IS '^^i^^B • > '■ ■ "^ • 4 ki P UC-NRLF $B SEfi flMb 11 a: »i Wrl r- % lA m-^« \ .2^ ? --w •^*,v i,>^ ,^v »^ ■» •. v* K^ .!# 'J m :• • r^^" -&i^^ y^ BERKELEY LIBRARY V^ UNJVERSITY OF til 9^4^ ^ TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS. BY THE Rev. T. G. HALL, M.A. PROFESSOK OF MATHEMATICS AT KING'S COLLEGE, LONDON; AND T.ATE FELLO'iT AND TUTOR OF MAGDALENE COLLEGE, CAMBRIDGE. SECOND EDITION, ALTERED AND ENLARGED. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS. LONDON: JOHN WILLIAM PARKER, WEST STRAND. M.DCCC.XXXVII. ip/\3o3 HA IS31 PREFACE. A KNOWLEDGE of the principles of this bi'anch of the Pure Mathematics is absolutely necessary, before any one can success- fully undertake the perusal of works on Natural Philosophy, in which the effects of the observed laws that govern the material world are reduced to calculation. For Students deficient in this knowledge, yet anxious to obtain as much information as may enable them to master the chief ana- lytical difficulties incident to the study of Elementary Treatises on the Mixed Mathematics, this book has been written: with the hope, too, that by its means a subject of high interest may be rendered accessible to an increased number of I'eaders. The ample Table of Contents which accompanies this woi-k will sufficiently exhibit its plan — and a very hasty glance will at once shew that its chief object is to treat of Functions of one Vari- able; at the same time the Theory of Functions of two Variables, and its application to questions of Maxima and Minima, is fully explained. But the Chapters on the Integral Calculus contain rules for the Integration of Explicit Functions only. A few words may be here added in order to explain the prin- ciples adopted in laying down the definitions. By a method, similar to that of M. Poisson I have shewn that ?/j -fix + }i) can always be put under the form u + Ah + Uh~, whence we obtain the equation ii,-u = Ah+Uh\ The term Ah is defined to be the differential of u, and A, or the coefficient of h, is called the differential coefficient. And from these definitions, the Rules for Differentiation have been in general derived. 290 IV PREFACE. But as the algebraical labour of finding A may sometimes be greatly diminished, if, after dividing both sides of the equation Ui — u = Ah+ Uh' by h, we make h — 0, this method is in a few in- stances made use of. The symbol -j- for the differential coefficient of u =f(x), in- CI jc vented by Leibnitz, and used almost without exception by the continental writers, is here retained — I mention the fact, since the notation d^u for the same term has lately been revived by some Cambridge Mathematicians. — I do not pretend to decide the ques- tion which of the two -7- or d^u estimated by its power of best representing the differential coefficient ought to be preferred, but I see that the latter is, to say the least, an imperfect notation, and is liable to the important objection that the suffix x,' in the cal- culus of finite differences has a meaning entirely diffei'ent from that indicated by the a: in d^. But the most important objection is that already alluded to, that when the proposed notation has been learned in our own elementary works, the eye must become familiarized with that of Leibnitz, before the works of Lacroix and Laplace can be read with advantage. Lastly, if it be considered necessary to offer an inducement to any one to enter upon the study of a science — which is the result of one of Newton's most brilliant discoveries, let him know " that it is a high privilege, not a duty, to study this language of pure unmixed truth. The laws by which God has thought good to govern the universe are surely subjects of lofty contem- plation, and the study of that symbolical language by which alone these laws can be fully decyphered, is well deserving of his noblest efforts*." * Professor Sedgwick on the Studies of the University. PREFACE TO THE SECOND EDITION. To this edition;, many examples have been added, and some alterations made in the early chapters, intended to facilitate the labour of the reader. The Integral Calculus is terminated by a chapter upon the Solution of Differential Equations of Two Varia- bles. The theory of these equations is of necessity briefly explained. Upon the whole^, there will be found nearly sixty additional pages, which it is hoped, will increase the usefulness of the work. King's College, London. TABLE OF CONTENTS. DIFFERENTIAL CALCULUS. Chapter I. DEFINITIONS.— PRINCIPLES, 1—4. I'AGE To prove that ii^ ^/(^ + h) ^u + Ah+ Uk^ 4> Definitions of Differential and Differential Coefficient 7 Rules for finding the Differential Coefficient, when u is an Algebraical Function of .r 9 Examples of Differentiation 15 Chapter II. Tan^ , sin A 1, •<. p j. « ,« — ; — and —J — , are each unity, it n = I9 k n Differentiation of Circular Functions 20 If«=/(x), P=^ 24 •^ ^ '' dx ax du Differentiation of Exponential and Logarithmic Functions 28 ^„ „, . , , . du du dz If «=/(.). and. = ,^(x), ^=^.^ 30 Examples 33 Chapter III. Successive Differentiation 38 Maclaurin's Theorem SQ Examples 40 Series for the length of a Circle, and for the Computation of Logarithms 43 * Series dependent on the Circle 52 * Multinomial Theorem 56 Examples 57 Vlil CONTENTS. Chapter IV. rACE dill (III, __ If«,=/(^ + /0: ji'ji -'9 Taylor's Theorem — Examples 60 Approximation to the Root of an Equation 64 d" u * Value of J— ^ investigated 68 * Chapter V. Failure of Taylor's Theorem — Explanation of the Term 72 If /(a + h) contain Positive Powers of h as far as the w**", and afterwards Fractional Powers, the first n Coefficients may be found by Taylor's Theorem 75 Limits of Taylor's Theorem 78 Separation of Symbols of Operation from those of Quantity ... 83 Chapter VI. Vanishing Fraction — defined 85 Their values may be Finite, Infinite, or Nothing 86 Examples 89 Chapter VII. Maxima and Minima defined 9* If the Coefficients A^, A^, A^, &c. be all finite, h may be so assumed that AJi shall be > A,h- + A^h^ + &c 95 • ' du ^ If u =f(x) be a maximum or mmimum, j- = 9o Rule for determining the Maximum or Minimum 97 Examples 99 , , du , , „ *Case of Maximum or Mmimum when -y- = oo iicJ Chapter VIII. Equations to Curves — Definitions of Terms 115 Equations to a Point 1 16 The Straight Line 118 The Circle 122 CONTENTS. IX PAGE Transformation of Co-ordinates 123 Parabola— Ellipse— Hyperbola 127 The General Equation of the Second Degree 133 Cissoid — Conchoid — Witch — Logarithmic Curve— Quadratrix — Cycloid — Trochoid — Spirals — Involute of Circle 137-14-4 Chapter IX. Tangents to Curves — Equation to Tangent 145 Equation to Normal 1 46 Values of Tangent, Sub-tangent, Normal, and Sub-normal 147 Asymptotes 151 Examples 153 Chapter X. The Differentials of the Areas and Lengths of Curves l64 If A be the area of a Curve, -j- = y l65 66 If* be the length, ^ = ^1 +^, 1 dV If V be the Volume of a Solid of Revolution, -y- = 'Try" l67 If S be the surface of the same Solid, . S-^.'/xA^: '«« Value of Perpendicular on the Tangent in Spirals 170 Differential Coefficients of* and A in Spirals 172 Tangents and Asymptotes to Spirals 174 Examples 175 Chapter XI. Curvature and Osculating Curves 178 Order of Contact 179 Osculating Circle — Radius of Curvature 180 Evolute and its properties 183 Radius of Curvature, and Evolutes of Spirals 188 Examples 192 X eoXTKNTS. Chaptek XII. PAGE Singular points in Curves 201 A Curve is Concave or Convex, according as y and y4 '^^.ve different or the same Signs 202 Points of Contrary Flexure 203 Multiple Points 209 Conjugate Points 215 Cusps 217 To trace a Curve from its Equation 220 * Chapter XIII. Change of the Independent Variable 229 * Chapter XIV. Functions of two or more Variables 240 Expansion of f{x + h, y + k) 241 Differentiation of Functions of two Variables 245 Implicit Functions 250 Elimination by means of Differentiation 255 * Chapter XV. Maxima and Minima of Functions of two Variables 26l Examples 263 Lagrange's Theorem 267 Equation to a Curve touching given Curves 273 INTEGRAL CALCULUS. Chapter I. PACE Definition of Integration 278 Integration of Monomials 279 Examples of Simple Integration 281 Chapter II. Rational Fractions 285 Roots of the Denominator, (1) all different, (2) some equal, (3) some impossible 286 Integration by parts 293 * Integration of — — ~ , and -^- — — 299 ^ x" ±1 j:-" - 2a;' . cos a + 1 Examples 309 Chapter III. Irrational Quantities 311 Integration of known Differential Coefficients 312 Binomial Differential Coefficients 321 Examples o'i'6 Chapter IV. Integration of Logarithmic and Exponential Functions oo"^ Examples 344- Chapter V. Integration of Circular Functions , 345 Examples o5^ Xll COXTEXTS. Chapter VI. P-VGE Application of the Integral Calculus, to determine the Areas and Lengths of Plane Curves, and tlie Volumes and Sur- faces of Solids of Revolution 358 * Chapteji VII. Differential Equations, order and degree of 3Q3 Equations of the first order and first degree 394 Equations of the first order and of the n^^ degree 411 Clairaut's Formula 415 Equations of the second and higher orders 417 Lagrange's Variation of Parameters 427 Note. It is recommended that the Articles and Chapters marked *, be omitted at the first perusal. THE DIFFERENTIAL CALCULUS. CHAPTER I. 1. One quantity u is said to be a function of another ,r when the value of the magnitude of u depends upon the variation of x. Thus the area of a triangle is a function of the base, when the altitude remains unaltered, since the area will increase or decrease with the increase or decrease of the base. And if u = aaP ^hx^ where a and h are constant quan- tities, and X a variable one, w is said to be a function of , M[ 2 3 4^ -^ j _ , «/ lL • « « and u = sm x = x 1 &c. to infinity, 2.3 2.3.4.5 are transcendental functions of x. 4. The equation u-f(x) expresses the relation between the function u and the single variable a?, and the values of u solely depend upon the change that may take place in x : but if we have an equation between three unknown quantities, such as u = ax~y — bx'if, where x and y are independent of each other, i. e. not con- nected together by any other equation ; then the value of u depends upon the change, both of x and y, and u is said to be a function of two variables ; this is expressed by writing As an instance, we may again take the area of a triangle the magnitude of which depends upon the rectangle of the base and the altitude, which lines are totally independent of each other. It is obvious that there may be functions of three, four, or of n variables. • M = l +,r + .i'^ + .r^+&c. to infinity is an algebraical function oi x, since the sum of the series is expressed by -j , < PRINCIPLES. 3 5. Let us however return to functions of one variable, and let u = f {oe) express the general relation between the function and its independent variable oc. Let X increase and become x + A, then the value of u will most probably be altered. Let the new value be repre- sented by Wi, then Ui=f(x + h), and u =f(x), by hypothesis. •■• ih-u=f(x + h)-f(x). Now Ui — u, or the difference between the functions of X + h and x, must depend upon h, and we shall first shew that it may be expressed by a series of the form Ah + Bh'+Ch^+kc. or that Ui=u + Ah + Bh~+ Ch^+ kc. where the powers of h ascend : the primary object of the Differential Calculus is to find the value of the coefficients A, B, C, kc. 6. We will first shew that u^ may be expressed by a series of the above form by a few particular examples. (1) Let u = x^; .'. Ill = C^ + hy = x^ + Sx^h + 5xh^ + h* = u + Sx^h + 3xh^ + ¥, which is of the required form. (2) Next, let 71 = x"; (n — l) .-. u, = (x + hY=x''+nx"-'h + 7i^ ^v''-^^ + kc. by the Binomial Theorem. Or, putting m for x", n - 1 u, = 7/, + nx^'^h + n x'"~"h^ + &c. 2 a series with ascending powers of h. A 2 * PRIKCIPLES. (3) Let u = Aw"" +Ba/''+ CxPw + kc; .-. Wi = J {x + /*)'" + B{x+ hy +C{x + hy + &c. (tn — l) = J (a?'"+ ma;"'-^h + m w'^-'^M + &c.) + 5(.r''+wa?"-'A + w-^^?^^^.r''-2A2^ &c.) 2 + C (j;-^ ^pxP-^h+p ^-^^ .r^-2/i2 + &c.) 2 + &c. = Ja?" + ^.r" + CccP-ir &c. + (m Jo?'"-' + tiBx"-' + &c.) A , (w - 1) C7^ — O 2 2 ^ + &c. + &c. = u + ph + qh^ + &c. by writing u for its value, Aw"' + Bx" + &c., and putting p, 7, &c. for the coefficients of h, h^, &c. (4) It may also be shewn that «''+*, log (x+h), sin {x+h), can be expanded into series of the form u + Ah + Bh^+Ch\ Sec. but we proceed to demonstrate the following general Propo- sition. 7. Prop. If u =f{x), and u^ be the value of u when x becomes x + h, then Ut^ = u + Ah -\- Uh^, where u is the original function, and Uh^ represents all the terms that follow Ah. (1) Wj or f(x + h) can contain only such powers of h, as have positive indices. For if u,= M+ Ah"+ Bh-^ + &c. = J/ + JA" + -^ + &c. h^^ when A = 0, Wj instead of becoming = w, would be infinite. I'RINCIPLKS. 5 (2) The first term of the expansion must = u. For let w, or f{x + h) ^ M + Ah'' + &c. then let A = 0; .-. f{,v) = u = M, i. e. M = u ; or Uj = u + Ah"- + Sec. Let therefore 71 ^ or f{x + h) = u + Ah" + Bh^ + &c. where a is the least of all the indices of h, and /3 the next in magnitude, and Al, B, &c. are functions of x. Now whether cV becomes x + h, or h becomes 2h, u^ will become /(,?? + 2 A), and the expansions upon either suppositions will be identical. (1) Let // become 2h or h +h, and let u^ be the value of ? + ^,A^> + &c. {B) = B + Bih"^ + B,hP^ + &c. 6 PRINCIPLES. then multiplying (A) by h" and (B) by A^, and substituting we have 7/2 = w + ^A" + -S/i^ + &c. H- J/t" + ^1^"+"' + &C. + Bhf^ + &c. = u + 2Jh" + JiA"+«' +2Bh^ + &c...(2). Equating the coefficients of the same powers of h in series (1) and (2), 2j = 2°J, i. e. 2 = 2°; .-.0 = 1, and Ml =f{x + h) = ii + Ah + fi/i'^ + &c. whence it appears that the second term of the expafision of f{x + h) contains the first power of h only. From this it follows that aj = 1 for A^h"' is the second term of the expression for A when x becomes a? + A ; and therefore W2 = M + 2 Ah + A^h^ + 25/i^ + &c. from (2) = u + 2Ah +2ft.Bhl^ + kc (1) Now, since in series (2) a term is found involving h~, some corresponding term must be found in series (l); and as /3 is less than any index that follows it, j3 must = 2. And there- fore, w, =f(.v + h) = u + Ah + Bh- + Chy + &c. = u + Ah+{B+ Chy-^ + kc.)h' = ti + Ah + Uh^. 8. The second term of the expansion, or Ah^ is called the differential of u : differential being the diminutive of difference. For Ah is the first term of the difference between ?) _ 1 i_idp ^ dx dx ^ dx 2\/p whence this rule. To find the differential coefficient of the square root of any quantity, divide the differential coeffi- cient of the quantity under the square root by twice the square root of the quantity itself. Ex. Let u = v a + bx + ex'- ; o dp .-. p = a + bx + cx^ ; .-. — = b + 2cx; dx , du b + 2cx and — = d^ 2 \/a + bx + ex- 24. u = — . Here -— may be deduced from the general p"" dx ^ ^ form w = I?" ; but as its form ought to be remembered, we shall deduce it separately. EXAMPLES. If M = — ; .-. u.p''^ 1. jjn du , dp p". -— + nu.p'' •-r- = dx dx But up"-' =-; P 1 du n dp • «" , _ - . dx p dx . du n dp and -— = — -—- . dx p" + ' dx dA — Cor. Up \x" I n "'"'■• dx x"^'' 15 Examples of the differentiation of algebraical functions. du _ l-i 4 (1) u = x^; .-. — - = 3.|.c^' =7x\ ^ ^ dx ^ (2) u = x^ + x^ + X + 1, du . — = 3x^ + 2x + 1. dx (3) u = (x + a) . {x + b), du ^ ^d(x + b) . ,^ d(x + a) ^ ^ ^ — = (^ + a) -^ ^ + {x + h) . ^ ^ . Art. (19) dx dx dx = X + a + X + b = 2x + (a + b). (4) U = X(l + X^) (1 + ,37^), du . ,^. .^dx d{l+x^) d{\+x^) dx dx dx dx = (l + x") (l + a?^) + a? (l + a?^) . 207 + a? (] + .r") Sa?" = 1 + 3./ + 4.r^ + 6a?\ 16 EXAMPLES. (5) u = ^=a^-% du , n da? w"+^ .„ 1 du 1 II u = - , — = X dx x^ X + a (6) u = -; ^ X + d{x + a) , d(x + b) W„ ^"^ + *) • —^i - ^^ + ""^--^ du dx dx dx (c^ 4- b)'- X + b — {x + a) b — a {x + bf {x + bf (7) u = a?" (^ + 1)" dx ~ {x + ly" (x + I) . mx'"'^ — mx"' mx"'~^ X + 1 1'"+^ .r + l] m + \ (8) u = a/i +x^=1 + a?^]S -— = 1.(1 -t-.i?0"-'-2.l' = (9) w = Vx + -s/l + a?-, , d(x -h\/l + x^) 'va? a^ + 'v/l+d?^ and — ^ — = 1 + ■ / = j^^^.^- . dx c V I + x^ V 1 + cT^ EXAMPLES. dll J V ,?• + -x/l +.1?- (10) u = (2n.v + ,vy (11) u = du d(2n.v + .r~) —- = m(2a.v + .v-)'"-^ . — ^ dx d.v = 2m . (2a,v + .r-)'"-' . (a + ,r). V ff" — .t"'^ d V «" + *'" .3^ , d \/ a' — tr"" since = , and X d.v a/o-^ + ,T'-' ' d.v y/a'-x'^ / iV / ;^ X du V r/- + .r- V " - •^' {a/ - ar)v + (cr + x-).v ~ [(a^ + 'V^)^ {a^ - x^')i 2a-.v (12) n = a- + X y{a - X )a V'l + *• + \/i - .V _ {\/i + .!■ + V 1 - •^^)' x/l + X -\/l - sin h < tan A, or k> nh + Nh^ a + Nh"-'^ < -7 + . + &c. -2 b2 20 DIFFERENTIATIOX OF It is proved, see Trigonometry^ Art. 53, that h > sin h < tan /^, or A lies between sin h and tan //. If therefore sin h , sin A , tan h = 1 ; .-. also = 1, and - =1. tan h h h sin h cos h \ , , Now = = - = 1, when It = 0, tan h 1 1 sin// , tan A , .• i i .-. — and are also respectively equal to unity- h h 2(5. Let 11 = sin w. For X, put x + h-^ therefore u becomes u + -— A + Uhr. ax d 11' and u + --h + UJv = sin (r + /?), ax and ^* = sin x ; — A + Uh~ = sin (.r? + /i) - sin x dx h\ . /* = 2 cos ( cf + - . sin - a + Nli"-'^ But .. ,„ = I, when /; = ; a + —zi^ + &c. .•. j ^also=l, when« = 0; a . .-. y = 1, 1. e. a = 1; .•. sin h = h + Nh", and . = 1 + Nh'-^ = 1, when A = 0, ^'' « , tan h , ^i" „ . , . « and tan A = ft + --^+ &c. and — r — = 1 + •:7+ &c. = 1, when n = 0. CIRCULAR FUNCTIONS. 21 . A-B A + B Since sin ^ — sin 5 = 2 sin . cos sin - du ( /t^ h U a = cos ' + -'^ ^^ ^ 6^ + &c. ^ ^ 2 2.3 (x^ x\ (x^ x^ x\ ,.^ = \ +xh + { ]f>' + {- + -]h' + hc. V2 2/ \6 2 3/ ,' If h' \ (f)' b' \ , = \ \\h- 4- &c. .J' + + &c. .r-'+ &c. V 2 3 ; V2 2 / = 1 + ^tf + B.v' + &c. EXPONEXTIAI, FUNCTIOXS. 29 Ir h"' where A = b 1 &c. 2 3 and B is also a function of (^ — 1). This being proved, dx = «"■.{! + Ah + BIr + &c.} by the Lemma ; .-. equating the coefficients of //, du — = Aa'. d.v CoR. Let e be that value of a whicli makes ^=1, or (^ - 1) - i (e - 1)^ + ^ie- 1) - kc. = 1 ; dcV e is found to be = 2.71828, &c. and is the base of what is called the hyperbolic system of logarithms. dtV 37. Next let u = log w; .-. x = a"; .-. — =A a'' = A , .t ; du dw, _ 1 _ 1 1 d.v dw A X dn dii 1 diV If the base be (e), A = \ and — = - , or d . log w = — . dx X X diif 38. We next proceed to find the value of — when dx u = f{z). where ^r is a function of x, so that z = (p{x). In fact, the labour of differentiation is often much lessened bv the substitution of z for some function of x, when we want to find the differential coefficient of a com- plicated function. 30 DIFFERENTIATION OF As in Art. 34, let hi, ^^, ^a% be the corresponding differentials of it, z, x. A A C Then since 15 = 7; x i; • Jo c -O ^U ^u ^z ^.v ^z Sec Su du Sti du Sz d% But -^ = -J- , ^- = 7- r and ^ = — . hx doc dx dz dcv dx du du dz dx dz dx Or if m and h be the increments of % and x, the co- efficient of h in f{x + h) = the product of the coefficients of m and h in the expansions of /(^4-m) and (^(x + h). But we give another proof of this important proposition. Let u hecome f{z) where z=(p{v), and let z + m be the value of ~ when x becomes x + h. Then since 71 is a function of x as well as of z, the value of u will become du „ n + -—// + Uh-, dx and / iz) becomes u + — — m + Zm ; dz du du „ „ .-. -r-h+ Uh^ = -—m + Zm" (l). dx dz dz But z + m = (h(x + h) = z + -— h + Z^h^; ^ dx dz , „ ,, .-. m = — h + Zih^, dx where Z,^" represents the terms after — . h, dx EXPONENTIAL FUNCTION'S. 31 substituting this value of m, in equation (l), -r-h + Uh^ = -—.-— h + ■-—. ZJf + Z {-— h + ZJr \ , da,' dz dx dx \da? J equating the coefficients of h. du du dz d,v dz d,v Ex. Let 11 = 9.az -{- bz-, where z = \/ 1 + .^7^, du dz X = 2a -f 2hz, and dz dx \/i + a?^ du 2(a + bz).x 2{a + b\/l + x^)x 39. Next let u = sin z, where ^ is a function of x. du du dz 1 hen — = — . — , dx dz dx du du dz But — - = cos;^; .•. -— = cos«.— — dz dx dx du 40. Let u = cos z ; find — . dx XT ^^ Here — = - sm ^ ; dz du du dz . dz — = — . — = — sin ;^ . — dx dz dx dx 4L Let u = tan z. du ^ .. s du . ^ s^^ — = (l + tan-^) ; .-.-— = (i + tan''^?) — - dz ^ ' dx ^ dx 32 Similarly, EXAMPLES. jf du dz n u = sec . ^, = sec ;jr . tan ^ . — , dx da,' du . dz u = v . sin z, — - = sin z . — , d.v da- ' du , , ^ dz w = cot . ^, = _ (1 + cot-^) . , dv dx du dz u = cosec .z, — - = - cosec ^ . cot ^ . — d.x dx ._, A . .p , du 42. Again ir u = a\ find — . dx du Since — = A «', du du dz . dx \ sina7j (is) 7« = e^ = e~, if i^ = e'" ; du dz .'. — z= e~ .-— = e~ .e^ = e e^. dv dx (14) 11 = is^'\ where .?, ??, and y are functions of ,t?. Let v-'- = u^ ; .•. w = ^''', d7< f dvj V, d%\ and — = .^' ' •( n. 1. ;j: . -^ ~ •~r( • dv \ dx z dx] ^ dv, f, , dy y dv\ But ••• V, = v^' ; .'. — '- = vy . h. 1. 1' X -^ + - . — > ; dx \ dx V dx) du „ f , , /, , dy y dv\ v'^ dz] dx [ \ dx V dx} z dx] {dv V dv I dz] dx V dx z dx) = Z . V--' (l5) 71 = h. 1. tan x ; die 1 + tan^.T sec'^ du U/U/ -13 -— =4.a? + a ; .-. (/i = 4a' dx ' -— 2 = 4.3a7 + a| ; .'. 6'2 = 3.4a, dx rfu dx' = 2.3.4 (.r + r/); .-. fl^ = '2 . 3 . ifl^ EXPANSION OF FUNCTIONS. 41 —- = 2.3.4; .-. fj, = 2.3.4, dw^ d^u —- = 0; .