiii. 1), but the results scarcely belong to this chapter. 154 EXACT THEOREMS. [CH. VIII. Limits of the Remainder of the Series for Su x . (Boole.) 16. Representing, for simplicity, u x by «, we have „ „ f , 1 Br du ,,,... Bm-i d'"-H expression we si its value. Now by (9), page 109, The second line of this expression we shall represent by B, and endeavour to determine the limits of its value. 2 »=.J_ 1.2...2r (2irf- ■"«•=! m* Therefore substituting, r=«+l „=1 (27r) ar »S ar *C ar - 1 = 2S" = "s' =- ( " 1)F " 1 ***" . —i f-»+i (2m tt)* dar" -1 ' Assume ,-» (- 1)'- 1 «P-'u _ '="+ 1 (2mir) 2 ' da! 1 "- 1 ' And then, making jt — = e e , we are led by the general theorem for the summation of series (Diff. Equations, p. 431) to the differential equation t + — ' d/ in u it being remembered that by (9), page 109, the coefficient of ^-^ in (1) is, in ^21,-1^ d in+1 u the limit, a mean proportional between the coefficients of „ n-1 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 St* affords a sufficient approximation, except when the first differential coefficient in the remainder changes sign within the limits of integration. ( 157 ) 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 x, a dependent variable u x , and any successive differences of u x , as Au x , A"u x ...A 7 'u x . 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 A"u x can be expressed in terms of u x , u x+1 ... u^. Difference-equations are said to be linear when they are of the first degree with respect to u x , Au x , &?u x , &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 u x , w wl , u^, &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,u x ,c) = (1), connecting a dependent variable u x with an independent variable x and an arbitrary constant c, and from the derived equation AF(x,u x ,c) = (2), we obtain, by eliminating c, an equation of the form (x,u x ,Au x ) = .' (3). Or, if successive values are employed in the place of dif- ferences, an equation of the form 1r( x > u *> O = (4). Either of these may be considered as a type of difference- equations of the first order. In like manner if, from a complete primitive F(a>,u x ,c v c 2 ,...c n ) = (5), and from n successive equations derived from it by successive performances of the operation denoted by A or E we elimi- nate c v c 2 ,...c n , we obtain an equation which will assume the form 4>{fe, «„ Au x ,...A n u x )=*0 (6), or the form ■f (x, u x , M i+1 ,...M I+B ) = (7), according as successive differences or successive values are employed. Either of these forms is typical of differerce- equations of the n* order. ABT. 3.] DIFFERENCE-EQUATIONS OF THE FIRST ORDER. 159 Ex. 1. Assuming as complete primitive u x = cx + c", we have, on performing A, Am, = c, by which, eliminating c, there results u x =xAu x +(Au x )\ -the corresponding difference-equation of the first order. Thus too any complete primitive of the form u x = cx+f(c) will lead to a difference-equation of the form u x = xAu x +f(Au x ) (8). Ex. 2. - Assuming as complete primitive u x = ca* + c'b*, we have Hence u^ca^+c'b*", u x+i = ca M + c'b^. Therefore or « w i — au x = c' (b — a) V, «*„-<"**« = c' (&- a) 6* +1 . u^ - au^ - b (!( W1 - au x ) = 0, u^-{a + b)u M + abu x = (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 ouly 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 AX we obtain AX =/(*.. u„,An.... A-'fO (10). Performing A we get A"*X = some function of x, u x , Au x . . . &u% and on substituting for A B w* from (10) this will reduce to an equation of the form A"X =/ t (x, u x , Ah, ... A"X) (11). Continuing this process we shall obtain A n+ X =/„ (*, u,, Au x . . . A"X) (12). But u r =E r+n u_ tl = (l + ^"u_ n = ^ +( „ + , )c , + (i|rr c , +&c . + ^p' (n + r) w c »-i + — I - — /( - n > «- c v . . . e_,) n + &o. +f r {-n, M _, Cl) ... c H _ v ) (13), where c v c 2 ... c n _ t are the values of Aw_„ ... A"~V„, and with the value of u^ form n arbitrary constants in terms of which and r the general value of u r is expressed. Thus (13) con- stitutes the general primitive sought. It is evident that it satisfies the equation for AX for all values of p, since it is derived from these equations. 5. Though this is theoretically the solution of (6) it is practically of but little use. On comparing it 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 contains ; — as a factor), so that the first portion of the series 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 factor - — i — - — ■ is never less than unity. \m J Having shewn that we may always expect a complete primitive with n arbitrary constants as the solution of a difference-equation of the n th order*, we shall take the case of equations of the first order, beginning with those that are Jdso of the first degree. Linear Equations of the First Order. 6. The typical form of this class of equations is Ux+1 -A,u x = B x ..... (14), where A x and JB X are given functions of x. We shall first consider the case in which the second member is 0. To integrate the equation V-4m« = '' (15), we have whence, the equation being true for all values of x, u r+1 = A r u r . * 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, u^=A x A x _ l A x _ 2 ...A r u r (16), r being an assumed initial value of x. Let G be the arbitrary value of u x corresponding to x — r (arbitrary because, it being fixed, tbe succeeding values of u x , corresponding to x — r + 1, x = r + 2, &c, are determined in succession by (15), while u r is itself left undetermined), then (16) gives u x ^ = GA x A x _ 1 ...A ri whence u.= G4„-4„...^ (17), and this is the general integral sought*. 7. While, for any particular system of values of x differ- ing by successive unities, G 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 u , the values of u v w 2 , w 8 , &c. are rigorously determined by the equation (15). Here then G, which represents the value of u , is an arbitrary constant, and we have u x+l =GA x A x _ l ...A l) . Suppose however the initial value of x to be \, and let E be the corresponding value of u x . Then, whatever arbitrary * There is another mode of deducing thia result, which it may be well to notice. Let u x =e t . Then «» +1 =e (+ , and (15) becomes .-. e At -4«=0, whence At = log A x , t=XlogA,+ C = log A _! + log A*+ + &c. + C =log n (A_J + O, following the notation of (18). Therefore «,= £ logII K-i) +c -C l n(J.. 1 ) as before. ART. 8.] DIFFERENCE-EQUATIONS OF THE FIEST ORDER, 163 value we assign to E, the system of values of u., u &c. will be rigorously determined by (15), and the solution becomes ^ M = EA x A^...A l , The given difference-equation establishes however no con- nexion between C and E. The aggregate of possible solutions is therefore comprised in (17), supposing therein to be an arbitrary periodical function of x completing its changes while x changes through unity, and therefore becoming a simple arbitrary constant for any system of values of x differing by successive unities. We may for convenience express (17) in the form u=GYL{A^) (18), where II is a symbol of operation denoting the indefinite con- tinued product of the successive values which the function of x, which it precedes, assumes, while x successively decreases by unity. 8. Resuming the general equation (14) let us give to u x the form above determined, only replacing G by a variable para- meter G x , and then, in analogy with the known method of solution for linear differential equations, seek to deter- mine C x . We have u x = GJi. (A x _ x ), «^=a +1 n(A). whence (14) becomes C„U(AJ-A m OJL(A l ^=B l , But 4jn(4j=n(4j, whence (C X+1 -C X )U(A X )=B X , or, (M X )IL(A X ) = B X) B. whence A (7,= n (A j ' c - 4 4 +c (19); 11—2 164 DIFFERENCE-EQUATIONS OF THE FIRST ORDER. [CH. IX. ••-.= n M s nfq +c } ^> the general integral .sought*. Ex. 1. Given u M - {x + 1) u x = 1 . 2 ... {x + 1). From the form of .the second member it is apparent that x admits of integral values only. Here A^x + 1, H (A^) =x (x- 1) ... 1, :1, 2^=*; 11(4.) *' -11(4.) .-.■u x = x (x — l)...l x (x + G), where C is an arbitrary constant. * The simplest method of solving the equation BtfH- A,u,=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 porfect differential, and found that it was given hy solution of the equation Ax ^ In the present case suppose U', to be the factor which makes the left-hand side a perfect difference, i.e. of the form v t+1 u^j - v,u x . Then v^. l = C,B.ndLV,=A,C,. Thus v M =-r = ; 'A, U(A.)' 1 Multiplying by v, +1 we get as above, putting the arbitrary constant equal to unity, since we only want one integrating factor, not the general expression for such. A K«,)= J ' n(A,y B, ■"-'•■— n(A,) •■•"«= n M s iifb + 4 V.) ART. 9.] DIFFERENCE-EQUATIONS OF THE FIRST ORDER. 165 Ex. 2. Given u x+l — au x = b, where a and b are constant. Here A x = a, and II (A x ) = a x , therefore I a" S^ + Pha- 1 1- a ~+C I- 1 a ■H-Ctf 1-a where C^ is an arbitrary constant. We may observe, before dismissing the above example, that when A x = a the complete value of II (A x ) is a x multiplied by an indeterminate constant. For ^{A x )^A,A x _ v ..A r = a . a . a. . . (x — r + 1) times, = a*~ +1 =a r+1 xa x . But were this value employed, the indeterminate constant a' r+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 u x+1 -A x u x = B x (21) consists of two parts, one of which contains the arbitrary con- stant and is the solution of Ux+l -A x u x = (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-EQHATIONS OF THE FIRST OBDEK. [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 Au x + 2u x = -x — l. Replacing Aw x by u x+1 — u„, we have Here A x = — 1, B x = — (x + 1), whence u x =c{-ir-i-\. -r. , a* Ex. 4. «_. , - au, — We find u =aT x (* + l)- . t dw^ +G -^^ + ^...4 + 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, hut 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 OP 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. u x = xAu x +f(Au x ). A solution of this is evidently u x = cx+f(c), which gives Au x = c. Ex. 5. u = xAu x + Auj* gives u = ex + denoting by yjr 2 (x) the result of performing yjr on i/r (x). Continuing we shall have M ^» = V W> or if u r = a > u r^ = V (a)- 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. Ux+l = 2u„' ; .-. u x+2 = 2 (2m/) 2 = 2\\ and continuing we obtain 168 DIFFERENCE-EQUATIONS OF THE FIRST ORDER. [CH. IX. C. Equations homogeneous in u. The type of such equations is ,(*.. -).a Solve for -s* and we obtain an equation of the form -* 4 " 1 = A„ which leads to a linear equation in u x . Ex. 7. u x+1 '-3u lf+1 u x + 2u x 2 = (23). Solving u x+1 = 2u x or u x , hence u x = 2*Cor 0. We shall examine further 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. u x+l u x - a x (u^ - u x ) + 1 = 0. Here we have 1 u rA _, — v„ ^ 1 + u * + x u * Now the form of the second member suggests the trans- formation u x = tan v^ which gives , 1 tan v.., — tan v„ a x 1 + tanv^jtan v. = tan(v 1 -vj = tan Ai^ whence f,= a+^tan- J -, a ■ EX. I.] EXERCISES. 169 -.1 . = tim(c + 2 tan -1 -V Ex. 9. Given u^u z + V{(1 - *0 (1 ~ Ol = «*• Let m x = cos u„, and we have a x = cos v x+1 cos « x + sin v x+1 sin v„ . = cos (y x+l - v x ) = cos Av x , whence finally , u x = cos (0 + 2 cos -1 aQ. 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)-- ^ - J . 3rd. u = ex + c'a". 4th. u = ca x + c 8 . _ ., , , /l-a\. .. a 2 * +1 5th. z^c'+cl:; }(-a) —-R %!• Solve the equations 2 - u x+x -pa* x u x =qaf. 3. m^, — atf, = cos nx. 4. %„«„ + (x + 2) m i+1 + xu x = - 2 - 2x - x*. 5. m„. +1 — u x cos aa; = cos a cos 2a . . . cos (« — 2) a. 6. «a+i + av x + b = 0. 7. v w - era* + b = 0. 170 EXERCISES. [EX. 9. 9. u^sin xd — u x sm (x+1) = cos (x — l)0-cos(3#+l)0. 10. u M - au x = (2x + 1) a'. 11. « w -2«.*+l-0. 12. (x + iy(u x+1 -au x )=a*(x*+2x). 13. ( M ^r = 4( % ) 2 {W 2 +ll- 14. M^ = »»(«,)". 15. AX=(^-K) S . 16. M I Aw a! = a;AMj 1! + l. 17. t^l 3 - SaVu^* + 2aV^| 3 = 0. 18. If P K 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, ft are the roots of the equation a? = (n - 1) \x + 1). [Smith's Prize.] ( in ) 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 n*" degree in the form (A«)- + P 1 (A«)- , + P i (Au) ,rt ...+P - = (1), P t P 2 ...Pn being functions of the variables x and u, and then by algebraic solution reducing it to the form (Am - Pl ) (Ah -p,) . . . (Aw -p n ) = (2), it is evident that the complete primitive of any one of the component equations, Am-^ = 0, Au-p a = 0... Au-p n =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 (Biff. Equa- tions, Chap. VII. 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 respect to -S- and solution of the component equations, may without loss of generality be replaced by a single complete primitive. (lb. 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 V v V v ...V n , 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 succeeding 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 (Au x Y-(a + x)Au x + ax = (4), which is. resolvable into the two equations Au x -a = 0, Au x -x = (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 V,- 0-^ = 0, V^-G^=Q, &a., can be replaced by the single primitive ( V T - C) ( Vj - C)... =0 is merely that the last equation exactly expresses the facts stated by all the others (viz. that some one of the quantities V v V 2 ,... isconstant) and expresses no more than that. Inaprecisely similar way the primitives of a difference-equation of the same kind, being represented by f x (x, u„ C X )=Q, / 2 (x, u„ C 2 ) =0, &c, may be equally well re- presented by f x (x, u x , G) x/ 2 (x, u„ G) 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 Au,. It is not loss 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 Au x shall be equal to a when a; is even, and to x when on is odd, is satisfied if we assume _<■+? + (_!,.«=_•, the solution of which is ^ = - + £i^i) + ( _ ir (-_«_i) + (7 , and it will be found that this value of u x 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 Au x 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 Au„ 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 % for which that term becomes the equivalent of A«„ and to for all other values. The forms of the coeffi- cients may be determined by the following well-known proposition in the Theory of Equations. Pkop. If a, 8, y, ... be the several n th roots of unity, then, x being an integer, the function - —^ is equal to unity if a; be equal to n or a n multiple of n, and is equal to if * be not a multiple of n. Hence, if it be required to form such an expression for Aii x as shall assume the particular values p lt p it ...p^m succession for the values x=l, x=2,...x=n, and again, for the values x=n + l, x=n + 2,...x=2n, and so on, ad inf., it suffices to assume Au x =P,_ 1 p l +P,_ s p 1 ...+P^p„ (6), where P ._£±£±2£i-, n o, ,8, 7,. ..being as above the different n ,h 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, u x 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 abscissae 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 u x 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 Aw 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 FIRST 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 = xAu + (£m) 2 has for a complete primitive u = ex + c" (8), an indirect integral is required. Taking the difference on the hypothesis that c is constant, we have Au = c; and taking the difference of (8) on the hypothesis that c is an unknown function of x, we have Am = c + (so + 1) Ac + 2cAc + (Ac) 2 , Whence, equating these values of Am, we have Ac(* + l + 2c + Ac) = (9). Of the two component equations here implied, viz. Ac = 0, Ac + 2c + a; + l=0, the first determines c as an arbitrary constant, and leads back to the given primitive (8) ; the second gives, on integra- tion, , c-(7(-l)--|-i (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. 176 NATCKE 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 »«{c(-U--i} , - f 5 (11). 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 + &c then represents a transition from one such system to another. If such change leave unchanged the values of u and of Am 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(x, 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 = xAu x + (Au x y (12), and taking the difference of both sides, we have At*. = At*. + *AX + AX + 2Au„A«i*. + (AX) 2 5 . •. AX (AX + 2Au x + x + 1) = 0, which is resolvable into AX=0 (13), AX + 2A« a + a; + 1 = (14). The former gives, on integrating once, Au x = c, and leads, on substitution in the given equation, to the com- plete primitive (8). The second equation (14) gives, after one integration, Au x =C{-iy-l-\, (15), and substituting this in (12) we have on reduction u ^\c(-iy-l\*-^, .(16), 4 ' 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 solving differential equations by differentiation (Boole, Biff. Eq. Ch. VII. Art. 9), where we form by differentiation a dif- ferential equation of the second order, (of which the given equation is one of the first integrals,) obtain by integration the B. F. D. 12 178 NATTJBE OF SOLUTIONS OF EQUATIONS [CH. X. other first integral, and eliminate -?■ between them. Thus if ° ' dx we have we obtain dx dx* ' an integral of which is and hence the solution of the given equation is y = 4 gives us the important limitations under which the proof on page 160 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 Au n+r , which will of course affect A n+r u, 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 [OH. X. restraints on the values of Ay that (21) does, since the first member of the series permits it to equal p,, the second per- mits it to equal p it 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 (Ay-a)(Ay-b) = (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-b=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 % (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 which we are talking, and is wholly distinct from a constant that determines y, i.e. the position of the point on some one oi those ordinates which form our working-ground. .(25), 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 {Ay-4- ft -°-^(-ir}{A y -if 8 + ^(-ir}-o, 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 dx Pl U ' dx P * dx Pn being supposed to be V t -C t = 0, F 2 -<7 2 = 0... F n -C n = (27), where C x , G v ...G n are arbitrary constants, the single solution { V 1 -G)(V 2 -G)...(V n -G) = Q (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 G lt C 2 ... G n 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 V lt 7,, &c. V„ are all linear. The equation obtained by performing A on (28) will generally be of the (n — l)"* degree in G and of the re" 1 in A.y. On eliminating C between it and (28), we shall in general obtain an equation of 184 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. the n 2 degree in Ay instead of the equation of the n" 1 degree from which we obtained (28). But it may also be seen geometrically thus. Suppose we stand at a point and choose G so that (28) contains the point in virtue of V 1 — C = 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 F 1 -C = 0, F 2 -<7=0... F n -C=0, no one of which except the first contains the point at which we start. Take now the value of G which causes V, 2 — (7=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 Ay 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 G 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 = ex + c 2 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 G variable, but assum- ing that the variation of G will not affect A#. 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 notice here. We shall see better its meaning and scope by working out fully an example. 18. We have seen that the equation u x = cx + c? (29) leads to the difference-equation u x = xk.u x + (Am,,) 2 (30). Kepresenting, as before, by u x 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 Am,,., 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 he 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 it, derived or indirect integrals, according as they do not or do leave to Ay 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 A I x 2 x l~ , x 2 x ,«,., . Ai*.=y «„ + £-£, or = -y M+^-g (31), / x* where a/ u x + j- is taken to represent the wwmen'caZ 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 ea sier than to disting uish between curves satisfying the equations y= + Jc*-x* and y= - Jc 2 - a; 2 . It is true that they will not be what we are accustomed to call complete curves, but they will be perfectly definite. A nd with this understanding it will be evident that the equation Au x = + /V/^a.+x - 5 gives a unique value of Au x 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 / a? the one we have made about the sign to be taken with a/ u + -j- remove all indefiuiteness geometrically, it also (as must necessarily be the case) removes jt arithmetically. As an instance take the theorem in italics. The next value of iC a x . / ■ T - 5 M+ V«: + \/« I -t (x + 1) 2 t + Au x + K -^-L- x+1 "~F~ = + \A / X s X :+V M *+T-2 + (x+iy x+i 4 2 = + \A , X 2 , / X* 1 x+1 + 4 2 =+ v "* x 1 1 x+1 / x? x + T + 2-~ =+ V"* + 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 expressions as + */ u + -j- - = 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 + -r- . Equation (29) gives us the same values for c. And the result of performing A on (29) tells us that Au x = c, in other words The point-system ob- tained by taking at each step ^,=+V M «+ J- will keep the latter function wholly unaltered, and thus the solution of this equation is V w * x x + 4-2- In a similar way the solution of / x* X' X 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 . / Ax + 1) 3 x+l ,» f / (x + lY x + 1 =-( aJ +i)-|+y« -H+ i-^L — _ = -(i»+l)-C, 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 V« a + y" — a? to be taken partly with a plus and partly with a minus sign, and thus are partly solutions of + 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 no point an 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 = 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 -5- = 1 to -j-= — 1. This difficulty, which 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 pathswe 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 Vw, where u is a rational integral function of x and y, so that u = is the singular solution, since it renders equal the two values of ~. This is a very common case, and the treatment will apply to other more complicated cases. Let AET. 29.] OF THE FIRST OBDEE. 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 = at x, y, the value of u at x + dx, y + dy must be of the second order (and hence Vm is of the first order) in dx and dy. Now take Vit and x as new variables, r\, x, expressing y in terms of them, and draw the curves represented by the primi- tives when x and t) 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 finite, since 77 was of the first order along a primitive in the neighbourhood of 17 = 0. Thus the primitives seem to cut 17 =5 at an angle. In fact near u = 0, du was of the order \ldx excepting for small displacements in the direction of Q/n u = at the point. Thus -r- is generally infinite for 97 = 0, or the distortion produced by the new representation is so great that all curves cutting 97 = 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 ^- is of the order — = for all directions of dis- ax Vcfe placement but one coinciding with r = 0, we shall see that a solution of the equation dr_dy^ Q dx dx must have the direction given by r = 0. So considered, the dy* du apparent absurdity of saying that -j -# = is satisfied by r = 0, -r ^ 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 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. 30. We might have expected (IV.)- The equation for -j4 , obtained by differentiating the differential equation after solving for -jf- , must give the value of -t4 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 -^- whose values are not unique, since such functions will naturally have unique values on the singular solution. Thus we should expect that the equation for -j4 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 -zjL in the polygon tends as Ax 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 ir 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 Ay curve. Here then the limit to which -^- approaches in the polygon is not the same as the value of -— in the curve. 200 NATURE 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 Ace 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. 34. The principle admits also of analytical expression. Assuming h as the indeterminate increment of x, let y x , y x# , y x+2h be the ordinates of three consecutive points of the polygon, let (j> be the angle which the straight line joining the first and second of these points makes with the axis of x, tjr the corresponding angle for the second and third of the points, and let yfr — $, or 6, be called the angle of con^ tingence of these sides. Now, tan(ft = yaH -"~ y * , tan-ft ^*"*"^ , Zfx+jh Vx+h Vx+h Vx . , h h tan & = . -1 I Vx+!t Vx Vx*4l> Vx+H + h ' h = h + h h Now, since h = Ax, we have y*v»-tyx^ + y x =A 2 y x , ART. 36.] OF THE FIRST ORDER. 201 Therefore replacing y x by y, tan0 = -—-s — j — 75- (A). "*" \Ax) ^ Ax Ax Now the principle of continuity demands that in order that the solution of a difference-equation of the first order may merge into a solution of the limiting differential equa- tion, the value which it gives to the above expression for tan 8 should, as Ax approaches to 0, tend to become infini- tesimal ;• since in any continuous curve or continuous portion of a curve tan is infinitesimal. Again, that the above ex- pression for tan0 should become infinitesimal, it is clearly A' 2 v necessary and sufficient that -~ should become so. 35. The application of this principle is obvious.- Sup- posing that we are in possession of any of the complete primitives of a difference-equation in which Ax is indeter- minate, then if, in one of those primitives, the value of Ax AV being indefinitely diminished, that of -^ tends, independ- ently of the value of the arbitrary constant c, to become infini- tesimal also, the complete primitive merges into a complete . AV primitive of the limiting differential equation; but if -r^- tend to become infinitesimal with Ax only for a particular value of c, then only the particular integral corresponding to that value merges into a solution of the differential equation. 36. We have seen that when a difference-equation of the first order has two complete primitives standing in mutual re- lation of direct and indirect integrals, each of them represents in geometry a system of envelopes to the loci represented by the other. Now suppose that one of these primitives should, according to the above process, merge into a com- plete primitive of the limiting differential equation, while the other furnishes only a particular solution; then the latter, not being included in the complete primitive of the differential equation, will be a singular solution, and retain- 202 NATURE OF SOLUTIONS OF EQUATIONS [CH. X. ing in the limit its geometrical character, will be a singular solution of the envelope species. Hence, the remarkable con- clusion that those singular solutions of differential equations which are of the euvelope species, originate from particular primitives of difference-equations ; their isolation being due to the circumstance that the particular primitives of the difference-equation, obtained from the same complete primi- tive or indirect integral by taking other values of the arbi- trary constant, not possessing that character which is required by the principle of continuity, are unrepresented in the solu- tions of the differential equation*. 37. Ex. The differential equation y — x -± + (-^-\ has for its complete primitive y = cx + is analogous to the singular solutions of difference-equations spoken of in Art. 21. ( 208 ) CHAPTER XI. LINEAR DIFFERENCE-EQUATIONS WITH CONSTANT COEFFICIENTS. 1. The type of the equations of which we shall speak in the present chapter is ^+A^ 1 + fe + A«,=i (i), where A 1 , A 3 , A n are constants and X is a function of the independent variable only. This form will manifestly include the form . A"u x + A l A«-\ + &c, + A a u x = X. (2),- and may be symbolically written f(E)u x = X (3), where f(E) is a rational and integral function of E of the n & 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=e D , so that we might write (3) in the form /(«>. = X (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 Ce mx where m would be deter- mined* by the equation f(e m ) = ; and to this we should have to add a particular integral obtained either by guess or ART. 3.] LINEAR DIFFERENCE-EQUATIONS. 209 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 / (E) 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 V v V v ...V n , connected together by no linear identical relation, then will u a =C l V l +C t r t + &o.+ G n V n (6) be a solution, and in virtue of containing n arbitrary constants B, F. D, 14 210 LINEAB 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 X=0. 4. Let f(E) = have as roots m v m 2 m n ; E being treated as a symbol of quantity. Then we know that f(E) = {E-m 1 ){E-my..(E-m n ) (7), whether E he & symbol of quantity or of operation, so that we may write (5) thus, (E-mJ(E-mJ...( k E-mJu. = (8), where E—m„&c. denote successive operations the order of which is indifferent. But if we solve the equation (E — mj u x = we obtain a particular solution of (8), since the operation (E—m^) (E — m 2 )...(E — m ti _ 1 ) 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 V t , V s ,...V n (which give us by (6) the general solution) by solving n separate equations of the form (E-m)u x = (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 in K and we take the full solution of the equation (E-m.yu.~0 (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 (#-m)X = (11). But (see note page 73) f(E)cfu x =a*f(aE)u, (12); hence (E - m) r u x = m x {mE - m) r m*u a = m"* r A r (nr ! u !t ) = by (11) ; .-. A' (m~X) = 0, .-. m"X = G + 0$ + C^ + ... G r _#T x since the r^ difference of such a function vanishes ; and thus u x =(G +C 1 a>+... + G r _ 1 aT l )m x (13). Thus the general solution of (5) is u x = Z(G a + O t x + ... G r _fT)m x (14), where r is the number of times the root m is repeated in the equation / ( E) = 0. 7. We will illustrate the foregoing by an example. Let the equation be u x+3 - Su x+1 - 2u x =0, (15), or (E a -3E-2),u x = 0. This is the same as (E+l) 3 {E-2)u x = 0, and thus the solution of (15) is u.~{C +o JB )(-i)'+Ofr (i6>. 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 JCH. XI. are impossible. This may be avoided thus. I mposs ible roots occur in pairs so that with any term Cut ? (a + fi J — Vf in. the solution, corresponding to a root (a + /3 J — 1) repe ated a t least (r+l), times, there will be a term CV (a — fij — 1)", Assuming a + /3 J^l = p (cos 5 + J~^l sin 0), which gives p = J7+&, tan0=|, the terms become «y { (cos a:0 + J^-l sin a;0) + 0' (cos 006 - V 3 ! sin x6% . or x r p x [Mcos xQ + if sin xB], where M and N are still arbitrary constants. Thus the part of the solution of f(E) u x = Q that corresponds to the pair of impossible roots a + yS J — 1 repeated r times in/ (i?) = is 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 X v X t ,...X n 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 u x = vL l X l + &c. + iM a X K (27), where X v X^. . . are the solutions of. (E — raj u x = X and kindred equations, and /i v /t 2 . . . are the coefficients of the partial fractions into which - *,m 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 : — u x ^~ s x= J-=A- + &c - + ■a Ba —\ x ( 28 )- f(E) \E-rn, E-mJ ' X ^E^, <»), But, since X t is a solution of (E — mj u x = X, X '•i .:u x = f . l X l + ^X 1 + ^X a ...-. (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)'u x =x will give r constants, and (E -m)'~ 1 u, = x will give r - 1 additional ones, while there should only be r in all. But since all the solutions of the equation (E - m)'-hi z = are solutions of the equation (JB-m)'« I =0, and all the terms which we are considering come from these last- equations, we neither gain nor lose in generality whatever solution of (E - nCfhi^—Q we take, provided we take the full solution of {E -itifu,=0 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)u x =(E-l)(E-a)u x (31), but (E — a) -J-, u x does not equal ^ (E - a) u x . . .(32), 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 u x , the latter is only so when we stipu- late that the constant in the term Ga x , resulting from the performance of ^ , shall be taken zero. One difficulty 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 u x 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 X; 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. -£r — ,the opera- ART. 15.] "WITH CONSTANT COEFFICIENTS. 217 tion (E— a), which is one of those composing f(E)*, is absorbed m 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),it must be actually equal to it"f\ Ex.2. ^-5^ + 6^ = 5*; or (E- 3) {E-2) u x =h x ; 5* .". u. = (E-3){E _M L_l 5 - -2) [E-3 E-2) = \ 5* 4- GT - \ 5* + (72* = I 5* + CB X + C'2*. Z 6 fa 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 f(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/(B) j .,^ 1 +&o. I X=X whatever Xmay, it is by (M — m^ ) no means true that j ^' - +&C. [ f(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 : cos ma (D-a) oosmx _ - m sin true - a cos mx _ -msinmaj-acosma; D+a ~ D 2 -a 2 ■ _ D 2 -a a ~ -m?-a?. * The first step with the interpretation afforded by the second is wholly inadmissible:, . It should be thus cosmas D+a -^- a ) W_&\-v ~ "i -W^tf -m»-a* cosma: (cosma;) ,_, . cosma: — m sin ms - a cos m* 218 LINEAR DIFFERENCE-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 a". Since/ (E) a" =f(a) . a" we obviously have -^7-™ a x = -ttt-t a". (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 f(E) when expressed in terms of A assume the form A r (A 4- BA. + &c), we must evaluate -r? or % r X before apply- ing this method, or may omit the factor A"', apply the method, and then perform % r on the result. (III.) When X is of the form a x tf> (as), where tf> (x) is a rational and integral function of x. Here the formula / (E) a" (x) = a*f(aE) $ (x) gives us which comes under our second rule. Sin tux 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- b) u x = a'. Here f(a) = ; .\ -??-. =» 00 . * Its determinatenesa will serve as our warrant for its truth. •ART. 17.] WITH CONSTANT COEFFICIENTS, 219 But we may in this case proceed thus : a x m . 1 u* = (E -a){E-r-b)~ a " {aE - a) [aE - b) by ^^ 1 3J-1 ® = a aA(a — 6 + aA) a-6+aA' .which comes under (II.). Ex. 4. (#- 2) s (E -l)u x - afT. This will be done in a precisely similar way ; 2 8 A 3 (2.#-1) ~ 1+2A (V B > (4) "I = sr*t % (a 2 - 4>x + 6) = 2 1 " 8 j^ - ^ + a*J- . 17. In a short note in Tortolini's Anmli (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 E, but the results do not seem of much value. EXERCISES. Solve the equations : 1. u x ^-^u x+l -4iu x = m x . 2- u x+2 + iu x+i + 4>=x. 3. u x ^+2u x+1 + u x = x{x-l)(a;-2) + x(-l) x . 4. u^ - 2mu x+1 + (m 2 + ra 2 ) u x = m x . 5. Au x + A\, = x + sin x, 6. u^ - 6u x+ , + 8u x+l - 3u x = a? + (- 3)*. 7. A*u x - 5 Au x + 4m x = 2 x (l+ cos so). 8. Ay t+1 -2AX ; = a' + r. 220 EXERCISES. [CH. XT. 9- Uxu, + n a u x = cos mx. 10. ^ n 14 i2n'<, + »\ = 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 ig planted as soon as produced. Find the number of grains at the end of the a;" 1 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 «*+» + A & 0) M *+»-i + A *4> ( x ) $(?-V) «*♦*-* + -4rf(*)^(«-]L)^(«-2)if^, + *c. = Jr (1), where A l A a ... A n are constant, and $ (%) a known function. These may be reduced to equations with constant coefficients by assuming ■ u x = $ (x-n) (a -n-1)... <}>(l) v x (2). For this substitution gives »W = 0( fl, )0( a '- 1 )*(*- 2 )-0( 1 ) ! W' «*+«-! = (« - 1) (* - 2 ) • • • C 1 ) Vl and so on ; whence substituting and dividing by the common factor (x) (x — 1) ... <£ (1), we get, «w + A v **+ + A *y*+v* + &c - = 0(n,)^^-i)...^(i) ' (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 u x = (x-n) are virtually included in the above class. For, assuming $ (x) = a", they may be presented in the form *V» + A4> G») *Wt + A" $ (*) (* ~ 1) iWi + &c - = x. (4). Hence, to integrate them it is only necessary to assume (x-n) (x-n+1) = « 2 % (5). 2. By means of the proposition in the last article we can solve all linear binomial equations. Let the equation be Ux+n + A x u x = B x (6). Assume A = VJ> x _ 1 ...v x ^ 1 (7). Take logarithms of both sides and let log v x _ 1l+1 = w x , then we have w x+n . 1 + w x+n _ s + &c. + w x = logA x j (8), a linear difference-equation with constant coefficients. Solving this we obtain w x and thence v x , 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 u r , iir+n' M »-+2n & c -> an d leave these last wholly unconnected with intermediate values of u. We should therefore assume x—wy 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 u X¥1 u x + a x u x+% +b x u x = c x (10) can be reduced to linear equations of the second order, and, under certain conditions, to linear equations with constant coefficients*. Assume u = %i- a v x Then for the first two terms of the proposed equation, we have < u * +a Hz- a ^ "Whence substituting and reducing, we find «*u + (K ~ a *«) v *+i ~ OA + o x ) v x = (11), a linear equation whose coefficients will be constant if the functions b x — a x+l and ajb x + c x are constant, and which again by the previous section may be reduced to an equation with constant coefficient's if those functions are of the respective forms A4>(w), Bj>(x)4>{x-l), 4. Although linear difference-equations with variable coefficients cannot generally be solved, yet, in virtue of their * Should c„ be zero the equation is at once reduced to a linear equation of 1 the first order by dividing by u x u x+1 , and taMng — as our new dependent as variable. . < 224 MISCELLANEOUS PROPOSITIONS AND EQUATIONS. [CH. XII. linearity, they possess many remarkable properties akin to those possessed by linear differential equations, and which under certain circumstances greatly facilitate their solution. One of these properties is stated in the following Theorem. Theorem. We can depress by imity ih$ order of a linear difference-equation «.♦.+ ^«W* + £*tW* + &c. = X (12), ' if we know a particular value of u x which would satisfy it were the second member 0. Let v x be such a value, so that V» + ^V»-i + -B*'W* + & c - = <> '......(13), and let u„ = v x t x ; then (1) becomes W-» + A J>**JL»r+ + B * V **J^ + &c - = X - Or Vx+n E% + A x v^E»-% + B x v x+n ^E n -H x . . . = X. Beplacing E by 1 + A, and developing E n , E n ~\ &c. in ascending powers of A, arrange the result according to as- cending differences of t x . There will ensue + FAt x + QA% ...+ ZA\ = X. P, Q,...Z being, like the coefficient of f, functions of v x , v^, &c. and of the original coefficients A x , B x , &c. Now the coefficient of t x vanishes by (13), whence, making At x = w x , we have Pw x + QAw x . . . + ZA^w x = X, a difference-equation of the n — 1 th order for determining w x *. This being found we have t x = tw x ; .'.u x = v x tw x . * That the supposition u x =v x t x would lead to a difference-equation of the (n-lV h order for At x is obvious from a priori considerations. For the complementary function of (12) contains a term Cv„ hence the full value of t x contains a term G, and thus the full value of At x contains only n - 1 arbitrary constants, and it must therefore be given by an equation of the (»i-l) ,h order. That this equation will be linear, follows frfim the fact that the full value of t x is linear in the constants of integration. ART. 5.] SIMULTANEOUS EQUATIONS. 225 5. We shall demonstrate the Theorem of the last Article by another method, which shews more clearly how the pro- perty in question depends on the linearity ef the equation ; and this second method will teach us how to extend the Theorem to the case in which more than one solution is known. It was shewn in the last Chapter that linear difference- equations of the w' h order had solutions, of this form : u x = C 1 U x + C 2 V x +&c....+I (14), where G v C 2 ,... are arbitrary constants, X v JT 2 are functions of x, and /is a particular integral; also, the part involving the arbitrary constants is the solution of the equation formed by putting for X in (12). Change ■as into x + 1 and eliminate C x between the equa- tions, obtaining j#-^} % =-c 2 F; + & c .+r (io), suppose. Call -~: = M l where M ± is of course a function of as. Proceeding as before we shall at length obtain {E-M n ){E-M n J...{E-M 1 )u x = a quantity depending on I alone, and therefore = X.... (16), for the left-hand side must be identical with the first member of (12), since, when equated to zero, they have exactly the same solution. Thus every operation denoted by an operating factor of the form E n + A x E n ~ 1 + &c.+N x can be split up into n consecutive operations, denoted by factors of the form E—M r ; and this, can be, done in many B. F. D. 15 226 MISCELLANEOUS PROPOSITIONS AND EQUATIONS. [CH. XII. ways, for if we change the order of elimination we shall find that we get wholly different operational factors. Now suppose we know the first r of the quantities U x , V x , then we know the last r operational factors. Assume (JE-M r )(E-M^)...[E-MJu l ,=v m (17); then v x is determined by the equation of the (it — r)" 1 order, (E-M n )(E-M n J...