-. f7, = 0, and f7„, U., &c. all = 0; dx^ 3.4,, 2.3.4 , 2.3.4 ^ ^"^ 1.2 1.2.3 2.3.4 = a' + 4 a''' a? + Gfr.i?" + 4ac??^ + w*. (2) Expand (a + 6.r + cx^y, u= (a + bx + cx^y ; .'. Uo = a", — = n.{a + hx + cxy-'{h + 'icx) ; .-. L\ = w6a"-\ da? ^ = n{n-\){a + hx+cx~y'^ {h-^2cxf+2cn . {a + hx+cx^-]', dx" .-. U2 = n.(n- l)a"-~b~ + 7i . 2c . a" -1 da?' = w . (w - 1) . (w - 2)(a + fea? + c.^'^)"~^(6 + 2cxy + 2n.{n - l)a + bx + ca?^]"""^2c. 6 + 2ca?] + 2c. w(w -!).(« + 6c'P + ca?^)"~^(6 + 2cx) ; .-. f/g = ^i(w - 1) (w - 2) . a"-^6^ + 2 . 3?i. (w - l)a"-^6c, &c. = &c. (n — 1^ + /« ■ (" - IJ (^^ - a) ^..3 J3 ^ „ , („ _ 1) . „.-4,j a?^ + &C. (3) Expand sin x and cos a? in terms of the arc 1, . A - 45"= tan-^ 239 ,1 ,1 .-. 45"= 4tan-' -tan"' — 5 239 = 4^ .- + -.--&C.} [5 3 5"^ 5 5' j -I 11111 1 . + - • — &C./ 239 3 (239)'' 5 (239)' j 47- The logarithm of x cannot be found by Maclaurin's Theorem, since if a^ = 0, Uq, Ui, f/o, &c. become infinite: but 71 = log (1 + x) may be easily found. Suppose the logarithms to be hyperbolic, i. e. let -4 = 1. «^ = h. 1. (1 + x) ; .-. f7o= 1^- 1- (1) = 0, = = 1 — X + X~ — X + X — X + &c. dx 1 + X by division. But from the theorem, du ^^ ^^ U^x- U^x^ U^x^ -—= Ui+ Uox + + - — + — -— + &c. dx 2 2.3 2.3.4 COMPUTATION OF LOGARITHMS. 47 therefore equating coefficients, C7,= l, f/o=-l, f/3=2, U,= -2.3, U,= 2.3.4>; .v~ 2a?^ 2.3x' 2.3.4>x^ .-. 7* = log; (1 + cv) = x - — + 1 \- &c. ^^ '' 2 2.3 2.3.4 2.3.4.5 or of' x^ af = cc ^ + &c. 2 3 4 5 CoR. Had a been the base of the system, then — = — . , where A = (a - \) - \a -lY + \a - \\~ &c. dx A x+1 V / ^ \ d 1 x^ x^ and locr (i -\- x) = — (x 1 &c.) ^ ^ A 2 3 = -hyp. log. (1 +x). A Hence, if we know the hyp. log. we may find the log to a base a by mviltiplying the hyp. log. by — . The factor — is called the modulus. 48. The series for log (l + x) does not converge, and is useless in actual computation ; but from it a number of series may be derived which are rapidly convergent. Let - X be written for x in the series for log (l + x) ; o Q 4. "j X~ X" X X .-. log; (l - x) = -X &c. ^^ -' 2 3 4 5 subtract this from log (l + x)^ then since log re- log b = log - /I + x\ . X^ X' log = 2 \x + — + — + &c.|, '^ \1 -xl ' 3 5 ' 1 + X M M - N for put —: ; .-. x = — — , 1 - .^ ^ N M + N ^ , M \M-N , /M-Ay ^ 1 and log — = 2 { — + i — — - + &c. > . a 48 SERIES FOR THE Suppose M=^N+z; .-. M + N=2N+z; ''^ ^ * \2N+z ^(2N+zy ' {2N+zy j and finally, if z = 1, W(iV+l) = logiV+2|— +i7-T7 — 77-3 + &C.I, a formula from which logarithms may be calculated. Thus, since log 1 = 0, fl 1 1 1 1 1 log 2= 2^- + -.- + -.- + &c.>, ^ [3 3 3^ 5 3^ j fl 1 1 1 1 1 log 3 = log 2 + 2 <~ + - . - + - . - + &c. > , fl 1 1 1 1 1 log4 = log5 + 2 <- + -.- + -.- + &c.>, &c. &c. 49. Expand a^ in ascending powers of x. u = a" - &c. = |^ = 1, and c* = 1 + ,r + h -\ + &c. 1.2 2.3 2.3.4 CoR. 2. To compute A. i 1 Since e = a^ •■, .'. e+^ = «, and e~^ = --^ a .-. — A = log I - ) in the Napierian system. 004 O) W OB Now I02: ( 1 + -^i?) = .r" ! h &c. ^ 2 3 4 for 1 + w put n ; .-. \ogn = {n - I) - ^{n - Vf + \n - l]-' - &c. ; D 50 EXPOXENTIAL FUNCTIONS. Let a = 10. A = (.9) + 4 (.9)^ + 1 (.9)^ + &c. = 2.302585 and — = .43429448. A This is the number by which we must multiply the Napierian logarithms to obtain those calculated to a base 10, or Brigg'^s logarithms. 50. In the expansion for e"^' or e^' = 1 + X + h h + + &c. ; 1.2 2.3 2.3.4 2.3.4.5 put successively for x, a?v — 1, and —xy/ — !; x^ . . e^''-' = 1 + x"^ + + — ~ &c. 2 2.3 2.3.4 2.3.4.5 I — / — x^ x^^/ -1 x^ x^\/ -I 2 2.3 2.3.4 2.3.4.5 by addition and then by subtraction. gW-i^g-rV3r^2 ji -- +~^ &c.^ =2cos^. *- 2 2.3.4 ' gxVri x^ x^ -e-^^-i = 2 V -1 ]x + hc.\ 2.3 2.3.4.5 = 2\/-l sin.r. Again adding and dividing by 2, gxv-i _ cos.r + v-l . sin.i'. Also by subtraction and dividing by 2, g-x\/-\ _ cos, J? - v— 1 sin a;. T-OGARITTIMTC SERIES. 51 Cor. 1. Hence cos cos 0.' 1 tt' >i TT 3V^ sin X X x' 1 1 X- 1 ti'- 2^,r^ S^^tt"-^ 2x 2+i^+^ + ^' ON THE CIRCLE. 55 1 ^ + + 12_02 ^2_Q2 ^2_Q2 1 TT 20"' 20tan7r0 * • , *^ ^o ^ y J 2cr 2^\/-i Again, let — = -0-; .-. a? = tt^V - l, and — = -21' V - 1 TT , + 2Tre , COS A' 1+6-==^^-^ > l+e" / and -; — = „ ,-^ V — 1 = r:n v - I ; 20-\/-ir 1 1 1 TT Oy/ - 1 I -e 1 + e^ire 1 ^ + r>2 ^2 + &c.= TT 62^^^+ 1 2 * gS-TrO _ J 20'" Again, cos . = (^1 - ^] (. - ^, j (^1 - — j &c. h. 1. cos 7r~ TT sin X 1 1 - g-'^'^V"^ 1 J _ gTrO g'n-e _ 1 cos 1 1 „ TT e^^-1 + ., ... + — — ^, + &c. = l2+f^2 3.^^. g.^^i 40 e-^^+l Other similar series may be readily deduced. 55. From the expression ^ x' e" = 1 + .2? + 1 h &c. 1.2 2.3 Lagrange in the Calcul. des Fonctions has derived an expres- sion for the general term of the polynomial (a+6+c+c?+&c.)'". Thus for X put {a-\-h + c + d + &c.) w ; _.^ g(a +6 + r + ,? + &c.)a- ^ 1 + ^,^ ^ ft + c + &C.) X 1 .2 {a + h + c + d+ hc.y . x 1 O Q -»j9 + &c. m - + &C. But e'*'^''"*'' + ^'-''' = e""" X e'^ x e*-'* + &c. a^x^ a^x^ . = (1 + a A' + 1- + &c.) ^ 1.2 2.3 ^ , b-x' b\v' ^ ^ X (1 + bx + + + &c.) ^ 1 .2 2.3 C^X^ (?X^ X (1 + ex + + + &c.) ^ 1.2 2.3 &c. MULTINOMIAL THEOREM. 57 Now the m'^ term of this expansion will be the product of aP.xP h'l.x'i c\w' X X X &c., 1 .2 ...p 1 .2 ... g 1 .2 ... r where p + q -\- r + &c =m, whence (« + 6 + c + d+ &c.)"' will consist of terms included under the general expression 1 .2.3 ...m X aP.b'i.c^,. I .2 ... p X. 1 . 2 . . . (/ X 1 . 2 . 3 . , . r X &c. subject to the condition that p + g + r + &c. = m. EXAMPLES. + ^n.{n-l). {ti - 2) x"-^ +n{n-}){n-2){n-3). x"-^} . d^u (2) If u = e'' sm X, ——; = 2 e^ cos x, ^ ^ dx' d^u d^u d^^u -— = -2w, -— = 4?/, — — = -8?^. dx^ dx- dx^~ (3) Shew that Sx' 1x^ (cos .xy = i V — - &C. ^ ^ 4. 8 4 6 (tan . xf = x'^ + - x^ + - x^ + kc. ^ ^ 3 5 0^ 30?* Sx^ e^'"'- = 1 + .T + &c. 2 2.3.4 2.3.4.5 58 EXAMPLES. (4) Expand iim(a + bje + ex"), and log (a + b ,v + c a;-) according to the powers of a\ (5) If cos (m) - cos (m + y) = x, shew that ,V f X \'^ y = 4 cot w . + &c, sin m "^ \sin m) (6) If sin u = m sin x, prove that m (m^ - 1) .r^ m {9m'^ - lOm^ + l) x^ u = mx + + 1- &c. 2.3 2.3.4.5 CHAPTER IV. Taylor's theorem. 56. If u=f{w), and u^ be the value of u when a; becomes oo + h, du d^u h^ d^u ¥ d"u h" Ml = 11 + -— h-^ -— + -— + &c. + -— . + Sec. dx d.v^l.2 dx^2.3 d.v" 2.3....n The proof of this theorem may be made to depend upon the following proposition. If «.=/(.. + A), ^ = ^, or the coeffieient of h is the same in the expansion f(a; + 2h), whether we suppose in f{x + h), x to become x + h, or h to become h + h, i.e. 2h. Let x + h = Ofi; and — — = some function of Xi = (b (xi). dxi ' dU] dui dx du^ dh dx^ dx dxi dh dx^ dxi But '.■ Xi = X + h, —~ = 1, if /t be constant, dx dx^ and — — =1, if .r be constant; dh dui dui dx dh 60 TAYLOll's THEOREM. Hence also it is obvious, that dx dh d^Ui d^7ii 1. e. = , dw' d¥ and that = dx'' dh" Let .-. Ml =f(w + h)=u + — .h+ Pie + QhP + Rhy + &c. dx du dx in order and magnitude, beginning with the least ; restoring the terms after -^— A, and arranging the indices of h du^ du d^u dP dO ^ dR .-. -^ = — + 3— > + -- /i" + — - /i/^ + -— hy + &c., dx dx dx^ dx dx dx , dui du „ , ^ o , and —i = — + aPh''-' +BQh^-' + yRhy-' + &c. ; dh dx .-. taking away the common term — — , and dividing by h, dx we have d^u dP ^ , rfQ « , dx^ dx dx = aP/i«-- + (iQhft-^ + 7i?/i^-^ + &c. Now since in the upper series there is a term — - independent of /i, there must also be a term in the lower series equal to d^u — - , that does not involve h ; let this term be the first, as we dx^ have supposed the indices a, /3, 7, &c. to be taken in order of magnitude and increasing; Taylor's theorem. 61 ■ — -=aPh''-^, and a-2 = 0; .-.0 = 2, d.v- (Pu ^ dru 1 and 2P= -;-;;; .-. P=-7-^. dcP a a?" 1.2 Similarly, — h'^'\ or- — A, must = /3Q/i^""; da? da? i.e. — =j8QA/^-^ .-. /3-3 = 0, and jS = 3; ax and^A/^-' = ^/r = 7i2A>'-^ dtV dw .'. —^ = yRhy-'^; .-. 7-4 = 0, or 7 = 4; dx , dQ d^u 1 • ^ = i_: = * dw dx^ 2.3.4< and similarly may the other coefficients be found ; dit dru /r d^u h^ .-. u^ = II + -— h + --T, H- — — 7 1- &c. dx da'- 1.2 d.'j?''2.3 d"^^ A" d,»" 1 . 2 .3 — »z a theorem which will give the expansion of /(a? + A) in all cases, if x remain indeterminate. CoR. We may now deduce the theorem of Maclaurin which we have proved by an independent process in the preceding chapter. 62 TAYLOR S THEOREM. For by making a? = 0, 7/, becomes f(K) and u, — , , dx dw^ d^u — — , &c. become U^^ Ui, U^, U^, &c. d.v* .'. f(h) = Uo+ U,h+ U,— + Us^+ &c. 1.2 2.3 or putting a? for h, in which case u may be put for /(r) = Uo+ UiX+ Uz- — + U3 + &c. 1.2 2.3 the theorem required. E X A 31 P L E S. 57. To expand sin (x + A), cos (x + h), log (.r + h) and (a- + hy, by Taylor's Theorem, c?«« - d'u h^ d^u h? Ui=u + —-h + -— ^ + — — + &c. dx dx" 1.2 d.r^ 2 . 3 (1) u = sincT-; du d^u . d^u d*u .'. — = cosy'-\ A^-'... (1 +(hhf'-\ &c &c. we shall have by addition the coefficient of //", which niul- d"u tiplied by \ .2 . 3...n, will jrive — — . Now by the Binomial Theorem, the coefficient of ofA-MnT^:^^'-^= ^ ^ ^^^ ^ ;^^ ; ^^_^ -Vyr-...-(2), ^ (2r-4).(2r-5)...(2r-n + l) „ ^ .^ r,2 (7 6 therefore, substituting — for 7,. — - for e,', and multiplying "^ 2y^ 4./>- (2) by r x e,'-, and (3) by e,\ &c. and the sum of (1), (2), (3), &c. by \.2.S...n.f, d'u f2r.(2r-l).(2r-2)...(2r-n+ 1) g" d:^=^--'-"^^lTT-2-: s-t:::::^ ^rT" " ^ r . (2 r - 2) . . . (2 r - w -f 1 ) 7" e' . "1 1 . 2 n-2 2"jO" q~ ] = 2r.(2r-l).(2r-2)...(2r-r^ + l)(i) .p- \' + ^--^^yZ7)-^ i ^ o o r. (2r-- l)..,2?- -3 7' ,..(,._!)(,. -2) n. (n-l)...(n-5) e« -, — ; • - "T" Cv^» (■ • 1.2 .3 2r (2r-5) r/' 70 EXAMPLES. Ex. 1. Let u = - , the example given by Euler. VI - w- Here r = - 1, a = 1, 6 = 0, c= -I, p = 1 - x\ q= -2x, d^il-x-yi 1 2? (* = i , e'= -4; d.37" 1 .2...W. a?" . J 7i, (w - 1) 1 (1 _ ^^sy^t^ • i^ + S- ^[7^ -"2 1.3 n.(n-l)...(n-3) I 1.3.5 yt.(M-l)...y^-5 1^ 2.4*1 .2 ... 4 ' x'^ 2.4.6' 1 . 2 ... 6 \v^ the law of which is obvious. Ex. 2. Let u = ^/i - a!\ Here r = ^ ; .-. 2r — 1 = 0, and some of the coefficients will be - . The example however can be easily put under a proper form. For, since u = \/ 1 — a;\ du — ai dx Y^i _ ^2 d~u 1 dw~ (1 - X') i d-\/l + X' 1 or — = — 7 :77a ' dx' (1 - x-)i 1 = .(1-X-) 5 ; dx" dx"-^ and writing 7i - 2 for n, and - - for r, we shall have the required term. EXAMPLES. 71 Ex. 3. Let u = V cos ^, cos ^ - 1 — 2 sin^ - = 1 - .27^, by substituting a/' for 2sin^-; .'. W = V cos ;^ = Vl — X^, which is reduced to the preceding case. EXAMPLES. (1) Tan (,v + h) = tan x -{- h . sec^x + h~. tan c^ . sec^a? + &c. (2) Sin ' {x + /O = sin-^j? + -7 ^, + — r ^ + -^ ^ + &c. 2 . 3 (1 - cl?~)2 (3) Prove that if u=f{a;) fx\ du X d?u X- d?u x^ -^ [2) ~^^ ~'dx2 '^ dx^'^T^ dx^' 2.3.2^ ( X \ du x^ d^u x"^ d?u x^ f I 1=7/ - -1 J- &c -'Kl-^xj dx'l+x dx^'2{^l+xY dx^'2.3{\+xy (4) Find the &-^ differential coefficient of Vcos.t?. (5) Approximate to a root of the equations (1) x'^-\2x - 2S = 0. Ans. .1? = 4.302, (2) x'^ ■{• X- 3 = 0. Ans. .2? = 1.165. CHAPTER V. FAILURE OF TAYLORS THEOREM; LIMITS OF THE SAME THEOREM. 61. By the Theorem of Taylor we are enabled to ex- pand the f{a; + h) into a series of the form f(x) +'ph + qh-+ rh"'+ &c. where the powers of h are integral and ascend. Indeed Ave may prove a priori, that so long as x retains its general value, the expansion of f{x + h) cannot contain any fractional powers of h. For, suppose that / (,v -^ h) = U+ P\/h + &c. where U represents the sum of the terms involving integral powers of h. Then since a; + h enters f(.v -\- h) in the same manner as ,v enters /(•r), it is plain that both functions (undeveloped) have the same number of values, and that the developement of/(ci' + /i) ought to contain no more than /(cx) orf(,v-i-h) does. Now if particular values be given to .r, wliich will neither make P infinite nor evanescent ; then to each value of F there will correspond two values of P\/h, since \//i has two values + a or — a ; and consequently the expanded function will contain twice as many values as the unexpanded one ; and there- fore twice as many as f{w), which is manifestly contradictory. Similar reasoning will apply when the index of // is — '/t Taylor's theorem. 73 62. If then we give such a value a to x in f(x + h) as will make the unexpanded function f{a + h) to contain frac- tional powers of h, we cannot expect that Taylor's Theorem will give the required developement. Now the hypothesis that X = a introduces a fractional index of h into /(tV + h), supposes that in the original function there must have been some such terms as (71 — a)", which becomes (x — a + h)" in ni Wi, or /i" when ,v = a. In such a case it is clear that some of the differential coefficients will become infinite, when x = a. As an illustration, let us suppose that m y = b + {x - a)" ; du m --1 .■. — = - {x - ay , ax n , d^ti m (m \ --2 and — T = 1 {x - ay- . ax' n \n I — m — - 1 and Wi r= 6 + (,r - a)" ^ . {x - a)" h n + -. --l].(x-a)" + &c. n \n J 1.2 m fm \ im \ ^~p h^ + -. — 1 ... _p + l (.r-a)" &c. m where if — < p, that term and all that follow will become w infinite when x = a. This circumstance of the differential coefficients becoming infinite when x = a is called the Faihire of Taylor's Theorem, an improper term, since it rather may be taken as an index that the function cannot be expanded according to the integral powers of h. 63. Again, as the general expansion of f(x + h) can never contain negative powers of //, for if f{x + h) could = A + Bh"" + kc. 74 FAILURE OF if h = 0, f{o(! + 1i) instead of becoming /(a?), would be infinite, we may be led to expect that if x = a introduces into the un- expanded function f{x + h) a term involving h~", the ex- pansion by Taylor's Theorem will indicate some absurdity. Now it is clear that to have such a term dependent on h~", M we must orio-inally have had such a term as ; for " -^ {w- ay putting X + h for a?, ilf , MM — becomes : r- = t- , X - a^ ('^ + /i - «)" ^" when X = a. M not being supposed to vanish when x = a. Here all the differential coefficients of are in- {x - af finite when x = a. 64. The theorem therefore fails whenever x = a makes some radical disappear from u = f(x), and therefore introduces into «*i =f{x + h), some term involving a fractional power of h ; or when x = a renders the original function infinite. As a simple example of the first case, let u = b + \/x — a ; .-, Ui = h + ^/ X + h — a make x = a ; .-. 21 = b, Ui = b + V //, and the expanded function contains infinite terms. 1 As an example of the second case, let u = ; X — a 1 I h ^' o X — a -^ h X — a (r — ay {x — a)' where 7^ = co , ?7i-l, but - A - «)"' A'" Let then /(a + h) = Ah'" + &c. (/ f(a + /i) - ?w^ .-. -^^ = + &c. dh h"'^' d^f {a + h) - m (m + 1 ) (m + 2)...(m + n - l). A and dh'- h m + n d"f{a + h) .„ , where it is manifest that if h = 0, — ^ — -; \vA\ become dh finite. in From this reasoning it is obvious that if the n^^ differ- ential coefficient become infinite when x = a, the true expan- sion contains a fractional pov/er of h lying between n - ! and {7i) and that if x = a makes f{.v) = oc thc> true expansion contains negative powers of h. Thus, let u =/Cr) = b+ (,r - o)?, find f(a + A), d?i , , , dx 3-: = T • T 7 ^:i ^^''1=^''^ = *= ^^ •''' = " ^ dx- (.p - a)-i .', the fractional index of Ji is <2>1. EXAMPLES. 77 But 7^1 = 6 + {v -r h - a)i = h + h% when w = o, and 4- lies between 2 and 1. Again, if u — b.v'" + c{.v — a)'' , du , cp . --^ doc q d^u = W (777 - 1 ) . . . (m - 77 + ] ) ftcT'"' "" ?^ /z' ,^ (p „ . ,^ .i + r .-.--- 1 .. . - 77 + 1 1 . c7 q ^q I ^q and let -<77 but > 77 - 1. Then - — is the first differen- q dx" tial coefficient which becomes infinite, and there ought in £ the true expansion to be a terra involving ^' which there is, for by putting w + h for x, and afterwards writing a for .r, p we have f(a + h) = b.{a + h)'" + ch" . If m 0. I .2.3...{ti+ I) dx dx" l...n Now both these quantities vanish when h = 0, since then Therefore by the Lemma, their first differential coefficient will also have the same sign. Now differentiating with respect to h, du, (du d'u d"u h"'^ ] mh" — - - { — + .h + &c. + .- } > 0, dh [dx dx'' dx"" \...{n-\)] 1.2...W ^ Mh" dui du d~u h d"u h"'^ and ■ + — -\ + &c. + — — — - > 0. l...n dh dx dx^ \ aa?" l...(w — 1) Again, considering these expressions as functions of h which vanish when h = 0, for then — - = — ; their first differential dh dx coefficients will have the same sign as the functions have, or be both greater than zero ; whence again differentiating, we have taylor''s theorem. 81 d'u, j d'u d"-'u li"-~ \ _ nth"-' 'dh^~\d^''^ ^' ^ dx"-^ T72...(n - 2) j 1 . 2...n - 1 ^ ' Mh"-' dhi, d'u d"-~u A" -2 and r— + — - + &c. + -7---^ . > 0, \.9....n-\ dh^ dx^ rfcr"-"* 1.2...W-2 which are both functions of h, which vanish when h = 0, since then. div dw' Now if this process be continued (« + l) times, we shall at length obtain dh"" + 1 - m > 0, and M - —j—r > ; d"+^u, d'+^u, or since = , dA"+' dx"^' ,, d''+'u^ , d''+'u, M > 0, and ; — m > 0, a condition which is satisfied by taking M equal the greatest value of the {n + l)'^ differential coefficient, and m equal the least value. d"+'f(x + h) , d" + 'f(x) or M = '- , , and m = — r—-, — , dx'' + ' dx"+' and therefore the true value of u^ lies between du d'u h^ d"+'f(x) h"+' u + ^h + -— + &c. + -^ ^ dx dx^l.2 d , [ 1.2 1.2...n 1 .2.3...n+l \ and the error committed by omitting the terms after the n! 1.2...(w + l)^ ^ < ^n^-fi .n _ 1)^ if u^ be the w*^ term. w + 1 Again, if U„ be the first term that converges, U„ . n+ 1 > 1, 1. e. — TT— > 1- th U„+i ' Ah Let (n+l) = 2 Ah, U U '-^ therefore error < — . (a^ - 1), < — . (a '^ - 1). taylor"'s theorem. 83 Ex. 3. Let u = logo?; and the error, by omitting the terms after the w*^, is 71. From this reasoning, it is plain that there is some one term which is exactly equal to the sum of the terms after the 7i"'. Let N be this term, therefore /(^r + h) becomes du drzi /r (d"u Nh \ A" w + -r- /i + -^TT, r-r + Sec. + ( -;-^ + 1 ^—_ , dw dcG^l.2 \dw" n + l)l.2...n and to find the value of h, which shall make da;" 1.2...n greater than the remaining terms of the series, we must merely have d"u Nh d"u n + l -z — > 5 or h < ~ — . — -— , dx" 71+1 dx" N it is not necessary that N should be known, we may substitute for it a greater quantity, as M. 72. We may here add some remarks upon a method of notation, by which the Theorems of Taylor and Maclaurin may be put under very simple forms. We have hitherto considered the letter d prefixed to u, as in du, d~zi, d^u, &c. to be a symbol of operation and not of quantity, thus d, d\ # &c. indicate that u has been differ- entiated, once, twice, &c- But we may separate the d and its powers from ic ; and if we treat it as an algebraical quantity, no error can arise, so long as we bear in mind its original sijrnification. F2 84 LIMITS OF TAVLOr's THEOREM. Thus suppose in Taylor''s Theorem where we have du d?u h^ d?u W" Ui = u + —- .h + -— h dx da^\.9. da^2.3^ we look upon c? as a factor of u we shall have d , d' /r d' Iv" ' dx da^ ] .2 dx^^.S ' let -— = t^ and .-. — — „ = f, &c., dx dw" .-. ?^. = w |i + th H + + hc.\. 1.2 2.3 ' = we' ; for e"* when expanded will produce a series of the required form, and so long as we take care that the powers of d be referred to operation and not to quantity, no error can be produced, and thence Taylor's Theorem may be concisely written "../, M, = 7^g'*^ Again since Maclaurin's Theorem is u = Uo+ UvX + 1- + &c. 1.2 2.3 if we may be allowed to treat the coefficients f/oj ^u ^25 ^3» &c. as powers C", ^\ W, (7^ &c. we have u= 1 + Ux + + + &c. 1.2 2.3 Nor can error arise, if we keep in mind the original meaning of the coefficients L\^ U^, U2, &c. and if when we expand e^'^ we change the indices of U into suffixes, putting U(, instead of unity. But the utility of this method of notation will be chiefly apparent when the reader enters upon the study of the Calculus of Finite Differences. CHAPTER VI VANISHING FRACTIONS. 73. Sometimes the substitution of a particular value for the unknown quantity, will make both the numerator and denominator of a fraction vanish, such a fraction is called a vanishing fraction. x^ — 1 Thus becomes =- when .v = 1, but since bv divi- X - 1 a;~ - 1 sion = (x + 1), the true value of the fraction when .7? — 1 .r = 1 is 1 + 1 = 2. In this exaraple the numerator and denominator vanish when X = 1, because both contain the factor (x — ]), which is = on the supposition of .??= 1. 74. We proceed to shew that the values of these frac- tions may be finite, nothing or infinite. Let u = — he the fraction, and let x = ahe the value of x, Q which makes P = and Q = 0. Then P and Q must be both divisible by (x - a) or the powers of (x — a), let P = p.x - a\\ and Q^q.x- n\' ; » X — a u = ~ . ■■■ 9 X — a 86 VANISHING FRACTIONS. (1) Let m = n; .-. u = - ^ and = — when x = a, which is finite, since neither p nor q contain (a? — a). P 1 (2) L.etm>n; .'. 71 = - .x - ar'" = 0, if cP = «. pi 1 .p (3) Let nKn; .-. zi = - . = - = x , 11 a? = a. 75, From the preceding example it appears that the true value of the fraction is found bv getting; rid of the factor (,v — o)'", which is common both to the numerator and denomi- nator. When m and n are whole numbers, the value may be found by successive differentiations. For since P and Q are each functions of x ; when x be- comes X + h, the fraction m will become P + dP.h + d'P +(/'P + &c. f{x + h) 1.2 2.3 ^^'' "^ ^'^ Q + dQ.h + drQ^ + ^^Q^ + &c. ' , f/P dQ writing dP, dQ, &c. for — - , — — , &c. a,2? ax Let a? = a; .