(E-M w ) = X. (18). This last equation we shall now shew can be obtained from our knowledge of U x , V x , &c. Let (17) when expanded be (E' + P 1 E r - 1 + &c. + P r )u x = v x (19); or, what is the same thing, let the equation whose solution is o 1 u m +o,r m +...c r z. (20) be (E r +P 1 E r - 1 + &c. + P r )u x =0. And let (18) when expanded be (E^ + Q 1 E n ^ + &c. + Q n _ r )v x = X. (21), Q v Q 2 , &c, being the coefficients that we are seeking. Substitute for v x from (19), we must obtain (13) thereby, and by equating the coefficients of u^ u x+l .. f of the result- ing equation with their coefficients in (13) we shall obtain n equations for the n — r unknown quantities, Q v Q 2 .... We shall thus obtain by algebraical solution of these equations the coefficients Q v Qp.-.Q^. Thus v x is made to depend on a linear difference-equation of the (n — r)^ order. When v x is known, u x can always be found, for the equation con- necting it with v x is in its resolved form, and can thus be solved by successive steps, each consisting of the solution of a linear equation of the first order. If n— 1 independent solutions be known the equation is reduced to one of the first order, and can therefore be fully solved. Thus we obtain the more general Theorem. ART. 5.] SIMULTANEOUS EQUATIONS. 227 Theorem*. We can depress by r the order of a linear difference-equation «*. + 4 A*** + £.«*-! + &c. = X (22), if we know r independent solutions which would satisfy it were the second member 0; and if we know n — 1 independent solu- tions we can solve the equation fully. Ex. If a solution of u x+i +A x u x+l + B x u x = (23) be U x , it is required to solve fully the equation «*« + -4w«« + -B 11 ,tt» = X (24). By the last Article equation (23) must be of the form {E-P x ){E-^)u x = Q (25); and on comparing the two forms we obtain P x . —^ = B x , and therefore (24) may be written ( M *lB (>-%>*= X - (26) " The first step in the solution gives us (,_^_„(^)[ S * + a] -.%n ( 2u[s^j + o] (27). Dividing by ZT^, summing, and multiplying by ZZ,., we obtain "•- u 4xK, u{B " } ^TO +c l tCT - (28) - * Tardi gives a proof of this theorem (Tortolini, Series i. vol. i.), and especially considers the latter oase. 15—2 228 MISCELLANEOUS PKOPOSITIONS AND EQUATIONS. [CH. XII. 6. Certain forms of linear equations can be solved by performing A upon tbem one or more times. Take, for example, tbe equation (a + bx) AV.+ (c + dx) Aw x + eu x = (29) and perform A" upon it. By tbe formula at tbe top of page 21, we have {a + b (a + n)} A n+ X + nb& a+1 u x + {c + d(a> + n)} A n+ X + n^A\ + eAX = 0...(30); and if we take « = — -?, supposing that to be an integer, we have a linear equation of the first order for A n+ X- Ex. xA*u x +(x-2)Au t -u x =0 (31). Performing A on it 'we have (ar+l)AV + »AX = 0, which gives &* AX = | (32); .-.A«. = Sr| + e' (33). Substituting from (32) and (33) in (31) we obtain A more general form of this solution would be M » = r^ + ^- 2 ){ s i>TT) + c j ( 35 >- The method is due to Bronwin (Gamb. Math. Jour. Vol. III. and Gamb. and Dub. Math. Jour. Vol. n.). AKT. 7.] SIMULTANEOUS EQUATIONS. 229 7. The solution of two very remarkable non-linear equa- tions has been deduced by Prof. Sylvester from that of linear equations with constant coefficients. Let u * +? +Pi M * + »- l + & c. +p n u x = (36) be any such equation. Then writing it down for the next n values of x *Wn +j>i«*4.» + &o- +P«u x+1 = 0, &c. = 0, Eliminating the quantities p v p s , &c. we obtain .u„ u u ,* .... u ,» = 0. .(37), an equation which must be satisfied by every solution of (36). Now the solution of (36) is u x = Acf + Bp' + &c. ton. terms (38), where A, B, ... are arbitrary and a, j3, ... depend on p v p 2 , ... 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 2m 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^, p v ...p„_ v we obtain U. .M .. u . =-J>. U x, ««.-.-!• = (-!)>» *x+n-V ">*n,-4V (39), 230 MISCELLANEOUS PROPOSITIONS AND EQUATIONS. [CH. XII. or calling the last determinant P x P I+1 = (-1)>„P,... (40), the solution of which may be written P, = {(-!)>„}* (41). Thus the solution of the equation (writing n + 1 for h) u „ ,,.u "««• =Cm x . •(42) is u„ = Ao.* + Bfi* + &c. to n + 1 terms (43), where A, B, &c. and a, /9, ... are arbitrary constants limited by the two equations of condition m = afiy. . . and C = the determinant P for some value of x. If we take this last-named value to be zero, it is evident that C = Aa n Aa? Aa, B/3, Cy, A, B, 0, .... 1, 1, 1, = ABG... 1, 1, 1,. a»/3, 7>- »(M-1) x(-l)-i- = ABO ... product of squares of differences of a, /3, 7. . . taken with the proper sign. Ex. The equation «V*«.-««"=0- .(44) may be supposed to be derived from the equation which gives also ART. 8.] SIMULTANEOUS EQUATIONS. 231 Whence eliminating p we have ^x+i — w x u x+2 = u* — u x _ji x+l and .\ = constant, since it is equal to its consecutive value. Hence u x = A A "+ Bfi x , where a/3 = 1 , and ( AaT 1 + JB/3T 1 ) (Aol + 5/3) - (A + Bf = C ; .-. ABa. 2 + AB/3 2 - 2ABa^ = Oafi, or C=AB(a-/3f. 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. MxM-a'-ra^Oj Ex. 1. Vi' From the first we have u M -a*(x+l)v x+l =Q. Hence eliminating v I+1 by the second u^-a'x (x + 1) u, = 0, the solution of which is u= \x-l {Cof + G'{-a) x }, 232 MISCELLANEOUS PROPOSITIONS AND EQUATIONS. [CH. XII. and by the first equation v x = ^p=t^l{CE} u x = - 2Ea* ; or (E+iyu x = -2a** 1 . This gives and from the first equation 2v.„ = - Au„= {2G+ a (2x + !)}(- l^ + ^Ti^ "xfl (<• + !)" 9. On the subject of linear equations -with variable coefficients the student should see a remarkable paper by Christoffel (Grelle, 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 !• M w. 2 — xu x+i+ ( x — 1) u x = sm x > °De portion of the com- plementary function being a constant. 2. «W* + tW, + &c. + «« = 0. 3. u x =x(u^ 1 + u x _ 2 ). EX. 5.] EXERCISES. 233 5. m m - 2 (x - 1) u M + (so - 1 ) (x - 2) u x = \^_ 6. ^4<=«- 8. Integrate the simultaneous equations w M -v x =2m(x + V) I % i i-u x = -^m(x + l)\ • »« + (-i)"«.=o. 10. ^ - «., = (Z - m) a; ' 11. 1^ + 2^-8^ = ^ 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 by assuming (Art. 3), shew that the two constants which appear in the value of v x effectively produce only one in that of u x . 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 x+2 -a(a x + l)'u x+1 + a"* 1 u x = is u x = 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 «.««««« = «(«* + u *±x + O may be integrated by assuming m., = //a tan v x . 16. Shew also that the general integral of the above equa- tion is included in that of the equation u x+3 — u x = 0, and hence deduce the former. 17. Shew how to integrate the equation 18. Solve the equations u M = {n-m")v x + u x ,\ ««. = (2m + !)».+ ««, J and shew that if m be the integral part of Jn, — converges as x increases to the decimal part of Jn. 19. If a t be a fourth proportional to a, h, c, h 1 a fourth proportional to b, c, a, and c, to c, a, 6, and a 2 , 6 2 , c 2 depend in the same manner on a lt b lt c l , find the linear equation of differences on which a a depends and solve it. 20. Solve the equation x(x + l) A\ + k (1 — x) Au x + ku x = 0. 21. Solve the equation considering specially the case when G is zero. 22. If v , v x , v % , &c. be a series of quantities the succes 1 sive terms of which are connected by the general relation and if v , v t he any given quantities, find the value of v n . [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 M«4« M «4.»-i ••• M » = a(%H> + M * +M -i + ■■• u) is included in that of and is consequently where a is one of the imaginary (w + 1)"" roots of unity, the n + 1 constants being subject to an equation of condition. 25. Solve the equation Pn +1 = P. + iVl-P, + P^P, + &C. + P^ + P„ and shew that it is equivalent to 4m — 6 n+t n 26. Shew that P = _ )Lp n »■ [Catalan, Liowoille, in. 508.] ^ +1 = - + «,-i can be satisfied by m^ = « 2a . +1 or m^^, and that thus its solu- tion is _ r 3.5.7... (2«-l) r , 2.4.6 — 2a> ^ 2. 4. 6... (2a:- 2) + "1 .3.5 ... (2a -1) ' 3.5.7... (2a>-l) ^ 2.4.6-(2s-2 ) ^-i u '2.4.6...(2a;-2)" t " " 1. 3.5 ... (2as-3)' and deduce therefrom the solution of %n = u x + (a; 2 - *) «*_!• [Sylvester, PM. _¥«#.] ( 236 ) CHAPTER XIII. LINEAE 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 (Biff. Equations, Chap, xvii.) 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-r-J and 2. It has been shewn that if in a linear differential equation we assume x = e e , the equation may be reduced to the form . (1). U being a function of 6. Moreover, the symbols -^ and e 9 obey the laws, "> \ ma ran n I & d0 +m ) u .(2). 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. ABT. 1.] LINEAR EQUATIONS, &C. 237 Now -J3 and e 9 are the equivalents of x -5- and x, and it is proposed to develope in this chapter the corresponding theory of difference-equations founded upon the analogous employ- ment of the symbols x -r— and.xE, supposing Ax arbitrary, and therefore A(/> (x) =(x + Ax) - (j> (x), Ecp(x) = (x + Ax). Prop. 1. If the symbols it and p be defined by the equations 7r = x Ax> p=:xE @)' they will obey the laws f{ir) P m u = p m f^ + m)u\ f^)p m =f{m)p m ) W' the subject of operation in the second theorem being unity. 1st. Let Ax = r, and first let us consider the interpretation of>X. Now pu x = xEu x = xu x+r ; ••• A, = pxu f+r = x(x+r) u x+2r , whence generally p m u x = x (x + r) ... {x + (m - 1) r} u^, an equation to which we may also give the form p m u x =x(x + r) ...{x + (m~l)r}E m u x (5). If u x = 1, then, since u x+mr = 1, we have p m l=x{x+r) ...{x+(m-l)r}, to which we shall give the form p m — x (x + r) . . . {x + (m — 1) r], the subject 1 being understood. 238 LINEAR EQUATIONS [CH. XIII. 2ndly. Consider now the series of expressions irp m u x , T?p m u x ,...Tr n p m u x . Now irp m u x = x-^x{x + r) ... {x + (m- 1) r) u^ _ (x + r)...(x+mr)u x+{mA . l) -x...{x+(m-l)r}u x+mr r = 03... {«+(m-l)r}- 2W1V «mr ^.■.{ e +(« W -l)r}^- a ^-^- mr) 3 = /3 CT (7r + OT)M a ,. Hence Tr'/At, = wp'" (tt + m) M, = p m (7r + m)X, and generally •jr*p m u x = p m (7r + myu x . Therefore supposing/ (77-) a function expressible in ascend- ing powers of it, we have f(7r)p m u = P y(-rr + m)u (6), which is the first of the theorems in question. Again, supposing u — 1, we have ART. 1.] WITH VARIABLE COEFFICIENTS. 239 But 77-1 = x £- 1 = 0, tt s 1 = 0, &c. Therefore f(7r)p^l = py(m)l. Or, omitting but leaving understood the subject unity, /W/>"=/(m)/)- (7). Prop. 2. Adopting the previous definitions of "k and p, every linear difference-equation admits of symbolical expres- sion in the form /,W»,+/ 1 ( 1 rK+/ 1 W/\...+/.W / r\ = Z......(8). The above proposition is true irrespectively of the parti- cular value of Ax, 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 ■*.»«. + -*>««••• + X A = 4> (*) (9)- Here it is most convenient to assume Ax = 1 in the expres- sions of 7r and p. Now multiplying each side of (9) by x (x + 1) ... (x + n — 1), and observing that by (5) sea »i = P u " «(* + !) M *« = j°X &C-. we shall have a result of the form & (*) «. + & (*) pu x .- + £. (») /X = ft (0 • • • (10) • But since A# = 1, 7T = a;A, p = #i? = aA + x. Hence ft (*) = & (- t + /»), &c These must be expressed in ascending powers of p, regard being 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 Fo {ir - p ) = F (*■) - F t (*■) p + F 2 (*■) ^ -^W r x3 + &c , (11), where F l (ir), F 2 (tt), &c, are formed by the law {Biff. Equations, p. 439.) The equation (10) then assumes after reduction the form (8). Secondly, suppose the given difference-equation presented in the form X u x + X 1 u^ 1 ...+X n u^ n =X. (12). Here it is most convenient to assume Aso = — 1 in the ex- pression of 7r and p. Now multiplying (12) by x (x-1) ... (x — n + 1), and ob- serving that by (5) ««*-! = P U *> « (® - 1) «_ = p\ , &c, the equation becomes & ( x ) u * + 4>i ( x ) />"*••• + &, (*) PX = x> but in this case as is easily seen we have x = it + p, whence, developing the coefficients, if necessary, by the theo- rem F (ir + p) = F (tt) +F t (tt) p +F 2 (*■) ^ + &c....(13), where as before ^W=^- 1 W-i ?T OT - 1 (T-i), 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 X , X t ... X n is such as to render this multiplication unnecessary; or the requisite factors may be introduced in another way. Thus resuming the general equation X u x + X 1 u x _ 1 ... + X n u x _ n = (14), assume ^ = 1727^- We find Xfl m + Xjm m ^...+XjB (x-1) ... (x-n+1) v x . a = 0...(15). Hence assuming A, 'Ax' where Ax = — 1, we have X v x + X lP v x ...+X„p"v x = (16), and it only remains to substitute ir + p for x and develope the coefficients by (13). 3. A preliminary transformation which is often useful consists in assuming u x = p?v x . This converts the equation J> !C + ZX_ 1 ... + X A _ n = (17) into A*"Z,«. + a *- 1 Z 1 b m ...X^ = (18), putting us in possession of a disposable constant p,. 4. When the given difference-equation is expressed di- rectly in the form X A" m +X 1 A m m...+X„ W =0 (19), it may be convenient to apply. the following theorem. B. F. D. 16 IT = :.'■ _ p= xE, 242 LINEAR EQUATIONS [CH. XIII. Theorem. litr = x -r-, p = xE, then l\x •tt (tt — 1) ...(^r — n + l)u = x(x + Ax) . . . {x + (n-l)Ax} (A)" M (20). To prove this we observe that since F (tt) p n u = p n F (tt + n) u, th erefore F (tt + n) u = p^F (tt) p"u, wh ence F{ir-n)u = p n F (tt) p^u. Now reversing the order of the factors tt, -jt — 1,. . .it — n + 1 in the first member of (20), and applying the above theorem to each factor separately, we have (it — n + 1) (it — n + 2) . . . iru = p 7T/3 p 7Tp . . . ITU = P»(p-V)V But p"V = (asE)"^ A = ^* a -i a i\ = E ' 1 'i\' .: ( ff -„ + l)( ff - Il + 2)... T = p»(rAJ But p"w = x (x + r) ... {x + (n — T)r] F"u, whence / A V (tt— n+1) (it— n+2) ... Tru = x(x+r) ... {x + (n — l)r} (-r-J u, which, since r = Ax, agrees with (20). When Ax = 1, the above gives tt (tt-1) ... (tt - n + 1) = x (x+ 1) ... (x + n- 1) A" ... (21). 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 t (as) 7T (tt - 1) . . . (ir - n + 2) u + &c. = 0. It only remains then to substitute x = — ir + 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 « +/i W P u +/, W ^ ■ • • +/. W P" M = (22), and assume w = 'Zajf 1 . Then substituting, we have 2 {/„ (w) a mP » +/, (tt) a m p'" +1 ...+/„ (tt) — } = 0, in which the aggregate coefficient of /j m equated to gives /o («•) o. +/, (m) V, ...+/. (m) d„,_ B = (23). This, then, is the relation connecting the successive values of a m . The lowest value of m, corresponding to which a m is arbitrary, will be determined by the equation /,W=o, and there will thus be as many values of u expressed in series as the equation has roots. If in the expression of ir and p we assume Ax = 1, then since p m ^x(x + l)...(x + m-l)... (24), 16—2 244 LINEAR EQUATIONS [CH. XIII. the series ta^™ will be expressed in ascending factorials of the above form. But if in expressing ir and p we assume Ax = — 1, then since p m = x(x-l)...(x-m + l) (25), the series will be expressed in factorials of the latter form. Ex. 1. Given (x - a) u x - (2x - a - 1) u^ + (1 - q>) (x - 1) u x ^ = ; required the value of u x in descending factorials. Multiplying by x, and assuming ir = x -r— , p = xE, where Ax = — 1, we have x (x — a) u x - (2x - a — 1) pu x + (1 - (x) =x (x+1) ... (x + a- 1) $ (x + a), -a,, \ $(x — a) P * W - („, + i) (a, + 2) ... (as + a) ' 7r-ty (at) = («A)> (a) = 2-2-. ..<£(*); X X T ' the complex operation 2 - , denoting division of the subject 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 x, x(x + 1), x (x + 1) [x + 2), &c. ART. 6.] WITH VARIABLE COEFFICIENTS. 247 to replace these by p, p\ p\ &c. and then interpret (27) at once by the theorem = {/o( m )}" 1 *(* + l)...(« + m-l)...(29). As to the complementary function it is apparent from (28) that we have (7r-a)- i = / )V- i 0. Hence in particular if i = 1, we .find (tt - a)" 1 = p'lr- 1 * =p a 2,x- 1 = Cp° = Cx(x+l) ...(x + a-1) (30). This method enables us to solve any equation of the form x (x + 1) ... (as + n - 1) A n u + A l x(x + 1) ... ... {x+n-2) A^u ... + A n u = X (31). For symbolically expressed any such equation leads to the monomial form {tt (tt — 1) . . . (it — n + 1) + Ajt (it — 1) . . . ...(ir-n + 2)...+A n }u = X (32). Ex. 2. Given x(x + l) A?u-2xAu+2u=x(x + l) (x + 2). The symbolical form of this equation is •jr('7r-l)u-2Tru+2u=x(x + l)(x+2) (a), Or (7T 2 - 37T + 2) M = p\ Hence u = (tt 2 - 3tt + 2)" 1 p s = (3 2 -3x3 + 2)-y 248 LINEAR EQUATIONS [CH. XIII. since the factors of 7r ! — Sir + 2 are ir — 2 and ir — 1. Thus we have X (x + 1) (x + 2) , ri / , n \ , ri fL\ u = -± ~- — - + (tt — 1) p*u ... + aj> (tt) (tt - 1) ... (-ir-n+l)p a U= U ART. 7.] WITH VARIABLE COEFFICIENTS. 249 may be resolved into a system of equations, of the form u — qcj) (ir) pu = U, q being a root of the equation < f + a iq ^ + a i (TT)p' t }- 1 U, and this is a particular case of the more general form, u = F{(j> (ir)p*) U. (35). Thus the unknown function u is to be determined from the known function Uhy the performance of a particular operation of which the general type is. Now suppose the given equations transformed by some process into a new but integrable binomial form, v + i|r (w) p n v = V, V being here the given and v the sought function of oc. We have v={l+f(7r)p«}- l r, which is a particular case of F{ty (it) p n ) V, supposing F(t) to denote a function developable by Maclaurin's theorem. It is 250 LINEAR EQUATIONS [CH. XIII. apparent therefore that the theory of this transformation must depend upon the theory of the connexion of the forms, Let then the following inquiry be proposed. Given the forms of

(Tr)p»} X (Tr)X = x('*)I'{f{T)p n }X (36), irrespectively of the form of X ? Supposing F(t)=t, we have to satisfy 4> to p n x to * = x to * to p % X (37). Hence by the first equation of (4), to x (w - «) />"-£ = ^ to % to -p n X, to satisfy which, independently of the form of X, we must have •f to % (tt) = # (tt) % (w - ») ; ••■ x to =£|^x ("■-»)■ Therefore solving the above difference-equation, Substituting in (37), there results, or, replacing U n j^^J ZbyX,, and therefore X by fll. jM^jT.^, ^w«= n .|^hw^ n »{ ^(tt) z, ART. 7.] WITH VARIABLE COEFFICIENTS. 251 If for brevity we represent 11^ y ^ ' ■ by P, and drop the suffix from X t since the function is arbitrary, we have (V) p n X = Pf (tt) p n P^X. Hence therefore {4> (tt) p n fX = Pf (tt) p^P-'P^ (tt) p n p- 1 X = P{^(7r) j0 TP- 1 X, and continuing the process, { (tt) p % } m Z=> P {f (tt) p^p-'x. Supposing therefore F (t) to denote any function develop- able by Maclaurin's theorem, we have F {cj> (tt) p n } X=PF{^ (tt) p"} P~ l X. We thus arrive at the following theorem. Theorem. The symbols it and p combining in subjection to the law f(Tr)p m X = py(Tr + m)X, the members of the following equation are symbolically equi- valent, viz. ' I* W A - n. {*$} 'it W *>1 n. $$■■■ (38). A. From this theorem it follows, in particular, that we can always convert the equation u + (tt) p n u = U into any other binomial form, T, • TT f*W by assuming u = n o -i - |i|r(7r) 252 LINEAB EQUATIONS For we have «={1 +WpV U = n, \* Ml {1 ^WpT'n, •wmj y -MM u - u - a - [CH. XIII. " IV WJ " W> W, whence since « = {l + +("VP7 r . it follows that we must have In applying the above theorem, it is of course necessary that the functions <£ (ir) and y M he so related that the continued product denoted by II J2_A_JL should be finite. The conditions relating to the introduction of arbitrary con- stants have been stated with sufficient fulness elsewhere (Differential Equations, Chap. XVII. Art. 4). B. The reader will easily demonstrate also the following theorem, viz. : F {<£ (tt) p"} X = p m F {cp {it + m) P n } f*X, and deduce hence the consequence that the equation u + cj> (tt) p n u = U may be converted into v + p w '~ w ' (e) ' a whence we should have by (A), v x = ti\L-2.Jw x , F. = n7-2^y. (/). In the latter case we should assume as the transformed equation v, ' + ^.Ip w '" W ' W> AET. 8.] WITH VARIABLE COEFFICIENTS. 255 and should find f\w„, TT.-nYL-ljF. (h). The value of W x obtained from (/) or (A) is to be sub- stituted in (e) or (g), w x then found by integration, and v x determined by (/) or (h). One arbitrary constant will be introduced in the integration for w x , and the other will be due either to the previous process for determining W x , or to the subsequent one for determining v x . j> Thus in the particular case in which —r is a positive inte- ger, we should have w "= {l 77- +2) (^ + l - x ) - <"■ + vY°> a particular value of which, derived from the interpretation of (ir+-j) and involving an arbitrary constant, will be found to be . Substituting in (e) and reducing the equation to the ordinary unsymbolical form, we have H {ax + b)w x + (A~ pa) xw^ = ^- , and w x being hence found, we have v ' = ( 7r+ 2J ("" + 2 ~ *) '■• ("" + ^ w * for the complete integral. 2ndly. Let yu, be determined so as if possible to cause the second term of (c) to vanish. This requires that we have 2afi + c = 0, (b — a) fi + e = 0, and therefore imposes the condition 2ae + (b - a) c = 0. 256 LINEAR EQUATIONS [CH. XIII. — C Supposing this satisfied, we obtain, on making fi = — , lb n J- —X+2 V or, representing any particular value of the "second member by V, «. — t — KP».=r, where ,_ V(c 3 -4a/) h c * an equation which is integrable if - be an odd number whe- ther positive or negative. We must in such case assume h 2 *'- *>- l) pltg " = ' F " and determine first W x and lastly v x by h. To found upon these results the conditions of solution of the general equation (a), viz. (ax + b)u x + (ex + e) u x _ t + (fx + g) u^ = X, assume foo + g=f(x , -\), u x = tj. Then + {c X ' + e - c l±tj t^ +f y _ i) ^ = X ', comparing which with (b) we see that it is only necessary in the expression of the conditions already deduced to change , . , , a (1 + a) . , c (1 + a) b into b *■ ' yj , e into e > , i,) . f J ART. 9.] WITH VARIABLE COEFFICIENTS. 257 Solution of the above equation when X=0by definite integrals*. 9. If representing u x by u we express (a) in the form -— 2— (ax + b)u+(cx + e)e dx u+ (fx+g) e" dx u = 0, or <* _ 9 JL -A 9 A x(a+ce~ dx +fe <*»)w+(& + ee dx + ge~ dx )u-Q, its solution in definite integrals may be obtained by Laplace's method for differential equations of the form *(s)» + + G5)-°. each particular integral of which is of the form ;--CJ 6 h® at, the limits of the final integration being any roots of the equation See Differential Equations, Chap. xvm. The above solution is obtained by assuming u=je* e f(t) dt, and then by substitution in the given equation and reduction obtaining a differential equation for determining the form of f{t), and an algebraic equation for determining the limits. Laplace actually makes the assumption u = St x F{t)dt, which differs from the above only in that logt takes the place of t and of course leads to equivalent results (Theorie Analytique des Probabildtes, pp. 121, 135). And he employs this method with a view not so much to the solution of difficult equations as to the expression of solutions in forms convenient for calculation when functions of large numbers are involved. * See also a paper by Thomse (Zeitschrift, sit. 349). B. F. D. 17 258 LINEAR EQUATIONS [CH. XIII. Thus taking his first example, viz. ti x+1 -(x + l)u x = 0, and assuming u x = JfF(t) dt, we have /f +1 F(t) dt- [x + l)JfF(t) dt = 0. (i). But (x + l)jFF[t) dt = JF(t) (x+1) fdt = F(t)r +1 -ff+ 1 F'(t)dt. So that (i) becomes on substitution / f +1 {F (t) + F' (*)} dt - F(t) f* 1 = 0, and furnishes the two equations F'(t)+F(t)=0, F{t) t M = 0, the first of which gives , F(t)=Ce-', and thus reducing the second to the form Pe-'f* 1 = 0, gives for the limits t = and t = oo , on the assumption that x + 1 is positive. Thus we have finally J e-'fdt, the well-known expression for V (x + 1). A peculiar method of integration is then applied to convert the above definite integral into a rapidly convergent series. Discussion of the equation (ax" + bx + c) u x + {ex +/) u^ + gu^ = (a). 10. Let u = ,, ^ * ; then x 1.2. ..x' y? {ax* i-bx + c)v x + fi (ex+f)xy x _ t +gx (x - 1) Vs = °- ART. 10.] WITH VARIABLE COEFFICIENTS. 259 Whence, assuming ir = x -r- , v = xE, where Ax = — 1, we have p? (ax 2 + bx + c)v x + fi (ex +/) pv x + gp\= 0. Therefore substituting it + p for x, and developing by (13), p 2 (air 2 +bTr + c) + p, {(2ap + e) it + (b - a) p. +/} pv x + (p?a + px+g) P \ = (6). First, let p. be determined so as to satisfy the equation ap? + ep + g = 0, then p (owr 2 + bir+c)v x +- {(2ap + e)ir+ (b — a) p. +/} pv x = 0. Whence, by Art. 5, v x = ?,a m x(x-V) ... (x-m + 1), the successive values of a m being determined by the equation p* (am 2 + bm + c) a m + {{2ap, 2 + ep) m + (b - a) p 2 +fp) a„_ t = 0, (2ap + e)m + (b-a)p+f „ or ct_ = — -, — 5 — j r (i„ ,. * p, (am 2 + bm + c) m_1 Represent this equation in the form a * = -f( m ) °W and let the roots of the equation am 2 + bm + c = be a and /3, then v x = G {«" -/(a + 1) £c !a+11 +/(a + 1) / (a + 2) ( «+ 21 - &c.} + C'{xW-f(/3 + l)xW ) +f(J3+l)f(J3+2)x« i+i) -&c.}...(c), where generally a;*' =#(o;-l) ... (x-p+1). 17—2 260 LINEAE EQUATIONS [CH. XIII. One of these series will terminate whenever the value of m given by the equation (2afi + e) m + (b - a) /i +f= exceeds by an integer either root of the equation am" + bm + c = 0. The solution may then be completed as in the last example. Secondly, let /a be determined if possible so as to cause the second term of (b) to vanish. This gives 2a/M + e = 0, (6-a)/*+/-0, whence, eliminating /j, we have the condition 2af+(a-b)e = 0. This being satisfied, and /x being assumed equal to — „- , (b) becomes (air* + for + c)v x - a Jl^A p * Vx = o. 6 Or putting h- W-W , h* , . IT + -7T+- a a and is integrable in finite terms if the roots of the equation 2 ■ b . C n m +- m+ - = a a differ by an odd number. Discussion of the equation (ax* + bx + c) A\. + (ex +/) Am. + #m. = 0. 11. By resolution of its coefficients this equation is reduci- ble to the form a (x - a) (x - 0) A 2 m. + e (oo - 7) Au„ + gu x = 0... (a). ART. 11.] WITH VARIABLE COEFFICIENTS. 261 Now let x — a. = x + 1 and u x — v x >, then we have a(*+l)(*+i-j3 + l)AV + e (x + a - 7 + 1) Aiv +gv x > = 0, or, dropping the accent, a (x + 1) (as + a -/3+ 1) A\ + e(aj + a-7+l) Aw„ ! + gfi> a . = 0...(6). If from the solution of this equation v x be obtained, the value of u x will thence be deduced by, merely changing % into x — a — 1. Now multiply (b) by x, and assume 7r = X AxP = xE ' where Aa? = 1. Then, since by (20), x (x + 1) AV,. = ir (w - 1) »,., we have a (x + a. — /3 + 1) ir [ir — 1) v x + e{x + a-y+l)TTV x + gv x = 0. But x = — ir + p, therefore substituting, and developing the coefficients we have on reduction ■7T {a (ir — a + /3 - 1) (ir — 1) + e (ir - a + y - 1) +g] v x -{a(7r-l)(7r-2) + e(7r-l)+ 5 r} /3 ^ = 0...(c). And this is a binomial equation whose solutions in series are of the form v x = ta m x (£C+1) ... (x + m- 1), the lowest value of m being a root of the equation m{a(m—Qi + t3-l) (m -1) + e (m - a + y -I) + g] = 0...(c£), corresponding to which value ct m is an arbitrary constant, while all succeeding values of a m are determined by the law a (m — 1) (to — 2) + e (m — 1) + g m [a (to — « + /3 — 1) (to — 1) + e (m — a. + y— 1) +g] 'm—i m 262 LINEAR EQUATIONS [CH. XIII. Hence the series terminates when a root of the equation a(m-l) {m-2)+e(m~l)+g = (e) is equal to, or exceeds by an integer, a root of the equation (d). As a particular root of the latter equation is 0, a particular finite solution may therefore always be obtained when (e) is satisfied either by a vanishing or by a positive integral value of m. 12. The general theorem expressed by (38) admits of the following generalization, viz. j>. * W rf - n. (££> )*!„,+ W „ n. (±$) . The ground of this extension is that the symbol w, which is here newly introduced under F, combines with the same symbol it in the composition of the forms H n I T?S . J , II n ( J, ^ J external to .F, as if 7r were algebraic. And this enables us to transform some classes of equations which are not binomial. Thus the solution of the equation /. (*0 u +/i W W p« +f 3 (ir) (tt) ^ (ir - 1) p 2 w = Z7 will be made to depend upon that of the equation /„ W » +/.W *«/» +/» t W ^(T-i)^-n, (^j tf by the assumption 13. While those transformations and reductions which depend upon the fundamental laws connecting it 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 t=-tq, p = e e > there appear to be three primary integrable forms for binomial equations, viz. , wrr + b „ TT ART. 14.] WITH VARIABLE COEFFICIENTS. 263 hr-|)(w-n) 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 7r and p. In difference-equations but one primary integrable form for binomial equations is at present known, viz. 1 TT u H -t pu = t/, air + b r 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 xxxiii.) on the class specially consi-' dered in this chapter, viz. when the coefficients are rational integral functions of the independent variable, Libri (Crelle, in. 234), Binet (Wemoires de VAcademie des Sciences, xix.). There is also a brief solution by Zehfuss (Zeitsehrift, 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] A 2 m + oo Aw — n*u = 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 XIV. MIXED AND PARTIAL DIFFERENCE-EQUATIONS. 1. If u xv be any function of a; and y, then Ax ** Ax A -u. ■(!)• Ay *■* Ay These are, properly speaking, the coefficients of partial dif- ferences of the first order of u XiV . But on the assumption that Ax 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 u xy . On the same assumption the general form of a partial dif- ference of u x „ is ~x,y ■(& m m\ co. {Ax) m (Ay) n "*■«" ° r \Ax) \Ay) "*■»' When the form of u xv 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 A 2 It is evident that the operations -r— and -r— in combination r Ax Ay are commutative. ART. 1.] MIXED AND PARTIAL DIFFERENCE-EQUATIONS. 265 Agl being A (J Again, the symbolical expression of -r— in terms of -y- HCC (XX- A# 1 ***** 1 " ,x + nQkX,y nUx +x,yH-(®+y)'U' X , y = Q, are, on the hypothesis of Ax and Ay 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 -r— , -j— , and by operations of the form -*- , -7- , singly or in succession. Thus Ax xiV y dy XiV 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 .F(A.,A,)« = 0, F{E x ,E t )u = 0, F(A„E,)u = 0, F{E x> A y )u = Q. Now the symbol of operation relating to x, viz. A x or E x , combines with that relating to y, viz. Ay or E y , 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 — a^X, upon which the solution of ordi- nary difference-equations with constant coefficients is ulti- mately dependent. For (Ghap. xi. Art. 11) {E-a)^X=d^t l a x X, whence (E-a,y i = cr i Z i = a rt (c + c i *... + 'c I1 _ 1 a^ 1 ), 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 u x+1 — au x = 0, of which the symbolical form is E x u x - au x = 0, being u x = Oaf, the solution of the equation u x+1 _ y — u X:y+1 = 0, of which the symbolic form is E x u XiV -E y u XiV = Q, will be t^(^)-0(y). To interpret this we observe that since E y = e* we have Ex. 2. Given u x+li y+i - u Xi ^ -«*,„ = 0. This equation, on putting u for u xy , may be presented in the form E y A x u-u = (1). Now replacing E by a, the solution of the equation a A x u — u=0 is u={l + a- 1 ) x C, 268 MIXED AND PARTIAL [CH. XIV. therefore the solution of (1) is U = (i+E;y(y) .•(«). where $ (y) is an arbitrary function of y. Now, developing the binomial, and applying the theorem E y -"$(y) = <}>{y-n), we find M = (b)db. J — CO As $ (b) may be discontinuous, we may practically make the limits of integration what we please by supposing

(b) db + xf b»-* +1 (b) db J -CO J —00 + ^iD J" V- (b)db + &c (7). J -co Now yjr (8) being arbitrary if $ (b) is ; hence which agrees with (4) »*, = *fo - x ) + x f (V ~ x + !) + i72 * & ~ x + 2 ) + &c - 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 A*,^, = d', Vl . Replacing u x = o ) or &;js,-*;eju=o. But A. = ^„-l, A, = .0,-1; therefore (E.'E, + E V - E;E X -E x )u = 0, or (EJE,-l)(E a -E,)u = 0. This is resolvable into the two equations (E x E y -l)u = 0, {E x -E y )u = Q. The first gives E x u-E v ~ 1 u = 0, of which the solution is u = (E~r(y) = $(y- x), The second gives, by Ex. 1, u = ty(x + y). Hence the complete integral is u = $(y-x)+f &/ + «). 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 d a u d 2 u _ drf dj*~ 0. ART. 5.] DIFFERENCE-EQUATIONS. 271 and, thence, by obvious transformation, in that of the equation dx~*~ a ~dt*~ Vt 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 B 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^, A s , the same. principle of solution is applicable. A symbolic solution of the equation F(x,A x ,A y )u = will be found by substituting A„ for a and converting the arbitrary constant into an arbitrary function of y in the solu- tion of the ordinary equation F(x,A x , o)«=0. And a solution of the, equation F(x,y,A x ) = 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, A x and A, -are involved together, this principle is no longer applicable. For although y and A^ are constant relatively to x and A x , 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 involves no difficulty. Ex. 4. Given u. „ - xu^. y . = 0. 272 MIXED AND PARTIAL [CH. XIV. Writing u for u xy the equation may be expressed in the form u-xJS-'E-'u-O (1). Now replacing EJ* by a, the solution of u — axE x ~* u = or v x — axu^ = is Cx(x-l)...l.a x . "Wherefore, changing a into ^, -1 , the solution of (1) is u = (E^Tx(x-l)...l.(y) = x(x-l)...l.(Ej")(y) = x(x — 1) ... 1 . # (y-a;). 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 i M <, + -B*,,«»Ht.»-l + *.V U x-2,** + &C - = V*.,,, A Xiy , B xy , &c, being functions of x and y, let y =x — k; then substituting and representing u XtV by v x , the equation, assumes the form X v x + X^ x + X 2 v x _ 2 + &c. = X, X , X t ...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 ART. 7.] DIFFEEENCE-EQUATIONS. 273 V X = CX (iB — 1) ... 1, M x,„ = * («- !) •••. 1 • £ (*-#). 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^^- (1 -p) »„, m = -0. Here % + y is invariable. Now the integral of «.-Jw.«-(l-p)«U = is v x = c + c'{-^). 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 B 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' a win- ning it ? Let «,, „ be the probability that A will win it, any positive values being assigned to x and y. B. F. D. 18 274 . MIXED AND PARTIAL [OH. XIY. Now A'a winning the game may be resolved into two alternatives, viz. 1st, His winning the first step, and after- wards winning the game. 2ridly, His losing the first step, and afterwards winning the game. The probability of the first alternative is jpw^,,^,, for after A'a winning the first step, the probability of which is p, he will have w + 1 counters, B, y — 1 counters, therefore the probability that A will then win is M mH . Hence the pro- bability of the combination is /»«« 4 . t>H j. The probability of the second alternative is in like manner (1 -p) «„.„«• 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 M *.»=P«*fi.»-i+(l-p) «*-!.»« (1). the solution of which is, by the last example, M*y= 4> (» + y) + (-y 2 ) + (* + v)- It remains to determine the arbitrary functions. The number of counters oc+y is invariable through the game. Represent it by m, then ,=*w+(y>w. Now A'a success is certain if he should ever be in possession of n counters. Hence, if x = n, u xs = 1. Therefore 1 ~P\\ l^W + (y)"fW ain, A loses the game if ever he have only m-n counters, since then B will have n counters. Hence .-*(«) + (—£] f(m). •(2), •(3), ART. 8.] DIFFERENCE-EQUATIONS. 275 1 — V The last two equations give, on putting P = " , i ptn— n 1 whence z,J/ THn—x-y -l _ {y-(i- J >r^j J " which is the probability that !A will win the game. Symmetry therefore shews that the probability that B will win the game is {{\-pf- : °-p n - x }p n \\-pf- x -*-p tn -° r v 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 r-" 1 , 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 «„,,„..., so that M = Sw m ,„...»V—. then making x = e 9 , y = e 9 , &c. we have -S{ty(w,n..0t*- ft .-*..} «"•»+"'■•■ (1). 18—2 276 ; MIXED AND PARTIAL [CH. XIV. Here, while X denotes summation with respect to the terms of the development of >w, 8 denotes summation with respect to the operations which would constitute the first member a member of a linear differential equation, and the bracketed portion of the second member a member of a dif- ference-equation. Hence it follows that if we have a linear difference-equa- tion of the form 8 [m, n ...) u m - P , n - q ... = ,...,..(2),, the equation (1) would give for the general determination of the generating function u the linear differential equation Mi- »•••)-**— ° » But if there be given certain initial values of u mtn 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 ^ {id' iff") f+'-^rfa »•••) w- 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. AKT. 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 -= a Aw — b -=- + abu = being presented in the form d (I-;)(a-*)«-o, the complete value of .m will evidently be the sum of the values given by the resolved equations - — au — 0, Am — bu = 0. Hence u = 0^ + 0,(1 +5)", where c 1 is an absolute, c 2 a periodical constant., Ex. 8. Again, the equation Ay^Ay + ^Ay being resolvable into the two equations, dz /dz dm \dxj ' . dz /dz\* Ay = z, z = x-j-+l- we have, on integration, s=cx + c 2 , y = tz = CX{X - 1) + d {l + M -h = *> 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 ^h» - a ^ + (* + n) %„ - cum x = 0. This is the most general of Poisson's examples. Taking first the lower sign we have L= — a, M=x—n, N= — ax. Hence the condition (2) is not satisfied. But (3) and (4) give (E — a)u = v, dv . . u = - an whence (E-a) dv , . an = v, ABT. 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 dw,. , dw x , _ -df- a -W +{CW *«- aXW * = > which, being presented in the form (5+-) (*-«)•--* 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 {^'(S\'' x ^ m ^~ H£)x - w- (Differential Equations, p. 445.) The following is the simplest proof of the above theorem. Since +{jl) xu =+{i ; + i) im ' if in the second member -=- operate on x only, and -j- on u, we have, on developing and effecting the differentiations which have reference to x, . 282 MIXED AND PARTIAL [CH. XIV. Let + (5) •"-•*" (S" + * / (is)* ^(£) M=s; ' then , Hi? +(sM+(!)} ,,= *cs or if ^r J-j- J be replaced by e , ££ 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 du x ■(*). n being either positive or negative. Now putting u for u x (d *- -1 £■ Let then we have {5+^^-^} « + xz = 0. AST, 11.] DIFFERENCE-EQUATIONS. 283 Or, d . we* Hence, / , d , ne* Y+ n e — a and therefore by (1), = (e* - o)V W *"*« <*> («* - a) - " (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 " = (el - fl) " e " f (3" e?(el ~ ar0 (C) ' Now « = (e*" — a)" 1 *, therefore substituting, and replacing e 5 * by #, «-(tf-aj"e-*(^" 1 eV-«rO (4). Two cases here present themselves. 284. MIXED AND PARTIAL [CH. XIV. First, let n be a positive integer ; then since (E - o)-" = a x (c„ + c 1 x...+ c n _, x"-% (E-ay* = (& + l-ay-\ we have u = (A + 1 - a)" -1 e~ » {(7 + Je T a 1 (c + eye . . . + c„ 1 af 4 ) &} (<*). as the solution required. This solution involves superfluous constants. For inte- grating by parts, we have Je T y = F„ Treating -j- as a constant, the symbolic solution is 2 having reference to a;. No constants need to be introduced / d X~ x in performing the integrations implied by It- J . Ex. 13. Given u xw - 3« ^j=» + 2« (0 - 1) ^%= 0. Let M a = 1 .2 ... (#-2) »„ then • {^- s£ 4 +2 (l)l"-- ' whence by resolution and integration M . = 1.2...(,-2){(|)^(,) + 2« (!)>(,)}. Ex. 14. «„, - 8 ^hi +2^=7, where 7 is a function of a; and y. 288 MIXED AND PARTIAL DIFFEBENCE-EQUATIONS. [CH. XIV. Here we have =l(*.-4/^l(*.-4)~v The complementary part of the value of u introduced by the performance of 2 will evidently be But in particular cases the difficulties attending the reduc- tion of the general solution may be avoided. Thus, representing V by V x , we have, as a particular solu- tion, which terminates if V x is rational and integral with respect to y. The complement must tben be added. Thus, the complete solution of the given equation when V = F(x) + y, is u = F(«-2) + y + 3 + 2*(£f$(y) + (^y f(j/) ; EX. 1.] EXERCISES. 289 EXERCISES. Solve the equations : 1. k x u Xiy -a-^u XtV =0. 2. u^-a^u^+b-^u^^Q. 3. u^ v -u Xt „ x = x + y. 4. «*«,»*, -«*,- »*""- 5. W "M-2,» _a X»« = - 6. M s+8.» _ ^ M aiM,»+l + 3 U x+t.V*l ~ M *.»+S 10. Determine u Xit from the equation d 2 where A affects x only ; and, assuming as initial conditions A do' 1 u Xi9 = ax+b, ^^,=0'/, shew that ^.-^vV+,0, where A, X and /tt are constants (Cambridge Problems). B. F. d. 19 290 EXERCISES. [CH. XIV, 11. Given ««h.« + (a-x-2y-2) u mm + (x + y) «*,„= with the conditions «„.,_,= 0, «„,(, = 0, and %,*+, = (), find u Xit . [Cayley, Tortolini, Series II. Vol. II. p. 219.] 12. u ZiV = u x _ tl _ 1 + u x ^ i2 + &c....+u x _ lltll . [De Morgan, Gamb. Math. Jowr. Vol. IV. p. 87.] ( 291 ) CHAPTER XV. OP 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, {yjr (x)} is usually expressed in the form tj>y]rx, brackets being omitted except when their use is indispensable. The expressions 66x, AAAx are, by the adoption of indices, abbreviated into d/x, dfx, &c. As. a consequence of this notation we have °x = x independently of the form of . The inverse form A' 1 is, it must be remembered, defined by the equation (jxjr 1 x = x... (1). Hence A' 1 may have different forms corresponding to the same form of . Thus if x = x 2 + ax, we have, putting x = t, x =dfH = -' and djT 1 has two forms. 19—2 a± V(« 2 + 4i) 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 yjrx to determine that of ifr'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 ifrx, required that of y%. There are cases in which this problem can be solved by successive substitution. Ex. 1. Thus, if -tyx = x", we have and generally ifr'x = of". Again, if on determining -ty^x, ^'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 yjrx = 1 — a;, we have TJr*X = l — (l—x) = x. Hence iffx = 1 — x or x according as n is odd or even. Ex. 3. If -drx = = , we find yfr*x= , ty*x = x. ART. 2.] OF THE CALCULUS OF FUNCTIONS. 