-. P = 0, and Q = 0, and the fraction, by dividing each term by h, becomes ^, ^, dP + d'P-^ + d'P.— +kc. ,_, f(a + h) 1.2 2.3 dP ^ ^ ' ^ ^ = — , when A = 0, ^ dQ + d~Q + a^Q. + &c. 1.2 2.3 P which is the value of u = —; when x = a. Q f VANISHING FRACTIONS. 87 Should, however, x = a make all the differential coefficients of P to the «z"^ order, and those of Q to the ?^*'^ order dis- appear, we have + &c. f{a + h) dx'" 1.2.3...m (pia + h) d''Q h" — . + &c. da.'" l.2.3...n If m = n, dividing numerator and denominator by A™ , and then making h = 0, 1 .2.3.,.m d'"P da;"" dx'" If m > w, w is = 0. If m < w, 21 is = - . » c» 76. If m be a fraction, this method is inapplicable. / Since oa = a will make some one of the differential coefficients / infinite. ' X a^ rr., .. {x'-a^)^ y y— dP Thus, if u = — . = V v + « = v2«, — - = - ^—- 's/ X — a dx \/x" dQ 1 and — - = 7= » a-2? 2\/a?-a both of which become infinite when x = a. In such a case we must have recourse to a method, which is perfectly general, and not difficult in its application. p Let — be the fraction, where P and Q both vanish when Q. X = a. For x put « + A, and let the numerator and deno- 88 VANISHING FRACTIONS. minator be expanded according to the powers of h, the indices increasing, so that the fraction becomes J,h"^ + B,hi^^ + C^hy^ + kc." which is of the proper form, since when h = the fraction becomes - . There will obviously be three cases, a = a, , a> a^, and a < a, . (l) If rt = ai divide each term by '''-'% and we have A + Bh^-^' + Chy' + kc. A " A, + B,k(^^-"+C,hy^'" + kc. Jj which is infinite. (2) a> a^^ then the fraction ^/i"-«i + 5A^-°' + &c when h = 0, A, + 5,A^i-"> + &:c. (3) a< a^, we have then = 0, when h = 0. A+BhP-'^ + kc. A = — = CO, when h = O P cc Cor. 1. If ti = — becomes — , when x = a it may be Q cc reduced to the form - . 1 1 p" 1 Q » For —=! — =:— = — ■= - , when r = a. Q Q \ I o p r ^ VANISHING FRACTIONS. 89 Cor. 2. If u = — = ,orcc-cc, when .r = a, P Q 11 may be reduced to the form — . 11 Q-P i or = — = - , when a? = a. P Q PQ o' CoR. 3. IfPxQ = Oxco, when .r = a, it may be put under the form - . For Q = — , if Qi= 0, when .v = a ; (all 1 P .^ Qi Q. o' ^•^ 1 Ex. 1. Find the value of u = , when .t = 1, ^ ^ dP . . P = a?'- 1 ; .-. = 3W^= 3, when x = 1, dQ Q = ,v^+2x'-,v — 2 ; .'. = 3.v^+4>a! - I =6, if w = I ; dcp 3 1 6 2 Ex. 2. Find the value of , when .r = 0, .T" 1-10 ?* = = - , when X = 0, P = a^ - h% and Q = ,r, —— = rt"" log a -If log ft = log a - log 6 = log - , when x ~ 0, and -— = 1 ; . . u - log - 90 EXAMPLES. af-w Ex. S. M = — 7. = - , 11 .«• = 1, 1 — c1? + log X "P =. of — cT, and Q = 1 - a? + log «, V ff" — .t?" is impossible, let h^+ A^h^^ kc. <2Aih; .-. A.jfi^ + A^h^ + he. < Jih ; and in the same manner may A-Ji^ be shewn to be greater than Asff' + A.ik* + kc. We have here supposed the series to proceed to infinity : if it extend to n terras, it is evident, a fortiori that any one term is greater than the sum of all that follow it. 80. Prop. If ?/ =f(^x) be a maximum or minimum . . du when cV = a. Then on the same supposition, = 0. dx Let Ui=f(x + h), and Uo—f(a; — h). Now at a maximum or minimum, u =f{cc) must be greater or less than both f(x + h), and f{x —A), or greater or less than both u^ and u^. Hence, u^— u and 7*2 — ^* 'imst both have the same alge- braical sign. du d^u h" d?u h^ But U^-U= —-k+ —— + — -r -— -i- &C. dx dw^ 1 . 2 dx^ 2 . 3 du d^u K' d^u h^ and .-. U2-U = ---h -ir — — --^ + &c. dx dx^ 1 . 2 dx^ 2 . 3 by writing — h for h in the value of Wj — u. Hence, since the first term of the expansion can be made greater than the sum of all the terms that follow it, (if the supposition of w = a, does not make any of the diff'erential du coefficients infinite,) it is clear that so long as the term — — h dx exists, so long will u,^ - u and 7*2 - '< have a different alge- MAXIMA AND MINIMA. 97 braical sign : i. e, u^ and tCo cannot be both greater or both less than u. Therefore, if there be a maximum or minimum, du , „ — = 0, therefore dcV u^-u = -——. + ^— ; + &c. dx' 1 . 2 dx^ 2 . 3 d-u h^ d'u h^ and iio -u - —-— 7-^ 1- &e. dx'' 1 . 2 dx^ 2 . 3 d^u , . /. Now if cr = a does not make - — = 0, the sign of w, - u and dx^ Wo - u, since fi^ is positive, will depend upon that of -~ . CL X dru If therefore — - be positive, u^ - u and u., - u are posi- dx^ tive If be negative, u^ - u and u.^— u are negative. dx"^ If therefore — ^ be positive, u-^ and u^ are both greater than u, or w is a minimum, and if — 5- be negative, then dx~ Ui and U2 are both less than m ; or w is a maximum. Hence this rule: to find whether u=f{x) contains any maxima or minima, put — = 0, substitute the values of x thus found in , if the results be positive, we have minima; if nega- dx^ tive, maxima. d^u 81. Should however = when x = a, dx'^ Uy-u = + -—- + &c. dx"^ 2.3 d^ti h? u-i-u = - —^ 1- &c. dx^ 2.S G 98 MAXIMA AND MINIMA. and ««j - u and u.^^ - u have again different signs ; and there- fore there will be no maxima or minima if exist. Hence it is obvious that we can have a maximum or minimum only when the first differential coefficient that does not vanish is of an even order. CoR. 1. If w = maximum or minimum, any constant multiple of M is a maximum or minimum. du dii . For if — = 0, a — IS also = ; due d,v and therefore if w = maximum, au is also a maximum. Cor. 2. If f{.x) be a maximum or minimum, /{xy^^ where n is integral, is also a maximum or minimum. For let u =f(x), and U=fix)]"; dw if ?/ be a maximum or minimum, dU and -- = w.7(^"-V0^)=O; •.•/(J?) = 0; dx and therefore f/ is a maximum or minimum. CoR. 3. If u=f{x) be a maximum or minimum, logw is sometimes a maximum or minimum. dJ] 1 du Let XJ = log 71 ; .*. -— - = -.-—. dx u dx I ^ du dU / But -- = 0; .-. 3- = 0, dx dx or U '\% 9. maximum or minimum, unless .r = a makes u = 0. MAXIMA AND MINIMA. 99 1 . . , Cor. 4. If u = maximum, - is a minimum, and con- u versely . 1 dv I du For let 1' = - ; •'■ -r = ? 1" » u dw u aw drv 2 du^ 1 d^u 1 d^u . = + — . . — - = . , when u = maximum. dw^ y? da^ vr doc- u^ dx^ Therefore, if — - be negative, is positive, or it u be dw^ d(xr a maximum, — is a minimum. u EXAMPLES. (l) Let w = .T^ - 6ci;- + ll.T - 6; find the values of se which make u a maximum or minimum. du — = S/F^ _ j2_j, + 11 = 0; dx .-. .r- - 4,^ + 4 = i ; .-. .r = 2 ± —7- = 2 ^ V^S ~" S ' dl^u ^ — - = 6.r - 12. dc^ Let .r = 2 J ; .". = 2 v .1 indicates a minimum., 3 dsc^ ■\/3 d^u /- .r = 2 ; .-. =-2v3 a maximum. 3 dcc'^ a? (2) Let t/ = .rtan^ ; find .r that y may be 4Acos~0 maximum or minimum. C?7/ ^ X -^ = tan0- -^-^^, rf.r 2^ cos 6^ f/^V 1 g2 J / 100 EXAMPLES. d u From -— = 0, a; = 2h tan 9 cos- = 2hsm0 cos 6 ; a A' , «^'2/ . . . . also -— IS negative ; .-. ^ is a maximum, and y = 2h tan . sin cos ; r-r — ^ 4/i cos^ e This equation is that of the path of the projectile, and the maximum value of y is the greatest altitude above the horizon- tal plane. (3) w = sine??]'". |sin (a - ct)}"; find o; that u may be a maximum or minimum. du -|m-l . I sin a - ^ ax ' = m sm a? 1 sin a - ^ T . cos ,t — w sin oeY . sin a — a?]""' cos (a - a?) = ; '. m sin (a - a?) . cos ; .'. DA = \/.v' - a' ; OA =a. Now DB : PB :: DA : 0A\ DB ioc + a) X a .-. PB = —-OA= ,— — - .-. ADPQ = PBxDB a(x + a) , . X ■\- (V^ X (<2? + a) = a ^ = \/x^ — a^ y/ CD — a minimum. Whence, if u= , ■r = 2a, and w = a^.3v3. CO — a (12) Find the greatest area that can be included by four given straight lines. Let a, 6, c, d = the four lines, Q the z included by a, 6, ^ ... / c, df, i) the diagonal subtending the two angles, and dividing the quadrila- teral into two As; ah . sin0 cdsin ax = maximum, whence x = a Vi (2) Let u = (mx + n) . {ny + m) be a maximum, and let a"'\b'"-^=c-, find x. Here — = m {ny + m) + n {m.v + n) — = ; dx dx and because w.r log o + w?/log6 = logc, — may be found, loff and X = c6" a" log(a-0 (3) Inscribe the greatest rectangle in a given triangle. AD = a, BC = b, AN=x; .'.Pp = — ; a f) V /7 1 .-. u = Pp . TVD = — (a - ,r), whence r = - , and u = - ABC (4) Inscribe the greatest isosceles triangle in a given circle. liCt r/ = radius, the triangle is equilateral, Ride = (7v3, area «= EXAMPLES. Ill (5) Inscribe the greatest rectangle in a semicircle. CN = cr, CA = a, NP = \/ a' - CD BN .'. u=2PM .CM=2a)'\/a^-w^\ .-. ,v= a an d M=a^\/2. ? Cor. The same construction applies to any curve. Let AC = h, AM = a? ; .-. PM = f{w), and u = 2(h-a') .f{x). Ex. 1. BAD a parabola; then y = 9,\/mx, and u = A^{h — x)\/ mx. Ex. 2. BAD a segment of a circle; AM = <3?, radius = a ; .*. PJ/ = V 2 aa? — .r^, and m = 2 (6 - .r) . v 2 « a? — x^. (6) Inscribe the greatest ellipse in a given isosceles triangle. IfZ>a=2.r, cb = y; .'.u = 7r.yx. Let AD = a, DB = b. Now ciV = 2 2 c^ a — X BD- nN = ax — 9.x DN: a — X ax a — X n y But . AN^ = PN^= -. {Na . ND) ; a — 9.x a — X = y (a — 2 -r') a {a-xf ' whence t/~ = — (a — 2 a?) ; a .-. iryx^ irb J a .x\/a — 9x% .'. .r = - Va 112 EXAMPLES. (7) Inscribe the greatest parabola in a given isosceles triangle. (8) Cut the greatest parabola from a given cone. (9) Required the least triangle TCt^ which can be described about a given quadrant. u = ^CT X Ct = ma, CA = «, CM =w, CN = y; a' : CT = - X y and if ?« = maximum, f{o), if c be positive, determmes the point P, y = + tj EUUATIONS TO CURVES. 117 Cor. 1. The distance of a point P from the origin, or AP = ^/AN''+PN'' = y/w'+y-. CoR. 2. The distance between two points P and Pi is thus found. ' A the origin. AN = x, AN^ = .t?, , NP = y, NP, = y,; y ^ ^ f ^...-^ 'trv s .-. PPi = distance = \/Pm^ + Pj m~ = V{AN, - AN)-' + {P,N, - PN)' 88. If in the curve CPQ the re- lation between PN and AN be known, and PN be a function of AN, the equa- tion to the curve is said to be known ; and from this equation, the curve itself may be drawn. Sometimes the equation y = f(x) is to be found from some given property of the curve ; as in the circle, of which the property is that all lines drawn from the centre to the circumference are equal. To questions of this description our attention in this Chapter will be solely directed. Since AN and NP are drawn at right angles to each other, the equation y=f{x:) is called an equation to rectan- gular co-ordinates. 89. Many curves however cannot be expressed by an equation between rectangular co-ordinates. Such are the Spirals, which may be conceived to be de- scribed by the extremity of a straight line of variable length, which revolves round a fixed point, called the pole of the spiral. The revolving line (called the radius vector) may be con- sidered as a function of the angle described. ' 118 THE STHAIGHT LINE. Thus if ^S" be the pole, SP the radius vector, SA the original position of 5'P, and z ASP = e, r =f{0) is the equation to the spiral. r and $ are called polar co-ordinates. We now proceed to investigate the equa- tion to the straight line. line. 90. The equation to the straight ^ A'V, Ay the two axes of x and y. AN-^w] NP=y\, iPCA = e\ AB = h. ek B7i \ to PN, r / Pn BA Then -zr- = ^rz tan 9. Bn LA or = tan = (i, bv writing a for tan 9 ; .-. y = a,v + b. Cor. 1. If the line be drawn through a given point, le£ a and /3 be the co-ordinates of the point. Then, when c-v = a, y = (3 ; .•. /3 = oa + h, and y = aw + h; '. y - (6 = a(x - a). Cor. 2. If the line be drawn through the origin, AB = ; and .-. h = 0; and y=aoe is the equation to a line drawn through //, THE STRAIGHT LINE. 119 91. If two lines intersect, find the point of intersection. Let y = ax + b, and y = a^x + b^ be the equations of the two lines. Then, at the point of intersection, the values of the co-ordinates are the same for both lines ; .-. ax + b = a^x + 61, b,-b and X = a ~ tti a&i — ab abi — a^b and y = V b = a — a, a — a. 92. Find the equation to the line which passes through two given points. Let y = ax + b be the equation to the line where a and b are to be determined. a and /3, ui and /3i tlie co-ordinates of the two points; .-. j3 = aa +b (1), and j3i=f/ai+/> (2); .•. (i — j3i= a . (a - tt]) ; .-. a = . a — oi But •.• y = ax + 6, and (i = aa + b; .-. y - 13 = a.(x - a) = ^ • (x - a). a — Oi 93. To find the angle which two y straight lines make with each other at the point of intersection. y = ax + b, and y = «i<.r + 61 , the equations to the two lines. 120 THE STRAIGHT LINE. PQR and P,QR, the lines. From A draw An parallel to Pi?, and Am parallel to P'R'; .'. ^nAm= PQP'; .'. PQP' = 71 Ax - mAx = tan"' a - tan~*aj, and tan PQP' = 1 + ffflj CoR 1. If the lines be parallel, PQP'=0, and «-«i=0; •. «j = a, and y = ax + b] . ,, , ,. ,• , are the equations to two parallel lines. y = ax + bi] CoR. 2. If the lines be perpendicular, 1 « — «i tan PQP' I + aa^ J 1 '. 1 + «rtj = 0, and the co-ordinates of B. BC = /3 J AN = X, BM = X,, NP=y, MP=y,. Then x = x^ + a^ and y = yi + l^- Substitute these values for x and y, and the equation i?- transformed, and the co-ordinates are measured from B. 124 TRANSFOItMATION OF CO-ORDINATES. (2) Let the axes be changed but still rectangular. Aw^ Ay, the old axes, Axx, Ay I, the new ones. AN = x, AM = Xi, ia!Axi = 6, NP = y, MP = y,. Draw Mm perpendicular to PiY, and Mn perpendicular to Ax; .-. X = An - Nn = c^i cos - yi sin 9, y = Nm + Pm = t^i sin B + y^ cos Q. For Z m PM = Q. (3) New axes not rectangular, but the origin the same. Ayi^, Ax\, the new axes, Ly^Ax^ = A, z Xi Ax = 0, PM parallel to Ay^, AM = Xi, AN = X, MP=y„ NP^y, X = All, +nN = x^ cos + yi cos (A + 0), y = Nm + Pm = ci'i sin -v y^ sin {A + 0). TRANSFORMATIOX OF CO-ORDINATES. 125 Cor. 1. If we wish to transform from oblique to rect- angular. Since X = jb\ cos Q -\-yi cos {A + 6), # y = x\ sin d + y^ sin ( J + 0) ; .*. ^ sin = x^ cos sin + ?/j cos {A + 0) sin 0, 1/ cos = c^i COS sin + ?/j sin (^A + 0) cos ; .', 2/ cos — a? sin = 2/i {sin {A + Q) . cos — cos {A + 0) sin d\ = 2/j sin ^ ; y cos — X sin Again, a? sin {A + G) = x^ sin (^ + 6) cos + y^ sin (^ + 0) cos (A + 9), y cos (^ + 0) = x^ cos (J + 0) sin 9 + y^ sin (J + 9) cos (^ + 0) ; .*. tP sin (A + 9) — y cos ( J + 0) = a^j sin A ; iT sin (A + 9) — y cos (^ + 0) sm^ Let (0, = ly.Ax) ; .-. J = (Oj - 9), and J + = 0j ; ^ cos 9 — X sin •'• ^^ " sin (0, - 0) ' X sin 9^ — y cos 0j and ci? = ; — r — — . sm (0 - 9 J Cor. 2. If J = 90", cos {A + 9) = - sin and sin (A + 9) = cos 0, and X = cVj cos 9 — y^ sin 0, 2/ = tX-j sin + ?/i cos 0, as in the preceding case. 126 TRANSFORMATION OF CO-ORDINATES. t^ (4) If the origin and inclination of the axes be changed. Let a and /3 be the co-ordinates of the origin, and then we must put x = a + a?j cos 6 + yy cos (A + 9), 2/ = /8 + cVj sin + 2/i sin (A + 6). 100. To transform rectangular co-ordinates into polar, the origin being the pole. AP = r, zPAN=e; .'. w = r COS0, y = r sin 0, which put for x and y and the equation will be transformed. But if the point S be the pole, y draw SB perpendicular to Ax, and Sm perpendicular to PN, AB = a, SP = r, BS = fi, ^PSm, = e; .-. X = AB + BN =. a -V r cos ^, y = BS + Pm = /3 + r sin 9. Ex. 1. Find the polar equation to the circle round a point S, co-ordinates a and j3, a^ + y' = a^ ; .-. (a + r cos 9y + (/3 + r sin 9y = a~ ; .-. r^ +2r.{acos9 + fi sin 0) + a' + /3' - «- = 0. Ex.2. Transform {x^ + y^y = a" {x^ - y") into polar co- ordinates, the origin being the pole, x'^ + y~ = r~, and x = r cos 9, y = r sin 9 ; .'. r'' = a^r^ . (cos^0 — sin-0) ; .•, r'^ = a' cos 20. THE PARABOLA. 127 Ex. 3. Transform the equation or ~ y^ = a? into another, the co-ordinates of which are rectangular, but the axis of y is inclined at an Z 45*^ to the axis of a?, w = Xi cos — 3/1 sin 9, y = c^j sin + w^ cos 0, = 27r-45; cos 9 = cos 45 = V^ -1 and sin 9 = - sin 45 = — ^ ; V2 .•. .t' = '^i + Vx a/2 — ? y = v/2 X'' —y— — = = 2tV^y^ = d' ; a •^'1^1 = - THE PARABOLA. 101. If from a fixed line QDq perpendicular lines, as QP, are drawn intersecting lines equal in length, but drawn from a fixed point S, the locus of P is the parabola. Draw SD perpendicular to Qq. Bisect SD in J, then the curve passes through A. Let SA = AD = a, AN=w, NP = y. 128 THE ELLIPSE. Now QP or DN = SP ; .-. nA+AN= ^NP' + SN'' ; .-. a + a; = y/y^ + (a? - a)- ; .-. (a+x)' or (x- af + 4^aa; = y^ + (x - a)'; .-. y" = '^•ax. Cor. Let SP = r, and iASP = e. Then r = DN =2a + SN = 2 a + r cos PSN = 2a — r cos G ; 2a .-. r = a 1 + cos ~ 7^ cos- The polar equation. THE ELLIPSE. 102. If from two fixed points S and H two lines SP and PH be drawn and intersect, and SP+PH=a constant line, The locus of P is an ellipse. Let SP+PH = 2a. Bisect SH in C, and take CA = CM = o, the curve passes through A and il/. THE KLLIPSE. 129 Through C draw BCb perpendicular to SH. With centre S and radius = a cut this line in the points B and b the curve will pass through B and 6, since HB and Hh each = a. Let CS : CA :: e : 1 ; i*. CS = ae, which is called the eccentricity. Make CN=a^, and CB = b, SF = D, NP = y HP = D,., .-. D^= SN'+ NP'={ae + xy + y\ Df = HN~ + NP~ = (ae - x)' + y"; .-. D'+ D^'=2(a''e'+x^+y'), and D^- — Z>,~ = 4ae.i\ But Z> + Z), = 2a; . . D -/>i = 2e<2?; .•. D = a + ex, and D^= a — ex., .-. D^ + i>i' = 2 a"^ + 2 e^i- = 2 {a^e~ + o?'^ + ?/') ; .-. y'=a'.(l -e')-x'.{l - e") = (1 - e") (a^ - ,2?2). CS^ a^-CS- SB'-CS^ b' But 1 -e'= t - a^ a^ a- a^ ' and 1- — = 1. 6^ a^ Cor. 1. If A be the origin. Make AN = x^ ; .*. .t?j =s rt + .r, or ,r = a?, — « 130 THE HYPERBOLA. a Cor. 2. If S be the pole, and ASP = 9, and SP=r; ... (2a - r)2= HP' = ^iV^+ iVP2 = (2ae - SNf + r' snr0, and SN = r cos PSH = -rcos9; .-. 4a^- 4ar + r^= (2ae + r cos0)^+ r^ sin^0 = 4a^e^ + 4aer cos + ?'^; .-. r.(\+e cos 0) = a (l - e-), « (1 - e^) r = 1 + e cos 9 CoR. 3. If C be the pole, CP = r, and PC J/ = 9. Then cr = r cos 9, and 2/ = r sin ; ••• — 4- — = r^ /cos^e sin2 0\ [ a' ^ b' j = 1; a6 ab ^'^ If :^ y/¥co&'9 + a' sin^^ \/«' (l - e~) cos' 9 + a^sin^^ 6 V 1 — e^cos^0 103. If the difference between SP and PH be constant, the locus of P is the hyperbola. Let the difference be 2 a. Bisect SH in C. Take CA = a = CM, and the curve passes through A, CN = w NP N- S Let CVS' = e . CA = ea, where e > 1. THE HYPERBOLA. 131 Then HP' = HN' + NP- = (ea + a^f +y'= D,\ SP- = SN~ + NP- = {ea - xf +y'=D-; whence Df+D^=2. {a-e^ + x'' + y~), and Di' — D' = i^aex. Also D^- D = 2a,, .-. D^ + D = 2ex; .". D^= a + ex, and D = ex — a\ .'. 2a' + 2e^x'^=2{a'e^+x^+y'^), and y2 = (e^' - 1 ) . x^ - {e^ - i) a- = {e'- \).{x''-a^) a' {x'-a"). Making h-=a-{e-- 1) ; y~ x~ 7 9 9 o~ a" Cor. 1. \i A he the origin, and AN = x^^ .J? = .Vj + a ; .-. a? + « = ,rj + 2a, and X — a = x^\ .'. x'^ - d^ = x^ + 9.ax^^ and y^ = —(2 ax^ + x^^). Cor. 2. To find the polar equation, S being the pole, SP^r, L ASP = e. 12 132 THE HYPERBOLA. Then (2 a + rf - HP' = PN'+ HN' = PN'+i2CS -SNf = r^ sin^ + (2ae - r cos 0)^; ... 4-a' + 4-ar + r^ ^ r^ + 4a' e^- 4aer cos0; .-. r . (1 + e cos 0) = a (e- - l), «(e-'-l) 1 + e cos 6 Cor. 3. If C be the pole, CP = r, z PC A = ; .-. X = r cos 0, and r/ = r sin 0, a- 0" cos-0 sin'f? a~o~ = 1 r=^ = 6' cos'^0 - a" sm"^' ah b r = y 6- cos^e - a~ shTe \/e' • cos^ 0-1* 104. The asymptotes being the axes, and the centre the origin, find the equation to the hyperbola. The asymptotes are lines, as CO and Co, drawn through the centre, making an angle = tan"' - with the axis of the hyperbola. CN=x, CM=iv,, and OCA=oCJ=ti, NP = y, MP = y,. Draw Mn perpendicular to CAN, and Pm perpendicular to Mn. THE HYl'KRBOLA. 133 Since MP is parallel to Co, and Pm is parallel to CAN, .-. AMPm = e. Now X = Cn + nN = a^^ cos 9 + yi cos 6^ = (oe^ + y,) cos 0, y = il/w — ilf m = Xi sin t? - t/i sin t? = (tti — ^i) sin ; oe' y' {x, + y,y {.v, - y ,y . cos- 17 — sin' t7 = 1 . 2 But tan ^ = - ; .-. 1 + tan^^ = = -^t^ a cos^ a'* cos^ 9 1 and sin- 9 cos^ 1 V d^ h^ + a^^ {w ^ + y^f — i^.- -y^y 1; b' + a- i. e. A^x^y, = a;' + b\ a' + b- Cor. If the hyperbola be rectangular, b = a, and Xiy^ = a' 105. The curves whose equations we have just investi- gated are termed Conic Sections, since they may be supposed to arise from the intersection of a cone by a plane. The Conic Sections, exclusive of the straight line, are also called curves of the second degree, since the sum of the indices of the unknown quantities does not exceed two. The general equation of the second degree is of the form Ay" + Bvy + Cr + Dy + Ex + F = 0. 134 THE GENERAL EQUATION. Now if the centre be the origin, the equation to the curve is the same when (- x) and {- y) are put for x and y: con- sequently the origin of the co-ordinates of the general equation is not in the centre; since Dy and Ew will both change their signs, when (- y) and (- w) are put for y and w. To get rid of these terms, transform the equation to the centre by putting w -\- a and y + fi for w and y, and making the coefficients of cV and y respectively = 0, we shall have two equations for determining a and /3 ; 2AE-BD and a = /3 = B'-4AC ' 2CD-BE B" - ^AC The equation is now reduced to Ay~ + Bxy+C.v- + F, = Q. Next, to get rid of the term Bxy ; let the axes be changed to others, making an angle 9 with the axis of x^ by putting X = .V cos — ?/ sin 0, and y = X sin 6 + y cos Q. Therefore the coefficient of wy becomes 2 A sin Bco%e + B (cos^ Q - sin- 0) - 2 C sin cos 9 = 0, or (y/- C) sin20 = - 5cos20; •*• tan 20= , an equation which is always possible, since the tangent passes throuo-h all degrees of magnitude from zero to infinity. The reduced equation finally becomes My- + Nar + F, = 0, which may be made to coincide with the equations to the circle, the ellipse, or the hyperbola, by giving proper values to M, N, and F,. OF THE SECOND DEGREE. 135 Cor. 1. If B"^ = 4fAC, a and /3 are infinite, and the curve has not a centre. The equation without the term B^vy becomes My' + N/v"" + Px + Ry + F = 0. Now 4.M.N=4.AC -B' = Oi therefore either M or N = 0. Let iV = ; then the equation becomes My^ + Px + Ry + F==0. Again, to get rid of the terms Ry and F, make X = X + a, and y = y + b, and we have If/ + (2 Mb + R)y + Px+ Pa + Rb + Mh' + F = ; to determine a and b, -R let 2Mb + R=0, or b = — - , Rb + Mb"' + F and Pa + Rb + Mb^ + F=0; .-. a = ^ ; and the equation becomes My^ + Px = 0, the equation to the parabola. If M = 0, then we shall have Nx~ + Ry=^0. Cor. 2. If My^ + Nx' = F...(l) be an ellipse, find the axes. The equation to the ellipse is a^y' + b'x~ = a"'b' (2); let h be such a quantity as multiplied into the equation (l), will make the terms identical with those of equation (2) ; 136 CURVES OF THE SECOND DEGHEE. .\hM=a\ hN=b\ and hF=a'b-; F .-. h'MN=a'b~ = hF; .-. h = MN F If f If N N M M 106. We have assumed that to prove this we must find M and N in terms of A, C, and R. By putting X = X cos 9 — y ainO in the general equation, and y = r sin 6 + y cos ; Jf = ^ cos- - 5 sin cos a + C sin'^ ; A^ = ^ sin^ + Bsinecos9+ C. cos^ ; .: M + N=A+ C, M- N=(A -C). cos 20- 5. sin 20. -B But since tan 20= -, A - C cos 20= — 7 , and sin 20 = ^(^ - c)^ + B' ' a/(j - cy + B' ' ^_^^ (A-Cy + B^- ^^^j_^y^s.; V{A - cy + B' .'. 2M=A + C+ \/(A - cy + B\ 2N=A+ C - V(A - cy + B^ ; .-. 4MN = (A + cy - {A - cy - B' = ^AC- B\ Whence, if 4^ AC > jB^ M and N have the same sign, if 4-AC< 5% different signs, = B^, either M or N must = 0. EQUATIONS ro CURVES. 137 CISSOID. 107- ^Q-S is a semi- drcle. Take AN and BM equal. Draw the ordinates NQ, MR. Join AR cutting NQ in P. The locus of P is the cissoid AN = .V NP = y AB = 2 a B Now AN' AM' AM-' AM NP' MR' AM. MB MB' X 2a — ,v or — = f- X ^3 2a — A' CoR. The Polar Equation. AP=r, LPAN=e, X = r cos 0, y = T sin 9, y^ sin- Ox r cos 9 x^ cos'9 2a — X 2a —r cos 9 ' 2 a sm^9 = r cos 9 (sm^9 + coh'9) ; sin 9 r = 2a cos 9 sin 9 2 a tan 9 . sin 0. 138 EftUATIONS TO CURVES. THE CONCHOID OF NICOMEDES. 108. The line CP revolves round a fixed point C, cutting the line ARN : RP is always of the same length ; then the point P will trace out the conchoid. Let RP = AB = a, AN = x, CA = b, m = y. MP'- AR~ RN' RP'-NP CM' CA' NP- NP~ w (b + yf y' ... y*+ ohy'+ (b'+ x' - «'0 y--2a~by - an'= 0. Con. Let CP = r, ^ PCM = 6, r==CP=PR+ CR = a + cosO 109. AQB is a semi-circle, and NP is taken a fourth pro- portional to AN, AB, and NQ. The locus of P is the " witch." AN = w, AB = 2a, NP = y; .-. NQ = \/2a.r - x', and .V : 2a :: \/2aoc -x^iy; 2 a \/ 2 a CO - x' y= =2a sj'- 2a - X X EQUATIONS TO CUllVES. 139 110. The Logarithmic Curve. In this curve, the abscissa is the logarithm of the ordinate, or if a be the base of the system, the equation •" to the curve is ?/ = a"*, .-. AB = a'= 1, m or the ordinate through the origin is always unity. * It is obvious that as the abscissa increases arithmetically, the ordinate increases geometrically. 111. The Quadratrix of Dinostratus. While the ordinate RN of the quadrant AQB moves uniformly from A to BC^ the radius revolves from CA to CB, cutting ^A' in P: the locus of P is the curve required. AN = w, CB=l, NP = y, z QCA = e. Then e TT ,v : 1 ; .-. e = ttx PN CN — tan Q ; y TTA' or = tan — ; I - oc 2 y = (\ - x) . tan TT.V Cor. When x = ], y = Cb = - •K 112. If RN move as before, and a line as QPM parallel to AC move uniformly from AC, the intersection P of RN and QM will trace the Quadratrix of Tschirnhausen. 140 EftDATIONS TO CURVES. Here AQ = ~, and NP = sin JQ . ttA' '. y = sin is the equation. 113. The Cycloid is the curve described by a point in the circumference of a circle, which is made to roll along a horizontal line. Let BQD be the circle, the centre ; and when it's di- ameter is perpendicular to the horizontal line at A, let the point P, which generates the curve, also be at A- Then Ab must = Ph, since each point of Pb has been in contact with each successive point of Ab. Let AN=.v, BD = 2a, NP^y, ^ QOB = e ; .-. X = Ab - Nb = aO - a sin 6 ^ a (0 - sin 0) ; .•. y = bm = a ver. sin = a (l — cos 0) ; 6 cannot be eliminated between these equations. CoR. 1. To find the differential equations. dtV d^="""^' d0 = o ( 1 — cos 0) ; dy d0 sin doc dec 1 — COS0 d0 a sin y 5in(9 = V^I -cos'a = \/(l-cos0)x(l+cos0)= \/- x ^^^— ^ ; KttUATlONS TO CUllVES. 141 . a sin d = vS ay — y' ; dy s/^ay-f div y CoK. 2. To find the equation from D. DM = 00^, .-. Xi=2a-y, MP = :v, , y,= AB - a ; dyx dx f y 2 a - .r Cor. 3. The equation from D may be also found from the properties of the curve. Join Ph and QB^ then these being equal and parallel, PQ = Bb = AB -Jb= AB -Pb= DQ. For AB is equal to the semi-circumference DQB. Let DOQ = (p. Then y = PM = MQ + PQ = a sin ^ + «^ = « ((i + sin 0), ■r = i>ilf = a ver. sin = a (l - cos 0), and eliminating d) by differentiation we have the equation previously obtained. THE TROCHOID. 114. The trochoid is the curve traced out by a point B in the circumference of the circle BRb, which is carried through space by the rolling of the outer circle AQ upon the horizontal line. 142 EQUATIONS TO CURVES. P a point in the trochoid. Thi-ough P draw a horizontal 1 L joR = e. A. ]sr line MRPm. Take O and o the centres of the circles. Draw ORQ and oP. Then Pm = RM, and z AOQ = Z A^oP. Let OA = a, AN = c?;] OB = b, NP = y\ Then it is obvious that arc^Q=^^^; .-. w = AA^ - NA^ = a9 - h sin 9,- y = NP = Oil, + om = a — b cos 9. ^ b Let - = e ; a .'. ,v = a (9 ~ e sin 9), y = a{\ — e COS0). If e = 1, that is 6 = r/, the trochoid becomes the common cycloid and their equations coincide. 115. SPIRALS. (l) The spiral of Archimedes. In this spiral the radius vector varies directly as the angle described, or roc0; .-. >• = a^ is the equation- EaUATIONS TO CURVES. 143 Its equation may be found from the following mechanical con- struction : Let the line SA revolve uniformly round S^ while a point P moves uni- formly from S along ^S*^, then P will trace the spiral of Archimedes. Let Z ASP = e, SP = r; and let a = value of r when = 2 7r; 9 a TT a . r = — 9 = m9 by putting m = a ■.TT •ZTT (2) The logarithmic spiral. Here the angle described is the logarithm of the radius vector, its equation is r = a^. This curve is also called the equiangular spiral, since the angle at which it cuts the radius is constant. (3) The hyperbolic spiral. In this spiral as the angle increases the radius vector decreases, and its equation is a r = - , or 6^ >• = a. u (4) The lituus so called from its form, 1 ^ «•- Here r oc — ~= , or = — . ^9 r (5) The spiral of Archimedes, the hyperbolic, and the lituus, are included under the general equation r=a9", as we shall see bv putting n = 1, - 1, or -, 144 EQUATIONS TO CURVES. (6) The involute of the circle is described by the extre- mity of a string which is unwound from the circumference of a circle*. A the point from which the string began to be unwrapped, QP the string once coincident with the arc AQ^ and therefore = JQ; PY a. tangent to the curve JP or to the involute, SY perpendicular to the tangent, join SP. SP=r\ .-. SQ = PY = ^/SP~ - SY' ; a- = r- — P 2 . p~ = r- - a^, is the equation. CoR. If = sec-' - = PSQ, and 9 = i ASP, a e + (p = Vr^ — a" a ; .: = y/r^ — cr — sec a * The figure is drawn inaccurately, AP should be perpendicular to the circlfe at A. CHAPTER IX. TANGENTS TO CURVES. 116. Def. a TANGENT is a line which has a point in common with a curve, and which, of all the straight lines that can be drawn through the point, approaches nearest to the curve. PPi the curve. QPT the tangent of which the equation is required. AN = NP A the origin of co-ordinates. Ay and Ax the axes of y and cc respectively. >, and y -fi-v) the equation to the curve, and 2/1 = AoTi + B the equation to the line ; • •• y = Ax + B, because it passes through P; •■• iVi-y) = A.- (a?i - dx QfV and .•• (yi - y) = -^ {x^ — x) is the equation to the tangent. Ct 00 CoR. 1. From P draw PG perpendicular to the tangent and meeting the axis of x in G, it is called the normal, and since if y = ax + hhe the equation to a line, y = x + hy is (Jv the equation to a line perpendicular to it ; d 00 .-. t/i = = = dx a [ \/2acr + ,j?«J \/2ax +a^ h s= 6 if ci? = CO ; V 2a 1 + — X dx llax -t X ax a AT = y X = X = = = a, dy a + X a + X a ^ 1+- X when 0? = CO ; .•• ^ 7^ = -^ major-axis, or T and C coincide. Join CD, it produced, is the asymptote. 121. This method is frequently difficult of application., and the following is more generally useful. If possible, let the equation to the curve be put under the form y = Ax + B + - + — + - + kc. X X X then it is obvious, that as x increases, the terms after S decrease ; and when x becomes infinitely great, they vanish, and the equation to the infinite branch of the curve is y = Ax + B. ASYMPTOTES. 153 But this is the equation to a straight line cutting the axis of 7/ at a point y = B, and x = 0, and making an angle = tan" 'J, with the axis of x. Hence it appears that the infinite branch of the curve is coincident with the line de- termined by the equation y = Ax + B., C D .'. if y = Ax + B + 1 — - + &c. be the equation to a curve, X x"^ y = Ax + B is the equation to the asymptote. CoR. If the form of the expanded f{x) be D E y = Ax^ + Bx + C +— + — , + &c. •^ X x^ the asymptote is a parabolic curve, of which the equation is y = Ax^ + Bx + C. EXAMPLES, (l) Find the equation to the tangent in the ellipse. The centre being the origin. if x^ W ' "' dy^ dw h^ X ar y 154 EXAMPLKS. a' y ^ + feS _ y» ; A 4' ••• yyi-y o^ a^ a^ •'• yyi = — - (« - -^-^i)' 6 a^ — cVcVi ^' a y/d'-x' Let 2/i = ; 4:* .'. CT xCN= CA~, (See Cowic Sections.) 2 2 and iV^jT = CT - CN = ^ ~ ^^ = sub-tangent, or iVT X CiN^= (a + .r) (a - w) = A,N x AN. b.(a — x) Cor. 1. Make Xi = a; .'. y^ — AD^ V a^ — w^ b.(a + x) y/ a- — a- .-. AD.A,D, = b' = CB\ Cor. 2. The equation to the tangent may be written a^yy^ + h\vx.^ = a^b^. In the hyperbola, the equation is a^yy^ — b'^xw^ = — a^b-. (2) Find the sub-tangent and sub-normal, &c. in the cissoid. TANGENTS. 155 Here y~ = or 9.a — aa dy J 3x'^.{2a- x) +00^ _al^ (6a -Sa?) (2a - w)'^ 2 ' (2a -a?)=^ »rfS = 4 .-. sub-normal = y dy x~ {3 a — x) dx (2 a -xY d 11 dividing y — ^y y^-, CLW I dy 2a — X x^(3a-x) (Sa — x) y dx x^ (2a — xy X . (2a — x). X {2 a — x) •. sub-tangent Also •.• lia — X - dy x~. (3a — x) dx y. (2a — xy The equation to the tangent is x^ 3a — X (yi - y) — (tPl — X), y (2 a -x) 2 or (3 a — x) or 2/2/1 - y (2 a - x) Y' • \^\ ~ ^)''> x~ ••• yyi = \(3a—x') (x^—x) 2/1 2a — X \ x^ (2 a -xf \/ X 2a — X -1 (2 a - xp Making y^ and x^^ successively = 0, {(3a — I??). a?i — ax^ ; \(3a — a?) a?i — ax^. ^^ a .X -, .^ [ X \ AT = , and JZ) = -Vi = a. 3a — X \2a — x) Note — PTD in the figure should be a straight line. 156 TANGENTS. dy If !■* • • tc — and a? will be infinite when b — a«* = 0, or ^ = \/ - , and then y = xz = x \/ - , a is the equation to the asymptote. 160 ASYMPTOTES. (11) Find when the curve, which is the locus of the general equation of the second order, has an asymptote. Ay~ + Bjcy + Cv^ + Dy + Eoo + F = is the general equation, or y^+ 2 (ax + b) y + cw^ + ex +/= 0, dividing by J, and making the proper substitutions ; .-. y^+ 2(ax + b)y + (ax + bf = (a^-c)x^+ {2ab -e)x + b^-f, and y=- {ax + 6) ± \/(a^- c)x^+ (2ab - e)x + (b^-f) ; J(„ + ^)i ^/(„=_e) + ?^ f 6 r~ — f 1 2a x{a+ - i Va'^-cn + — 7- [ a? [ 2x {a b-e V'-n + X X' ab-e A B\ — c) x^ a^ / ,^ab—e A = - \ax + b i^ \/ d~ - c {x + ^ — ^ — + — + &c.) | ; and therefore the equation to the asymptote, which is of the form y = mx + w, is / 2ab - e\ y=-(a^^^^)x-[b^--^, B' C which is possible when a^ > c, or —— > — , or B^-^AC>0, which is the case in the hyperbola. Cor. If a''^=6', the equation is of the form 2/ = - (ffc^ + 6) ± V mo?. \/l+-... which cannot be reduced to the form y = Ax + B. TANGENTS. 161 122. Find the locus of the intersections of perpendiculars drawn from the origin upon the tangent, with the tangent. Let y =f{x) be the equation to the curve; •'• iVi - y) = i^\ ~ ''^') ^^ ^^^ equation to the tangent, fj If* and Vi = x^ is the equation to the perpendicular from the dy origin upon the tangent. J dy J, Between these three equations eliminate y, x, and — — , and ^ d.v the resulting equation will contain y^ w^ and constant quan- tities, which will be the equation to the curve required. Ex. Let the curve be the hyperbola, and the origin the centre ; y- x^ dy W X b' a^ dx a' y and .-. Wj = I'-x^ from equation to perpendicular; h" h* x" h^ \o'^ x'j b^ b"x" '' y^ a' y- a' fh' h~\ a? a .'. X' — X a~x{' — o yi a^x^ 1 ^' Vi -%i and f/ = . — X = a Xi \/arxx — h^yx xx^ yy^ out — 2 W~ ^ ■'^ equation to tangent; cC' b' '\/ c?x^ — ly^yi .'. Orf + vf )- = a^xy^ - b'^yil L 162 TANGENTS. Cor. If a = 6, or the hyperbola be the equilateral, (,^1 + iff = «' {^\^ - y\)- The equation to the lemniscata. Prob. P any point in a curve, PG a normal; let Pp make with PG the angle pPG = z APG. A. ST Find the equation to Pp AN = .%^ NP = y G p a?i and t/i the co-ordinates of Pp ; dx = p •'• {y\-y) = ^ • ("^1 ~ '^) ^^ ^^^ equati on, where J = tan Ppx. But Pp.r = APp + P^G = ^APG + P^G, and 27iPG = 2(7r - PJG - JGP) ; .-. 'iAPG + P^G = Stt - PAG -2 AGP., V 1 or P«.'r = 27r-tan-^- - 2 tan'^ - = (Stt - M) ; .-. A = tan (27r - ilf ) = - tan iW A' 1 F_ 2 ?/ P x' 1 y 2» - (2) Let the curve be the witch : NT = ; NG = -. a X' (3) The focus of a parabola is in the centre of a given circle, its vertex bisects the radius, find the point and angle of intersection of circle and parabola. (4) Shew that the normal to the curve defined by y- = 'i-ax^ 4 is a tangent to the curve defined by y" ^ (.r-Sa)^; and that when the curves intersect x = 8a. (5) If ?/'- = 4a (cJ7 + a) be the equation to a parabola, the origin in the focus, shew that the points of intersection of the tangents, and perpendiculars from the focus, are determined by the equations y x^ = - a, and ?/i = - • l2 CHAPTER X. THE DIFFERENTIALS OF THE AREAS AND LENGTHS OF CURVES : OF THE SURFACES AND VOLUMES OF SOLIDS OF revolution: spirals. 123. One of the applications of the Integral Calculus is to find the areas of curves included between given ordinates/ the lengths of their arcs, and the surfaces and contents of solids. The solids of which we shall treat are called solids of revolution, since they may be supposed to be generated by the revolution of a plane figure round a line, thus termed an axis. Hence it follows that every section perpendicular to the axis will be a circle, the radius of which is the revolving or- dinate, and every section made by a plane passing through the axis will reproduce the original area. Considering the areas and lengths of curves, and the contents and surfaces of solids, to be functions of one of the quantities x or y, we can, by the Differential Calculus, find equations between the differential coefficients of these functions, and expressions containing x or y^ by which we shall hereafter obtain the values of the functions themselves. We shall find it useful first to establish the truth of the following Proposition. 124. If J + Bx, Ai + BiX, and A + hx, be three alge- braical expressions taken in order of magnitude, viz. A^ + BiX< A + Bx, but > A + hx, then shall A^ — A. DIFFERENTIAL OF THE AREA. 165 For if A do not equal J,, since A + Bx > Ai+ B^x. and Ai+ BiX>A +bx; .-. A -A,+ (B,-B)x is>0, and Ai - A + (B^- b) x is > 0, whatever x be ; but if we make x = 0, we have A - ^, >o, and A^- A >0; or both {A - A^) and - (A - A^) are at the same time > an absurdity, unless Ai= A. 125. Let AP be a curve, and y=f(x), the equation to it, where AN = X, NP = y ; and let A = area ANP. Then — = y. dx Let NN-i = hi- Complete the paral- a lelograms QN^ and PN^. Then the area P.PNN, is >CDPN„ y chord PP' < PM + MP, . 7 72 7 2 But arc PP' =AP,- AP=(h {x + h) -d) (x)=—h^-^ -^ + he ' dx dx- I . 2 chord PP'= y/Pm'+ {P'mf^=\/K'^ {ph + PJrf = h ^/(i +f) + opph+ P~Ji\ PM = ^Pm^ + Mm'"^ - y/lf+pVi' = h\/r+p\ MP, = MN, -N,P, = (y +ph)- (y +p h + Ph^) =~Phr; whence, dividing by h ds d^s h -r- + d'lc dx'' 1 . 2 + &c. >y/l +p-+2Pph + P-h- < ^/l + p-- Ph Pp >\/l +p^+ / - h^Uc.<\/\j^rr-Ph ; .V --i = ^i+^- = V 1 + ^ c?.r ,2 • P, TO = P, iV, - PA^ = y + ;>/; + Ph- -~ y ^ ■ph ■\- Ph\ Mm = Pm . isn MPm = h . im PTX = h . — . '' ^ '■^^'- + iUL +^)i-t^ * u7^ f^^t^ V^»^ Y VOLUME OF A SOLID. 167 127. If V be the volume of a solid of revolution APp, Let AN = w\ dV d~ V^ h^ NP =y); .-. JPp =/(,» + /,) =F+—/i+ + &C Then the solid Ppp^ P^ is > cylinder PMm^p, < cylinder RP^p^r ; . dV d'V Ir ^ „j 1. e. -T-h + ^-^, : — - + &c. > Tryh, IX dx- 1 . « <7r(y + ph + P/ry^h, dV d'V h , or ^ + -^^ z — 7 + &c. > Try- < iriy +ph+ Ph), dx daf 1 . 2 or > Try- < Try- + ^Ttpyh + &c. whence — = Try. dx Prop. The surface of a truncated cone, of which the radii of the greater and smaller ends are a, h, and the slant side 6', = 7rs{a + 6). Let / = length of cone, radius of the base = a, I.- h; 168 SURFACE OF A therefore, surface* of frustum = Trla - irl^h = tt {sa + l^{a -h)\, but Z or /jH- 5 : /j :: a : 6; .'. s : l^ :: a — b : by .'. sb = l^{a — b) ; .-. surface of frustum = tts . (a + b). 128. If .S'= surface of the solid of revolution JPp, dS da) = 27ry d,v^ AN =.r^ NP =y JP =s NN, = h] Draw the tangent P3I, and chord PP'. * The surface of a cone when unwrapped coincides with the sector of a circle, the centre of which is the vertex of the cone, and radius the slant side, and arc or base, the circumference of the base of the cone. But area of sector = ^- '- = ^ circumference of the base of cone x slant side ; or if (.?) be the slant side, and (a) the radius of the cone's base, convex surface of rone = h . 27r«r.< = Trrc. SOLID OF REVOLUTION. 169 Then, surface generated by arc PP' will be > than that by the chord PP' , < by PM and MP'. Now chords PP^ and PM generate truncated cones, of which the surfaces respectively are 7r{PN +P, N, } PP', and tt { PiV + MN, ] PM ; and MP' will generate a circular zone = iriMN^ — NiP^) -, and the surface generated by arc PP, dS d^S h" = ^r-h->r-— + &c. ax d.v~ 1 . 2 But {PN+PiN,\PP' = (2y + ph + Ph') . \/A' + iph + Phy = (2y + ph + PIr) !i\/l +p- + Mh, Mh = terms involving h ; and (PN+MN,)PM • = {2y + ph) . \/h^ + p'li' = {2y + ph)h \/l + p^, also MN,'-N,P,' = (y+phf-(y+ph+ Ph")- =-Ph~(2y+ 2py+ Ph^) = - N/r, by substitution ; •. + — ^ + &c. > 7r(22/ + ph + Ph^) \/l + p^ + Mh dx dx 1 . 2 < 7r(22/ +ph)^y\ + p^ - Nh > 'iTvys/ 1 +^" + ii//i+.terms involving A, < ^Tryy/l +p' + ph\/l +p^-Nh / yMh > 2 7r2/V H-JM- + — ^=^ + &c. •s/i +p' < 27ry\/l+p' + ph\/l +p^ - Nh:, 170 SPIRALS. dS n/ 1 + p-" = 2 7rtJ 129. The expressions that we have just obtained, and those of the preceding Chapter, are only applicable to the cases where the equation to the curve is known in terms of the rect- angular co-ordinates; we shall now find corresponding ex- pressions for the perpendicular upon the tangent, the area and length of a curve, &c. when referred to polar co-ordinates ; that is, when r = f{0), or p =f{T), p being the perpendicular on the tangent, r the radius vector, and the angle traced out by r. First, to find the expression for the perpendicular on the tangent in polar curves. SN = X, SP = r, NP = y, SY = p, and lASP = 0. Now, Art. 117, SV = p dy dec das'^ But ,x = SP cos PSN = - r cos =/(0), y^SP sin PSN = + r sin = (0), dy ana — = i -^ i — = , dx \ddj \dxj dw dO dw d0 P dw dy •^ d9 dd /d.^ df de^ dO' ^ dw , ^ ^ dr But -—= + r sni f? _ cos . — - , du dv dy dr and -f- = r cos + sin (?.-—; dO cI9 SPIRALS. 1^1 •'• 2/ ^7^ - -^ ^T^ = '■" sin-0 + r cos-0 = r, ~r A • P whence p may be found in terms of ?• and 9; but the formula may be put under a form more convenient for practice. Thus, _ 1 ^ dO' 1 1 c^r^ I 1 du dr 1 ^ du~ f dff' Example. Find the value of p in the Conic Sections. m r = 1 + e cos ' where m = X latus rectum ; 1 e .". U = (- — . COS0, m m du ^ • n. -— = . sm ; dd m du^ 1 ^*' + 3^o = — -{l +2ccos0 + e'i — , . (2w? ?t - 1 + e-) ; •.• e COS = mri - \ , m 172 SPIRALS. 1 (2m - r(l -e')l .-. p^ m^. r 2m — r(l — e~) mr (1) In parabola, e=l; .-. jf = — , and m = 2;SA; . SY' = SP.SA. (2) In ellipse, e 1 ; e^ - 1 = — : • '• p = 7-; ^ = ~ ' 2m + r(e^-l) 2a + r and therefore in ellipse and hyperbola, SY^ = — jz^ — Cor. 1. Since if * = arc of a curve, dx dor If 5, X, y be functions of 0, ds ds dd dy dy dd — ^ , -, • ";; — , and — = ~~rz • ~^ •> dw dO dx dx dd dx d^_^di^^^r;;d? which is the differential coefficient of s=f(9)- SPIRALS. 17*^ Cor. 2. Assuming the expression p = 1 1 \ dr- li = :7. + r X/ r" + - — ^2 ^2 ^A ^01^ ^^' .Wl n ..