293 1 X— 1 Hence yjr"x = x, = or according as on dividing JL — $C 36 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 ^"x 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 r*=t n , r^=t a+1 (2). Then, since ^jr"* 1 x = -^r\jr , 'x, we have *« = *(0 (3)- The arbitrary constant in the solution of this equation may be determined by the condition t l = yjrx, or by the still prior condition t = yjr°x = x (4). It will be more in analogy with the notation of the other chapters of this work if we present the problem in the form : Given ijrt, required yjr% thus making x the independent vari- able of the difference-equation. Ex. 4. Given tyt = a + bt, required yjft. Assuming •^•"t — u,, we have ««« = <*+&**«, the solution of which is Now tt = yfr't = t, therefore a 294 OF THE CALCULUS OF FUNCTIONS. [CH. XV. Hence determining c we find on substitution Vo^+W (5), the expression for ^"t required. Ex. 5. Given -Jrt = = — - , required -Jr*t. b + t Assuming yfr't = u x we have a x+1 b + u.' or U x u x+1 + bu x+1 = a. Assuming as in Ch. XII. Art. 1, %+&=>*, v x we get v xvi - bv^ - av x = 0, the solution of which is "v^cjf + cjr; a and being the roots of the equation m 2 — bm — a = 0. Hence M * = V + c> ~ 6; or, putting for — and a + /3 for &, and reducing, "*=-"£ a . + Cf r ■■■■■ ■••.(6). Now u = yfr" t = t, therefore t = -aP 1+0 1 + ' ART. 2.] OF THE CALCULUS OF FUNCTIONS. 295 whence = — — — ; and, substituting in (6), the expression for ■yfr x t required, a Since in the above example i/r£ = ^— , we have, by direct substitution, a a & + ^rf and continuing the process and expressing the result in the usual notation of continued fractions, . x _ a__ a a a T b + b+b + ...b + f the number of simple fractions being x. Of the value of this continued fraction the right-hand member of (7) is therefore the finite expression. And the method employed shews how the calculus of finite differences may be applied to the finite evaluation of various other functions involving definite repe- titions of given functional operations. Ex. 6*- Given yjrt= . , required ty't. Assuming as before ■ty x t = u x , we obtain as the difference- equation eu x w x+1 + cu x+1 -bu x -a = (8), and applying to this the same method as before, we find %_ b * =1, equations which require that b should be any a;" 1 root of unity except 1 when a is not equal to 0, and any a;'" 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. _J_ ]*■/■ Ex. 8. Required the conditions under which is a c + et periodical function of the a;" 1 order. In the following investigation we exclude the supposition of e = 0, which merely leads to the case last considered. AKT. 3.] OF THE CALCULUS OF FUNCTIONS. 299 Making then in (16) ^r x t = t, we have t — fi +i»tan (tan -1 — — —a; tan" 1 -] (18), or — = tan ( tan -1 — — x tan -1 - 1 , v \ V flj an equation which, with the exception of a particular case to be noted presently, is satisfied by the assumption x tan -1 - = itr, i being an integer. Hence we have -=tan-; (19), fl x or, substituting for v and fi their values from (13), (b + c) 2 x whence we find & 2 -26ccos — + c a e = * (20). 1 4a cos — x 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 = makes the expression for t given in (18) indeterminate, the last term assuming the form 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 x = as the condition of periodicity, a condition which however is satisfied by all functions what- ever, in virtue of the equation Hi = t. The solution (9) expressed in exponential forms does not lead to any condition of periodicity when a, b, c, e are real quantities. 300 OP THE CALCULUS OF FUNCTIONS. [CH. XV. We conclude that the conditions under which — ■ — -. , when c + et not of the form A+Bt,isa periodical function of the x* 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 $f$~H is also a periodical function of the same order For let 4>fpH = ft, then fH = ffl*$fpH And continuing the process of substitution ft -#"*-*. Now, if ft be periodic of the n th order, ft = t, and Hence ifr't = (jxjfH = t. Therefore ifct is periodic of the ft* 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 hjft. Then if j>t = f , Itt = >Jt, #r*~*. These are periodic functions of the second order ; and the number might be indefinitely multiplied. The system of functions included in the general form < kf4 > ~ 1 & nave b een called the derivatives of the function ft. * I am not aware that the limitation upon the integral values of i has been noticed before. (1st Ed.) ART. 5.] OF THE CALCULUS OF FUNCTIONS. 301 Functional Equations. 5. The most general definition of a functional equation is that it expresses a relation arising from the forms of functions ; a relation therefore which is independent of the particular values of the subject variable. The object of the solution of a functional equation is the discovery of an un- known form from its relation thus expressed with forms which are known. The nature of functional equations is best seen from an example of the mode of their genesis. Let f(x, c) be a given function of so and c, which con- sidered as a function of x, may be represented by (fa;, then x =/(*, c\ and changing x into any given function tyx, x, ^vs)«=0 (1). This is a functional equation, the object of the solution of which would be the discovery of the form , those of J? and ^ 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 tyx. Thus if we assume as a primitive equation $(x) = cx + - (a), and change x into — x, we have A (— x) = — ex + ~ . c Eliminating c we have, on reduction, 302 OF THE CALCULUS OF FUNCTIONS. [CH. XT. a functional equation of which (a) constitutes the complete primitive. In that primitive we may however interpret c as an arbitrary even function of x, the only condition to which it is subject being that it shall not change on chang- ing x into — x. Thus we should have as particular solu- tions 1 f[w) = x cos x H , cos a; (x)=x a + ^, these being obtained by assuming c = cos x and a? respectively. Difference-equations are a particular species of functional equations, the elementary functional change being that of a; into x + 1. And the most general method of solving func- tional equations of all species, consists in reducing them to difference-equations. Laplace has given such a method, which we shall exemplify upon the equation F{x,4tyx,$ X x)=0 . (2), the forms of ty and % being known and that of sought. But though we shall consider ithe above equation under its general form, we may remark that it is reducible to the simpler form (1). For, the form of ■x/r being known, that of i|r _1 may be presumed to be known also. Hence if we put yjrx = z and yty'^z = i/r^, we have and this, since ty' 1 and yjr t are known, is reducible to the general form (1). Now resuming (2) let ■f * = U t , x x — Vi 4»fx = v t , 4% x = •"« Hence v t and u t being connected by the relation v, = fu (4), the form of ^ will be determined if we can express v t as a function of u t . .(3). ABT, 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 => • (5), u, «+i the solution of which will determine u t , therefore yjrx, 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 v t , the solution of which will determine v t as a function of t, therefore as a function of u t since the form of u t has already been determined. But this deter- mination of v t as a function of u t is equivalent, as has been seen, to. the determination of the form of . Ex. 9. Let the given equation be ^ (mx) — a (x) = 0. Then assuming x = u t , mx = u M \ .. (x) = v t , (f> (mx) = v^J '■" w ' we have from the first two m^ - mu t = 0, the solution of which is u t = Crn' (b). Again, by the last two equations of (a) the given equation becomes whence ,.f t = CV (c). Eliminating t between (6) aiid (c), we have log«<-logg v t = C'a lo s m ' . _ logC Hence replacing u t by x, v t by x, and G'a~ losm by C lt we have , ,'. log a; (^|) - a [x) = 0. Assuming 1+05 1-x "m> (ai) = v t , $(j±?j =Vt+i , we have _l + u t or u t u m -u t+1 + u t + l = 0. The solution of which is « t = tan( C+^tj. Again we have whence v t = C'a<. Hence replacing u t by x, v t by (x), and eliminating t, (x) = C 1 a' AKT. 6.] OF THE CALCULUS OF FUNCTIONS. 305 Oj being any function of x which does not change on chang- 1 + x ing x into z. . J. "~~ SO 6. Linear functional equations of the form ffl'x + a^-f^x + a s W*x ...+aJ>(x)=X (6), where tfr (x) is a known function of x, may be reduced to the preceding form. For let nr be a symbol which operating on any function $ (x) has the effect of converting it into $ijr (x). Then the above equation becomes ir"<£ {x) + ay^ 1 ^ (»)... + aj> (x) = X, or (■jr° + ay-*...+a„)(x) = X ,(7). It is obvious that it possesses the distributive property expressed by the equation it (u + v) = iru + irv, and that it is commutative with constants so that irau = aim. Hence we are permitted to reduce (7) in the following manner, viz. £(z)=(7r* + a 1 7r"- 1 ...+aJ- I X = {N 1 {,r- mi r^N^-my...}X (8), m 1 ,m 2 ... being the roots of m n + a 1 mr l ... + a, n = (9), and JV,, JV 2 ... having the same values as in the analogous resolution of rational fractions. Now if (tt — nif 1 X — j> (x), we have (tt — m) (x) = X, B. F. D. 20 306 OF THE CALCULUS OF FUNCTIONS. [CH. XV. or (pifr (x) — m (x) = X, to which Laplace's method may be applied. Ex. 11. Given $ (m 2 x) + a$ (mx) + b (x) = x n . Eepresenting by a and j8 the roots of a? + ax + b = 0, the solution is „ log a log? (*) = » X , z. + Cx loem + C'x ioem , YK ' m M +am n + b ' C and C 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 ir, the complete value of $ (x) consists of two portions, viz. of any particular value of x, fyjrx) = (10), and let yjrx be a periodical function of the second order. Then changing x into -tyx, and observing that -^x = x, we have F(yjrx, $^X, a:) =0 (11). Eliminating i]rx the resulting equation will determine (f>x as a function of x and tyx, and therefore since tyx is supposed known, as a function of x. If -frx is a periodical function of the third order, it would be necessary to effect the substitution twice in succession, and then to eliminate (fttfrx, and (pyffx; and so on according to the order of periodicity of yjrx. ART. 7.] OF THE CALCULUS OF FUNCTIONS. 307 Ex. 12. Given (<£a) 2 -^— - = a 2 x. 1—x The function ^ is periodic, of the second order. Change 1 — x then x into = , and we have 1+x /\l-aV, e l-« 1 —a; Hence, eliminating ■ , we find ^-oM^) 1 as a particular solution. (Babbage, Examples of Functional Equations, p. 7.) This method fails if the process of substitution does not yield a number of independent equations sufficient to enable us to effect the elimination. Thus, supposing yfrx a period- ical function of the second order, it fails for equations of the form F(x, x and tytyx. 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 (**>' + {* g -*)}*= 1. This is a particular case of the more general equation (<£«)* + mW|-»U =l + nxx, m and n being constants which must be made equal to 1 and respectively, and %x being an arbitrary function of x. 20—2 308 OF THE CALCULUS OF FUNCTIONS. [CH. XV. Changing x into ~ — x, we have {,£(£-*)JVm{ (x)Y= 1 -m + nWx-mxi^-mji . Therefore Now if m become 1 and n become 0, independently, the fraction = , becomes indeterminate, and may be replaced by an arbitrary constant c. Thus we have {j> (x)Y= ^ + ex (x) - c X (| - i/r (x) = V t , f\^-X} = V« ART. 7.] QF THE CALCULUS OF FUNCTIONS. 309 we have or «m + M «=|. »« + ».==!• The solutions of which are c u t = Cl (-iy + l, v t = Ci {-\T+\. Hence 1 *'~2 o 8 7T C, Therefore ^=|+c(« e -f), *<-H + 0(— i)- Therefore *«-{HH)}* in which C must be interpreted as a function of a; which does not change when x is changed into -^ — x. It is in fact an arbitrary symmetrical function of % and -^ — x. The previous solution (12) is included in this. For, equating the two values of (%) with a view to determine C, we find 310 OF THE CALCULUS OF FUNCTIONS. [CH. XV. x( x )-x(J- x ) •7T - Xfo) | xg— ) it IT IT 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<£ (x) 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 (MScanique, 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. 4> + y) + £ (» - y) - <£ (») $ (2/). Now, differentiating twice with respect to x, we have f {x + y ) + ^{x-y) = 4>" (x)$(y). And differentiating the same equation twice with respect toy, f (x + y) + $' (x - y) = <}> (x) 4>" (y). Hence £M = £M H <£ (y) ART. 8.] OF THE CALCULUS OF FUNCTIONS. 311 I It I \ Thus the value of , ) : is quite independent of that of x. "We may therefore write *<") ± ' m being an arbitrary constant. The solution of this equa- tion is $ (x) = Ae mx + Be~™°, or <£ (as) = A cos mcc + B sin mx. Substituting in the given equation to determine A and B, we find (a,) = e mx + e"™", or 2 cos mx. 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 ( = ) = 1. This will be found to require that the second form of (x) be taken, and that m be made equal to 1. Thus (f>(x) = 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

{x)(y) = (0). "312 EXERCISES. [CH. XV. Differentiating with respect to x and with respect to y, we have Therefore -£W - £M- x (x) y {y) • Hence we may write x (a?) a differential equation which gives (x) = Ce™*. The condition that $ ( (x) = 2x? - 1, determine " (x). 3. If i/r (t) = jj^ t and yfr x (t) = -^7^ , shew, by means of the necessary equation T/nJr* (t) = -^r x ^r (t), that ud = .g <7-J a e c — 6 ■EX. 4.] EXERCISES. 313 4. Shew hence that ty x (t) may be expressed in the form a + bj b x -b + c -k-et' the equation for determining b x being & A« + cb^ - bb x -ae = 0, and that results equivalent to those of Ex. 5, Art. 2; may hence be deduced. Solve the equations 6. f(x)+af(-x) = x". 7- f(x)~af(-x)=e x , 8. f(l-x)+f(l + x) = l-x\ 9. f{x)=wf{x)+f{f{x)}. 10. Find the value, to x terms, of the continued fraction 2 2 1 + 1 + &c. 11. What particular solution of the equation /<•)+/©-* is deducible by the method of Art. 7 from the equation f(x) + m ff±\ = a + n (x) ? 12. Required the equation of that class of curves in which the product of any two ordinates, equidistant from a certain ordinate whose abscissa a is given, is equal to the square of that abscissa. 314 . EXEBCISES. [CH. XV. 13. If wa; be a periodical function of x of the n a degree, shew that there will exist a particular value oif(ir) x expres- sible in the form a + ajrx + ajfx ... + a^ir^x, and shew how to determine the constants a , a v a 2 ... cf n-1 . 14. Shew hence that a particular integral of the equation ^(l^l) -a *^ =a; will be 1 x-1 , , x a 3 / 1 1 + « 1.1 r w 1 — a \ al—x ax a x + 1 15. The complete solution of the above equation will be obtained by adding to the particular value of x the comple- 4tan~ 1 jr mentary function Ca w 16. Solve the simultaneous functional equations (Smith's Prize Examination, 1860.) 17. Solve the equation F(nx) =f(x) +f(x + 1) +/{x +l) + &c. +/(* + ^1) . [Kinkelin, Grimert, xxn. 189.] 18. Solve the equation *(*)+* (y) = * {*/(y) +#/(<# [Abel, CWfe, n. 386.] w. EX. 19.] EXEECISES. 315 Magnus (Crelle, v. 365) and Lottner (Crelle, xlvi.) have continued the investigations into this and kindred functional equations. 19. Find the conditions that (x, y) + J— 1 ■yfr (x, y) may- be of the form F (x + y J-l). [Dienger, Grunert, x. 422.] 20. Shew that _ d"u , d^u " dx n : dx™ satisfies the equation dz„ a dx n "' 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 z n , the ratio, multiplied by a certain constant, of the distances from the centre of the feet of perpendiculars drawn from the 71 th and (n — l) th angles, counting from A, on the diameter through the fixed point, prove that z n is a function which satisfies the equation. 21. If (z) = [ 1 = u, , then we have n \ n J the complete integral of which is « =a + C,cos — + a cos — ... + (/„_. cos — . 318 GEOMETRICAL APPLICATIONS. [CH. XVI. Hence we find r=a+C 1 cos9 + C 2 coa2d ...+ C n _,cos (n- 1) 8, the analytical form of any coefficient G ( being O l = A+B l cos nO + i? 2 cos %nQ + &c, + E t sin nd + E 2 sin 2nd + &c, A, B t ,E lt &c, being absolute constants. The particular solution r = a, + b cos ff gives, on passing to rectangular co-ordinates, {x*-hx + y y = a 2 (x* + tf), 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 y x the ordinate corresponding to the ab- scissa x, we shall have the mixed difference-equation y 'dx ay ^ dx ~° W" Let 2k-Jf = %,, then «* - a u *-i - o ; .'. u x =Ca x , whence *>%-<>* < 2 >- Hence integrating we find y. = '• It is easily shewn that twice the angle which the normal at any point of the curve makes with the axis of x 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 a; is - . The tangent of the angle which the normal makes with the axis of x is — , and the tangent of twice that angle is _2 p 2p f Hence the tangent of the angle which the ray reflected from x, y makes with the aiis of x is 2 P y 1-p 1 x ^xp-yjl^p 1 ) + l-pfx 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, 2x'p'-y'(l-p'*) 2xp-y(l-p*) _ x'[\-p*) + 2yp' x{l-p 2 ) + 2yp w ' ■while for the equation of that ray, we have 2xp-y{l-f) a s x(l-p 2 ) + 2yp K ' w • Now, regarding x and y as functions of an independent variable a which changes to s+1 in passing from the first- point of incidence to the second, the above equations become 2«y-y(l-ff a ) * x (l-p*) + 2yp u * * y x(l-f) + 2yp^ The first of these equations gives 2xp - y (1 -f) p oc{\-p i ) + 2yp { >' whence by substitution Ay = CAec. Therefore y = Gx + C. Here and C are primarily periodic functions of z which do not change when z becomes z + 1. Biot observes that, if G be such a function,

(G), and, restoring to G its value in terms of x, y, and p given in (4), we shall have AKT. 2.] GEOMETRICAL APPLICATIONS. 321 y a;(l-/) + 22/p" 1 " ? l*(l- i ) i! ) + 2ypj w " This is the differential equation of the curve. Although Lacroix does 'not point out any restriction on the form of the function , it is clear that it cannot be quite arbitrary. For if G = ^ (z), we should have C'=W(z), and then, giving to (f> some functional form to which yjr is inverse, there would result G' = z, so that C would change when z was changed into z + 1. From the general form of periodic constants, Chap. IV., it is evident that a rational function of such a constant possesses the same character. Thus the differential equation (5) is applicable when indicates a rational function, and generally when it denotes a functional operation which while periodical itself does not affect the periodical character of its subject. If we make the arbitrary function 0, we have on reduction (f-a?) F + xy(l-p*) = 0, the integral of which is x* + y* = r>, 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 + (e), 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 foEowing particular solution, viz. {y-Af+ (*-B)»=0, A and B being given by the equation } , -+{-+^}-« a)- Differentiating, we have *'(*)+*(*) ^Wf+^}(l + ^)==0, which is resolvable into the two equations, 1 + ^ = (2), *>)+*' {*+±^)} = (3)> AET. 3.] GEOMETBICAL APPLICATIONS. 323 The first of these gives on integration yfr (as) + a? = ax + /3 (4). Substituting the value of i|r (x), hence deduced, in (1), we find as an equation of condition a=0, and, supposing this satisfied, (4) gives f+x* = ax + j3, 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 a = 0. To solve the second equation (3), assume x+%f (x)=x( x )> and there results tf(x)-2 x (x)+x = (5). To integrate this let x = u t , % (x) = %,_., and we have w (+2 -2w (+1 + « e = 0, whence u t = G+ C% C and C being functions which do not change on changing t into t + l. If we represent them by P(t) and P.fy), we have u t = P(t) + tP.it), u^P^ + it + ^PS), whence, since u t = x and m w = % (x) = x + ^ty' (x), we have x = P it) + tP.it), If (x) ~P.it). 21—2 324 GEOMETRICAL APPLICATIONS. [CH. XVI. Hence f ( from which, when the forms of P(£) and P^t) are assigned, £ must be eliminated. If we make P (t) = a, P x {t) = /S, thus making them constant, we have x = a+/3t, y*=J/3*dt = /3H + c. Therefore eliminating t and substituting e for c — cc/S, y* = fix + e. Substituting this in (1), we find Thus, in order that the solution should be real, a must be negative. Let a = — h s , then fi = ±2h, and y*=±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 EXEECISES. 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 (f> 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 $ 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 + (p (m), m being a variable" parameter. ( 326 ) ANSWEKS TO THE EXAMPLES. CHAPTER II. 6. Obtained from the identity A n (0-l)(0-2) (0-w)0* = 0. 9. e (l + * + *» + ! f 8 ). 14. (x — 6a; 8 ) cos x — (7x 2 — x*) sin as. 16. (2) «.-^-V. CHAPTER III. 1. 2-3263359 which, is correct to the last figure. 2. x*-$a? + V7x + §. _ - 3^ + 10^ + 5^-21). _ -2v + 5v 1 +10v i -Sv lt 6 - v * ~ 10 ' v ' 10 13. It will be so if (x) = have one root, and $(x) = have no root between 1 and k. , CHAPTER IV. . . (2n-l)(8n + l)(2n + 3)(2n + 5)(2n + 7) 21 l - W 10 2 " (2) A_ i V"/ on i 90 6(2w+l)(2ra + 3)(2ra+5)' ,„. (2ra - 1) (2n + 1) (2ra + 3) (2n + 5) (8» + 43) 129 W 55 +-8~- ANSWERS TO THE EXAMPLES. 