^'-P r* = r . d0^ \p' rV p^ dO_ P dr ry/r'^ — p^^ * whence given 0=f(r), we may find p = (p(j). ds ds dr Cor. 3. Smce 37^ = 3- x -rr. ^ dfj dr du ds / ^ dr' dr / ^ dG' ar a r r- - p dA r~ Cor. 4. If J = area ASP, -rr = — d6 ~ For ASP = ANP - ^iVP = ANP - ^ ; 2 dA dx ^ ( dx dy\ , ( dx dy\ = lr". (Alt. 129.) 174 SPIRALS. 130. To draw a tangent to a spiral. P the point to which the tangent is to be drawn. y T S the pole. Join SP. Suppose PT to be the tangent. Draw SY perpendicular to PT, and ^7" perpendicular to PS. ST is called the sub-tangent. And ^'r = ^P.|^=-7==^ = alsor-.^. PY s/r-j)^ dr Find therefore from the equation to the spiral ^ , or clB r^. — , according as the equation is 6 =f(r), or p =f(r). dr Draw ST perpendicular to SP and equal to either of these values. Join TP, it is the tangent. CoR. Since *S'r= =tr-—==F^, "^ X^ O^ O-t dr dii J 1 ldu\ . ^t^ ^d$ ST' KdOJ ' dr d^ 1 „ {du\- 1 1 = w + -77; = ^^^. + SY-' \d0) SP"- ST' 131. Asymptotes to Spirals. If ST remain finite when SP is infinite, a tangent may be drawn which will touch the curve at a point infinitely distant from S, and is therefore an asymptote. And since those lines are said to be parallel which coincide only at an infinite dist- ance ; the asymptote must be drawn parallel to the infinite line SP. EXAMPLES. 175 dB Hence to construct, we must find 6 and r^ — when r is dr infinite. Draw SP at the angle found by making r = co , ST perpendicular to SP, and from T draw TP parallel to the infinite radius vector, TP produced is the asymptote. EXAMPLES. o" (1) Find the equation between j) and r, when Q = —^ 9= — = a"u"; J." ti"^ a" du 1 ^'1 1 -. • — = = by substitution ; dO na\u"-' 6" '' U- + du^ _ 1 r-»-''^_ 1 |6^"+7-^"l ^ = 7' + ^i« - 7' \ 6'" J ^ dO b".r ••• 7^ = a/ 6-" + r^" (2) 'Draw a tangent and asymptote to the spiral ; where a 9 = - = mi; r 1 du 1 ; .. iS I = a; ST dd a or the locus of T is a circle radius = a. Since ST is constant, and = tT when r = CO . Produce SA indefinitely. Draw ST perpendicular to it and = a. Then a line from T parallel to SP will be the asymptote required. 1^6 EXAMPLES. (3) If 7'^= a^ COS 20 which is the polar equation to the Lemniscata ; find equation between ]} and r. Here m^ = •. cos 2 a^ cos 2 ' 1 2 2 ■ d0 1 du li^a^ sin 20^ and sin 2 d0 1 du u\/ a^ u'- 1 1 -U- + ldu\^ \d0] ~ u' + a = a' *• p-- ^a'' (4) Let r = a^, the equation to the logarithmic spiral ; .-. h. 1. r = h. 1. a = A0; dr dO p 1 or dr r\/r"-p^ ^i' y/r'-f A PY A that is, Z .S'PF is constant, and on this account the curve is called the cquianoular spiral. EXAMPLES. 177 Cor. 1. Since = A; .'. — = 1 + ^*; P p" P 1 " = sin SPY = , ; r V 1 + J^ r '• p — — ; ■ = mr., by substitution. Vi + J:' Cor. 2. The radii including equal angles are propor- tional. Let SP and SPi including an / a, and SQ and SQ^ include the same angle. Let z ASP=0, and ASQ=-(p; .-. SP = a\ SQ = a'i>, SPi = a^+«, SQ, = 0^^+" ; SP, ^P = a", and SQ, -=a"; ^'P SQ SP, SQ, ' or SP : SP, :: SQ : SQ,. Cor. 3. Given the ratio of SP and SP„ which include an angle a, find a. Let SP : SP, :: I : 1 + c. But SP=^a\ and 5Pi = o^+"; ^SP = 1 + c = a" ; a .-. h. 1. (1 + c) = a h. \. a = aA = , if /3 = constant Z SPY, tanf5 or a = tan /3 . h. 1. (l + c). M CHAPTER XI. CURVATURE AND OSCULATING CURVES. 132. When two curves, as QPQ^, RPP], cut each other in the manner represented in the figure, the values of y and x are the same for both curves at the point of intersection ; i. e. if y=f{oc) be the equation to the curve RPP^, and y = (f> (x) the equation to QPPi, and AN = a, and NP = b, the values «, and b put for X and y will make the equations b = f (a) and b = (p (a) true equations, and .-. /(a) = ^(a). 133. But if for .r, a + A, be written, (or as we shall put it, a? + h,) the values of the ordinates of the two curves no longer become equal, and their difference, which is represented in the figure by PiQi, is equal to the difference between f{x + h) and (p (x + h), and will therefore be some function of h, and its value will depend upon the relations existing between the differential coefficients of f{x) and d) (x). For, let y, = iV,P„ y, = N, Q,, z =/(.<), and v = (p {x); d. d~ X h~ z h' dx dx'1.2 dx^2.H + &c. dv d^v h^ d^v h^ and 2/2 = 2/ + -7- /i + -;— , + ^-r, + &c. ; dx dx^ I .2 dx^ 2 . 3 (dv (^^\ J /^^^ d^ z\ h~ \dx dx) \dx'^ dx'y 1.2 or putting A^ A^A^, &c. A^ for the coefficients of A, /i^, h^, &c. THE CONTACT OF CURVES. 179 The distance A between the curves, or the difference between the ordinates, is represented by a series with ascending powers of h, so that A = AJi + AJi^ + A^lv" + AJi"" + &c. + AJf + &c. d V d ^ Cor. 1. First, let J^ = ; .•,— = —, or the first da: cLv differential coeflScients are equal. ^ dv dis , . -But -r- and — represent the trioronometrical tangents dx d.v t^ » i' of the angles which the tangents of the two curves at the point P make with the axis of w. Hence at such a point the ordinates are equal, and the tangents are coincident. This is called a contact of the first order. CoR. 2. Let not only Ji = 0, but Ao = 0, therefore we have This is called a contact of the second order. And in o-eneral the curves are said to have a contact of the n^^ order when the first power of h, in the expression for A is h"'^^ ; ii e. when all the differential coefficients as far as the (n + 1)"^ are respectively equal in both series. 134. To find the degree of contact which a proposed curve of given species has with a given curve of known dimen- sions. Let y = f{.x) be the equation to the given curve, and y^ = (p (,r,) the equation to the proposed curve, which is sup- posed to contain 7i arbitrary constants. m2 180 OSCULATINCi CIRCLE. Then, to determine these n constants, we must have the n equations dy dy, a^y d'y, d"-'y d^-'y, y=''- di^=d^,^ ^=d^' ^"^^:^ = d;^' or the contact must be of the (w - 1)'^ order. Thus, let it be required to find the degree of contact which a straight line may have with a given curve ; we observe that the equation to the line is y^ = aw^ + 6, and contains two arbitrary constants a, b, or the contact may be of the first order. / Next to determine the line which has a contact of the first order with a curve. In this example — = = a ; and .-. y = yi, and .r = a?i, d.v dx^ dy .-. y = ax + h^ or b = (y — ax) = y x ; dx therefore substituting for a, and 6, dy dy d u or y^ — y = ~- (jx-^ — a), which is the equation to the tan- gent, or the tangent has a contact of the first order, with the curve which it touches. 135. In the circle of which the equation is R'={oo,-ay+{y,-(iy there are three arbitrary constants, the radius R and the co-ordinates of the centre a and /3. The circle therefore may have a contact of the second order, and the constants may be determined by means of the equations dy dy^ d^y d"y, dx dxi dx' dxx 2 RADIUS OF CURVATURE. 181 The circle so found is called the circle of curvature, and its radius the radius of curvature of any point in a given curve. For since the curvature in the same circle is uniform, while it varies inversely as the radius in different circles, and that curves are geometrically said to have the same curvature, when at a common point, they have the same tangent, and ultimately the same deflection from the tangent, which con- ditions are both fulfilled by the circle that has a contact of the second order ; this circle is assumed to be the proper measure of curvature, and curves are said to have the same or different curvature, according as the radii of these circles are the same or different, and the curvature in general oc radius of curvature The circle of curvature is also called the osculating circle. 136. To find the radius of curvature, and co-ordinates of the centre of the osculating circle to any proposed curve. Let y =f{aj) be the equation to a given curve, R"- = (w^ — a)'' + (yi — fiy' the equation to the circle ; ••. o = {.v,-a) + {yr-fi).p^ ..(1), and0=..g.(,,-/3).g (.). 1. , ' ■ ^fydy, d'y d'y, But y = yi, 5' the pole. PO a normal, and O the point of ul- timate intersection of two consecutive nor- mals. O is the centre of the circle of curva- ture. SY = p, SO = rj Now SO' = SP- + PO' ~2P0. PN, or r,^ = r"- + R^-2R.p for PN = SY. Then since SO and OP remain constant, while SP and SY vary, and since p -/{r) ; dp r-R dr R = r,-~ dp dr dr dp If OM be drawn perpendicular to PS, or PS produced, PM = ^ the chord of curvature through S, 1 „,, SY dr p dr and P3f -^ PO x-— = r .—-.-= p .— . SP dp r dp 143. E volutes to spirals. The point O will trace out the evolute, and PO is always a tangent to it, and SN is perpendicular to PO, we must therefore find the relation between SO and SN. Now r^ = r^ -\- R~ - 2 Rp . id pi = PY = V y^ - p^ an( 0), (2), SPIRALS. 189 and p=fir) (S), and R = r.4- (4), dp between these equations p, r and R may be eliminated, and the resulting equation will involve r^, pi, and constant quantities, which will be the equation required. Ex. Let the spiral be the equiangular. Here p = rsm (3 = mr ; rdr T dp m and pi = vr^ - p' = rvi - m^ ; R = m V 1 — «r and p = mr = m K = But r{' = r + R^-2Rp; m'^ 2 Pi' :Pi' 2;?,^ r/ = + ^ 1 - to"^ ?72^ (l - m~) 1 — W" .-. pi = mri, or the evolute is a spiral similar and equal to the original, and described round the same pole S. 144. When two curves intersect, we have seen that the distance between them, measured along the ordinate is, (when oj becomes a' + h) expressed by the equation A = A^h + Azfi^ + Ash^ + AJv" + AJv' + &c. If therefore we put {- h) for /i, we shall have an expression 190 CONTACT OF CURVES. for the distance between them at a point where the abscissa is ,v — h : let A] be this distance ; .-. Ai = - AJi + AJi^ - A^h^ + AJi' - A-Jt" + &c. Now assuming that h may be taken so small that any one term shall exceed the sum of all that follow it. We observe First, that if J, = 0, A and Ai have the same sign, or that in a contact of the first order, the curves touch, but do not in- tersect. Thus the tangent does not cut the curve, unless Ao = 0, or at a point of contrary flexure. Secondly. Let both J, = and Ao = 0, or the contact be of the second order. Then A= A^Ii" + AJi' + he. A, = - A^h^ + A,h' - kc, which have different signs, and therefore if the osculating curve be below the given curve at a point where the abscissa is ,27 + //, it will be above it at a point Avhere w becomes x — h. Hence the circle of curvature both cuts and touches the curve. There is an exception to this rule, which is when the radius of curvature is a maximum or minimum ; for then A^=0, and the expressions for A and Aj have the same sign. If the contact be of the third order, A = Aji^ + AJi" + he. A, = AJi'-A-Ji' + kc; that is, A and A, have the same sign, and therefore the oscu- lating curve does not cut the given curve. From this reasoning it is obvious that, when the contact is of an even order, the osculating curve both touches and cuts the given curve, but when the contact is of an odd order, it merely touches it. CONTACT OF CURVES. 191 145. When the radius of cvirvature is a maximum or minimum, the contact is of the third order, or Ao= 0. dy d'y d^y (1 + p')^ Then R=- ^-^ . dR But if R be the maximum or mmimum, — — = 0; dx / 7, (1 + p1^ 9 d^y 3pq" or r = -— - = 5 . dx^ 1 +/> But from the circle, and if there be a contact of the third order, we may differen- tiate this equation again, and put the co-ordinates of the curve for those of the circle; .-. Qpq +pcj + (y - I3)r ^0; .-. 3pq = - (y - fi) . r, and I + ir= - (y - (3) .g ; 3pq r " 1 +p^ 7 ' ^ 3pq' I + p~ The same result as before, and therefore when ^3, which is the difference between the third differential coefficient of y =/(a?), and of R~= (x - a)" + {y - /3)', equals 0, or when the contact is of the third order, the radius of curvature is either a maxi- mum or minimum. 192 RADIUS OF CURVATURE. EXAMPLES. (1) Find the radius of curvature and evolute of the common parabola. dy dx 2a + — , y d'y dx^ 2a X dy dx J df 4a2 4( 9 '^ x- + y~ 4a (a +x) dx' '%' ^" f W 1 + df\l _3 „ / "^ ^j {^a:{a^x)\l 2(a + a?)5 xJut XI = ~ d'y 4a^ Va dx" \/ a dy^ /3) = // = d y ^a^+ y~ -if (a^ + aa?J dx^ yx -y-^ a ' •• ^" a "4a^' ^^ / o\ 2a y (x + a) y a (a — 2a '*-«=-/• (2/ - i3) = - — „ = - 2 (a is impossible, e e EXAMPLES. 195 •. d' R o / ; o d J and --- = - .S e: v d^ - e'x~ + ^e-x -— v « — e^x^ dce^ ax = — 3e^a, if x = 0; therefore jR is a maximum, when x = 0, or ?/ = ± 6. Hence, at the extremities of the minor axis the circle of curvature touches the ellipse. (4) To find the equation to the evolute. ^ dx^ (a- - e^x^) \/ d^ — x^ y {or — e^x'^) d^y ha h^ dx^ (a' - e'x' ] ye\ ^ ^3 (aey ^ -^ dy (d~ - e^x^) y/ d^ - x^ b x x-a= -(y-(3). dx ha a y/d^ — w^ (a^ - e^x^) e~x^ a^ or a?V - a X- (a)3 n> ae~ a^ algs •J 9 X' If K"t - + V-, = 1 ; a- h~ 2 y, /-Iv 2 a^ (6/3)3 • y- - = 1 or (a a)' + (6/B)' = (ae)^ = («■ - b^)l n2 196 EXAMPLES. (5) Radius of curvature of cycloid. AN=x , AB = 2a\ '^ 'vy aa!~ y dy d^y -a dy dx dx' y'^ dx ' ^y a dx^ y^ ' laY y 2a ^ /2a y^ / \y I a y y a ^ Evolute. ^ 2a ?/ a/2 aw - 2/ -2/ 2/ = - 2\/2ay - y-y dtf x-a=-{y-(i)~=-2y dx .-. a = X + 2 \/2ay - y^. Take therefore, AM = X + 2 \/2 ay - y- = AN + 2 Rn, and MO = NP, and O is a point in the evohite. Its identity with the cycloid may be thus shewn: ~=1+ ~(^-y) ^ ^ 1 2 (g - y) ^ 2a-y d^ \/2ay -y'^' dx y y ' da _ da d(i da dy da /2a —y dx dfi' dx d(i' dx dj3 y . ^ - A A^-y _ . /2a-(-^) • AOD in the figure should be a curve. EXAMPLES. 197 Take Am = .i\ = — j3, and mO — a = y^; da dy^ Ila — x^ y/lax^ — cc^ d(i dx^ .r-j Xy The equation to a cycloid, of which the vertex is A, and the diameter of the ffeneratino- circle = 2a. (6) Find the chords of curvature drawn through the centre and focus of an ellipse. Since by Conic Sections CD'+CP~=AC'+BC\ and CD^ PF= AC. BC. If CP = r, and PF = p, a~b~ p-— d^+h- - T- is the equation between p and r, measuring from the centre ; .-. 2 h. 1. p = h. 1. d'h^ - h. 1. {a? +b'- r') ; dp r pdr «""'+ b'~ — r" ' , , , , 2pdr 2(a^+b^-r^) qcD' .-. chord tlirough centre = = = , ^ dp p CP dr 2 (a" + b- - r-) 2 CD^ diameter = 2r — = = . dp p PF Chord through the focus. Here p' = 2a — r 2h.l.j9 = h. 1. &2+h.l. r-h.l. (2a -r); 2dp 1 1 2a = - + pdr r 2 a — r r>(2a— r)' dr _ r (2a - r) _ SP. HP _ CD\ ''' ^dp~ a AC 'AC' '^CD- .. chord = --^. 198 CONTACT OF CURVES. (7) Find the form of the parabola y = a + hx + cx^ , which has a contact of the second order, with a given curve at a given point. Make the given point the origin : then the equation be- comes y = bco + cvr^; ♦ dy d'y .•. — — = 7> = o + 2cA% and = (I = 2c. dx dx~ But at the origin x = ; .-. b = p, and c = -; qx- q f 2p phjf .-. y = px + = -. [x^ + — x + ^] — ; ^ ^ 2 2 V q q~)2q The equation to a parabola, of which the axis is perpen- dicular to the axis of x, and the co-ordinates of the vertex , and — - ; 2q q 2 the latus rectum = -. Cor. The general equation to the second degree, or y-+ {ax -t b) y + cx~ + ex + f= o, containing five constants, may have a contact of the fourth order, with a curve. And should there be a point at which a^— 4c = 0, the osculating curve is a parabola. Immediately before and after this point, or must be greater or less than 4c ; and therefore the osculating parabola is in- termediate between an oscidating ellipse and hyperbola. EXAMPLES. 199 EXAMPLES. (1) Uy'~ + a;^ = aa;-ay; which is an equation to the a cu'cle, R = — ^ . (2) In the cubical parabola where a~y = x^-^ R= x-^ ; 6a x and in the semicubical parabola where ay^ = ar^; (4« + 9^^)^ R= - 6a (3) The equation to the hyperbola being y' = —(x'^-a') ; R = - — ^ ; and the equation to the evolute is ah (aa)^-(ft/3)^ = (a' + 6')i. (4) In the parabola the chord of curvature through the focus = 4>SP. (5) If yx = a\ R = - ~- , and equation to evolute is {a + /3)' - (a - /3)3 = (4a)i (fi) The equation to the catenary is 9.y = a\e'' + e "j ; shew that the radius of curvature is equal, but opposite, y~ to the normal. R = . a (7) If r = «(l + cosO) ; find equation between p and >• ; 2\/«r '~r and shew that radius of curvature = ; and chord = — . 200 EXAMPLES. (8) In the spiral of Archimedes, if r = —7=. , shew that the radius = the chord of curvature. (9) Find the evolute of the spiral of which the equa- 2 2 tion IS p = e . -5 . The evolute is a similar curve, e — a~ ,22 2 2 2 *'i ~ ^1 1 ^ Pi = e .-^ g- ; and a. = — . e — a e (10) Find the chords of curvature drawn through the centre and focus of an hyperbola. (11) If y \/ 1 + -7-^ = a, be the equation to a curve a lv (the Tractrix) ; the equation to the evolute is — = — ~ da a (the Catenary). (12) The length of an arc of the evolute of the para- bola is expressed by CHAPTER XII. SINGULAR POINTS IN CURVES. 146. If in the equation to a curve expressed by y =f{,v), where y is the ordinate, and x the abscissa ; some value of a? as a makes any of the differential coefficients 0, - , or - , •^ the point so determined is called a singular point. (l) Let the values of the first differential coefficient be considered ; dy Since — represents the tangent of the angle which the ax 1 (III tangent makes with the axis of x, if — — = 0^ the tangent is parallel to the axis of x, and this circumstance generally indicates a maximum or minimum value of the ordinate. If — — = - , the tangent is perpendicular to the axis of x. diV dy If M = when -— = 0, then the axis of ,r is a tangent ax ^ to the curve at the origin. If a? = when — = - , then the tangent passes through the origin, and is coincident with the axis of y. When — = - . Many branches may pass through the point, as we shall see in the succeeding pages. 202 SINGULAR POINTS IN CURVES. If — — have a real value when — ^ = 0, the ordinate is a ao!' ax maximum or minimum, as in the annexed figures. cPy Before we proceed to investigate the values of -— j at these points, we must establish the following proposition: 147- Prop. If the ordinate y be reckoned positive, a d"y ■ curve IS convex or concave to the axis, according as is positive or negative. In the annexed figures, let JN=x) NP=y M -Vi r > and y = f(.v) be the equa- NN, = h^ tion to the curve. A S N^ Draw the tangent PM, its equation is dy Now at the point Pj, the equation to ^ the curve becomes N^P^ = f{a; + h), or ~a — ?y M N, N,P,=y + -f-h^ T^ -- + -A + &c. ; dx dx- 1 . 2 dx-^ 2.3 and for the tangent, putting x + h for .?•-, and N^M for y^, d X POINTS OF CONTKARY FLEXURE. 203 therefore the deflection from the tangent, or MP^ in figure (l) = iV^ M -N^P, = --^^ - T^ &c. .„ figure (.) = ,V.P, - N,M^^^ _ ^_.^ _ , S.C. ; and since Ir is positive, and that Ji may be taken so small, that the first term of the expansion may be made greater than the sum of all the terms that follow it, the algebraical sign d'^y of J/Pj will depend upon that of doc' But we have seen that when the curve is concave to the dx' 2 axis, J/P, = — ^;^ - — &c. ; and when convex to the axis, d^y h- TT 1 . . . it = + ~ i- &c. Hence whea y is positive, a curve is dx' 1,2 J f ^ I . 1- d~y . . . convex or concave to the axis, according as — '— is positive or dx'- negative, or generally according as y and — - have the same or dx'~ different signs. 148. Sometimes the curve after being convex to the axis suddenly changes its curvature, and becomes concave, the point at which the change takes place is called a point of inflexion, or of contrary Jlexure. If the tangent at this point be produced, one branch of the curve will be above it and the other below it, consequently d" n on one side of the point in question — '- will be positive, and QjX 7 9 u~y on the other side negative. Hence at the point itself — '— dx" must = 0, or cc, for no quantity can change its sign without passing through zero or infinitv. There is not however a point of inflexion corresponding d~ u to every value of x. that makes — - = 0, for not only must dx~ ■^ 204 POINTS OF CONTRARY FLEXURE. this equation be satisfied, but — -^ must change its siffn after having passed through the point under consideration. Also if the same value of ,v that makes -~ = 0, also makes doc d?y 1 Ml 1 • r. -~ = 0, there will not be a point of contrary flexure aoo For since — — is a function of x, write .v + h and oo -h for oc. dCG^ d'y and then — -^ becomes, on these two suppositions, either dry d^y , d^y /r -^ ^ ' dar dx' dx' 2 (Py But at a point of inflexion = ; dx'^ .-. the deflections from the tangent at points .i' + h and x — h are respectively proportional to dry d^y K~ dx' dx^ 2 d''y dS/ /r and - — h + -f- &c. , fZ.r^ dx^ 2 which have contrary signs if — - do not = ; but if — - = 0, dx^ dx^ and — ~ do not vanish, the deflections before and after the dx'^ point will iiavc the same algebraical sign, and the branches are both concave, or both convex, to the axis. And hence in general there may be a point of contrary flexure, when the first differential coefficient which does not vanish is of an odd order. POINTS OF INFLEXION. 205 Hence, to find whether a curve has a point of inflexion, d^y 1 put =0, or - , and if a be one of the values of x so ^ dx' determined, substitute a + h, and a - h for .37 in the expres- sion for — ;. Then if — :; be affected with different signs, dx' dx' X = a gives a point of contrary flexure. Ex. 1. The cubical parabola ary = ^r', x^ y = a^' dy dx 3x' a' (Py 6x ; and if x = 0, y = 0, dx' a" d- y If X be positive or negative, y and -— are positive or negative ; the curve is therefore always convex to the axis. d~ 11 If ^v = 0, -4 = 0. dor If X = h d^y 6h . ' M = —^ IS positive. d'x a" If X = — /i. d'y dx' 6h a „ is negative. 2 O The origin is therefor^ a point of contrary flexure ; also, since x = makes -p- = ^ ^^^ V = ^^ the axis of x is a tan- dx sent to the curve. Ex. 2. The Witch, y = ^—y/^ax - or, X ax — X' dy I \/2ax - X' —^ = 2a dx \ x'" — \/2ax — x' -2a- X's/'iax — x~ 206 POINTS OF INFLEXION. \/'2.ax - r,„2 - X'' + ax — x' = 2a^ X (3 a- . {2ax d'y ^2 ax — x^ 2x) dx' -" " X' (2, ax -x^) -x')^' 3a which = if X = — , and changes its algebraical sign, when 3a 1 3« , . , /. f- A and n are successively put for x. There are therefore two points of contrary flexure when 3a - 2a X = — , and « = ± — ;= . 2 a/3 Ex. 3. In the trochoid, find the point of contrary flexui'e. y = a{\ - e cos 0), X = a(6 - e sin 6) ; dy . .-. 75 = «e sin0, au d T* and — - = a (l - e cos 6) ; dO dy e sin dx 1 — e cos 9 ' d^y e cos 6 (l - e cos 9) — e sin" 9 dO dx^ (1 — e cos 9)^ dx e cos 9 — e^ 1 X (1— ecos0)- a(l-ecos0) e (cos 9 - e) ~ a (1 - e cos 0)^ ' -4 = if cos0 = e, and cos (9 + h) is < e, and cos (^ - A) > e ; SPIRALS. 207 COS 9 = e gives the point of contrary flexure, and 2/ = fl (1 - e^) = a 1 — -A = a a 149. Points of contrary flexure in spirals. A^— X-^ Let there be two spirals, one concave and the other convex to the pole. Take two points P and P, in each near to each other, and draw SY and SVi perpendiculars on the tangents at P and Pj, and let SY=p, SP = r, and SP, = r + h, and p =/(>•) the equation to the spiral; therefore if A be the difference between SYi and SY^ we have in figure (l), where the curve is concave to the pole, dp d'p hr A =/(r + h) - f(r) = J-.h + -JL + &c. ; but in figure (2), where the spiral is convex to S., A=/(,-)-/(r + A)=-^.A-f?il-&c.; dr dr- 1 .2 dp and as h may be taken so small that -— h may be greater than all the terms that follow, we see that the spiral is con- dp . cave or convex to S, accordmg as -— is positive or negative. dr 208 POINTS OF INFLEXION. Hence at a point of contrary flexure — = 0, and changes its sign immediately before and after the point under con- sideration. EXAMPLES. Let ?* = a d'\ find the point of contrary flexure, h.l. r = h.\.a->r7i h. 1. 0; 1 _ de \ dd e ^ a r" r ^ 1 dr nr nr na^r l-iHr. Butf= P ^^ r \/ r~ - p- p r" \/r^ — p'^ no," ■2 2 2 r'' n'a" r" + n-a" ~.= 1 + 2-^-r 2 - 2 " i^n If n n + 1 r « vCI - 1. _ » - 2 dp n + I 1 \/~l 1. i+J r " drw ^/i 2-"» . ^ r " + omitting the denominator ; n '' r " + w^a" MITLTIPLK POINTS. 209 LI 11 .-. {n + l)r" (r" + n^a") - r" = o ; 12 2 .-. 7'" Inr" + n'(n + l)a"} = 0, whence r = 0, and >• = a| — yz .(??.+ l) p , n If — be a fraction with an even denominator, it is obvious that n . (n + 1) must be a negative number; 1 let n~ + n = - p; .-. w + J- = ^/^ - p ; n _ _ 1 h^'V^~P'-< ■'• P "lust never exceed ^^. If ^ = i? ^ = - i» and r = — 7= , or = —- the equa- tion to the lituus. MULTIPLE POINTS. 150. Whenever two or more branches of a curve pass through a point, it is called a multiple point ; and a double, triple, or quadruple point, according as two, three, or four branches pass through it. If the branches intersect, as in figure (l), I» / which represents a double point, there will be at P two tangents, inclined at different angles to the axis, and thus — will have ax two values corresponding to one of .v or y. Should however the branches pass through P, as in fig. 2. and touch each other, and the contact be only of the first order, there will be but one value fj ffj — of — - ; but as there are two deflections da? from the tangent, there will be two values of v^ A N d'y TV dx' O 210 MULTIPLE POI>JTS. 151. Problem. If u = f(cc, y) = be the equation to a curve, cleared of radicals, and there be a point where two or more branches intersect, — = - at the point of intersection. dx ^ Let the equation be differentiated, the result will be of the form ,^ ^y ^T , ,, du ^ ,^ du M.~-+N=0, where il/ = — and iV^= — . d) = -r- , II a? = and y = ; a^p .: p^ = 1, and p = i i, and the values of — - are + 1 and - 1, or — = tan 45, and dx dx tan 135. MULTIPLK POINTS. 213 Ex. 3. Find the same, when tr^ — ayai^ + hy^ = 0. Here 4tX'^ — ax'p — 2a yx + 3by"p = ; 9,ayx — 4cr^ .'. p = — ; — = - , when 0? and v = 0, Sby^-ax' ^ ' 2ap,v + 2ay -12off' 71. :; = ~ > n X = 0, bbpy -2ax 2ap + 2axq + 2ap —2^x d'y 6bp^ + 6byq-2a ' ^ " rf^ ' 4 1?, 1 d'^y 15 ,1 y = b + c {a + hy + h % and - — ^ = 2c H h^ ; dx~ 4' whence in consequence of the index 1, y and — '- have each d'y two values, but those of — ^ are both positive, and since CLOG 7 r 1 ^y 1 d~y a - fi put ior x makes y, — - and -~ impossible, the point €t/a<2a, y is negative, cT = 2 a, y = 0, x>2a, y is positive, a? = CO , t/ is CO . Again, let - /v, be put for a? ; X + 2a .'. y = — X X + a To draw the asymptote, is always negative. 1 - 2a y = x X a 1 X I 2a\ , a . 222 EXAMPLES. .-. y = x\\ ^ -&c.( X .V ' a 2 a- = X - a &c. ; X .-. y =z X — a is the equation to the asymptote. Take therefore AB = a, and AD = a, and the line BD pro- duced is the asymptote. Also take AC- 2(1. Then since y = 0, both when x = and x = 2a, the curve cuts the axis at A and C. Between A and B the curve is above the axis, and at B the ordinate is infinite ; from B to C the curve is below the axis, and from C to infinity is above Ax. Again, since if .v be negative y is negative ; the branch on the left hand of A is entirely below the axis. To find the value of dx' dy X -2a {x - a) - (x - 2a) dx X - a (.r - ay x~ — Sax + 2a2 a,v (x - a)~ ' {x - ay x" — 2 ax + 2d' {x - ay dy Let X = a; .-. — = cc ; dx therefore the infinite ordinate through B is an asymptote ; a? = ; • -^1- = « dy dx or angle at which the curve cuts the axis at ^ is = tan ' (2), ■J7 = 2a, and—— agam =2, or angle at winch the curve cuts the axis at C is = L at A. EXAMl'LKS. 223 Since (w- -2ax + 9a~) ^ (x - of + a^, the roots are im- possible; therefore — does not = 0, or there is no maximum dx or minimum ordinate. . d'y 2 . (.^ - af - 2 5 (a- - a)2 + an - 2 a^ Aa. dx' But X a <9.a^ y is negative ; and x>2a, y h positive; therefore from A to B, and from B to C the curve is convex, and from C concave to the axis. d'y 2 a" . • • 1 If 07 be negative, -— = -^ is positive, but y is negative ; therefore the branch from A to the left hand is concave to the axis. ., cr" + 1 x+1 Ex. 2. Let y' = = {v - .z + i) ; X — 1 X — 1 /X + 1 y- V .'V x~- X + I. ^ X - 1 A the origin; Ax^ Ay the axes. If 0/ = 0, y = - -_^= is impossible, xl, y is possible ±, .p = X , ;?/ is X ± ; therefore there are two infinite branches extending above and below the axis of positive abscissas. 224 EXAMPLES. V^ - 1 For X put - cT; .-. y = ± \/ '^^ -(x^ + x + l), which is X + J impossible, if a? be < 1, and =0, \i x=\. If a7>l, and increase to infinity, y is possible ± and increases to infinity ; therefore there are two infinite branches which meet the axis A.%\ in a point C, if JC= 1. To find the asymptote : y = ^\/lL^.Vi- 1 1 I — + O .77 x~ 1 - X 1 + — + Sec = ± . a' (1 - — + -— r + &c.) 1 2a? 2a^- 1 + &c. 2J7 EXAMPLES. 225 = i j 1 + - + &c. i .r ( I + &c.) 1 ^ o > = ± .-J? > 1 + — + __ + &c. J ; ■^ 2 a;' x^ ^ .-. y = i (a? + 1) gives the two asymptotes. Take AD = AD, =^, and JC = 1 Join CD and CD, , these | lines are the asymptotes, and if through B an infinite ordinate a be drawn, two branches of the curve will lie within the angular j space formed by the intersections of this line with CD and CD^ produced. For these branches of the curve will always lie above the asymptotes. Since the ordinate of the asymptote is always less than the ordinate of the curve. This may be thus shewn. Let y, be the ordinate of the asymptote ; .-. y~ =- , and ^1^ = o;-^ + /r + 1 ; ,v — 1 ••• y~-y^ = ^zr^ ' 3.V + 1 «r (y + yd (y - Vi) = 4 . (cr - 1) But y + y, is essentially positive ; therefore y — y, is posi- tive ; or y >y,. By a similar mode of reasoning it may be shewn that the branches which extend from C, above and below the axis Cx,, lie between the lines D,C, and DC (asymptotes) produced. dy To find the values of -— . clx 2 h. 1. 2/ = h. 1. (,v^ + 1) - h. 1. (,r - 1) ; P 226 EXAMPLES. dx 2 3x^ 2x' - 3x^ - 1 ^X'+l X-]J 2(X- l)t . y/x^ + 1 ' which is CO , if x = \, or a? = — 1. Hence the infinite ordinate through B touches the curve at an infinite distance from Ax, or is an asymptote : and the curve at C where y = cuts the axis at right angles. Also since the numerator 2x^ — Sx^ — 1 is = — 2 when x = 1, and is = 3 when X = 2, there is some value of x between 1 and 2 which will d 7 J make — = 0, or « a minimum. Take AM this value, and dx MP and Mp will be minimum ordinates. Ex. 3. As a last example, let the equation be between polar co-ordinates. Trace the curve defined by the equation r = a (l + cos0), p e=0; .-. r = ff (1 + ]) = 2a, TT 9=-; .-. r = a. TT Let $ = — \- a; ■'. cos 9 = - sin a, 2 and r = a(l — sin a), or r < a, "" /I a = — , or y = TT ; .". r = 0. 2 Again, let 9 = (tt + a) ; .'. cos 9 = — cos a, and r = a (1 — cos a), which increases as a increases, and r = a when a = PO. EXAMPLES. 227 Affam, let 6 = ha; .-. cos h a = + sin a ; '. r = a('l +sina), which increases as a increases, and when TT a = — , or (9 = 2 TT, ;• = 2 a. 9 It is obvious that the curves in the first and fourth qua- drants are the same, and those in the second and third resemble each other in every respect. Take AB = 2a, AC = AD = a, and the points at which it cuts the axes are determined This curve is called the Car- dioide. EXA3IPLES. (1) U y = a.v + b.T- + c.v^; there is a point of inflexion a' (~) V' = ~; -> '•• trace the curve : there are inflexions if .r = ± a/2 * /.v — 9 i^) y = {^ -~) \/ ^ ? trace the curve ; there is a w conjugate point if .i' = 2 ; and y is a minimum if cT = — - . (4) y~ = ax^- — x^, a cusp of the first species at the origin, 2fl an inflexion \f ,r = a ; a maximum ordinate if ,v = — . 3 (5) If y = , there are two points of flexure; the 1 + ,r~ curve cuts the axis at 45°, and the axis of x is an asymptote to the two infinite branches. p2 228 EXAMPLES. (6) (a?^ + 2/")^ = 4aa?2/ ; a double point at the origin, p = and = CO , (7) ('X^ + y^y = ^a^x^if ; a quadruple point at the origin. (8) tf + 2axif - a,v^ = 0\ a triple point at the origin, 1 jo = ± — —- , and = x . V2 CHAPTER XIII. CHANGE OF THE INDEPENDENT VARIABLE. 156. From the equation y =f{oc), we have in the pre- ceding pages derived the values of the differential coefficients ax ax considering a? to be the independent variable : but as it is left entirely to our own choice which quantity of the two we may assume to be a function of the other, let us see what sub- stitutions we ought to make for dy d~y ax dx^ when y is changed into the independent variable, and x be- comes the function. Again, it is frequently convenient to make a substitution for X in the equation y=f(x), such that x^= r" V a J r a d~r «' - r' „ a + r _ y ■ '^ -^ •. r^ + 2 r' :■ ' -' - ,,. ' dr d!^- a* - r* a* + r* d0' ' d&' • — •> ' 7'" Tt - . b) a" a"- 3 a* 3aS- 3r' Ex. 3. As another exemplification of the formulas of the change of the independent variable, let the equation d^y .X dy y • , • i i . . - — h — ^ — :, = 0, where a? is the inaependenr, dx'^' I — .v~ dx 1 — X' 236 CHANGE Oh THE variable, be transformed into one where = cos~'a; shall be the independent variable. a; = cos 6 ; .■.-—=— sm 0, and —~ = - cos 6 = - x, ad dd dy^ and — = — = - — ^ ^ dne dx sin dO^ d9 d^y dy d?x d^y l dy d^y d¥~d^'d^ dff'~^^'''''^^dd and -— - = — — ^_^— ^-^^^_^i_ ; -r - _^ a.T?^ aa?^ , jani?(J LV //^ tf dO^ d v d/" v .'. substituting for x, -— , - — , and for 1 -cr'^ = sin-0, dx dx'' 1 d^y cos 9 dy cos 9 dy y shi^ ■ d&' ~ sin' 6 ' d9 "^ sin^O ' d0 "^ iiii^ " ^ ' d-y an equation which is satisfied by making y = Acos{9+ B). 160. Find the radius of curvature, where the arc is the independent variable. V dx~j , ds'i dy^- R = — ^— T"^— 5 and — : = 1 + — -r d^y dx^ dx^ dx~ But if X and y be functions of 5, 'dyV dy- dsj _lds\^Udxy (dyY\_ds\ dx\~ \dxj \\dsj \dsj j dx~^ ds INDEPENDENT VARIABLE. 237 ••• U) ^ (i) = ^' and fl ^V'= f— V dy d^x dx d~y , d~y ds ' ds'~ ds ds^ and = ■ ; dx'^ fdx\-' ds 1 R = dy d'^x dx d^y o 5 ds ds'^ ds ds~ which, by multiplying numerator and denominator by ds^, ds^ may be written R = — — -— — -— , dy d\v — dx dy where dy, dx, d~y, and d~x are the first and second differ- entials of y and x with respect to *. 1 dy d^x dx d^y '^^^'"' R^Ts'd?~d^-d?' _ dx^ dij^ But — - + ~= I; ds~ ds- dx drx dy d'~y .'. — . — J + — .—^ = 0; ds ds~ ds ds' d^y ' dx ds ~ dy ds' ds' d'x' ds' fdyy d. vd~y s ds' dy \ds'j ds d^x and - ds ds'- ds ds~ 238 CHANGE OF THE INDEPENDENT VARIATsLE. dy d-jc dx d'^y dy \dsV \d.r ds ds~ ds ds^ ds d'^x ds~ dw' But ' ^^ ' ' ds~l dy' \ds~ ds~ d'^y\^ fd~x\- / id^yV' (d'^x\~ ds dy^ ' ds 1 . / fd'yV' ^ fd'xy R Kds'J \ds- and R = /(d-'yy uPxy' d\v\- /dx- dy-\ /d^x\ W^rf^/ fd'^\' [d7) ds') Kds'J f^yy \ds'J [dy\"- d-x \ or multiplying numerator and denominator by d^, and using d~y, and d'~x^ for the second differentials of y and x, R= '' Vid"-yy+{d\vy' Ex. Find the radius of curvature of the catenary- Tj /-TT- o , , , s + y/s'+c' Here a? = v ^'" + *~? ^"d y - e, h. 1. : EXAMPLE. 239 doo , and — :- = ds v^c-'+s'^ ds' (e2+.9^)i' d^^ , and ' — cs /^\- uPyy _c' + c's' _ c^ R = C-+ s a;- CHAPTER XIV. FUNCTIONS OF TWO OR MORE VARIABLES. IMPLICIT FUNCTIONS. 161. As yet we have only treated of functions of a single variable, we next proceed to the case in which u =f{xy), where x and y are independent of each other, and the value of u correspondent to new values w + /i, and y + k^ of x and y are required. Now if ti is a function of x and y, or u =f{^'vy), u may vary on three suppositions; 1st, x may vary, and y remain constant ; 2d, y may vary, and x remain constant ; and 3d, X and y may both vary together. Suppose 7i = xy^, and let x become x + h, and y re- main constant ; therefore if u be the value of u, 71^ = (.p + h) y- = xy'+ y'h. Next let y become y + k, and x be constant, and let Wi be the value of u ; .-. u^ = X (y + ky= xy- -\- 2xyk + xk~. Again, in the equation u = xy'~ write x + A, and y + k for ?/, and let U2 be the value of y, that is, Wg =fi^' + '*» V + k) ; .'. U2 = {x + K) (v + ky = xy^ + y-h + 2xyk + 2ykh + xk~ + k^h, the same result as would have been obtained had we put // + /(• for // in n\ or x + h for x in 7/,. FUXCTIONS OF TWO VARIABLES: 241 162. Next considering the question in a general point of view. Let u = f{v,y)., then if y remain constant, while x be- comes w + h, we have by Taylor's Theorem, du d^u h" d^u Ji" f{x + h, y) =ti + -—h + -— ; + — — + &c. ; "^ ■^ dx dw 1.2 dx-' 2.3 or, if X remain constant while y becomes y + A;, du d"u k^ d^u k"^ f{x,y + k) = u + -—k + — -^ + -— . + &c. -^ ^ '^ ' dy dy' \ .^ dy' 2 . 3 Suppose now that x and y both vary ; or x become x + h, and y become y + k; it is not possible to make both these assumptions at once : but if we use either of the two series, for f(x + k, y) or f{x,y + k), and in the former put y + k for y, or in the latter x + h for x, we shall in either case have f{x + h, y + /c), and its true developement. Assuming the first expansion, -^ , ^ du d^u K~ d^u h^ f{x + h.y) =u + ~-.h+ -—, + — ^ + &c. dx dx- 1 . 2 dx' 2 . 3 du d'u l5ut u=f(xy), and therefore -— , -^-~, are also func- dx dx'^ - . .- , „ - du d^u tions of X and y, it thereiore y become y + k ; u, — , — ^ , dx dx- &c. will become functions oi y + k, and may be expanded by Taylor's Theorem, x being considered constant. Let therefore y become y + k; du d'^71 k' d^u k^ .-. u becomes u + -y- . k +-^— x h -rr- f- &c (a), dy dy^ 1.2 dy^ 2.3 . du d~u . du and to obtain the values of — — , 7—^, &c. we must write ^— , dx dx' ax d^u . . . • / X , &c. for u in the series (a); dx^ Q 242 FUNCTIONS OF d {—\ d'' (—] du du ' \d.v) ' \dxl k' .'. -— becomes -r- + , k + — — ^— . \- &c. dx dw dy dif 1 • 2 d . (^'~ d'^u d^u ' \dx\ ^j-z, -j—. + 5 ./r+&c. d.v^ dx- dy d f— ] d^u d^u ' \dx") -n -1—. + ; .k + kc. dx"^ dx" dy (du d . \ — . d^u \dx ijut It has been agreed to write — lor , , ° dy . dx dy which expresses that the function has been differentiated twice, 1st considering x as variable, and then making y variable : \dx^j . . d^u , \dx"', . and =— — IS written —^ — —^ , and is written dy dydx'' dy" d'"'^".u — , denotino; the differential coefficient when the func- dy^.dx"" ° tion has been differentiated 7n times with regard to x, and n times with regard to y. Making these substitutions, and multiplying the expan- d'U d^ii h^ sion of — by h, that of bv ? &c. we shall have dx -^ dx^ -^ I .2 ^, , ,^ rfw , d~u k' d^u k? f{x + A, y + k) = ?i + -— k + -—- ~ " + -— + &c. '^ ^ dy dy' 1.2 dy^ 2.3 dti d^u d^u hk~ ^ k + —hk + , ., , -■ — + &c. dx dy.dx dy.dx 1.2 d-u h^ d'u h^k -I H . (- &c. dx^ 1.2 dy . dx'^ 1 . 2 d^u fir + -r-: + &C. dx^ 2 . 3 + &c + &c. TWO VARIABLES. 24 163. But this developement was obtained, by first sup- posing cV, and then y to vary ; but manifestly we should have had an equal result, had y first become y + k, and then 3) to have increased to x + h. On this supposition we have du d'^u k~ d^u k^ f{w, y + k) = 70 + —-k + -— ; + -— T, • + &c. ; •^ ^ -^ ^ dy dy-1 .2 dj/ 2.3 and then putting o! + h for x, dii dhc /r d"'u h^ u becomes u -f h + — — — \- — 1- &c. , dw \d£' 1 . 2 dx' 2 . 3 du du d^u d^u h~ -T- 3-+ 3 — r ''+ . V , • — - + &C. , dy dy dxdy dxdy 1.2 d^u d~u d^u h dy~ dy~ dxdy~ 1 +&C.; whence by substitution the total developement becomes du d'ti hr d^u h^ f(x + h, y + ^) = u + ^-h + ~~--^ + -—-—- + &c. dx dx^ I . 2 dx^ 2 , 3 du d^u d^u h^k + T-^^+ :r^r ^^ + tti + Sic. dy dxdy dxdy 1.2 d'^u k^ d"u k'h + -—7 + ~ — --, + &c. dy- 1 .2 dxdy- 1 .2 d^u W + -7^ + &c. dy 2 . 3 + &c. CoR. 1. Since the series are equal, the coefficients of the same powers of h and k ought to be equal ; d'^u d^u dydx dxdy 0.2 244 DIFFERENTIATION OF and dy-dx dxdy'^^ d'u d^u dydcV^ dcc^dy ' &c. = &c. d"*" . u d'"+"u dy'"d,v" da''dy" Whence it appears that the order of differentiation produces no alteration ; or that the differential coefficient of u diflPer- entiated m times with respect to w, and n times with respect to ?/, is equal to the differential coefficient when u has been first differentiated n times with regard to y, and then m times with regard to x. d^u d'^u CoR. 2. Again, since — — = - — — ; dy . dx dxdy • • ^^ therefore by writing -— for u we have dx dx \dx dy . dx dxdy d^u d^u or dydv^ dxdydx and dydx dxdy d^.u d^.u dydxdy dx.dy'^ du/ d fi d tjt 164. Since -— , ^— r, -; — , &c. have been obtained by dx a.r dx^ the consideration of x alone being the independent variable, such differential coefficients have been called partial differ- FUNCTIONS OF TWO VARIABLES. 245 du d^u ential coefficients, and for the same reason — - , — — , &c. are dy dy^ also called partial differential coefficients, and these partial differential coefficients are frequently included within brackets, thus ( — ) is the partial differential coefficient with respect \dxl to X, and ( — | is the partial differential coefficient with \dyl 'du\ _ , /du Jy differentials of u, with regard to x and y respectively. -, (du\ , - (du\ . respect to t/, and — a.r, and — jrf?/, are the partial du du , . , . 165. The term -p- /i + -7— ^? which involves only the dw dy first powers of h and k is called the total differential of w, and putting dx for /i, and dy for k, is thus written ; [du\ fdu\ du = — - ]. dx + {-—] dy, \dxj \dy I or the total differential of u ^/(xy) is the sum of the par- tial differentials. Ex. 1. u = x'"y". tind dti and shew that - — — = - - dydx dxdy du\ , (du\ dxj \dy I -. du = mo;'""' . 2/" • dx + nx'^y^'^dy = a?'"~^y'^ (jnydx + nxdy), ^ d^u d^u and - — 7— = 7imx"'~^ . «" ' = dydx * dxdy' Ex. 2. w =■ sin x'y, du „ du — =^ ^xycosx'-y, —-—x^cosx^y, dx ' dy d^u . . . 2 ^^^ = + 2' fdzi\ (du\ dz and thus obtain - — , and \dx') dydx \dy" We must consider these equations as functions of x^ y, %^ (du\ (du\ fdu\ , and that -— , H— and -— are also functions of the \dxj \dy ) \dzj same variables. 254 IMPLICIT FUNCTIONS. Let equation (l) be differentiated with respect to x, it must then be considered as a function of a- and z, and the fill dii differentials of — and — , may be derived from (l), by first doc dz ^ y J du J. du putting — — for ti, and then — for u in equation (l) ; d^-u\ d'~u dz d~u d z + -. ^--r- + doc'^ } dx.dz dx dz.dx dx d-u\ dz^ (du\ d'^z \dz' I dx'^ \dz I dx"' I'. d''zi\ d'~u dz (d''u\ dz" or ! — - + 2 . -, — - . -— + '\dx'' J dz.dx dx \dz' J dx~ Again, differentiate (2) with respect to //. We may obtain the differentials of — | and f — | from •^ \dyj \dz j (2) by first writing (-7—) for u, and then (-7—) for u in (2), whence we have ld~u\ d^u dz /d''u\ dz^ Kdy^ j ~ dzdy 4y \dz' j dy^ (du\ fd''z\ , ^ M,;^)-y-" --W Now either differentiate (1) with respect to ?/, or (2) with 1 • . 1 r. dz d'-z respect to x ; and since m the former case —— becomes — , dx dydx d'^ d % d iij and that in the latter -^— becomes - — 7— , and that dy dxdy dydx d^% , , .„ , . 1 . 1 , the results will be identical. dxdy IMTMCTT FUNCTIONS. 255 Let equation (l) be differentiated with respect to y\ and (du\ . (du\ ,. . -/si to do this put (— and (— tor w in equation (2), whence we have d^u d'u dz d'U dz + -, — 7--^- + dx.dy d.vdz dy dz ,dy dx fd'^u\ dz dz fdw\ d^z , ^ -f . — . — + — — =0 (.5). \dz^ I dx dy \dzj dy.dx d~z d'^z d~z From the equations (3), (4), (5), — — , , —7 and dx^ dy~ dydx may be found. By a similar process may the differentials of the third order be found. ELIMINATION BY 3IEANS OF DIFFERENTIATION. 173. We have seen that if a constant quantity be con- nected with the function by the signs ±, it disappears from the diff'erential coefficients. Should however it be multiplied into the function or any term of the function, it will still appear in the value of the diff'erential coefficient. Thus if M = be a function of x and y, involving a con- stant a, both 11 = and du = will contain a, but between these two equations it may be eliminated, and an equation will arise independent of a, which is called a diff'erential equation. Thus, let y = ax^\ dy I dy .-. — = Sa-r, or a = — . — ; dx 2x dx X dy 2 ' dx an equation from which a has disappeared. '^~2-dx' 256 ELIMINATION OF FUNCTIONS By differentiation also irrational and transcendental quan- tities may be eliminated. Thus, let y = {a? + x^Y '•> m ax n n{(r + x^) n {a^ + x^^ If there be two constants as a and h involved in the equation y =f(x), then to eliminate them, the equations u = 0, du = and d'^u = must be combined. Ex. 1. Let u = y — ax^ — bx =^0, or y = ax^ + bx; .-. — ^ = 2ax + b, dx d~y -4 = 2a; dx dy d-y dx dx- x^ d y dy „ d-y y ± -x-^ +.v^ — ^ 2 dx'^ dx dx^ d-y 2 dy 2y dx'^ X dx X' Ex. 2. Let y = a . cos mx + b. sin mx eliminate a and b. dy . , — = — ma sm mx + mb cosmx, dx — ^ = — m n cos mx — m^h sm m.x KY DIFFEllENTIATION. 257 = — ;//' ^a COS m.i + b sin 7nx\ = — my ; dw"' + m^y = 0. Ex. 'J. Let y = ne^^ sin {ii.v + 6) eliminate w an(! h. — = 2ae'--'' sin (.S.t- + />) + Sae~'' cos (."^a' + 6) dx = 2y + 3y cot (.'J.r + ft), — ^ = 4f/e-' sin (3.V + b) + Gae^'' cos (:i.r + ft) + Gae-'' cos (S.r + 6) - pr/^-' sin (r,,v + ft) = - 5?/ + 12?/ cot {3.V + ft) dy = - 5y + 'i ~ S // ; dv d'y dy .-. -^ -4 -^ + \3y -0. d,T'' dx 174. Again, if 11 =f{.Tyz) =0, or ^ =/(.riy). We may by means of the partial differential coefficients — and — eliminate two constants from z =f{ooy), and by dy d,v proceeding to the second differential, we have three other equa- d'z d'x ^ d'z 1 , J? /? . ^ tions for and , and therefore nve constants dx'- dyd.v dy' may be eliminated : and not only constants, but indeterminate functions. Ex. 1. Let X == f{a.r + by) ; eliminate the arbitrary function. Let {ax + ftt/) — r ; R 258 EXAMPLES. , dx dz dv and — = — . — , dx dv dx But -T-=f (v), and — = a; dv dx dz , dz dz dv and -— = —-.—- =/' (u) . b, dy dv dy •' ^ ^ dz f or 6.— = ahf {v), and a . — = ab ./' {v) ; .*. a — - = 0. a.2? ay As an example. Let ^ = sin (a<27 + hy) ; .-. — — = a cos (aa? + 6?/), and -— = h cos (a.2? + by) ; .•. a — = 0, dx dy and similarly if z = (ax + hyY, ox z ^ log {ax + hy), the differential equation will be verified. Ex. 2. Let z= {x + y)'"