327 W S" 4W+H 12 4 (2re + 1) (2n + 3) ' (5) Apply the method of Ex. 8. (6) Write 2 cos 6 = x + - and use (10) page 73. SO »(2re+l)(8w 2 +4w-7). m, sin— (2a:-l) cos-g (2«) sin-^ (2a;+l) ( 2sm 2J 2sin^ (2sin4rl 2 V 2/ cos £ (2a + 2) sin £ (2as + 3) ± ASJL/,v.\ i ^ A* }) vT - AW*) + jy- A*A(«) + &c ( 2sm 2J ( 2sm 2j 6. (1) cot|-cot2"- 1 ft . . 2 sin M0 cos (re + 1) 6 sin 20 ' » , -i, is ^ /v Iog2sin2"0 „ , 2" 7. tan * (n - 1) +(7, ^__ , <7+--. 8. Assume for the form of the integral (A+Bx + .-. + MaT^s* and then seek to determine the constants. CHAPTER V. 1. c-lf-^^!^. ' l(. + g 2(»+g 6(. + i) 8 30 ( W+ i)j' where (7= 1'0787 approximately and is the sum ad inf. 328 ANSWERS TO THE EXAMPLES. 1 _„ 1 1 57 . The sum ad inf. differs from that of the first nine terms by •0000304167. '{** + &Y & iy { 4. See page 71. 5. (1) Apply Prop. IV. page 99. If — a? be written for x* in the first series it can be divided into two series similar to the Example there given. _J +1 S >(as+l)(ar+2) 2 x {x + 1) (as + 2) (x + 3) I 1 . ** + &c l + 4 x (x + 1)' (x + 2) (x + 3) {x + 4) + ""• J • (z-x + l)z^' 13. See Ex. 7. Also page 115. CHAPTER VII. 1. - tan -1 a and =- . ff, , 2a 3. (1) Divergent. (2) Convergent (3) The successive tests corresponding to (G) are obtained by writing — Au x+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. L W M = 2^l(^ + 2^l)- ( 2 ) Thesame - (4) (^l) 2 + a "(^T- w ) = - & Thesame - * ^ 1 - pa „ „ .. . cos (x — 1) n — a cos nx 3. % = Ca* + — ^ — ^r 1 — ! — . 1 — 2a cos w + a *• > — cT2^ _fl5 ~ 5. %, = {(/ + cosec a tan (a: — 1) a] cos a cos 2a. . .cos (a;— 1) a. 6. Assume «„ = », + »» where m is a root of m B + am + & = 0, and there results a linear equation in — . 8. M^Ce^^' + ^e' 1 - 112 . . 330 ANSWERS TO THE EXAMPLES. 2 sina:0 sin [ x — = ) 8 in (x-^j 9. u x = ' ^ + G sin xd. sin 2 10. to^oT^+O}. 11. u x = cos2 x 0. 12. ..^{o+.-J^}. 13. «. = }(a* , -^ 1 v 14. M x = m"- 1 a-". 15. By writing «» + ~ = i>„ the equation may be reduced to v^ = 1^+0. When C = - 2 this gives «„ = 2 cos 2* ft 16. cu x = c'x + 1. 17. -^ = aa; or — 2ax. Hence two associated solutions (see Ch. x.) are u x = Ga x Tx and u x =*C(-2a) x Tx. CHAPTER X. „ a + 2b 2(a-b) v ( 1\ 1. ^ = a + -g-*-_ W -COB, r «COBg^- | ). 8. The two others are given by M * = ( C ^-^l)' where z is a root of /* 2 + /* + 1 = 0. ANSWERS TO THE EXAMPLES. 331 9. (Ay-J+a + l^+^.-O, A y=(|- a! -l){l + (-ir}-A { i_ ( _ ir }. CHAPTER XL 1. u x =G(-l) x +C'¥ + (m + 1) (m — 4) 2. „.= C(_4)- + «£L=«. 3. ^ = ((7+ 0'*) (-1)" + \ p - 6a 2 + ^ a> - 3l + 6 4. n, = (m 2 + m 2 )* {c cos (a tan" 1 £) + 6 sin x tan — - + —s . \ ml) n 2sm- 6. ^=(-3)^(7 + ^^+^+^+^ .(8) 960 + Sjfi{±*-28* + 28}. 7. The particular integral is obtained by (II) and (III) ge 218. It is any value of 2 p_a) ^E-E-2 ' 8. u s = ^- ( ^±p^ + C -2* +^+0^+0^+0^ 332 ANSWERS TO THE EXAMPLES. + w s cos mx + cos (x — 2) m , „ 9. % = 1 , g o , i + complementary func- 1 n + 2w 2 cos 2m + 1 r ' tion, which is „ izx , „, . irx] x G cos -5- + C7 sin — -}■ » , or Cri* + C' (—n) x , according as the upper or lower sign is taken. 10. rTu x = {C x + C,x} cos ™ + {C a + Crf sin ~ , C 1+ C^+{0,+ Cf»}(-l)'. 11. (a + JA:)-] — j— [■ — M; where A == , 10 1 f/ll + 3\/l7V /ll - 3 \/l7\*l 12 ' 3Vlf \[ 2 J - { 2 ) | • CHAPTER XII. 2. u I = ^sin(X+a)+5sm(2X + /3) + Csin(3X+7) + &c. to — ~— terms (supposing that n is odd) where X = . 3. «.-l£±i{s^ ! +OJ. i n f^v y» a? (# — 1) (#=»2Y 5. «,= |a-3 |a+ g f g+ v ± % $ •• ANSWERS TO THE EXAMPLES. 333 6. iog«. = (_„)-ja+C*}+J2^. 7. For (x + 2f read (x+2)\ The equation is then re- duced into a very simple form by substituting =-r 2 ^ for u x . 8. M^^ + C^-ir + ma; j 9. % - c„ + ^ (- iy, v x = o; - c (- i)". 10. u x + g a. = J 4 + Js C os— + C r sin-g-K-l)", and B a + &c, w x + &c. are obtained by writing x + 2 and x + 4 in the quantity on the right-hand side. 11. % = (A+Bx)r + (c + Bx)(-2y + a ^ ^- ) ? a ~ x ~\ and u^ (and therefore v x ) is given at once by the first equa- tion. 13. It may be written (E - cf x ) (E - a x ) u x = 0. x(x-l) x{x-l) 14. u x = a 2 {o+Vta 2 }. 15. u x = Va tan \ C 1 cos — ^- + C, sin — =- k 17. Compare with (15) after dividing by u x u xn u x+1 . 19. If log a n = m„ we have and the solution is i-(-2)» fie) 334 ANSWERS TO THE EXAMPLES. 20. See page 228. Perform A on the equation and linear equation in AV,. results. 21. u x = Pa +Qp° + By* where txfiy = 1 and C=P QB (a - /3) 2 (0 - 7) 2 (7- «)*• If G= 0, the solution becom'es 2 cos a cos ma — «„ sin (m — 1) a 22. If 1^ = 2008 a, fl m sin a 4 (I)-©} 1. u, CHAPTER XIV. •d\ m , 1 1 ii\ x = a* e «(|)«^). 2. u x ,„ = a* (|JM where a and /3 are roots of to 2 — am + & = 0. 3. v, Xty = x{y + x-l) + $(y + x). 4. w„„ = a 1 ^-! + r _ ( l sPi+ g r-(^i-«r , m _™ 1. #'w-^Pi tF1+arr+aF1 _. r . (»-n 2. ^•(fli)=g{(a> + 7?^l)- + (a!- iN /?^r)-}. (m = 2*). 5. f(x)=Cx. 8. /(*)-/(*)-/(2-*)+^2-. 2 ^-2(-ir 1U - 2* +1 + (_ 1)- ■ 336 ANSWERS TO THE EXAMPLES. 11. f(x) = Ui(x)-4,(± 12. y = ce^ ix ~ a) , (x) denoting an odd function of #. 13. Develope/(7r) in ascending powers of tt, and apply the conditions of periodicity. 16. (x) = siQmX sin (mx + c) ' sine ■Jr (x) = - , r sin (m% + c) 22. *(«) = {/(*)}- CHAPTER XVI. 1. r — a +f(a- ) ~f[~Q — ) where/ (as) satisfies the equa- tion A/(a>) = 0. 2. Write log r for r in the answer to the previous, ques- tion. CAMBBIDGE: PBIHTED BT O. J. CLAY, M.A. AT THE UNIVERSITY PUESS. May 1879. A CATALOGUE OF EDUCATIONAL BOOKS, PUBLISHED BY MACMILLAN AND CO., BEDFORD STREET, STRAND, LONDON. MACMILLAN'S CLASSICAL SERIES, for COLLEGES and SCHOOLS, being select portions of Greek and Latin authors, edited, with Introductions and Notes at the end, by eminent scholars. The series is designed to supply first rate text-books for the higher forms of Schools, having in view also the needs of Candidates for public examinations at the Universities and elsewhere. With this object the editors have endeavoured to make the books as complete as possible, passing over no difficulties in the text, whether of construction or of allusion, and adding such information on points of Grammar and Philology as will lead .students on in the paths of sound scholarship. Due attention moreover is paid to the different authors, in their relation to literature, and as throwing light upon ancient history,~with the view of en- couraging not only an accurate examination of the letter, but also a liberal and intelligent study of the spirit of the masters of Classical Literature. The books are clearly printed in fcap. 8vo., and uniformly bound in heat red cloth. The following volumes are ready : — /ESCHYLUS— PERSJE. . Edited by A. O. PmCKARD, M.A., Fellow and Tutor of New College, Oxford. With Map. ,y. ddz CATULLUS— SELECT POEMS. Edited by F. P. SIMPSON, , B.A., late Scholar of Balliol College, Oxford. 4*. dd. 50,000.5.79 2 MACMILLAN'S CLASSICAL SERIES. CICERO— THE SECOND PHILIPPIC ORATION. From the German of Karl Halm, Edited, with Corrections and Additions, by John E. B. Mayor, Professor of Latin in the University of Cambridge, and 1 Fellow and Classical Lecturer at St. John's College. New edition, revised. $s. THE CATILINE ORATIONS. From the German of Karl Halm. Edited, with Additions, by A. S. Wilkins, M.A., Professor of Latin at the Owens College, Manchester. New edition, y. 6d. . , THE ACADEMICA. Edited by James Reid, M.A., Fellow of Cains College, Cambridge. 4J. 6d. , t; . t ,, - DEMOSTHENES — THE ORATION ON THE CROWN, to which is prefixed iESCHINES AGAINST CTESI- PHON. Edited by, B. Drake, M.A., late Fellow r of Kind's College, Cambridge. 5^. ,'. ' '<-'• [Sixth and thoroughly revised edition iH May, ' HOMER'S ODYSSEY— THE NARRATIVE OF ODYS- SEUS, Books IX.— XII. Edite'd by John E. B. MAYOR/ M.A. J>art I. 3*. [To be complete^ l 'sAqr(iy. JUVENAL- SELECT SATIRES. Edited by JOHN E. B. Mayor, Fellow of St. John's, College, Cambridge, and • Professor of Latin. ' Satires X.and XI. ' %s. Satires XII.— '"XVI, y.<6d.~ LIVY-HANNIEAL'S FIRST CAMPAIGN IN ITALY, Books XXI. and XXII. Edited by the Rev. w, W. Capes, Reader in Ancient History- at Oxford,, With. 3 Maps. 5$. ~ ; SALLUST— CATILINE and JUGURTHA. Edited by C. Merivale, B.D. New edition, carefully revised and en- larged. 4j. 6.d. Or separately zi. 6d. each. * ■ r, -'~ r . '". J T TACITUS— AGRI COLA and GERMANIA. ,, Edited by A. J. Church, M.A and W. J. Brodribb, M.A. Translators of Tacitus. New edition. yZ6d. Or separately 2 s. each, 1 ? THE ANNALS, Book vi.' 'By the same Editors, is. 6d. CLASSICAL. TERENCE— HAUTON TIMOBVHENOS. Edited by E. S. Shuckburgh, M.A., Assistant-Master at Eton College., y. With Translation, 4J. 6d. THUCYDIDES — THE SICILIAN EXPEDITION, Books VI. and VII. Edited by the Rev. Percival Erost, M.A., Late Fellow of' St. John's College, Cambridge. New edition, revised and enlarged, with Map,-- . 5j. XENOPHON— HEL1ENICA, Books I. and II. Edited by H. Hailstone, B.A., late Scholar of Peterhouse, Cambridge. With Map. +r. 6aT. -.-,>.. The following are in. preparation : — iESCHYLUS— SEPTEM CONTRA THEBAS. Edited by A. O. Prick ard, M.A., Fellow and Tutor of New College, Oxford; CICERO— PRO ."LEGE MANILIA. Edited, after Halm, by Prof" A. S. Wilkins, M.A. [In the press. CICERO— PRO ROSCIO AMEEINO. From the German ol Karl Halm. Edited by E. H. Donkin, M.A., late Scholar of Lincoln College, Oxford, Assistant Master at Uppingham. DEMOSTHENES— FIRST PHILIPPIC. Edited by Rev. T. Gwatkin, M.A., late Fellow of St. John's College, Cambridge. EURIPJDES— SELECT PLAYS, by various Editors. ALCESTIS. Edited by J. E. C. Welldon, B,A., Fellow and Lecturer of King's College, Cambridge. BACCHAE. Edited by E. S. Shuckburgh, M:A., Assistant- Master at, Eton College. HIPPOLYTUS. Edited by Rev. J. P. Mahaffy, M.A., - Fellow and Tutor of Trinity College, Dublin. MEDEA. Edited by A. W. Verrall, M.A., Fellow and Lecturer of Trinity College, Cambridge. PHCENISSJS. Edited by Oscar Wilde, B.A., Magdalen College, Oxford. HERODOTUS— THE INVASION OP GREECE BY XERXES. Books VII. and VIII. Edited by Thomas Case, M.A., formerly Fellow of Brasenose College, Oxford. a 2 4 MACMILLAN'S EDUCATIONAL CATALOGUE. HOMER'S ILIAD--THB STORY OF ACHILLES. Edited by the late J. H. Pratt, M.A., and Walter Leaf, M.Ai, Fellows of Trinity College, Cambridge. HOMER'S ODYSSEY-Books XXI-XXIV. Edited by S. G. Hamilton, B.A., Fellow of Hertford College, Oxford. LIVY— Books XXIII. and XXIV. Edited by Rev. W. W. Capes, M.A. LYSIAS— SELECT ORATIONS. Edited by E. S. SHUCK- burgh, M.A., Assistant-Master at Eton College. MARTIAL— SELECT EPIGRAMS. Edited by Rev. H. M. Stephenson, M.A., Head-Master of St. Peter's School, York, OVID— SELECT EPISTLES. Edited by E. S. Shuckburch, M.A. OVID— FASTI. Edited by G. H. Hallam, M.A., Fellow of St. John's College, Cambridge, and Assistant Master at Harrow. PLATO— FOUR DIALOGUES ON THE TRIAL AND DEATH or SOCRATES, viz., EUTHYPHM, APO- LOGY, CRITO, AND phsdo. Edited byC. W. MquLE, M. A., Fellow and Tutor of Corpus Christi College, Cambridge. PROP ERTIUS— SELECT POEMS. Edited by J. P. Post- gate, M. A., Fellow of Trinity Coliege, Cambridge. TERENCE— PHORMIO. Edited by Rev. John Bond ? M.A., late Scholar of St. John's College, Oxford, and A. S. Wal- pole, late Scholar of Worcester College, Oxford. THUCYDIDES— Books I. and II. Edited by H. Eroadbent, M.A., Fellow of Exeter College, Oxford, and Assistant- Master at Eton College. THUCYDIDES— Books IV. and V. Edited by Rev. C. E. Graves, M.A., Classical Lecturer, and late Fellow of St. John's College, Cambridge. XENOPHON— MEMORABILIA." Edited by A. R. CLUER, B.A. Balliol College, Oxford. Other volumes will follow. CLASSICAL. CLASSICAL. iESCHYkUS*— THE EUMENIDES. The Greek Text, with Introduction, English Notes, and Verse Translation. By Bernard £>rake, M.A., late Fellow of King's College, , Cambridge. 8vo. . 3j. dd. AKISTOTiB-^Ar INTRODUCTION TO ARISTOTLE'S RHETORIC. With Analysis, Notes and Appendices. By E. M. Cope, Fellow and Tutor of Trinity College, Cambridge, Svo. i4-\ ARISTOTLE ON FALLACIES; OR, THE S0PHISTIC1 • : ELENCHI. With Translation and Notes by E. PosTE, M.A. Fellow of Oriel College, Oxford. 8vo,. Ss. 6d. . ARISTOPHANES— THE BIRDS. Translated into English Verse, with Introduction, Notes, and Appendices, by B. H. Kennedy, D.D., Regius Professor of Greek in the University of Cambridge. Crown 8vo. 6s. Help-Notes to the same, for the use of Students, is. (id. BELCHER— SHORT EXERCISES , IN LATIN PROSE COMPOSITION AND EXAMINATION PAPERS IN LATIN GRAMMAR, to which is prefixed a Chapter on Analysis of Sentences. By the Rev. H.. Belcher, M.A., Assistant Master in King's College School, London. New Edition. iSmo. is. 6d. Key, is. 6d. a SEQUEL TO THE ABOVE. EXERCISES LN LATIN iblOMS, &c. By the same author. ' \In May. ■BTU&.CSLI&-GREEK AND 'ENGLISH DIALOGUES FOR USE IN SCHOOLS AND COLLEGES. By John Stuart Blackie, Professor of Greek in the University of Edinburgh. New Edition. Fcap. 8vo. 2s. 6d. ClcmLO— THE ACADEMICA. The Text revised and explained by James Reid, M.A., Fellow of Caiiis College, Cambridge. New Edition. With Translation. 8vo. {In October. SELECT LETTERS,— After the Edition, of Albert Watson, M.A. Translated by G. E." Jeans, M.A., Fellow of Hertford College, Oxford, and Assistant-Master at Hailey bury-. 8vo. [In October. 6 MACMILLAN'S EDUCATIONAL CATALOGUE. classical waiters. Edited by J. R. Green, M.A. Fcap. 8vo. is. 6d. each. A Series'of small volumes'upon some of the principal classical writers, whose,works form subjects of study in our Schools. The main object of the Series is Educational, care being taken to impart information in a systematic and thorough way, while an intelligent interest in the writers and their works is sought to be aroused by a clear and attractive style of treatment. CICERO. By Professor'A. S. Wilkins. [In preparation. DEMOSTHENES. By S. H. BUTCHER, M.A. [In preparation. EURIPIDES. By Professor J. P.Mahaffy. [Ready. HORACE. By T-. H. Ward, M.A. [In preparation. LIVY. By Rev. W. W. Capes, M.A [In the press. VERGIL. By Professor H. Nettleship. [In preparation. Others ta follow. ELLIS— PRACTICAL HINTS ON THE QUANTITATIVE PRONUNCIATION OF LATIN, for the use of Classical Teachers and Linguists. By A. J. Ellis, B.A., F.R.S. Extra fcap. 8vo. 4;. 6d. GBDDES— THE PROBLEM OF THE HOMERIC 'POEMS. By W. D. Geddes, Professor of Greek in the University of Aberdeen. 8vo. 14/. GLADSTONE— Works by the Rt. Hon. W. E. Gladstone, M.P. JUVENTUS MUNDI; or, Gods and Men of the Heroic Age. Second Edition. Crown 8vo. 10s. 6d. THE TIME AND PLACE OF HOMER. Crown 8vo. 6s. 6d. A PRIMER OF HOMER. i8mo. is. Goodwin — Works by W. W. Goodwin, Professor of Greek in Harvard University, U.S.A. SYNTAX OF THE MOODS AND TENSES OF THE GREEK VERB. New Edition, revised. Crown 8vo. 6s. 6d. AN ELEMENTARY GREEK GRAMMAR. New Edition, revised. Crown 8vo. [In preparation. CLASSICAL. GREENWOOD— THE ELEMENTS OF GREEK GRAM- MAR, including Accidence, Irregular Verbs, and Principles of Derivation and Composition ; adapted to the System of Crude Forms. By J. G. Greenwood, 'Principal of Owens College, Manchester. New Edition. Crown 8vo. $s. 6d. HERODOTUS, Books I.— III.— THE EMPIRES OF THE 'EAST. ,\ Edited, with Notes and Introductions, by A. H Sayce, M.A., Fellow and Tutor of Queen's College, Oxford, and Deputy-Professor of Comparative Philology. 8vo. • - - [In preparation. k SCENES FROM. HERODOTUS. Translated, with -Intro- duction, by Oscar. Wilde, B.A., Magdalen College, Oxford. Crown 8vo. ; ■-, , [In preparation. HODGSON -MYTHOLOGY FOR LATIN VF.RSIFICA- TION.) A brief Sketch. of the -Fables of the Ancients, prepared to be rendered into Latin Verse for Schools. By F. Hodgson, B.D., late Provost of Eton. New Edition, revised by F. C. Hodgson, M.A. iSmo. 3*. HOMER— THE ODYSSEY. Done into English by S. H. Butcher, M.A., Fellow of University College, Oxford, and Andrew Lang, M.A. , late Fellow of Merton College, Oxford, Crown 8vo. lot. 6d. HOMERIC DICTIONARY. "For Use in Schools and Colleges. ' Translated from' the German of Dr. ;G. Autenreith, with Additions and ' Corrections by R. P. Keep, ' Ph.D. With numerous Illustrations. Crown 8vo. 6s. • HORACE— THE . WORKS OF HORACE, rendered into English Prose, with Introductions, Running Analysis, and Notes, by J. Lonsdale, M.A., and S. Lee, M.A. Globe 8vo. is.6d. THE ODES OF HORACE IN A METRICAL PARA- PffRASE.^ By R. M. Hovenden. Extra fcap. 8vo". 4J. ' HORACE'S LIFE AND CHARACTER. An Epitome of ■•" 'his Satires and Epistles. By R.M. Hovenden. Extra fcap; 8vo. 4J-. 6d. WORD FOR WORD FROM HORACE. The Odes lite- rally Versified. By W. T. Thornton, C.B. Crown 8vo. • Js. 6d. •" 8 MACMILLAN'S EDUCATIONAL CATALOGUE. JACKSON— FIRST STEPS TO GREEK PROSE COM. POSITION. By JBlomfield Jackson, M.A. Assistant- Master in King's College School, London. New Edition revised and enlarged. i8mo. is. 6d. JACKSON— A MANUAL OF GREEK PHILOSOPHY. By Henry Jackson, M.A., Fellow and Praelectotf in Ancient Philosophy, Trinity College, Cambridge. [In preparation. JEBB— Works by R. C. JEBB, M.A., Professor of Greek in the University of Glasgow. THE ATTIC ORATORS FROM ANTIPHON TO ISAEOS. 2 vols. 8vo. 25J. SELECTIONS FROM THE A TTIC ORA TORS BEFORE DEMOSTHENES. Edited, with English Notes. Beings companion volume to the preceding work. 8vo. [In. October. THE CHARACTERS OFTHEOPHRASTUS. Translated from a revised Text, with Introduction and Notes. ' Extra fcap. Svo. 6s. td. A PRIMER OF GREEK LITERATURE. i8mo. is.' A HISTORY OF, GREEK LITERATURE. , Crown 8vo. [In preparation. JUVBNAIr— THIRTEEN SATIRES OF JUVENAL. With a Commentary. By John E. B. Mayor, ., M.A'., Kennedy Professor of Latin at Cambridge. Vol,- ,1. Second Edition, enlarged. Crown 8vo. p. 6d. Vol, II, Crown Svo. lar. 6d. KIEPBRT— CLASS-BOOK OF ANCIENT GEOGRAPHY, Translated from the German of Dr. Heinrich Kiepert. • [In preparation. KYNASTON— GREEK IAMBICS FOR SCHOOLS. By Rev. H. Kynaston, M.A., Principal of Cheltenham College. [In preparation. LIVY, Books XXI.— xxv. Translated by A. J. Church, M.A., and W. J. Brodribb, M.A. [In preparation. LiOYD— THE AGE OF PERICLES. A History of the - Politics and Arts of Greece from the Persian to the Pelopon- nesianWar. By William Watkiss Lloyd. 2 vols. 8vo. 21 s. CLASSICAL. ,9 MACMIiLAM- FIRST LATIN GRAMMAR. By M. C. Macmillan, M. A., late Scholar of Christ's College, Cambridge, Assistant Master in St. Paul's School. iSmo. [In preparation. mahaffy— Works by J. P. Mahaffy, M.A., Pjrofessor of Ancient History in Trinity College, Dublin. SOCIAL. LIFE IN GREECE; from Homer to Menander. ', Third Edition, revised and enlarged. Crown 8yo. gs., RAMBLES AND STUDIES IN GREECE. With Illustrations. Second Edition. With Map. Crown 8vo. .... los„,fyl. '.'-,' A PRIMER OF GREEK ANTIQUITIES. With Illus- trations. i8mo. xv ,, ./ EURIPIDES. i8mo. is. 6d. MAEBKALl-i TABLE OF IRREGULAR GREEK VERBS, classified according to the arrangement of Curtius' Greek Grammar. By J. M. Marshall, M.A., one of the Masters in- Clifton College; 8vo. cloth. New Edition, is. MAYOR (JOHN E. B.)— FIRST GREEK READER, Edited after' Karl Halm, with Corrections and large Additions by Professor John E. B. Mayor, M.A., Fellow and Classical ,,; Lecturer of St. John's College, Cambridge. New Edition, revised. Fcap. ,8vq. 4j. 6d. ^ , •■ i., .'* H BIBLIOGRAPHICAL CLUE TO LATIN LITERA- ^yTURE* Edited after Hubneb, with large Additions by . Professor John E. B. Mayor. Crown 8vo. ,,6s. 6d. MAYOR (JOSEPH B.)— GREEK FOR BEGINNERS. By the Rev. J. B. Mayor, M.A., Professor of Classical Literature in King's College, London. Part I., with Vocabulary, is. 6d. ., .Parts II. and III., with Vocabulary and Index, 3*. 6d. com. plete in one Vol. New Edition. Fcap. 8vo. cloth, 4s. 6d. NIXON— PARALLEL EXTRACTS arranged for translation into English and Latin, with Notes on Idioms; By J. E. Nixon, M.A., Classical- Lecturer, King's College, London. Part I.— Historical and Epistolary. New Edition, revised and enlarged. Crown 8vo. 3s. 6rf. io MACMILLAN'S EDUCATIONAL CATALOGUE. NIXON Continued— A FEW NOTES ON LATIN RHETORIC. With Tables and Illustrations. By J. E. Nixon, M.A. Crown 8vo. is . peile (JOHN, M.A.)— AN INTRODUCTION TO GREEK AND LATIN ETYMOLOGY. By John Peile, M.A., Fellow and Tutor of Christ's College, Cambridge, formerly Teacher of Sanskrit in the University of Cambridge. Third and Revised Edition. Crown 8vo. ior. 6d A PRIMER OF PHILOLOGY. i8mo. is. By the same Author. , , PINDAR— THE EXTANT ODES Of PINDAR. Translated into English, with an Introduction and short Notes, by Ernest Myers, M.A., Fellow of Wadham College, Oxford. Crown 8vo. $s. VUMiQ—THE REPUBLIC OP PLATO. Translated into English, with an Analysis and Notes, by J. Ll.. Davies, M.A., and D. J. Vaughan, M.A. New Edition, with Vignette Portraits of Plato and Socrates, engraved' by Jeens from an Antique Gem. i8ino. #. td. PHILEBUS. Edited, with Introduction and Notes, by Henry Jackson, M.A., Fellow of Trinity College,Cambridge. 8vo " Lin preparation. PLAVTVB— THE MOSTELLARIA OF PLAUTUS.' With Notes, Prolegomena, and Excursus. By William Ramsay, M.A., formerly Professor of Humanity in. the University of Glasgow. Edited by Professor George G. Ramsay, M.A., of the University of Glasgow. 8vo. 14J. POTTS (A. W., M.A.)— Works by Alexander W; Potts, M.A., LL.D., late Fellow of St. John's College,' Cambridge; Head Master of the Fettes College, Edinburgh. HINTS TOWARDS LATIN PROSE COMPOSITION. New Edition. Extra fcap. 8vo. 3*. PASSAGES FOR TRANSLATION INTO LATIN PROSE. Crown 8vo. [In the press. CLASSICAL. EOBY- A GRAMMAR OF THE LATIN LANGUAGE, both Plautus to Suetonius, By H. J.Koby, M.A., fete Fellow of " ' St/John's College, Cambridge, fn Two Parts. Third Edition. Parti, containing: — BookL. Sounds. Book II. Inflexions. Book IIL Word-formation. Appendices. Crown 8vo. 8j. 6a!'. ... Part II. — Syntax, Prepositions, &c. . Crown 8vo. ior. 6d. , "Marked by the clear and practised insight of a master in bis art. A book that would do honour to any country."— Athen^um. SCHOOL LATIN GRAMMAR. By the same Authori" [In the press. RUSH- SYNTHETIC LATIN DELECTUS. A First' Latin Construing "Book arranged on the Principles of Grammatical Analysis. With Notes and Vocabulary. By E. Rush, B.A. With Preface by the Rev. W; F. Moulton, M.A., D.D. Extra fcap. 8vo. zs. RUST— FIRST STEPS TO LA TIN PROSE COMPOSITION. By the _Rev. G. Rust, M.A. of Pembroke College, Oxford, Master of the Lower School, King's College, London. New Edition. i8mo. is. 6d. RUTHERFORD— A FIRST GREEK GRAMMAR. By W . G. Rutherford, M.A., Assistant Master in St. Paul's School, London. Extra fcap. 8vo. Is. SEELEY— A PRIMER OF LATIN LITERATURE. By Prof. J. R. Seeley, [In preparation. SHUCKBURGH— A LATIN READER. By E. S. Shuck. burgh, M.A., Assistant Master at Eton College: [In preparation. TACITUS— COMPLETE WORKS TRANSLATED. By A. J. Church, M.A., and W. J. Brodribb, M.A. THE HISTORY. With Notes and a Map. New Edition. Crown 8vo. 6s. THE ANNALS. With Notes and Maps. New Edition. Crown 8vo. Js. 6d. >THE AGRICOLA AND GERMANY, WITH THE DIALOGUE ON ORATORY. With Maps and Notes. New and Revised Edition. Crown 8vo. 4?. 6d. 13 MACMILLAN'S EDUCATIONAL CATALOGUE. THEOPHRASTOS— THE CHARACTERS OF THEO- PHRASTUS. An Englisn Translation from a Revised Text. With Introduction and Notes. By R. G "Jkbb, M.A., Pro- fessor, of Greek in the University of, Glasgow. Extra fcap. 8vo. 6s. 6d. TURING— Works by the Rev. E. TURING, M.A., Head Master of Uppingham School. A LATIN GRADUAL. A First Latin Construing Book for Beginners. New Edition, enlarged, with Coloured Sentence Mapi Fcap. 8vo. 2s. 6d. A ' MANUAL OP MOOD CONSTRUCTIONS. Fcap. 8vo. is. 6d. A CONST-RUING BOOK. Fcap8vo. 2s,6d. VIRGIL— THE WORKS OF VIRGIL RENDERED tNIO ENGLISH PROSE, with Notes, Introductions, Running Analysis, and an Index, by James Lonsdale, M.A., and Samuel Lee, M. A. ,New Edition, Globe 8vo. 3;. (td. gilt edges, \s. (td. WILKINS- A PRIMER OF ROMAN ANTIQUITIES. By A. S. Wilkins, M.A., Professor, of Latin in the Owens College, Manchester. With Illustrations. l8mo. is. WRIGHT— Works by J. WRIGHT, M.A., late Head Mastcriof Sutton Coldfield School. .HELLENICA; OR, A HISTORY OF GREECE, IN GREEK, as related by Diodorus and Thucydides ; being a First Greek Reading Book, with explanatory Notes, Critical and Historical. New Edition with a Vocabulary. Fcap. 8vo. •'■ 3s. 6d. A HELP TO LATIN GRAMMAR; or, The Form and Use of Words in Latin, with Progressive Exercises. Crown 8vo. 4J-. td. THE SEVEN KINGS OF ROME. An Easy Narrative, abridged from the First Book of Livy by the omission of Difficult Passages; being a First Latin Reading Book; with Grammatical Notes. With Vocabulary, 31. 6d. [New and thoroughly revised edition, just ready. MATHEMATICS. 13 WRIGHT Continued— FIRST LATIN STEPS} OR, AN INTRODUCTION BY A SERIES OF EXAMPLES TO THE' STUDY OF THE LATIN LANGUAGE. Crown 8vo. 5^ . • ATTIC PRIMER. Arranged for the Use of Beginners. Extra fcap. 8vo. 4s'. 6d. A COMPLETE LATIN COURSE, comprising Rules with Examples,' Exercises, both Latin and -English, on each Rule, and Vocabularies. Crown 8vo. us. 6d. ) MATHEMATICS. AIRY— Works by Sir G. B. AIRY, K.C.B., Astronomer , Royal : — ELEMENTARY TREATISE ON PARTIAL DIE- FERENTIAL EQUATIONS. Designed for the Use of Students in the Universities. With Diagrams. Second Edition. Crown 8vo, $s, 6d. ON THE ALGEBRAICAL AND NUMERICAL THEORY OF ERRORS OF OBSERVATIONS AND THE COMBINATION OF OBSERVATIONS. Second Edition, revised. Crown 8vo. 6s. 6d. - . UNDULATORY THEORY OF OPTICS. Designed for the Use of Students, in the University. New Edition. Crown 8vo. 6s. 6d. ON SOUND AND ATMOSPHERIC VIBRATIONS. With the Mathematical Elements of Musici Designed for the r , Use of Students in the University. Second Edition^ Revised and Enlarged, Crown 8vo. gs. . , A TREATISE OF MAGNETISM. Designed for the Use of Students in the University. Crown 8vo. gs. 6d. AIRY (OSMUND)— .rf TREATISE ON GEOMETRICAL OPTICS. Adapted for the use of the Higher Classes in Schools. By Osmund Airyj B.A., one of the Mathematical Masters in Wellington College. Extra fcap. 8vo. y. 6d. 14 MACMILLAN'S EDUCATIONAL CATALOGUE. BAYMA— THE ELEMENTS OP MOLECULAR MBCHA- NICS. . By Joseph Bayma, S.J., Professor of Philosophy, Stonyhurst College. Demy 8vo. iar. 6d. beasley— AM ELEMENTARY TREATISE ON PLANE TRIGONOMETRY. With Examples. ByR. D. Beasley, M.A., Head Master of Grantham Grammar School. Fifth Edition, revised and enlarged. Crown 8vo. 3;. 6 J. BLACKBURN (HUGH) — ELEMENTS OF PLANE TRIGONOMETRY, for the use of the Junior Class in Mathematics n the University of Glasgow. By Hugh Blackburn, M.A., Professor of Mathematics in the Univer- sity of Glasgow. Globe 8vo. is. 6a5. BOOLE— Works by G. BOOLE, D.C.L., F.R.S., late Professor of Mathematics in the Queen's University, Ireland. A TREATISE ON DIFFERENTIAL EQUATIONS. Third and Revised Edition, Edited by I. Todhunter. Crown 8vo. 14?. a Treatise on differential equations. Supplementary Volume. Edited by I. Todhunter. Crown 8vo. is, 6d. THE CALCULUS OF FINITE DIFFERENCES. Crown 8vo. 10s. 6d. New Edition, revised by J; F, Moulton. .,,.■*: BROOK-SMITH (J .)— ARITHMETIC IN THEORY AND PRACTICE. By J. Brook-SMith, M.A., LL.B., St. John's College, Cambridge ; Barrister-at-Law ; one of the Masters of Cheltenham College. New Edition, revised. Crown 8vo. 41, 6d. CAMBRIDGE SENATE-HOUSE PROBLEMS and RIDERS WITH SOLUTIONS i— 1875— PROBLEMS 'AND RIDERS. By A. G. Greenhill, M. A. Crown 8vo. ,Ss. fid. • , , 1878— SOLUTIONS OF SENATE-HOUSE PROBLEMS. By the Mathematical Moderajx>rs and Examiners. Edited by J.' W. L. Glaisher, M.A., Fellow of Trinity College, Cambridge. i2s K MATHEMATICS. is CANDLBR-ZfflZP TO ARITHMETIC. Designed for the use of. Schools. By H. Candler, M.A., Mathematical Master pi Uppingham School. Extra fcap,,8vo, 1 2f. (td. CHEYNE— AN ELEMENTARY TREATISE ON THE PLANETARY THEORY. By. C. H. H.'Cheyne, M.A., F.R.A.S. With a Collection of Problems; Secdnd Edition. Crown 8vo. 6s. 6d. CHRISTIE— .4 COLLECTION OF ELEMENTARY TEST- 'QUESTIONS' IN PURE AND MIXED' MATHE- MATICS; with Answers arid Appendices on Synthetic '".iPiyisionj, and on the Solution of Numerical; Equations by Horner's Method. By James . R. Christie;'- F. R. S., , Royal Military Academy, Woolwich. Crown 8vo. Ss. 6d. CLIFFORD— THE ELEMENTS r OF 'DYNAMIC. An In. troduction to the Study of Motion and Rest in Solid and Fluid Bodies. By W. K. Clifford, F.R.S., Professor^ Applied Mathematics and Mechanics at University College, London. Part I.— KINETIC: Crown 8voi js. 6d. CUMMING— AN, INTRODUCTION TO TH~E THEOR\ OF ELECTRICITY. ' By LinnWs Cumming, M.A., one of the Masters- of- Rugby School With illustrations. Crown 8vo. Ss. 6d. .yv.v CUTHBERTSON— -£ UCLIDIAN GEOMETRY. By Francis Cuthbertson, M.A., LL.D., Head Mathematical Master of the City of London School.- Extra fcap. 8vo. 4*. 6d. DALTON- Works by the Rev. T. ' DALTON, M.A., Assistant Master of Eton College. • • ;;,i, r t,i;- RULES AND EXAMPLES IN ARITHMETIC, New .5, Edition. iSmo. as. 6d. c ;■ v , [Answers to the Examples are appended. ' ■- - * 1 , ■ v. : RULES AND EXAMPLES IN ALGEBRA. Part I. New Edition. i8mo. 2s. Part II. .: i8mo. 2s. 6A 16 MACMILLAN'S EDUCATIONAL CATALOGUE. DAY— PROPERTIES OF CONIC SECTIONS PROVED GEOMETRICALLY. Part L, ' THE ELLIPSE, with Problems. By the Rev. H. G. Day, M.A. Crown 8vo. . y. 6d. VOOGSON— EUCLID AND BIS MODERN RIVALS. By the Rev. Cj L Dodgson, M.A., Mathematical Lecturer, Christ Church, Oxford. Crown 8vo. ft. 6d, . < DREW- GEOMETRICAL TREATISE ON CONIC SEC- TIONS. By W. H. Draw, M.A., St. John's College, Cambridge, New Edition, enlarged, Crown 8vo. Sj. SOLUTIONS TO THE PROBLEMS IN DREWS , CONIC SECTIONS. Crown 8vd. 4s. 6d. EDGAR (J. H.) and PRITCHARD (G. S.j— NOTE-BOOK ON PRACTICAL SOLID OR pESCRIPTIVE GEOr , METRY. Containing Problems w)th help for Solutions. By J. H. Edgar, M.A., Lecturer on Mechanical Drawing at the Royal School of Mines, and G. S. Pritchard. New Edition, revised and enlarged. Globe 8vo. 3*,, FERRERS— Works by the Rev. N. M. FERRERS, M.A., Fellow and Tutor of Gonville and Caius College, Cambridge; AN ELEMENTARY TREATISE ON TR1LINEAR CO-ORDINATES, the Method of Reciprocal Polars, and the Theory of Projectors. New Edition, revised. Crown 8vo. , 6s.6d. . , AN ELEMENTARY TREATISE ON SPHERICAL HARMONICS, AND SUBJ-ECTS CONNECTED WITH THEM. Crown 8vo, r p. 6d. FROST— Works by PERCIVAL FROST, M. A., formerly Fellow of St. John's College, Cambridge ; Mathematical Lecturer of King's College. 1 AN ELEMENTARY TREATISE ON CURVE TRA- CING. By Percival Frost, M.A. 8vo. 12s. SOLID GEOMETRY. A New Edition, revised and enlarged ' ; of the" Treatise by Frost and Wolstbnhoxmb. In 2 Vols. Vol.. I. 8vo. 16s. MATHEMATICS. ly dobPRAY— Works by HUGH GODFRAY, M.A., Mathematical Lecturer at Pembroke College, Cambridge. A TREATISE ON ASTRONOMY, for the Use of Colleges and Schools. New Edition. 8vo. 12s. dd. AN ELEMENTARY TREATISE ON THE LUNAR Si THEORY, with a Brief Sketch of the Problem up to the time ' ! of Newton. Second Edition, revised. Crown 8vb. $s, 6d. HEMMING— AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS, for ' the U§e of Colleges, and Schools'. ByG. W. Hemming, M.A., Fellow of St John's College, Cambridge. Second Edition, with Corrections and Additions. 8vo. gs. JACRSON — GEOMETRICAL CONIC SECTIONS. An . Elementary Treatise in which the Conic Sections are defined as the Plane Sections of a Cone, and treated by the Method of Projection. By J. Stuart Jackson, M.A., late Fellow of Gonville and Caius College, Cambridge. Crown 8vo. 4s. 6d. JELLET (JOHNH.H TREATISE ON THE THEORY OF FRICTION. By John H. Jeixet, B.D., Senior Fellow of Trinity College, Dublin; President of the Royal Irish Academy. 8vo. Ss. 6d. JONES and CHEYNE— ALGEBRAICAL EXERCISES. Progressively Arranged. By the Rev. C. A. Jones, M.A., and C. H. Cheyne, M.A., F.R.A.S., Mathematical Masters of Westminster School. New Edition. i8mo. zs. 6a. KELLAND and T AIT— INTRODUCTION TO QUATER- NIONS, with numerous examples. By P. Keixand, M.A., F.R.S. ; and P. G. Tait, M.A., Professors in the department of Mathematics in the University of Edinburgh. Crown 8vo. •js. 6d. KITCHENER— A GEOMETRICAL NOTE-BOOK, containing Easy Problems in Geometrical Drawing preparatory to tlie Study of Geometry. For the use of Schools. By F. E. Kitchener, M.A., Mathematical Master at Rugby. New Edition. 4to. ss. b 18 MACMILLAN'S EDUCATIONAL CATALOGUE. matjlt— NATURAL GEOMETRY: an Introduction, to. the Logical Study of Mathematics. For Schools and Technical Classes. With Explanatory Models, based upon the Tachy- metrical Works of Ed. Lagout. By A. Mault. l8mo. it. Models to Illustrate the above, in Box, 12s. &/. WERRIMAN — ELEMENTS OF THE METHOD OF LEAST SQUARES. By Mansfield Merriman, Ph.D. Professor of Civic and Mechanical Engineering, Lehigh Uni- versity, Bethlehem, Perm. Crown 8vo. ,,■••••* SOUND: a Series of Simple, Entertaining, and Inexpensive Ex- periments in the Phenomena of Sound, for the use of Students ot every age. By A. M. Mayer, Professor of Physics in the Stevens Institute of Technology, &c. With numerous Illustrations. Crown 8vo. 3s. 6d. 1 Other volumes to follow. EASY LESSONS IN SCIENCE. HEAT. By Miss C. A. Martineau. Edited by Prof. W. F. Barrett. [In the press. LIGHT. By Mrs. Awdry. Edited by Prof. W. F. Barrett. [In, the press. ELECTRICITY. By Prof. W. F. Barrett. [In preparation. SCIENCE LECTURES AT SOUTH KENSINGTON. VOL. I. Containing Lectures by Capt. Abney, Prof. Stokes, Prof. Kennedy, F. G. Bramweia, Prof. G. Forbes, H. C. Sorby, J. T. Bottomley, S. H. Vines, and Prof. Carey Foster. Crown 8vo. 6s. VOL. II. Containing Lectures by W. Spottiswoode, P.Ti.S., Prof. Forbes, Prof. Pigot, Prof. Barrett, Dr. Burdon- Sanderson, Dr. Lauder Brunton, F.R.S., Prof. Roscoe, and others. Crown 8vo. 6s. .1 SCIENCE. 35 MANCHESTER SCIENCE LECTURES FOR THE PEOPLE. Eighth Series, 1876-7. Crown 8vo. Illustrated. 6d. each. WHAT THE EARTH IS COMPOSED OF. By Professor Roscoe, F.R.S. THE -SUCCESSION OF LIFE ON THE EARTH. By Professor Williamson, F.R.S. s WHY THE EARTHS CHEMISTRY IS AS IT IS. By J. N. Lockyer, F.R.S. Also complete in One Volume. Crown 8vo. cloth, zs. BLANFORD— THE RUDIMENTS OF PHYSICAL GEO- GRAPHY FOR THE USE OF INDIAN SCHOOLS ; with •ii'ji' a Glossary of Technical Terms employed. ByH.F. Blanford, F.R.S. New Edition, with Illustrations. Globe 8vo. zs. 6d. EVERETT— PHYSICAL UNITS. By Prof. J. D. EVERETT. ■' Extra fcap. 8vo. [In the press. 6EIKIB— OUTLINES OF FIELD GEOLOGY. By Prof. Geik'ie, F.R.S. With Illustrations. Extra fcap. 8vo. 3.?. 6d. OiOKOOVt—AN ELEMENTARY BOOK ON HEAT. By J. E. H. Gordon, B.A., Gonville and Caius College, Cam- bridge. Crown 8vo. zs. Xj-o M'KENDRICK— OUTLINES OF ' : ' PHYSIOLOGY IN ITS RELATIONS TO MAN. By J. G. M'Kendrick, M.B., F.R.S.E. With Illustrations. Crown 8vo. 12s. 6d. MIALI^-STUDIES IN COMPARATIVE ANATOMY. No. I. — The Skull of the Crocodile : a Manual for Students. By L. C. Miaix, Professor of Biology in the Yorkshire College and Curator of the Leeds Museum. 8vo. 2s. 6d. No. II.— Anatomy of the Indian Elephant. By L. C. MlALL and F. Greenwood. With Illustrations. 8vo. 5^. MTTIR— PRACTICAL CHEMISTRY FOR MEDICAL STU- , r DENTS. Specially arranged for the first M.B. Course, By M. M. Pattison Muir, F.R.S.E. Fcap. 8vo. is. 6d. SHANN— AN ELEMENTARY 'TREATISE ON HEAT, IN RELATION TO STEAM ANDTHE STEAM-ENGINE. By G. SHANN, M.A. With Illustrations. Crown Svo. 4s. 6d. ... -> c 2 ■ ,36 MACMILLAN'S EDUCATIONAL CATALOGUE. WRIGHT— METALS AND THEIR CHIEF INDUSTRIAL APPLICATIONS. By C. Alder Wright, D.Sc, &c Lecturer on Chemistry in St. Mary 's .Hospital Medical School. Extra fcap. 8vo. 3s. 6d. 1 HISTQRT. BEESLY— STORIES FROM THE HISTORY , OF, ROME. By Mrs. Beesly. Fcap. 8vo. 2s. 6d. . " The attempt appears to us in every way successful. The stories are interesting in themselves, and are told with perfect simplicity and good •feeling."— Daily News.'. i'_ - c ■'■ , FREEMAN (ED-WARD A..)— OLD-ENGLISH HISTORY. By Edward A. Freeman, D.C.L., LL.D., late Fellow of Trinity College, Oxford. With Five Coloured (Maps. New Edition. Extra fcap. 8vo.'half-bound. 6s. GREEN— A SHORT HISTORY OF THE ENGLISH PEOPLE. By John Richard Green, M.A., LL.D. With Coloured Maps, Genealogical Tables, and Chronological Annals. Crown 8vo. Ss. 6d. Sixty-second Thousand. " Stands alone as the one general history of the country, for the sake of which all others, if young and old are wise, will be speedily and surely set aside." — Academy. , < GUEST— LECTURES ON THE HISTORY OF ENGLAND. By M.J. Guest. With Maps. Crown 8vo. 6s. .. HISTORICAL COURSE FOR SCHOOLS — Edited by Edward A. Freeman, D.C.L., late Fellow of Trinity College, Oxford. I. GENERAL SKETCH OF EUROPEAN HISTORY. By Edward A. Freeman, D.C.L.. New Edition, revised and enlarged, with Chronological Tablfe, Maps, and Index. l8mo. cloth. 3-f. 6d. ,1 " It supplies the great want of a good foundation for historical teaching. The scheme is an excellent one, and this instalment has been executed in a, way that promises much for. the volumes that are yet to appear."— Educational Times. II. HISTdRY OF ENGLAND. By Edith Thompson. New Edition, revised and enlarged, with/ Maps. l8m0. ar. 6d. III. HISTORY OF SCOTLAND. By Margaret Macarthur. New Edition. i8mo. as. "An excellent summary, unimpeachable as to facts, and putting them in the clearest and most impartial light attainable." — Guardian. HISTORY. 37 HISTORICAL COURSE FOR SCHOOLS Continued^- , , . IV,, HISTORY OF ITALi. . : By the Rev. W. Hunt, M.A. l8mo. 3 j. "It possesses the same solid merit as its predecessors .... the same scrupulous care about fidelity in details Jt is distinguished, too, by information on art, architecture, and social politics, in which the writer s grasp is seen by tie firmness and clearness of his touch"— Educational ' Times. 1 ' ' V. HISTORY OF GERMAN^. By J. Sime, M.A. l8mo. - Jj. "A remarkably clearfand impressive history of Germany. Its great events are wisely kept as central figures, and the smaller events are care- fully kept, not only.^ubordinate and subservient, but most skilfully woven - into the texture of the historical tapestry presented to the eye." — Standard. ■' VI, HISTORY OF •AMERICA. By John A. Doyle. , With Maps. , l8mo, 4s. 6a. " Mr. Doyle has performed his task with admirable care, fulness, and clearness, and for the first time we have for schools an accurate and inter- esting history of America, from the earliest to the' present time/'— Standard. ' " EUROPEAN COLONIES. ' By E. J. PayNe, M. A. With Maps. i8mo. 4s. 6d. "We have seldom met with an historian capable of forming a more ^comprehensive, far-seeing, and unprejudiced estimate of events and ""peoples,' and we can commend this little work as one certain to rjrrjve o£ the highest interest to all thoughtful readers." — Times. • FRANCE. jBy Charlotte M. Yonge. With Maps. i8mo. 3J. 6d. GREECE. 'By EdwArd A. Freeman, D.C.L. ; [In preparation. ROME. By Edward A. Freeman, D.C.L. [In preparation. HISTORY primers— Edited by John Richard Green. Aiithorvof" A Short History of the English People." ' - ROME: By' the Rev. M." Creighton, M.A., 'late Fellow and Tutor of- Mertoh College, Oxford. With Eleven Maps. iSmo. Is. , "The author has been curiously successful in telling in an intelli- gent way the' story of Rome from first to last." — School Board Chronicle. ,;: GREECE., By C. A. Fyffe, M.A., Fellow and late Tutor of University, College, Oxford. W^th Five Maps. i8mo. is. "We give our unqualified praise to this little manual."— School- master. 38 MACMILLAN'S EDUCATIONAL CATALOGUE. HISTORY PRIMERS Continual— EUROPEAN HISTORY. By E. A. Freeman, D.C.L., LL.D. With Maps. l8mo. is. "The work is always clear, and forms a luminous key to European history." — School Board Chronicle. GREEK ANTIQUITIES. By the Rev. J. P. Mahaffy, M.A. Illustrated. iSmo. If. "All that is necessary for the scholar to know is told so compactly yet so fully, and in a style so interesting, that it Is impossible for even the dullest boy to look on this little work in the same light as he regards his other school books. " — SCHOOLMASTER. CLASSICAL GEOGRAPHY. By H..F. Tozer, M.A iSmo. is. " Another valuable aid to the study of the ancient world. ... It contains an enormous quantity of information packed into a small space, and at the same time communicated in a very readable shape."— John Bull. GEOGRAPHY. By George Grove, D.C.L. With Maps. l8mo. IS. "A model of what such a work should be .... we know o£ no short treatise better suited to infuse life and spirit into the dull lists of proper names of which our ordinary class-books so often almost exclusively consist."— Times. ROMAN ANTIQUITIES. By Professor Wilkins. Illus- trated. l8mo. is. " A little book that throws a blare of light on Roman History, and is, moreover, intensely interesting. "—School Board Chronicle. FRANCE. By Charlotte M Yonge. . i8mo. is. In preparation : — ENGLAND. By J. R. Green, M.A. michelet— A SUMMARY OP MODERN HISTORY. Translated from the French of M. Michelet, and continued to the Present Time, by M. C. M. Simpson. Globe 8vo. 4x. del. OTTE— SCANDINAVIAN HISTORY. By E. C. Otte. With Maps. Globe 8vo. 6s. PAULI— PICTURES OP OLD ENGLAND. By Dr. R, Pauli. Translated with the sanction Of the Author by E. C. Otte. Cheaper Edition. Crown 8vo. 6s. DIVINITY. TAIT— ANALYSIS OF ENGLISH HISTORY, based on Green's "Short History of the English People." By C. W. A. Tait, M.A., Assistant Master, Clifton College. Crown 8vo. 3*. 6d. WHEELER— ^4 HISTORY OF INDIA. By J. Talboys Wheeler. Crown 8vo. [In the press. YONGE (CHARLOTTE VI.)— A PARALLEL HISTORY OF FRANCE AND ENGLAND : consisting of Outlines and Dates. By Charlotte M. Yonge, Author of " The Heir -of Redclyffe," &c, &c. Oblong 4to. y 6& CAMEOS FROM ENGLISH HISTORY. —FROM ROLLO TO EDWARD II. " By the Author of "The Heir of Redclyffe." Extra fcap. 8vo. New Edition. fr. -: A SECOND SERIES OF CAMEOS FROM ENGLISH HISTORY— THE WARS IN FRANCE. New Edition. Extra fcap. 8vo. $s. A THIRD SERIES OF CAMEOS FROM ENGLISH HISTORY— THE WARS OF THE ROSES. New Edition. Extra fcap. 8vo. 5 \In preparation. HACLEAR- Works by the Rev. G. F. Maclear, D.D., Head Master of King's College School. A CLASS-BOOK OF OLD TESTAMENT ' HISTQRY* New Edition, with Four Maps. iSmo. - 4s. 6d. . A CLASS-BOOK OF NEW TESTAMENT HISTORY, including the Connection of the Old and New Testament. With Four Maps. New Edition. i8mo, p. 6a. DIVINITY.; < ... 