' (y - ax). dy • • r2 260 EXAMPLES. Differentiating a second time, d^% — arf" (y + aw) + arcj)" (y - aw). =f"(y + aw) + (p"{y-aw) df 1 d^z a^' dx^^ d^z o d^z .-. a2 _— = 0. dx^ dy This equation occurs in some investigations in Natural Philosophy. CHAPTER XV. maxima and minima of functions of two variables. Lagrange's theorem. 175. If zi =/(*', y) be an equation between the function u, and the two independent variables, x and y, there may be some particular value of a,', and also a value of y, which will make the function greater or less than the values which immediately precede or follow it. It is then a maximum or minimum. We proceed to find the relation between the differential coefficients, when this circumstance takes place. 176. Let «, be the value of ?/, when x + h and y + A', are written for x and y respectively ; and u.^ the value of u when X - h and y -k are substituted for the same quantities. d"u ^ ^ d'u , ^ . d^u Also put A for — - , B for - — — , and C tor --^r • 1 hen dx dydx dy' ,^ u + — k + ~k+i \AK' + 2Bhk+ CkH + &c. dx dy ■^ and w. = u- { ~ h -r — k] + ^ \AK- + 2Bhk + CkH - &c. \dx dy I ^ Now since the values of h and k may be assumed so small that, (as long as the differential coefficients — and — remain dx dy finite) the algebraical sign of u^ — u and u.^ — u will depend upon that of the term du , du,\ — h + — k \, dx dy J 262 FUNCTIONS OF TWO VARIABLES. it is manifest, that if this term exist, u^ — u and u.^ — u cannot be both positive or both negative, or there cannot be a mini- mum or maximum of u. „,, - . . du du therefore at a maximum or mmimum -— h -\ k must dx dy = 0. A condition which can only be fulfilled by making du , du —- = 0, and -— = 0. dw dy Therefore at a maximum or minimum, u,-u = \{Ah~ + 2Bhk + Ck') + &c. h" = \A + 2Bn + CnH + &c. 1.2 by putting k = nh. Therefore the sign of u^ — u, and also of u^ — ?^, will A' depend upon that of the coefficient of — , that is, upon J + 2B7i + Cn\ Hence, this term must not change its sign whatever be the value of n ; which it will not do, if it can be put under the form of the sum of two squares, as (.r + a)' + /3^ Now A + 2Bn+ Cn~ = - \CJ +2BCn + C'n^ = ^ {CA-B- + {B + Cny\ Lj^cj-B'.e-{?:,.n)y, which will be of the requisite form, if CA be not less than B^ : or to have a maximum or minimum of a function of two • 1 1 ^1 du ^ du , J variables, we must nrst have — = and — = ; and second- dx dy d'u d^u [ d^u \2 'y- d7' ' df ■"" '^^' "^'•"" irf^-) ■ EXAMPLES. 263 It IS obvious that — - and must have the same sign ; dx' dtf * and if they be both negative, ?^ is a maximum, if positive, «* is a minimum. If the second differential coefficient of u become = 0, when the firs^ does, there will not be a maximum or minimum, unless the third differential coefficient vanishes, and the fourth neither vanishes nor changes its sign whatever be the value of n. Ex. 1. Let u = x'^ + y^ - 3axy, du ^ cc^ -— = 3x~ -Say = 0; .:y = -, a X a du — =3y^ - Sax = 0; dy X* .'. — - aa? = 0; a, therefore x — 0, and x^ — a^ — 0; whence x = a, the other two roots are impossible, x^ a^ and y = — = — = a, or =0. a a d"u d~u drii Also --— = 6x, —— = w, and - — -- = -3a. dx'' dy dydx If 07 = 0, J = 0, C = 0, and 5= -3a. If cT = «, A = 6a, C = 6a, B=^-3a, AC = S6a\ and B^ = 9a% and X = a gives a minimum, and u = — a^, ,v = gives neither a maximum nor minimum. Ex.2. 11, = x'^y^(a - X -y), find the values of ,r and ;/ that ?t may be a maximum or minimum. 264 EXAMPLES. ci tl ~— = SoG^y'{u — a- - if) - .r^y' = 0. d.T du — = 2 art/ (a — .? — i/) — .r'y~ = ; dy ' .". o(a — = , V~ 9 f) 4 9 J 9 ' dy' ^ -^^ -^[8 6 8 3} 8 ' d'u - — — = B = G 5', and A is negative, and M = — X — x-= IS a maximum. 8 9 6 432 Ex. 3. ?^ = (,r + 1) . (y + 1) . (;;? + 1 ), where «'"6'''c' = A ; .-. A" h. \. a + y h. 1. 6 + ^ h. 1. c = h. 1. A. XT <^^ / ^ f ^ ^ ^'-1 Now 3"! = (.V + • ■[- + Jl + (•*■ + 1) • -^\ = O5 dw ■ I dcc\ du ^ ^ { ^ ^ dz\ d% h. I. ff dz h. 1. b and — = - r-j— , and -7- = - ry- ; da; h. 1. c at/ h. 1. c EXAMPLES. 265 h. 1. a .-. ^ + 1 - (.r + 1).— -— =0, n 1. c or {x -\- 1) h. 1. c = {w + 1) h. 1. a, and .-. h 1.6 Also ^ + 1 - (// + 1) . ,— , — = ; rr+' = a'+i. and h. 1. c Al so z+ I = (a-+ I). h. 1. a (!/+0 h.l.h -. X = y h. \.c ^'' ' ■^ h.l. c' (.v + 1) h.l. a — logc logo (,PH- ])h.l. a -h.l. 6 log h ,v log rt + {.V + 1 ) log a — log 6 -f (am- 1) log rt — log c = log A ; .-. Set- log a + 2log « - log be = log J ; log ^ fee — 2 logo •. x = .-. .T + l = y + 1 u = slog (a) log^6c I- log« log (Abe a) 3 log a S log a log (A be a) 3 log 6 ' (log^ftca)^ log (abcy and ;? + 1 = log (A ben) 3 log c Ex. 4. In a circle of given radius, inscribe the greatest tri- angle. R the radius. a, 6, c the sides. = z 5, (p= iC. BC X AD ft = maximum, 266 EXAMPLES. and 2jff X AD = JB .AC, (Euclid, Book vi. Prop, c.) or 2 Re sin 6 = cb, or b = 2R sinO ; sin .-. c = b.—i — ^ = 2R sm + Q) = max. .-. — = 2^^ {cos0. sin {(p + 0) + sin 0. cos {(^ + 9)^ sin (p = 0, and =2R" \cos(p . sin {(p + 9) + sin . cos{(p +9)] sin0 = O; Cv CD .-. sin {(p + 29) = = sin tt, and sin (0 + 20) = = sin tt ; .-. (p + 29 = Tr, and + 20 = tt; .-.0-0=0, or = ; .-. 3 0= TT, and 0-60", and ^ = 7r- 20 = 60", and the triangle is equiangular ; and u = 2R-.sin^60 = RK^^^ . 4 Ex. 5. Inscribe the greatest parallelopipedon within a given ellipsoid. Let 211?, 2 1/, 2z be the edges, 2ff, 2h, 2c the principal diameters of the ellipsoid; .-, u — 8.T?y^ is a maximum. ,- z^' w- xP' But - + -„+•;, = 1 ; c a' h' lagrangk's theorem. 267 _ ''' 2 7 2 a du dz .-. — = 8y% + 8y.v—- = 0, d.v dx du d% — = 8a>» + 8v,r -— = 0. dy dy _, d% cc c'^ - (?;5f y c^ But — = . — , and -— = . — : dec % a" dy ^ b" x^ & , y c- \ z . — = 0, and z t> = ^ ' z a' % b~ z^ c/ - z^ y^ x^ - = -, and - = 7i= ••• -2^ & a^ c~ a 3x- a — - = 1 , and X = —^ ; • *i i • • 71 = • — ^ , ^" _ 1 . c anci — — jj , . . ^ — — -7^ , . 8 a 6c and u = ^> x/s' LAGRANGE'S THEOREM. 177- Let u=f{y), where y = x + x(p{y), and z is in- dependent of X ; required ic or /(.y) in terms of x. By Maclaurin, u = r7„+ fr r + U,^^ + f/3-^ + &c. + '^ + &c. 1.2 2.3 1.2.3 ...w 268 Lagrange's theorem. du d'u where Uq, U^, U.^, &c. are the values of — , -, &c. ; d.T dx'^ w hen at = 0. First, if a? = 0, y = z ; .-. Uo=f(z). du du dy du du dy Now -— = — . — , and — = — . -— . dx dy dx dz dy dz But -l = ci.(p'{y).'^ + (p (y), where (p'y = ^^^ dx ' dx '^ ^ dy dy (p (y) ' / \ ' dx 1 - x(p' (y) ^y , , ^' ( \ ^y . ^y ^ d% " d% dz I — x

{y) ~ ^i" 2/> ^"^ /(^) = ^- Here Z = = 1 ; dz y = x + ^'^^l ■ dz 1.2 d!^ 2.3 (.^) = sin ;y = sin m if .r = ; .-. |<^(2r)}- = sin^^; .-. ^ ^ ^ ^^ = 2 sm :? . cos ^ = sin 2z = sm 2m it a; = 0, dz {(f) (z)'^ = sin-'^; {d(b{z)\' .-. -^^ — ^^— = 3 sin'^cos^, dz d^{••'. But /3 and a are the co-ordinates of the curve in which the centres of the circles are found; therefore ^i is a known function of a, or 13 = (fj (a) ; ••• \y-(p(a)\''+Cv-aY = r^ (1.) Now if we suppose a to receive an infinitely small increment, the equation (l) will belong to an equal circle, the centre of which is infinitely near to that denoted by equation (]), and the two circles will intersect at a point of whici) the co-ordinates are ultimatelv 7 '■J 2 ,v v o ir no ^^ =0, orf. = -; «•* Ir a Ir a h = a — , and ab = rri- = a? — \ y Ir = mr — \ cc x" y' 1 a b m- m^ .-. wy = — , 2 the equation to the rectangular hyperbola. Prob. IV. Find the equation to the curve whose tangent cuts off from the axes, two lines the sum of which = c, V'V + y/y = vc Prob. V. Find the curve which touches all the curves included under the equation y = X tan 6 - 4! h cos'- 9 being supposed variable : x'^ = 4 + 7 + '• + &c.) Jr\dx dx dx J Jidx ^ p + 7 + r + &c. EXAMPLES. 281 or the integral of the sum of any numher of differential coefficients = sum of the integrals of each differential coefficient taken separately. Ex. Let — = Ax'" -r Bx" + C wf + &c. ; dx .-. u = Jjlx'" + Bf^x" + Cjlx^ + &c. x'"-'+ ,r"+ ' + ,v^+ ' + &c. in + I 71 + 1 p + I (). If - — = -'". — , where ;^ is a function of .r, find ?(. dx dx Since if u = z"-+' + C, du , , d% d% «'"+' .-. Lz'\— = -+ C; •^ dx m + 1 or to integrate a function of this description, increase the index by unity, divide by the index so increased, and by the differential coefficient of the quantity under the index. EXAMPLES OF SIMPLE INTEGRATION. d u (t x^ (l) Let -— = ax^\, dx ^ du a (2) Let — = — = ax dx X- 1 »' du - (3) Let -— = ax" ; dx m+n (4) Let -^= (ax" -f hy\v"-\ dx . . U/ — 4 ax-' a -2 . . U = •» • _^ — ' - 1 X 71 m + n .-. U = . ax " . 282 EXAMPLES. Let z = ax" + h ; dz dx = nax" ' 1 dz .-. (aw" + hy . x"-^ = — .z'".-—; na. dx .-. u=~. L.z'". 1 - dz x^'" + ' na dx na . {m + 1) {ax" + by" + ' na . (m + 1 ) du , ,, (ax + by-^' (5) — = (ax + by ; .: u= — • ^ ^ dx a- {m + 1) (6) — ^ (ax" + by .x% m being a whole number. dx Expand (ax" + by by the binomial, and after having mul- tiplied each term by a?', integrate them separately. ('7\ — = '- , m and n beina; whole numbers. ^'^ dx (a + bxy' ^ z — a Let a + bx = z ; .-. x = — - — ; b x^ {z - a.y (a + bx)" b'\z" ' du du dz du dx dz dx dz du I du 1 (z - o)" dz ~ b' dx ~ 6'" + ' ■ ^ 1 r(^z- ay u 6'" + ' J, z" Expand (z - a)'" by the binomial, and integrate cacli term separately, first dividing by z". EXAMPLES. 283 (8) -— = — — -—r~, m and w beinfj inteoers. 1 doo 1 lor * put -; .-. — = , iz dz z^ du du 2 «""^" and ^ = — r- . z" = doc dz (az + 6)" ' .-. n = - r- ^m + « — 2 Jziaz+by' which resolves itself into the preceding case. , ^ du 1 11 (9) -r = d,v a + bx'- a b a 1 + - .v^ JLet z- = ~..v-^ .-. z— -v/ - • '*■) and — = \/ - a ^ a dx ^ a du du dz du /b 1 1 dec dz dx dz a a V ■\- z'"' 1 r 1 or 11 = : / . . y/ab -J-- 1 +-" T^ •/> , du 1 iJut it ?< = tan~\5:, dz 1 + ^"^ ' '^ , 1 fb .-. «=-—=. tan-'^ =—-= .tan-\t"\/ _. \/«6 \/a6 a du 1 1 /^ I'X. -— = ^; .-. z^ = — -^tan-^,r\/ _. dx 2 + 3x~ y^g ^ 2 c^w 1 , , (10) — - = ; .-. u = h. 1. (a + x). dx a + X J ^ (2 + ocy 7- These simple integrals being found, it will be con- venient to classify the remaining functions in the following order. (l) Rational fractions of the form Ax''-' + Bx'' + CxP + &c. J,x"' + 5, .a?"' + C,cr?'> + &c. (2) Irrational quantities. (:>) Exponential and logarithmic functions which are of the forms «'/(«'), log(.i>), log (/^), p'" log ((/). (4) Circular functions which are of the form sin/>, /(sin p), &c. 'I'hc melliodh for lIk' iiitcgraticju of such functions will l)c given in the four succeeding Chapters. CHAPTER ir. RATIONAL FRACTIONS. 8. Every rational fraction may be represented by Ax'"-' + Bx'"-' + Ca'"'-' + &c. A,x'" + B,x'"-' + C,a)"'-- + &c. ' m for it is manifest that the index of x in the numerator can by division be made less by unity at least, than that of x in the denominator. To integrate this fraction we must first separate it into fractions of a more simple form. Now the denominator may be composed 1st of simple factors all different. 2d. Some of the factors may be equal. .Srd. It may contain quadratic factors, the roots of which are impossible. 4th. It may be an assemblage of all these. 9. We shall first consider the case where the factors are all different. Let therefore — be a fraction where V is the product of n factors all different, so that V = {x - tti) {x - a.,) {x - cTg) ... (x - a,). . U A\ A2 A^ Aj, Assume — = 1- -- ^ + 1- &c. V {x - a,) (c^? - a.2) {x -a^) ' ' {x - a,) " •. U=Ay {x - fto) . {x - 0,3) ...{x- a,) + A... {x -a^) . {x - a.^) . {x - a ,) + Sec. + A„ . (x - a^) (x - a,,) ... (.c - «„_,) 286 INTEGRAL CALCULUS. Successively make cr=o, , a.^, «3, &c. ; and let f/„_, f7„.,, U,i.^ &c. be the corresponding values of T/; .-. U,^ = A, (a, - a^) (r/i - a,) ... (a, - «„), or ^i, = (gi - no) (r/i - a^) Similarly, Ao = — — , and A _ 3 3 (cfo-«,)(a2-cf3), &c. (o;,-a,)(ff3-a2)... fiv) =^' f-^— + A, f-J— + A, f-l— + hc. = Ax h. 1. {x - n^) + A2 h. 1. {w - a.^ + ^3 . h. 1. {x - a^) + &c. = h. 1. (x - a,)^i (x - a^Y^ {x - aiY^...{x - a,J^". 10. Let some of the roots be equal, viz. m of them = «, or let {x — a)"^ be a factor of V. Let r - (.1? - a)'" Q. U A B C P Assume — = 1- 1 1- &c. + — ; V {x - a)'" {x - a)'"-' {x - a)"-- Q .'. U=AQ+ \B . (x - a) + C. {x - af + kc.\ (^ + P (x - a)'". Let X = a, and let U^, Q^ be the values of U and Q ; .-. f/„ = ^Q,„ and^ = ^; .-. U--!i.Q = (x-a) \[B+ C.(x-a) + D (x - af + &c.] Q + P(x -ay"-'\. Hence, as the right-hand side of the equation is divisible by (x — a), the left-hand side is also, let the division be effected, and let U' be the quotient ; .-. f/' = \B + C. (x -a) + D (x - of + &c. \ Q+ P.(.v- ay-'. RATIONAL FRACTIONS. 287 Again, make x = a, and we have B = -~- , and proceed- ing in the same manner we at length arrive at P, which is either constant, or a function of w ; if the latter, the case is reduced to that of the preceding article. To illustrate these methods, we will take two examples*. du x~ — 7\/-^ RATIONAL FRACTIONS. 289 Substituting and making the sum of the possible quan- tities = 0, and also the coefficient of v- 1, =0, M and N may be found. , Or if P be first found, subtract P {{x - af + (i^] from each side of the equation ; .-. U-P \{w - af + /3^} = {Mx + N).Q. Divide both sides by Q, and then M X + N = ^^^^ — - is known ; Q rU _ r Mx + N rP ■ ■ iv~ Jr {x - af + /3' "^ X Q ■ ^ . du Mx + N - To integrate -— = ^ , let a? - a = « ; ® dx {x - ay + fr du du Mz + Ma + N daff dz ^ + j8' %'' Mz Ma + N + u + ^' z' + ^' ' = i/ h. 1. ^/;^^= + ^^^ tan- (I = M h. 1. v/(.^ - aY + /3' + ^2_L^ tan- M CoR. If a = 0, or i3 V /3 du Mx + N d^ " x' + fi" ' u = i¥h. 1. s/x' + /3- + -g *^»^'' ^ 290 INTEGRAL CALCULUS. _, ^ du X - 2 a; - S Ex. 1. Let — =— r = dw x-^ + 1 (x + 1) (x"^ - X + 1) X -5 A Mx + N Let — = + x^ -\- I X + \ (x'^ - X + l)^ .-. X -3 = A{^x^ - X + 1) -{^ {x + \) {Mx + N), 4 X = — 1 ; .'. —4' = 3A,orA= ; 3 4> , „ , 4a?^ - X - 5 (4 2% -— = --h.]. (^+l)+-h.l.(.^2-.,? + l)-— ^tan-^ — 7- , , ly/x''-x+ i\t 2 _ 2.i?-l = n.l. I 7= tan > — -^ . UATIONAL FRACTIONS. 291 Ex. 2. Let -— = — -TT^r^ , which includes d.r Or + l)(a?+ 2)*(a7-+ 1) the three cases. U A B C Mx + N ^ Let — = + + + — , V ct? 4- 1 {x + 9.f X + 2 x' + 1 \= A.{x^ 2)2 (.x^'^ + 1) + [5 + C . Or + 2)} (^2 + 1) (.r + 1) + {Mx + iV) . 0^ + 1) {x + 2)-, 0? = -2; .-. 1 =5.5.(1 -2) = -5jB, i. e. 5= --, 5 a?=-l; .-. 1 = J . 2 = 2 J ; .\ A = -. 1 - ^^ '-—1 L + 2 1—1 L = C . (ct? + 2) ix + 1) (a; + l) 2 5 + (ilf 07 + N) . Or + 1) 0^ + 2)', (5a?^ + 18 0?^ + 23a?2 + IScT + 8) or - ^^ = (ct? + 2) . (c27 + 1) 10 _. \C.{x' + 1) + {Mx + iV) G^ + 2)}. Divide both sides by {x + 2) . (a- + l), or ^r^ + Sa? + 2 5.r- + ^x + 4 10 C 0^' + 1) + {Mx + iV) Or + 2). Q 9 Letcr=-2; .-. --=5C; .-.0=- — ; 5 25 9 0r^+l) 5a?"^+3,r+4 (70?^+ 15^^ + 2) .,^ __. , . 11 L = _ -"^ L = (Mx + N) (x + 2), 25 10 . 50 (7 — c 4c'* (1) Let->-^g, or 4ffc>6^; =-■/" — ^ c Jz ^ 4a iac - o^ 4c'^ Jz z + a a 1 * tan"' — ; a 1 2c _, / 2csf •. u = - . / = • tan , , VA/4ac-62/ 2 , 2ca? + 6 . tan-^ a h^ , o ^^ - ^o-c (2) Let - < — -„ , make a' = 5 — ' ^ c 4c 4c .•. u c J^^-ot 2ca JA^-a % + aJ ^.h.i. '"-" 2ca \z + uj 1 2ca? + b - y/b^ — 4>ac , h. 1. 7- \/6'^ - 4ac 2ctr + 6 + V 6^ - 4aG Ex.1. Leta = 6 = c=l; .-. \/4ac - b^ = y/3, r 1 2 A 1 + .1? + ,r"' a/q 2 2a? + 1 —7= tan 1 — 7^- . V3 VS 298 INTEGRAL CALCULUS. Ex.2. Let c = fe = 1, and a = — 1 ; .". vfe~ — 4ac=\/5; ••• /-T-i— =4=h.i.f?iiil^^). 18. To integrate cC" (a + bx + coe^)" a?"' 1 0?" (a + bx + c\rO" c" / ^ 6 a U?' + - J- + - V, c c Let X + — = ^, or cT + a = ^, if a = — ; 2c 2c o b « , a *' / 2 . /32X C C C 4-0 ' ■ J.V (a + ba! + cvy ^?Jz (F±^ ■ Here are two cases : /XT a b~ r (i^ — a) 1 /> 11 ^^^ ^* c^4?' ^^" i.C^N^W ""^^ the method used in Art. 15. (2) Let -< ^' • . r ^^ " "^'" r ^" - «)'" must be integrated by the method of partial fractions. 19. Again, to integrate a?'" (a + bx + caf'')" 1 d^ I Let uV = ~; .-. — = ^— — z ; z dec iV du du d% ^du d.v dz dw dz ' z"'^^" ^du {az' + bz + cy dz ' RATIONAL FRACTIONS. 299 . 27??7r 2m(;-+l)7r . 2w?7r .-. Mn sin = 2 cos . sin , n n n ,, 2 /2m . (r + l) tt or M = - . cos n V w 2 . 2m . (r + 1) TT . 2W7r and iV = . sin sin n n n 2 2m (r + 1) TT 2m7r . cos — . cos n n n 2 ^mr . IT = — — . cos ; n n 2m (r + 1) TT 2m r . tt _, »T^ ci? . cos cos Mx -^ N 2 n n „ 2m7r n „ 2m7r .V'' - 2 cV cos — — +1 .r- - 2.3? cos + 1 n n Case I. Let n be odd ; .-. Q = * - l, and P = A; v" — 1 also w'- =f(Mx + N) . (x - I) + A / where /. (Ma; + iV) represents the numerator of the fraction, formed by reducing the fractions having quadratic denomi- nators into one. L.et X =z 1; .-. = n, and A = — . X - 1 n 304 INTEGRAL CALCULUS. .-. / «=-•/ + -• JxX^ - 1 n JxSo - \ n 2m.(r+l)7r ^mr.tr ' W COS COS n n „ 2mir or — 2cV cos 1- 1 n , . „ , - r Mw + N the latter integral is of the form / — ^ J^ x^ - 2ax + 1 = M .\\.\. y/ a^ - SacT + 1 + Ma + N putting for M and N their values ; tan -1 SmTT w — a y/l-d' 2m7r and since a = cos ; .-. v 1 - « = sin w n Ma + N 2 . 2m.(r+l).7r , Also — y = .sin — , we have Vl - a' n n r cc' 1 /• 1 2 f / = — . / 1 — { cos . J^a"- 1 n J {oo - I) n\ 2m (r + l) TT w h.l. \/c'i?'^ - 2,r 27n7r . 2m{r + 1) tt cos + ] — sin n n 2w7r /iV — cos tan -1 n 2nnr sm w where m must be taken from m = 1 to w = w- 1 CoR. 1 . If r = 0, we have / — — - J X — 1 I o Q/wtt / 2m7r = - . h. 1. (.r - 1) + - . cos . h. 1. V ''^^ - 2.1^ cos + 1 2 r« TT 2m7r ,t? — cos .sin n n . tan -1 n 2rmr sin n BATIONAL FUNCTIONS. 305 Cor. 2. If we add to the integral the constant quantity '2m7^^ '2m (r + 1) . TT^ — sin n — cos tan-V n 2 m IT sin 7» Since tan"^ {A- B) = w' 1 A-B 1 + J5 •. f— — = — .h.l. Or-l) 2 /2w. (r + 1) 7r> + — . cos n \ n j h.l. V a?2-2cr 2»W7r 2 . 2w (?• + 1) TT cos hi . sm n n n 2imr J? sin tan -ij 91 1 —W COS 2nnr n d / = - h. 1. (.J7 - 1) + - cos , , , /~ 2wx 2 . 2m7r h, 1. V^ ^^ - 2.17 cos +1 .sin 2m7r ajsm tan~^ X n \ — X cos 2m7r 1 1 92 Ex. f— = -h.l. (a?- 1) + 2 Stt - cos 5 5 AT 2^7r 2 . 27r , \/ a?^ - 2,r cos — 4-1 sin — tan ^{ ^ 5 5 5 U a? sin 27r 1 — ^ cos 27r 306 INTEGRAL CALCULUS. 2 ^'^u 1 a/~^ ^ + - COS — h. 1. 'v a?^ - 2x cos — + 1 2 . 47r _j — sin — tan ^ 5 5 .h.l. V x^ -2a; cos i W + 1 2 . (n- l).(2m + l)7r — — . sin n n tan -1 . {2m + 1) ' cr.sin TT n 2m + 1 \— W cos I TT + C. n 23. To integrate w'" - 2cV" cos a + 1 The quadratic factors of the denominator will be all com- • 1-7 9 /^ in 2w7r + a prised in the term x" — 2.v cos y + 1, where 6 = Let cV Mo! + N n + t;; a?2« _ 2^" . cos a + 1 c^ — 2x cos 0+1 Q .-. x'' = (c^?2 - 2a?cos + 1) Z' + (J/cT? + iV). Q. Let .x- = cos 4- V - 1 sin = ;^, and Q, be the value of Q ; Now 0?'