41 MACLEAR Continued — i J.'J'i ' _<..-; ' i4 'SHILLING BOOK OF OLD TESTAMENT "•'HISTORY, for National and Elementary Schools. With Map. igmo. cloth. New Edition. ' A SHiLLlNG BOOK OF NEW TESTAMENT HISTORY, for National and Elementary Schools; ^With J Map. i8mo. cloth. New Edition. ' . -'-'•' These works haiiet been carefully abridged from the .author's : larger manuals. ;:, i 1 ,.-. . ...■; Class-book 01 ''The' catechism of the CHURCH OF ENGLAND. New Edition. l8mo. cloth; is. 6d. ■■ • ■ . i. A FIRST CLASS-BOOK •■ OF THE CATECHISM OF THE CHURCH OF ENGLAND, with Scripture Proofs, ' for Junior Classes and Schools'." iSmo. 6d. '^New Edition. J -.,■-•' ' ' ■ . ' '" ■■■ ' •'.' .-.' i.l :. , -.. JJ .J j A [MANUAL OF INSTRUCTION F-QR CONFIRMA- TION AND FIRST COMMUNION. WITH PR A YERS -,,^AND DEVOTIONS. 3?mo. cloth extra, red edges, is. tH'CLELLAN— THE NEW TESTAMENT. A New Trans- lation on the Basis of the Authorised Version, from a Critically . ; revised Greek Text,, with Analyses,) copious References and i " Illustrations", from original ^authorities, -.New , Chronological and Analytical Harmony of the. Four Gospels, Notes. and Dis- sertations. A contribution to Christian 'Evidence. .iB^f, John Bkown, M'Cleli.an, M.A., late Fellow of Trinity s QoJlege, Cambridge. In Two Vols. . Vol. T.— The Four Gospels' with the Chronological and Analytical Harmony. 8vo. 30?." ' - ''One of the most' remarkable productions of recent" times*" says (he Theological Review, "in this department of sacred literature ; " and the .British Quarterty Review terms it \ 'a. thesaurus pf first-hand inyestiga- tiohs." .•-".. ^ '.' . . _ '•:< '■, > v, MAUBICB— THE LORD'S PRAYER, THE CREEDyAND THE COMMANDMENTS. Manual for Farentsand School- masters. To which is added the Order of the Scriptures, . By the Rev. F. Dbnison Maurice, M.A. i8mo. cloth, limp. is. 4« MACMILLAN'S EDUCATIONAL CATALOGUE. PROCTER— A HISTORY OF THE BOOK OF COMMON PRAYER, with a Rationale of its Offices. By Francis ' Procter, M.A. Thirteenth Edition, revised and enlarged. Crown Svo. 10s. td. ROCTER AND maclear— AN ELEMENTARY INTRO. DUCTION TO THE BOOK OF COMMON PRAYER. Re-arranged and supplemented by an Explanation of the Morning and Evening Prayer and the Litany. By the Rev. F. Procter and the Rev. Dr. Maclear. New and Enlarged Edition, containing the Communion Service and the Confirmation and Baptismal Offices. i8rao. 2s. 6d, PSALMS OF DAVID CHRONOLOGICALLY ARRANGED. By Four Friends. An Amended Version, with Historical Introduction and Explanatory Notes. Second and Cheaper : Edition, with Additions and, Corrections. Cr. 8vo. 8s, 6d. RAMSAY- THE CATECHISERS MANUAL;, or, the Church Catechism Illustrated and Explained, for the Use of Clergy, men. Schoolmasters, and Teachers. By the Rev. Arthur Ramsay, M.A. New Edition. i8mo. is. 6d. SIMPSON— AN EPITOME OF THE HISTORY OF THE 'CHRISTIAN CHURCH. By William Simpson, . M,A^ .■ New Edition. Fcap. 8vo. 3s. 6d. ' '?!*"' ' . TRENCH— By R. C. TRENCH, D.D., Archbishop of Dublin. ■' LECTURES ON MEDIEVAL CHURCH HISTORY. Being the substance of Lectures delivered at Queen's College, London. Second Edition, revised. 8vo. 12s. SYNONYMS OF THE NEW 7ESTAMENT. Eighth 1 Edition, revised. 8vo. 12s. WESTcott — Works by Brooke Foss Westcott, D.P.> Canon ' of Peterborough. a General survey of the history of , the canon of the new testament during the FIRST FOUR CENTURIES. Fourth Edition, Wift .Preface on "Supernatural Religion." Crown, 8vo. ioj. 6d. INTRODUCTION TO THE STUDY OF THE FOUR GOSPELS] Fifth Edition. Crown 8vo. 10s. 6d. MISCELLANEOUS. 43 WESTCOTT Continued— THE BIBLE IN THE CHURCH. A Popular Account of the Collection and Reception of the Holy Scriptures in the Christian Churches. New Edition. i8mo. cloth. 4?. 6d. ' THE GOSPEL OF THE RESURRECTION. Thoughts' on its Relation to Reason and History. New Edition. Crown 8vo. 6s. WILSON— TffE BIBJLE STUDENT'S GUIDE ,tp *e more .-Correct Understanding of the English, Translation of the Old Testament, by reference to the original Hebrew. By William Wilson, D.D., Canon of Winchester, late Fellow of Queen's College, Oxford; 'Second Edition, carefully revised.- 4to. cloth. 25*. YONGE (CHARLOTTE to..)— SCRIPTURE READINGS FOR SCHOOLS ANP.FAMILIES. By. Charlotte M. Vonge, Author ;of "The Heir of Redclyffe." First Series. Genesis to Deuteronomy. Globe 8vo. 'ils. 6d. With Comments, 3*. 6d. , ■•* Second Series. From Joshua to Solomon. Extra fcap. 8vo. is. 6d. With' Comments, y. 6d. . Third Series. The Kings and the Prophets. , Extrajcap. 8vo. is. 6d. With Comments, 3*. 6d. Fourth Series. , The Gospel Times, is. 6d. With ' Comments, extra fcap. 8vo., 3*. 6d.\ Fifth Series. Apostolic Times. Extra fcap. 8vo. is. 6d.. With Comments, 3*. 6d. MISCELLANEOUS. Including works on English, French, and German Language and Literature, Art Hand-books, &c, &e. ABBOTT—^ SHAKESPEARIAN GRAMMAR. An Attempt to illustrate some of the "Differences between Elizabethan and Modem English. By the Rev. E. A. Abbott, D.D., Head Master of the City of London School. New Edition. Extra fcap. 8vo. 6s. 44 MACMILLAN'S EDUCATIONAL CATALOGUE. ANDERSON — LINEAR PERSPECTIVE, AND MODEL' DRA WING. A School and Art Class Manual, with Questions . and Exercises for Examination, and Examples, of Examination : Papers. ■ By ..Laurence Anderson,,' With Illustrations. Royal 8vo. 2S. , B&X&ZEB.— FIRST, LESSONS IN THE PRINCIPLES OP COgKING. ByLADY^ARKER. New Edition. i8mo. is. BEAUHARCHAIS-li BARBIER DE SEVILLE, .Edited, with Introduction and Notes, by L. P. Blouet, Assistant Master in St. Paul's School. ' Fcap.8vo; y:6d. '■ BURNERS- FIRST LESSONS ON HEALTH: 1 Bj 'J. W ners. New Edition. iSmo. is. ' , , BLAKI8T0N— THE TEACHER, Hints on School Manage- ment. A Handbook for Managers, Teachers' Assistants, and ,Pupil Teachers. By J. R. Blakiston, M.A. Crown 8vo. ' \s. 6d. ' v '' < "'","'' " • ■" '- : ' r "'~ l '"" T '»* "•»''iOV " Into a comparatively small book he has crowded a great deal of ex- ceedingly useful and, so.urid advice. It is a plain, common-senge^iook, full of hints to the teacher on the management of his school and his children. — School Board Chronicle. , ; -, ^ , ; BREYMANN— Works by HERMANN BREYMANN, Ph.D., fPro- , fessorof Philology in the University, of Munich, c - t) A FRENCH GRAMMAR.BASEDON PHLLOLOGfCAL PRINCIPLES. Second Edition. Extra fcap. 8. vo. 4*. 6d. ' J FIRST FRENCH EXERCISE BO QIC. Extra fcap., 8vo. ^ 4t. 6d. "SECOND FRENCH EXERCISE BOOK. Extra' fcap. ~8vo. 2S. dd. .... 1 1 i : ;. 1 CAI.liBKWOOJi—HANDBOOXOF MORAL PHILOSOPHY. By the Rev. Henry Calderwood,„LL.D.-, Prttfesscfr ot Moral Philosophy, University of Edinburgh. Fifth Edition. Crown 8vo. 6i,' : ' f .."-*. < DELAMOTTE— A BEGINNER'S DRAWING JBQQIC,,,, By P. H. Delamotte, F.S.A. , Progressively arranged. New Edition improved. Crown 8yp, 3*. 6d. ENGLISH WRITERS— Edited" by JopN RICHARD, 1 ' fjRpEN. Fcap. 8vo. Price is. 6d. each. MILTON. By the Rev. Stoeford Ai Brooke. Others to follow. .•"'.ii_ MISCELLANEOUS. 45 FAWCETT— TALES IN POLITICAL ECONOMY* "By , Millicent Garrett Fawcett.. Globe 8vp, y. . ... FEARON— SCHOOL INSPECTION. By D. R. : Fearon, M.A,., Ag?istam; Commissioner of. Endowed Schools. Third Editidri. Crown 8vo. 2s. 6d. r ' GLADSTONE— SPELLING - REFORM FROM AN ■ EDU- CATIONAL POINT OF VIEW. By J. H. Gladstone, Ph.D., F.R.S.,' Member for the School Board for 'London. New Edition, Crown %vo.-"is. 6d. • ""' GOLDSMITH— THE TRAVELLER, or a Prospect of Society ; and THE DESERTED VILLAGE. By Oliver Gold- smith. With Notes Philological and Explanatory, by J. W. Hales, M.A. Crown 8vo. 6d. , GREEN— READINGS FROM ENGLISH HISTORY. Se- : ' lecte'd and Edited by John Richard Green, M.A.i LL.D., Honorary Fellow of Jesus College, Oxford. Three Parts. Globe, gy.py is.6d. each. I. Hengist to Cressy. II. ^ressy to Cromwell. III. Cromwell to Balaklava. HALES— LONGER ENGLISH POEMS, with Notes, Philo- logical and Explanatory, and an Introduction on the Teaching of English. Chiefly for Use in Schools. ... Edited by J. W. Hales, M.A., Professor of English Literature at King's" : College, London, &c. &c. New Edition. Extra fcap. 8vo. 4r. 6d. HOLE— A GENEALOGICAL STEMMA OF THE KINGS OF ENGLAND AND FRANCE. By the Rev. C. Hole. On Sheet, is. JOHNSON'S LIVES OF THE POETS. The Six Chief Lives (Milton, Dryden, Swift, Addison, Pope, Gray), with Macaulay's "Life of Johnson.".: Edited with Preface by Matthew Arnold. Crown 8vo. 6s. LITERATURE PRIMERS— Edited by JOHN RICHARD GREEN, Author of " A Short History of the English People." ENGLISH GRAMMAR. By the Rev. R. Morris, LL.D., sometime President of the Philological Society. • iSmo. cloth, ix. 46 MACMILLAN'S EDUCATIONAL CATALOGUE. LITERATURE PRIMERS ConHttUtd— ENGLISH GRAMMAR EXERCISES. By R. Morris, LL.D., and H. C. Bowen, M.A. i8mo. is. THE CHILDREN'S TREASURY OF LYRICAL POETRY. Selected and arranged with Notes by Francis Turner Palgrave. In Two Parts. i8mo. is. each. ENGLISH LITERATURE. By the Rev. . Stopford Brooke, M.A. New Edition. i8mo. is. PHILOLOGY. By J. Peilb, M.A. l8mo. is. GREEJC LITERATURE. By Professor Jebb, M.A..i8mo. is. SHAKSPERE. By Professor Dowden. i8mo. is. HOMER. By the Right Hon. W. ; E. Gladstone, M.P. tSmo. IS. l ENGLISH COMPOSITION. By Professor Nichol. i8mo. is. ■ ' ?r In preparation : — ; LA TIN LITERA TURE. By Professor Seeley. ,,', HISTORY OF ,7 HE ENGLISH LANGUAGE., By J. A. H. Murray, LL.D. ■ j \j- . MACMILLAN'S COPY-BOOKS— j Published in two sizes, viz.': — i. Large Post 4to. Price 4^. each. 2. Post Oblong. Price 2d. each. *i. INITIATORY EXERCISES &> SHORT LETTERS. *2. WORDS CONSISTING OF SHORT LETTERS. *3. LONG LETTERS. With words containing Long Letters — Figures. MISCELLANEOUS. , •" ■'■''"■' •■•" "' , % WQRDS CONTAINING LONG LETTERS: '\. f~ PRACTISING AND REVISING COPY-BOOK. For Nos. I to 4. % CAPITALS AND SHORT .HALF-TEXT. : Words '' beginning with a Capital. *6. HALF-TEXT WORDS, beginning with a Capital— ' Figures. *7. SMALL-HAND AND HALF-TEXT. With Capitals and Figures. ][ *8. SMALL-HAND AND HALF-TEXT. With Capitals arid Figures. ■"■ 8a. PRACTISING AND REVISING C0PY-BOOK. For Nos. 5 to 8. " ' V3 .,.!.■■'• *9. SMALL- HAND SINGLE HEADLINES— Figares. io. SMALL-HANb SINGLE HEADLINES— Figures. *n. SMALL-HAND DOUBLE HEADLINES— Figures. ^ ' 12. COMMERCIAL ■ AND ARITHMETICAL EX- AMPLES, &c. 12a. PRACTISING AND REVISING COPY-BOOK'. For Nos. 8 to 12. * 7/fojtf numbers may be had with Goodman's Patent Sliding , i . Copies. . Large Post 4to. Price 6<£ each. By a simple device the copies, which areprinted upon separate slips, are. arranged, with a movable attachment,, by whjch they are adjusted so., as to be directly before the' eye of the pupil at 48 MACMILLAN'S EDUCATIONAL CATALOGUE. MACMILLAN'S COPY-BOOKS ConUniued-~ -, all points of his progress. It enables him, also, to keep his own faults concealed, with perfect models constantly m view for imitation. Every experienced, teacher knows the advantage of the slip copy, but its practical application has never before been successfully accomplished. This feature is secured ex- . , delusively to Macmillan's .Copy-books under Goodman's patent. An inspection of books written on the old plan, with copies at the head of the page, will show that the lines last written at itht-bottom are almost invariably the poorest. -The copy has been too far from the pupil's eye to be of any practical use, and a repetition and exaggeration of his errors have been the , , result. : / MACMILLAN'S PROGRESSIVE FRENCH COURSE— By G. Eugene-Fasnacht, Senior Master of Modern Languages, Harpur Foundation Modern School, Bedford. I. — First Year, containing Easy Lessons on the Regular Ac- cidence. Extra fcap. 8vo. is. II. — Second Year, containing Conversational Lessons on Systematic Accidence and Elementary Syntax. With Philo- logical Illustrations and Etymological Vocabulary, is. 6d. MACICILLAOTS PROGRESSIVE GERMAN COURSE— By G. Eugene Fasnacht. Part I. — First Year. Easy Lessons and Rules on the. Regular Accidence. Extra fcap. 8vo. ix. 6d. Part II.^-Second Year. Conversational Lessons in Sys- tematic Accidence and Elementary Syntax. , With Philological Illustrations and Etymological Vocabulary. Extra fcap. 8yo, 2S. MARTIN — THE POET'S HOUR: Poetry selected and arranged for Children. By Frances Martin. Third Edition. i8mo. 2s. 6d. SPRING-TIME WITH THE POETS: Poetry selected by Frances Martin. Second Edition. i8mo. 3s. 6d. MISCELLANEOUS. 49 MASSON (G%STAVE)— A COMPENDIOUS DICTIONARY OF 'THE FRENCH LANGUAGE (French-English and ''■; English-French)'. Followed by a . List of - the: Principal Di- verging Derivations, and preceded by Chronological'' and ' Historical Tables. By Gustavk Masson, ' Assistant-Master and Librarian,, Harrow School. Fourth Edition. Crown' 8vo. . half-bound, ,6s. MOLIERE— LE MALADE IMAGINAIRE. ^Edited, jwith Introduction and Notes, by Francis Tarver, M.A., Assistant- : Master at, Eton. ,- Fcap. 8vo. ■ ; ■,.,,...-. - \In preparalfpti.- MORRIS— Works by the Rev. R. Morris; '• LL.D., Lecturer on English Language and Literature in King's. College School, HISTORICAL OUTLINES OF ENGLISH ACCIDENCE, comprising Chapters on the History and Development of the Language, and on Word-formation. New Edition. Extra fcap. 8vo. 6s. ELEMENTARY LESSONS IN HISTORICAL ENGLISH GRAMMAR, containing Accidence and Word- ■ formation. New-Edition. iSmo. 2s. 6d. 1. PRIMER OF ENGLISH GRAMMAR. i8mo. is. NIGOlr-HISTORY OF THE FRENCH LANGUAGE, with especial reference to the French element in English. By HeNry Nicol, Member, of the Philological Society. [In preparation. OLlPHANT— THE OLD AND MIDDLE ENGLISH. A New Edition of " THE SOURCES OF STANDARD j ENGLISH" reyised and greatly enlarged. By T. Kington Oliphant. Extra fcap. 8vo. gs. PAI.aRA.VE— THE CHILDREN'S TREASURY OF LYRICAL POETRY. Selected and Arranged with Notes by Francis Turner Palgrave. i8mo. ss. 6d. Also in Two parts: i8mo. I s. each. ,i PLUTARCH— Being a Selection from the Lives' which Illustrate Shakespeare. North's Translation. Edited, with Intro- ductions, Notes, Index of Names, and Glossarial Index, by ■the' Rev. Wi W. Skeat, M.A. Crown 8vo. 6s. d 5P MACMILLAN'S EDUCATIONAL CATALOGUE. PYIODET— NEW GUIDE TO GERMAN CONVERSA. TION: containing an Alphabetical List of nearly gop Fajniliar Words followed by Exercises, Vocabulary of Words in frequent use j Familiar Phrases and Dialogues ; a Sketch of German Literature* Idiomatic Expressions, &c. By L. Pylodet. iSmo. cloth limp. 2J. 6d. ''■■ A SYNOPSIS OF GERMAN GRAMMAR. Frpm the above. iSmo. (id. READING BOOKS— Adapted to the English and Scotch Codes. Bound in Cloth. PRIMER. l8mo. (48 pp.) 2d. BOOK I. for Standard I. i8mo. (96 pp.) 4* ■■■■„■■ II. » II. l8mo. (144 pp.) & „ HI. ii In: i8mo. (160 pp.) 6tt; „ IV. ii IV. i8mo. (ljr6ptS.) ■Sa. » v. i> v. l8mo. (380pp.)' is. „ VI. ii VI. Crown 8vo. (430 VV-', Book VI. is fitted for higher Classes, and as an Introduction to English Literature. ...,.' -'.'.v. - V They are far above any others that have appeared both in form and ■-- ' substance. . . . The editor of the present series has rightly seen that reading books must 'aim, chiefly at giving to. the pupils flhenpower of accurate, and, if possible, apt and skilful expression ; at cultivating in them a/ good literary taste, and at arousing a desire of 'further reading. This is done by taking care to select the extracts from true English classics, going up in Standard VI. course to Chaucer, Hooker, and Bacon, as well as Wordsworth, Macaulay, and Froude. .^ . . This is quite on the right track, and indicates justly the ideal' whicli we ought to set before us. — Guardian. ','■■• , ; 1 . SHAEESPEA3XE— A SHAKESPEARE MANUAL: ByF.G. Fleay, M.A., late Head Master of Skipton Grammar School. Second:Edition. Extra fcap. 8vo. ' 4s. 6A AN ATTEMPT TO DETERMINE THE CHRONO- LOGICAL ORDER 6P SHAKE SPEARE'S PLAYS. By the Rev. H. Paine Stokes, B.A. Extra fcap. 8vo. 4*. 6d. ■• ' ""■ - 1 -- THE TEMPEST.. With Glossarial and Explanatory Nptes. By the Rev. J, M. Jephson. Second Edition. i8mo. 11. MISCELLANEOUS. .51 SONNENSCHEIN and MEIKLEJOHN — THE .ENGLISH METHOD OF TEACHING TO READ. By A. Son- nenschein and J. M. D, Meiklejohn, M.A. Fcap. 8vo. COMPRISING: THE NURSERY BOOK, containing all the Two-Letter Words in the Language, id, (Also in Large Type on Sheets for School Walls. 5j.) THE FIRST COURSE, consisting of Short Vowels with Single Consonants. . 6d. THE SECOND COURSE, with Combinations and Bridges, consisting of Short Vowels with Double Consonants. 6d. ' THE THIRD AND FOURTH COURSES, consisting of Long Vowels, and all the Double Vowels in the Language. 6d. 11 These are admirable books, because they are constructed on a prin- ciple, and that the simplest principle on which it is possible to learn to read English/ ttSpectator. , ; TANKER— FIRS7 PRINCIPLE'S OF A GRICUL TURE. By H. Tanner, F.C.S., Professor of Agricultural Science, University College, Aberystwith, &c. ,i8mo. is. TAYLOR— WORDS AND PLACES; or, Etymological Illus- trations.of History, Ethnology, and Geography. By the Rev. Isaac Taylor, M.A. Third and cheaper' Edition, revised '■' and compressed. With Maps. ; Globe 8vo. 6s. A HISTORY OF THE ALPHABET. By the same Author. \In preparation. TAYLOR— A' PRIMER OF PIANOFORTE PLA YING. By Franklin Taylor. Edited by George Grqve. i8mo. is. TEGETMEIER — HOUSEHOLD MANAGEMENT "AND COOKERY. With an Appendix of Recipes used by the '■" Teachers of the National School of Cookery. By W. B. Tegetmeier.. Compiled at the request] of the School Board for London. i8mo. is. MACMILLAN'S GLOBE LIBRARY. : ' BeautUully>{Mnted oh toned paper, price y. 6d. each. Also kept in various morocco and calf findings, at moderate prices. . The Saturday Review says,:—" The Globe Editions are admirable ■ipt- their scholarly editing, their typographical excellence, their • compendious fdrm; and their cheapness." The DaUy Telegraph calls it " a series yet unrivalled for its com- ' binationpf excellence and cheapness." SHAKESPEARE S COMPLETE WORKS. Edited by W. G. Clark, M.A.,, and W. Aldis Wright, M.A. With Glossary. MORTE. If ARTHUR. Sir Thomas Malory's Book pi King Arthur and of his Noble Knights of the Round Table. The Edition of Caxton, revised for Modern Use. With an Intro- ' duction, Notes, and Glossary, by Sir Edward Strachey. •■ BURNS' S COMPLETE WORKS: the Poems, Songs, and • iietters. Edited, with Glossarial Index and Biographical , j Memoir,, by Alexander Smith. ROBINSON, CRUSQE. Edited after the Original Editions, with Biographical Introduction, by Henry Kingsley. ' SCOTT'S POETICAL WORKS. With Biographical and Critical : ■ r. Essay, by Francis Turner Palgrave. GOLDSMITH'S MISCELLANEOUS WORKS. With Bio- graphical Introduction by Professor Masson. SPENCER'S COMPLETE WORKS. Edited, with Glossary, by R. Morris, and Memoir by J. W. Hales. POPE'S POETICAL WORKS. Edited, with Notes and Intro- ductory Memoir, by Professor Ward. DRYDEN'S POETICAL WORKS. Edited, with a Revised Text and Notes, by W. D. Christie, M.A., Trinity College, Cambridge. COWPER'S POETICAL WORKS. Edited, with Notes and Biographical Introduction, by W. Benham. VIRGIL'S WORKS. Rendered into English Prose. With Intro- ductions, Notes, Analysis, and Index, by J. Lonsdalej M.A., and S. Lee, M.A. HORACE. Rendered into English Prose. With running Analysis, Introduction, and Notes, by J. Lonsdale, M.A., and S. Lee, M.A. MILTON'S POETICAL WORKS. Edited, with Introductions, &c, by Professor Masson. Published every Thursday, price 6d. ;'• Monthly parts, 2S. and 2s. 6d., Half-Ttearty Volumes, 15s. NATURE: AN ILLUSTRATED JOURNAL OF SCIENCE. Nature expounds in a popular and yet authentic manner, the Grand Results of Scientific Research, discussing the most recent scientific discoveries, and pointing out the bearing of Science upon civilisation and progress, and its claims to a more general recog- nition, as well as to a higher place in the educational system of the country. It contains original articles on all subjects within the domain of Science ; Reviews setting forth the nature and value of recent Scientific Works ; Correspondence Columns, forming a medium of Scientific discussion and of intercommunication among the most distinguished men of Science ; Serial Columns, giving the gist of the most important papers appearing in Scientific Journals, both Home and Foreign; Transactions of the prin- cipal Scientific Societies and Academies of the World, Notes, &c. In Schools where Science is included in the regular course of studies, this paper will be most acceptable, as it tells what is doing in Science all over the world, is popular without lowering the standard of Science, and by it a vast amount of information is brought within a small compass, and students are directed to the best sources for what they need. The various questions connected with Science teaching in schools are also fully discussed, and the best methods of teaching are indicated.