^" — 2 x" cos a + 1 = (a^' — 2 ^r cos 9 + 1) . Q ; do 2n. a?^''~^-2w.a?"-^cosa= (2^-2cosO) Q + (x -2xcos6+l) dx 2w^^"~' - 2w^""' cos a = {2z - 2 cos 9) . Q,, or 2ns;^" - 2nz"cosa = n(is^" - 1) = (Ssr^- 2^cos0). Qj n (z^" - 1) .-. Qi = u2 308 INTEGHAI, CALCULUS. %'^ - I 1 z z 1 ' or cos {(r - w + l)} + V^- 1 sin (r - n + I) = n \M (cos9 + y/ - I sin 6) + N] x - sin n6 sin t^ .cosOsmnd ,, / — . ^ smn9\ / - I . sm nd + N . — ; sin 6 J . ((r - n + l).(2m7r + a)] /,,cos0sinw0 ,, / — . ^ ,^ smn9\ = n. [M ^-- + J/V-l .sinn0+iV. — ; ; V sm (9 sin / , ^ sm <- ,, 1 sin (r - w + 1) ^ 1 I n .'. M = — . n sin w0 n sin a iV.sinw^ iVsina 1 . . J/, cos sin a and ^ — ;, — , or — -, — — - = - cos (r - m + 1) y -. — sin t^ sin t^ 71 sin (j = {—. cos (r - n + 1) 0. sin . sin (r - w + 1) 0. cos0> -; — — [n n J sin ] sin (r — n)9 1 sin {n — r) 9 n sin n sin !in - r) (2m7r + a)] n J n sin n9 n sin a and the integral is reduced to that of 'X . %m {r - n + \)9 + sin {n - r)9 « sm a . Jx 0^ — 2w cos 0+1 The integral of which is known. EXAMPLES. 309 EXAMPLES. 9,x + 3 1, Or-1)^ /• 17^ 1 r >f 1 ^ ^ .'^ {x" + 1) . (a?'^ + 4) 3 \ 2 J /- Sct-^ + .r - 2 1 1 5 1 (4) / = . J^{pc -\Y-{pc' ^\) 2(a?-iy' 2 a;-l \/ x^ 2 *' a? - 1 + 1 -1 — tan X . (1 + xy " V Sx\ 1 = a' + = u- + 2 / 1 + a- 2 3\ tan ' a?. I ^"^ + - ) ^ r— + log \/l + V 4/ (1 + xj ^ ^ X + tan"' X. 2,x^ X ^1 / 1 5 5.t7\ 1 5 -I .r. X b = -.log\/a + hx + c ^a + bx + car c & + fA_2)./- \ . \2c^ cl J a + bx + cx^ r X a? + 2 2 r X^ X^ . zi— (11) \~ = logV'^ + 1- ^ ^ Kx- +1 3 ^ 2 /2 b 1 y t) r 1 y/az" + bz + c A . / — . a 2a 'Jz^az^->r bz + c the integral of which depends upon a preceding example. du 1 30. Integrate -— = , dx {a +bx)'\/ c -\- ex Let z = y/c + ex ; dz e ' ' dx 2z ' Also X = ; e ae —be + hs^ .'. a + bx — , e du e du e — X — dx {ae-hc ■Yhz^)z dz 2z du 2 _ dz ~ (ae - be + bz^) , / 2 ae-bc \ ( ae-bc\ (1) Let ae>hcx 2 / 6 zy/b ••• ^ = 1 ^ V —.tan-^— ^ b ae- be y/ ae - be --,— . ^ . tan-' ( / - V c + ea? y/by/ae -he Xy/ae-bc IRRATIONAL FONCTIONS. 317 (2) Let aes — a and a? = h ' c6^ + e (i^ - o)^ ch" + ea^ — 2ae;s + ez- c + ex^ = 2^ = — . {^^ - 2az ■¥ z^\ by substitution ; dtt 1 du '■ -r- = jn = T~ • ^ ' 6 = — J-, . / , a known integral. Ve -''- ^ y/z^' - 2az + /3^ 318 INTEGRAL CALCULUS. „- _. du 32. Inteffrate — - = dx (a + hx^) y/c + e ar _ 1 dw 1 L.et a? = - ; .-. — = . ss dss z^ . 1 ^^ du dz „ du -^"^ 7 2 r: > „ = T- • -7- = - ^ • T- ' {az^ + b)y/cz^ + e dz dx dz du z dz (az- + b^cz^ + e Again, make \/cz^ + e = v, % 1 dv 's/ cz^ + e c dz , , , (v^-e\ , av^-ae + bc and az~ + b = a { ) + 6 = — ; \ c J c z 1 dv du dv (az^ + b) A^ cz^ + e ^'^^ — ae + be dz dv dz'' dti 1 dv av^ - ae + bc^ • u=- f ^ = ^ f ^ Jv av^— ae + bc aJj,t;^±/3^' which will be a circular arc, or a logarithm, according as the positive or negative sign is taken. _. du 1 33. Integrate — - dx (a + bx) '\/cx^ + ex + m (a + bx) y/cx^ + ex + m \/c . , x * / « e (a + bx) \/ ^v + —X + m c c IRRATIONAL QUANTITIES. e dx Let a; -i = z; .'. — - = 1. 2c dz he ^ And hx + a = hz + a = bz + fib. 2c X ^ ^ 2 -Let - = a ; c 4>c'' by/c ' ^^(z + (i)\/z' + a:' Again, make z + j3 = -; 319 Z' 1 "/? 12/3 1 + a-=— + a +p =-^ +5i' 1)^ V v~ V o 1 1 / ij^ ?* = — \/c' / 1 /l 2j3 1 ^/f - V -? + Yz hy/c' JvV^-'^fi^'v + 'o'' a known form. £ 34. Next, to integrate X{a + hxy, where ^ is a ra- tional function of x. Make a + hx = z'^; zi -a ^ dx q „_, .-. X = — - — , and -r- = 7 ^ » b dz b and f,X. (a + bxf=!,Z. ;^p. 1 5?'"' = | /, Z. srP+'-S , z'i — a . where Z is the value of ^, when — - — is put tor x. 320 INTEGRAL CALCULUS. 35. Again, to integrate A . (x + ■yl + = v''^'\ Also n is assumed positive, for if not, let 1 71 ,r= ; .-. x-" = ti\ Ex. 1. Let — - = .T^\/l + - . Having integrated these functions, we shall next integrate —r==-, and / , and / . ' ^.v -\/2ax — x^ ^.t x'" \/ lax — ai^ ^""y/ a + bx + cx^ d u x'" 39. — = - , , (m) an integer, dx ^i — o!^ X'" , X I ^,m — \ r X J ~7T^^ = J-^^'" X^ ■\/l -x^ ^/\ - dq X Here p = x'" ', and dx y/l _,^2' dp _ J - — — = (m - l)a;"' % and o = — -v/ 1 — x^ \ dx 326 INTEGRAL CALCULUS. a;'" / , = - .r™-' \/l - x' + (m - 1) /,.r'"-^ \/l - x^ = _ ,r"'-' Vl - .r- + (m - 1) { \ (m - 1) / , putting — -. for v 1 - .1?'' ; m / .' = - x"'-' \/l - cv^ + (m - 1) / y ^ -^,r a/ 1 _ ,t?2 -^ r a/ 1 - c-P J y^i _ ,^2 wi w X ^1 _ .r?8 ' and by putting m — 2, m — 4, m — 6, &c. for m, the integral will be reduced either to f , or f —. ; ^.r v 1 _ ^2 -'r V 1 - a?^ that is, to — V 1 - .r-, or sin~\r, according as m is odd or even. J p. ^^ ' — , be required. Here m = 4, « V 1 — X'' r ,r^ x^ y/ \ — X^ g r c-f" X y 1 - a7« ~ 4 ^ */, x/l -a?^ r P, = - 4 Qi + 2 ^o> where Po= sin-'cr ; (2n- I) .(2n-3) + 2n. (2n -2) {2n -4) (2n - 1) (2n - 3) (2w - 5) 3 . 1 Q2«-5+&C.V sin~'.'i?+ C. 2n . (2w-2)(2w-4) 4.2 If the integral be assumed = 0, when w = 0. Then C = 0, for Q2n-i, Q2«-3. &c. each = 0. TT If .r=l, Q2«-i5 Q2«-39 &c. each = 0, and sin ' a; = — ; from.r = Ol {2n-l).(2n-3).{2n- 5)...3.1 ir j- x~^ from tr = Ol ^^\/\ - .T^' to cr=lj .^1 _ ,-p2' tOcr=lj 2w . (2/i-2).(27i-4)...4.2 2 (2) Let m be odd and = 2n + I ; 1 2w 2W + 1 2n+ 1 1 271-2 ■* 2n-l = ~ TTT ^ Q2n-2 + TT"^ ^ ■* 2«-3J 271-1 271 -1 ^3 =-iQ2+f/*l, 328 INTEGRAL CAL(!ULUS. C CG , and P, = / —y = - V I -x^ f 1 2w [Sw+l (2W+1)(2W-1) 2w.(2w-2) \ + — ^^ Q.ln 4 + &C. > (2w + l)(2w-l)(2w-3) '""' j 2n . (2yt-2)(2n-4)...4.2 /—^—, (2w + l)(2w-l) (2w-3)...5.3 If -P2»n-i=0 when a?=0, since then Qa„» Q.^,,-^? &c>- each = ; 9.71 . (2w -2) (27i - 4)...4.2 ^ .-. = ^ '— + C (2w + 1) (2w - 1) (2W. - 3). ..5 . 3 whence by subtraction, 2w . (2w - 2) (2w - 4)...4.2 -^271 + 1 — (2n + 1) (2m - 1) (2n - 3). ..5 . 3 1 2W ^ „ I \2n + 1 "" (2w + 1) (2w- 1) Let iX' = 1 : ,,2«+i from cx = 0) 2w . (2w - 2)...6 . 4 . 2 \)(2n- 1)...7.5.3' ,^2n+i from TT 1 .3.5.7, &c. 2.4.6.8, &c. or 2 ■ 2 . 4 . 6 . 8, &c. 3 . 5 . 7 . .9, &C. ' TT 2.2.4.4.6.6.8.8, &c. or — = , 2 1.3.3.5.5.7.7.9, &c. which is Wallis's theorem for the length of the circle 41. Integrate IRKATIONAL VUNCTIONS. 329 d u 1 rf<'r ,r"'\/l + .r- 1 , rf(/ c^ Here p= r, and -— =^ . dp w? + 1 — , and q = \/l + -t'"; •■• / T = ; — + (m + 1) . / -s— ^ — + (m + 1). /■ -^=^ + (m + 1). 7=-; 1 1 \/l + a?' m r 1 For w + 2 put w? ; z' 1 1 VI + X~ OT-2 /- 1 a formula of reduction by which the integral may be reduced either to f 7---=- 5 "r f ^ . = 4- X' accordino; as m is odd or even, and { 1 = h.l. - x/cT? - + 1 = - \/l +.r^ .r 330 INTEGRAL CALCULUS. 42. Inteerate .r'" y/x" - 1 r \ r .V rid. \/ aP' — 1 -'■^ w'" \/w' - 1 ~ y, .r'" + > ^/.r2 - 1 ~ - ^ 'i'"' * ' ' d.v y/w'^ — 1 , ^ r y/.V^ — 1 ; — + (w + 1) . / , - (m + 1 ) . / > r 1 1 V c??' - 1 m r 1 .-. / -)--— — = . — + • / / ; therefore, writing m for (m + 2), 1 1 \/x- - 1 m -2 r 1 -l .r'" \/ar - 1 " w - 1 ■ cr'"- ' m-l' J.r .i-"' -^ V^^^^ ' a formula by which the integral may be reduced, when m is 7; odd, to / = sec '.r, and when m is even, to ^ X \/ X' — 1 ' .-2 / ..a V^.i?2 + 1 J .?7^ \/ X^ - 1 <2? 1 Example. Find / — , >■ ■■" x^ V .r*- — 1 / , /^: = 2 • 5 — - + i • sec ^x; 1 1.3 -.r.r''\/.r'^ - 1 "^ 00^ 2.4' .i?'-^ 2.4 sec ' X. IRRATIONAL FUNCTIONS. 331 43. Inteirrate —— = ^ .27'" r- .r"-' . (a - ct) + ax'"-' -^■T^y^ax — .v' • "^ \/ 'iax — rr rx"'-Ha-x) r x'"-^ ^ _ j +a. — — ..^^..^ . - '^ V lax — x~ '^■f \/2ax — x^ Now jlx"'-^—y=^.^^=x"'-^\/2ax-x^-{m-l).jl\/2ax-x~.x"'-~ .t'"'-' r X" --x'"-^ \/ 2a x-x^ -'i.{m -\)a . I ^ + {m-\). I — '^xy/2ax-x- 'Ji\/ 2ax-x- therefore, substituting x'"-^ m f - = - A'"'~ ' \/2ax-x'^ + (2m -I) a. j - ; ^■r^y2ax — x^ '^'\/2ax—x^ x"'''^\/2ax — x'~ 2m -I »,'" - 1 r X"' X'" "'V 2ax-x~ 2m -I r X'" - ' .-. / , ^ = - ~ + .«./—-==, '^.ry/2ax-x'^ m m ^\/2ax-x~ a formula by which the integral may be reduced to I = Fsin-^-. ^'•\/2ax — x^ ^ The last term = a'".- —^ Fsin"' - . m m — \ m—2 ... 2.1 a CoR. Suppose \he value of the integral be when x = 0, and its value be required when x = 2a. Then, since all the terms on the right-hand side vanish when 07 = ; therefore C = : and when .r = 2a, all the terms of the form ,r'"'~^ v Sa.r - ar vanish, but Fsin~' — = tt; 2a from a? = to 0? = 2 a. x^ r X- •^•^ V Sao? — \ .3.5. ..{2m -S). {2m -I) TT . a . 7 r 1 .2 . 3... (m - 1) m 332 INTEGRAL CALCULUS. 44. Let -— = doc .v"" \/2ax - .V- ^, , 1 dz I Make z = -; .-. -— = = — z' ; .V dx w du , du ^m + I dx dz -y/^az - 1 ' " 2a where /3 = — . ,vrT^=(.'-f ^^J V 1 - x^. [5 15 15J ic^' ijx^ 5x\ / „ 5 - + + — > V I - cr^ + - 6 24 l6j 1( sin '/» . EXAMPLES. 337 (12) f = -i + \\/l+.v^ ^'x''\/\^x^ V>^ 15^^ 15.1?] /.I f 1 _2_1 X /• 1 ' tT + a ''^ (2a.a7 + cr^)^ <^^ y/^.ax + a?^ /- 1 f 1 2 1 X + a "^^ {2ax + x^f~ \s{2ax+x'') 3a^] a^^/oax + x'' r 1 2. (207 4-1) (16) / = 1 ^. ''^ (1 + .J? + .r'^)5 3 V 1 + ^'i' + ^ - 1 f 1 81 2 (2.r + 1) J'^(l+x + x')^ ~ \l+a? + x^ 3} gy/i + X + x^ (18) f 5 ^ = -^lJL^ + tan - '■ (y/x). (ig) r-W^ ^ I - - 1 I 2 \/a? + 2 tan"^ \/ct7. ^ ^ J^l +x \3 J Rationalize the integrals ^^/ 7^ .r + 3x^ ' ^^^ X a?5 + ^(1 + a?) in (1) make x = »'^, and in (2) make (l + iv) = « 3 CHAPTER IV. INTEGRALS OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 49- These functions are of the form X (log cx)", -Y.log F, X.a"^ where X and Y are functions of w. 50. Integrate /, X . (log x)". Let/,X=P, f,P.~=Q, and f,Q.- = R. Then X ^ (log ^)" = P (log ^vf -n.iP. (log .r)" " ^ - , and /'-.(log.r)"-l = Q(log.r)"-^-(w-l)./,Q.(log.r)"-^-, Jx X '^ f - . (log xy-' = R (log xy -'-{n-2).f,R. (log xy\ - ; &C. . •. /^ ^ (log xy = P (log .1?)" -n.Q (log .r)«-i + n . (w - 1) . i? . (log xy-^ - Sec. Ex. /, cr-" (log .r)% /, .r*" (log xy = f:::!l(^g_^)! _ _^ . /^ .^'«+> . (log xy-^ . - a?'"+* (log.^?y' n m + 1 m + 1 /,.«-. (log*)"-'. j;.. „og „).-. = ?!:^m::: _ ^ ./„.- . oog ..)--, LOGARITHMIC FUNCTIONS. 339 and in this manner may the integral be reduced to faf^ = 1 , if n be a whole number, m 4- 1 and .-. Lv"' . (log wY = j (log wY (log x) w.(w-l), , „ , w. (w - 1) . (w - 2)....2. 1 Every term of the integral vanishes both when x = and -2 (w -!).(»- 2). (w- 3).. r'-^ (w - 1) . (n - 2) ^ ' -l).(n-2)..,.l ' X "^ ' and T— can be found by expanding a*. 1 + J c'P H 1 1- &C, „ a^' 1.2 2.3 For ^ = — 1 ^ J^l- J^.2?^ „ = - + ^ + — + + &C. X 1.2 2.3 /-a' , , , A'x^ A\v' A\v' .'. / — = h. \.x + A.x + s + z — o + — :r-:^ + &c. 1.2'^ 2.3' 2.3.r Ex. 1. Find / - log {a + bx). log (a + 6c2?) = log a (1 + - a?) = log a + log (l + - .v) (h b\v' b\v^ b\v' ^ \ = log a + (-X 2 + ^TT ~ :rT + &^- 1 ' ^ Va 2a^ 3a^ 4a' / .'. / - log (a -I- 6 ; (m + 1)- ^ ^ ^ (m + l)-" I .-. ^a?logcr = — .(loga?-l), /p3 r 2 . 1 1 /,.X" (log J?)^ = - I (log xY - I log a? + — I , x^ ( 3.2 3.2) £ (^ log <^)' = - I (log cX')^ - I (log xf + -^ (log x) - ~^\ , &c. and arranging the terms according to the powers of log x, nx^ 11? x^ n^x^ n'^x^ Lv"'' = X — + — — + — &c. •' 2^ 3^ 4>^ 5'' x"- nx^ n^x'^ ■^ '2 ~"F + -^-&<^-)^iog'^ «3 ^ „4 ^2 „5 7^^ (loga?)^ i /a?* nx n~x^ „ \ 'n/{\c + + -— - &c. — ^ V 3 42 5^ / 1 . V4 5^ 6^ / 1 + &c. Cor. Since x^ (log ~ w^ X(sin0)"-'' ' • ^ (w - 1) (sin 0)"-' V n-lj Je (sin 0)"-^ COS w - 2 + - 1 Je (sin 9)"- (n -\) (sin 9y-' n a formula by which n may be diminished. CIRCULAR FUNCTIONS. 349 r 1 Ex. Let u = -r-. — ;r-o be required ; Je (sin ey ^ therefore here n = 3; cosO r 1 cost) 1 r ^ Je (sin ey ^ ~ 2 (sin Oy "*" ^ X^ e sin 6 cosO , , / 9> + 2 (sin ey llog(^tan-j 61. If — - = — — , then, as in last article, ad (cos^)" r 1 r sin^G JeicosOy-^ "^ JdJcosOy ' , /• sin"0 sin 1 r cos and / ~~ = / ; : Je (cosOy (n-l). (cos oy-' n - l ^e(cos0)"-' sin0 u = sm t/ n — 2 r 1 {n-l){cosey-' "*" n- I Je (cos 0)"-^ * 62. Let — ^ = (sin 0)'" (cos Oy m and w both integers, (sin 0)'" (cos ey = (sin 0)™ cos e (cos f^)"-^ . /e(sin0r(cos0)"= ^'^"^^'"'"'^'"'^^" -+^:^/e(sina)'"^^ (cos^)-^ '' ^ ^ m+1 m+1 -^ ^ ^ (sinm'" + '(cos0)"-' w-1,,,. ^ , ^^ m-fl +^ti/e(^^"^r(cos0r-^-/e(sin0r(cos0rj; / yi-l\ w + Ti (sinm-^+Vcos^)"-' .-. 1+ ]u= u=- ^: ~ — \ m+ 1/ m+ I m + 1 m + I Jg 350 INTEGRAL CALCULUS. (sin 0)™ + ' (cos ey-' n-1 i L. ^ i — 4- . j (sm dy" (cos 8)" m + n m +n a formula by which the integral may be reduced to /e(sin0)"', or /e (sin 0)™ cos 0. Ex. Let m = 2, and w = 2 ; .5 5 (sin ey COS (sin Oy cos 2 cos 3.5 3.5' substituting the value of /0(sin0)^ from Art. 58. „ ^ du sin'"0 63. Let T^ = — T7,' a 6 cos" u = sin'" --^0 sine (sin0)'"-^ m-\ r(sme) /•sm'""'ysmy l^smt^j" m - i r Je {cosOy (w-l).(cos0)»-^ ~ n-i ^^6 (cos 0)""'^ ' a formula by which the integral is easily reducible to a known form. Let m = 3, and w = 4 ; (sin0)' (sin0)- 2 /• sinO .-. w = Je(cos0)* " 3 (cos 0)3 ~ 3 \' (cos 0)' (sine)^ _ 2 1 3 (cos 0)' 3 ' cos 1 ({siney 2 \ , }- cos-0> (cos0)M 3 3 I „ /e (tan ey = tan -^(^^) = tan - ; .-. /e (tan 9)' = — tan + 9. ^' Je (tan BY J^ CIRCULAR FUXCTIONS. 353 1 + tan'^O - tan^^ (tan O)"" Je (tan 0)'" d . (tan 0) d0 r 1 1 r 1 (tan 0)'" X(tan0)'"-2" (^j-i) (tan0)'«-' Je(tan0)"*-2' 1 r 1 and X (tan ey-' (m - 3) (tan 6)'"-^ Jd (tan 0) 1 m — i ' JeaandY' (tan^)"' (m-i)(tane) m-l 1 _ 1 "^ (w - 3) (tan 0)^^ ~ m-5 (tan 0)" -^ "^ vm-5 a formula by which the integral is reduced to 0, or /'-^ = h.l. (sin0). Je tan 9 ' 68. jxe"'" sin A\r. . do , Integrating by parts, and making p = sm kx, and —- = e in the formula li^p-r- = P9 - Jx9-r ■> ^^ "^^^ rtcF rfcr e^'^sinAr-r k . ,,^ Le"'' sin ka;= . Le"'' coskx (1), a ^ e"^ cos koJ l^ r ax ■ r and Le cos kx = + - . Le sm A; a'. 4' Multiplying by - , and transposing k^ r „ . ke'^'' co%kx k . .. , . . -. £e"" sinfcar = -— ~ + - - S.e"" coskx (2). d^ ^' a a ,a» i 354 INTEGRAL CALCULUS Adding (l) and (2), k\ r „r. . (a .smkx - k cos kx) e'" . „, , (a . sin AjcT - A; cos A;a?) e" /,e"^ sin kx = ^ ^ (a . cos /ca? + A; sin A;.r)e° Similarly, y^e co%kx = — ^ fC "T CI du 1 69. To integrate — - = r . ax a + . cos a? 1 - tan - 2 Since cos x = ^— — — 1 + tan^ - 2 Let ^ = tan - ; 2 .'. cos X = 1 -^^ 1 +z^ 4z d% sin a? = — -7- (1 + %y dx But sin ^ = V 1 - ( ; ) = - d% (1 + ;?^) 2af + « ■ * del? 2 1 du I +%^ 1-^2 d^ 2 a + fc 7, du d% a (1 + %") + 6 (1 - ^') ClllCULAR FUNCTIONS. 355 (l) Let «>6; dti 2 2 1 d% a + b+ (a -b)ss^ a + b a-b 1 + 7 . m^ a + b 2 a + b y/a^ - 6~ (2) Let a. \^ a + b 2j du ' d% {b- a) b + a b — a -z' /b + a V ; + % 1 a/^-^ki'^^"'' .-. W = . V :; n. 1. b -a b + a /b + a z b — a V! v/fe + a + \/b - a . tan - 1 2 . h 1. x/62 _ „2 ^/b + a-y/b-a. tan - . 1 70. Similarly may / r-^ — be found. * ^ '' Jwa + bsmx sm-r , r cos a? Also / ; , and / Jxa + b cos X Jx a + b cos X d . cos X r sin a? 1 / dx ^ i i / . /. „^o »,\ For / = - 7^ = - 7 . h. 1. (a + o cos .r), J « + 6 cos cr 6 a + b cos a? 6 z2 356 INTEGRAL CALCULUS. COS X r a + b COS X — a and r cos X r a + ' Jx a + b cos X J^r b (a {a -\- b cos x) 1 a \b b a + b 1 \ X a r 1 > cos x] h b Jx a + b cos x And to integrate . Let tan x = z ; a + b tan x dx dz 1 + ^^ ' /■ I . /• Jx a + b tan a? A (l + sr^) (a + 6») which must be integrated by partial fractions. EXAMPLES. / X r/- m5 nfCsin6>V 4(sin0)^ s) . ^((cos^)^ 5(cos0)' 5 cos 6)) 5^ (.) /e(cos^)^ = - ^{'-V ^ V^ ^ ^ ^ 16 • (3) /e (sin ey (cos 0)* = ^''"^''°'^^' + - sin^0 cos ^ • a a ^ sin t^ cos a + — ; 16 16 (4) /e(sin0r(cos0y^ = (^^%^)(sin0)^. (5) / r = - cos 0< h > + -logtan - ^ ' Je (sin ey U (sin Oy 8 (sin eff 8 ^ 2 r 1 . f 1 4 ^ I ^^^ Je (cos 0)" " ''^^ ^ |5(cos0f "^ 15 (cos 6^)=^ "^ 15.cos0r CIRCULAR FUNCTIONS. 357 r jsiney 1 f (sin ey 4(singy 8| ricosoy r, ^., 3 COS 01 1 3,^0 (8) / ^^ ~ = I (cos Of > , . ^,, - - • log tan - . ^^ Je(sin0)^ \^ ^ 2 J(sin0)« 2^2 ^'^^ ie (sin 0)~ . (cos Oy " l2(cos0)-^ ~ 2 j ^m^ 3 , /tt y\ + - . log tan — I- - 2 ^ V4 2/ 10") / = cot 2 0. ^ Je(sin0y(cos0)- 3 cos (sin 0)=^ 3 1 1) f^e^ . cos = 0' sin + 30- cos - 60 sin - 6 cos 0. (sin ^^y x\/\—ar . _ or sin~Vv = sin \v + — , Jf a; r X . , sin"^1? , /\ - x 13) / jTaSm-^o? = J + log V :; • *^x (1 - '^ )^ a/T^ 1 + ^ 14) r tan~^r = = ?/, or dw d0 o Ex. 1. To find the area of the circle. CN = ,v\ CA = a] • "• A = f^y = f^ y/ d^ — x'' ; •. area CBPN = {, y/ d' - x". But CBPN is a circular area, of which the cosine is CN^ and radius = CA. Hence f^\/ nr—,v"= a circular area, of which the cosine =.r, and radius = n. AREAS OF CURVES. 361 Again, let AN = w, then NP = y = v 2 a a,' - x^ ; .-. ANP = /, y = /, \/2 ax - x^. But ANP is a circular area, of which AN is the versed sine ; •*• X V 2atr — .T"' = a circular area, of which ver-sin = x. Resuming the expression for CBPN, we have O 9 o O , y r a- — X r a- r X' CBPN=Wa^-x^ = f-y=f^, = J-rr=. - I-7-r=. ■v /- ^ + - V a" — 0? ff" . , tT' xy — .sin-^- + -^ 2 a 2 = -.sin-i- + ACPN; 2 a .-. C^PiV - CPN = sector 5CP = - . sin' - = 2 a 2 Cor. Since C^PiV = - . sin"' - + — "^ , 2 « 2 let .r = a ; .'. area oi the quadrant ^CiJ = = — — ; ^ 2 2 4 therefore area of circle = trn^. (2) To find the area of an ellipse. h y Here y — - \/ rr — x^ ; a A = jly = - . fr \/a^ — x^ = - . circular area cos = ,r + C- a -^ a But ^ = when .r ^(\\ .-. C = ; 362 AREAS OF CURVES. 6 .'. ^ = - X circular area cos = x, and rad = a ; a therefore whole ellipse = - X circle radius a, = - ira^ = irab. a a (3) To find the area of the common parabola. y^ = 47WcX^ ; .-. y = 2 \/ mx. area = ^?/ = 9. j^y/mx = 2 y/m • | <2?^ + C And area = 0, if a? = ; .-. C = ; .*. area = — - — x^ = :^2\/ mx .x = ^yx = I of circumscribing rectangle. (4) To find the area of the Witch. 2a y = — y/ 2ax — x^ % . r\/ 2ax — x^ r 2a — X . . area = j^y = 2a = 2a / . = Js X '^^\/2ax - x' = 2a| \ . ' ^ + a \ —^==\ I ''^ y/ 2ax — x^ *'■? y/ 2ax — x^) = 2a < y/2ax — a^ + a ver-sin"^ - > + C I «j And area = 0, if .r = ; .-. C = ; a { y/2ax — ,r + a ver-sin~' — > . .-. area = 2 Let X = 2a\ .•. area = 2« x a. ver-sin' (2) = 2 7r«^ = 2 area of circle rad = a. AREAS OF CURVES. 36-3 (5) Find the area of the hyperbolic sector CAP. Sector CJP = ACNP - area JNP. L.etCN = a)] NP =y\-> •■• y = 7\/ct7"-a'-. CA = a b a x^ — a^ ANP= ,y^- ,V^'-a' = -' / -7= = - . \cV \/a;^-a'^-p.\/.v'—a~-a^. h. 1. (a) + 'V x~ - a^) | . |! y.^2 -a'--.h.l {.V + -y.??^ - a')\ + C, 6rt and = . h. 1. a + C. 2 For ANP = 0, if x = a; therefore, subtracting wy ba , , ftv + s/cv'^ — ANP = — .h.l. 2 2 \ a a~ = ACiVP ba (X + \/x^ — a^ -"•"■'■( a ba ^ IX y .-. sector CAP = — . h. 1. - + 7 2 \a (6) Find the area of the portion PNMQ, PQ being an arc of the rectangular hyper- bola. a- Here yx = — . Let CN=ci, ^ 2 and CM=(i, Ly = — f- = - . h. 1. + C, and C = since the area = 0, when 9 = 0; .-. area JSP = a^ Itan - + i . tan^ -i. I 2 3 2J (11 ) Find the area of a portion of the lemniscata. Here r = a^ cos 20; •*• jei»*^ = - •/ecos20 = — sin20 + C. There is no area when = 0; .-.0 = 0; «' • a .•. area = — sm 2 0. 4 Let = 45° ; o .*. 5^th of lemniscata = — ; 4 and therefore area of lemniscata = a 2 AEEAS OF CURVES. 367 (12) Find the area of the spiral of which the equation is r = a $". dA , , If A be the area, -7^ =2**- dA dA dr ^^^ 7e ^'d^'dO^ dA dA d6 , „dd dr dQ dr ^ dr^ dr Here G)' d0 1 Ir-i = -r . r dr '- dO _1_ , 14 2na" 4/-;^ = — T-/- "' 1 ^ '-ir^ ^ or area = — — r • '' + ^> ^ ^ 2n + 1 2W«" and C = 0, if area = 0, when r = 0. Cor. Let w = 1, or the spiral be that of Archimedes; .-. area = r— . o« But if R be the value of r when = 27r, a= -— ; 27r 27rr^ Trr^ . . area = —z-=r = — ;= • 6R 3R 368 AREAS OF CURVES At the end of the first revolution r = R ; ttR- therefore area of spiral in first revolution = . To find the area after two revolutions of the radius vector we must find r when = 47r. n n Now r = — = — 47r = 2i?. 27r Stt But before r = 2i?, it will have made two revolutions, and therefore have twice generated the area from r = to r = B. Consequently we must subtract the area described in the first revolution from that in the second ; TT.i^Rf TT.R' IttR' .'. area = = . 3R 3 3 And area intercepted between the arcs of the first and second , . 77rR' ^R' ^ _ revolution = = 2 7r/c . 3 3 At the n^^ revolution r = nR, (71- 1)'^ r^{n-l)R; TT (7iRy-{{n-l)RY area after n revolutions = 3 R irR' \n^ - (n- if] Area after (n + 1) revolutions = — ^ j(n + 1)^ - w^^ ; TT R' space between the arcs after n + l and w revolutions \(n + ly + (»? - 1)^ - 2w^| = .bn = 2mrR~ = n times the space between the first and second. AIIF.AS OF CUUVES. 369 (1.'}) Find the area of the curve of which the equation is 1/ - Saxy + cc^ = 0. If the curve be traced there will be found a nodus as JPMQ, to which the axes Ay and A,v are tangents. y Let y = xz ; .-. x = - = tan PAN; X .r'^' — Sax z + X = ; Sax 3az' .'. X = -, and y = 1 +z^ 1 + %■' And since x is = 0, for each of the branches APM and AQ,N^ this will happen if ^ = oo or = 0. dA dA dx dx dz dx dz ' dz^ dx 3a . \l + z^ - 3z^\ 1-2^ . dx ••• -4=J,y.— = 90 dz z~(l -2^^) '-- (1 + z') (1 + ^0 Sz"" 1 .4 (I + ^)') Aa 370 AREAS OF CURVES. Let .r = 0; .-. C = - 3« - - y and let «i = - at il/ ; X .'. area 2 [ •'^ (1 + z^"); 3 1 + ^i=*J Again, integrating between ^ = oo and z = z-^ for the branch APM, area {1 2 1 1 2(1 + ziy 3 1 +z-'\ the nodus APMQ = area APMm - area AQMm = Sa" (14) Find the area of the evolute of an ellipse D'n^|='' where CA^^ = a, and C^i = /3. Let y = xz ; 2 A'* = 2 /a — -,, where r=-; 1 + {czy P a ,v = , and y = az \\ + {cz)'.\i' " {i + {czy\i For the arc i?,^, the limits of x are and a ; thev are x and 0. •. of z. J AREAS OF CURVES. Now dJ dA d.v d.v dz dx dz ' dx d,v aci% 3 and — - = - ^>f' dx {l+{c%)i]i dx .. r (c^r)! C .\. (1 + v-y X (1 + vy " ~ 6 (1 + vy ^ 2 • j,, (TT^ • and /• '"' = - ^ + !/• ^ "^"'^ f 7^ ^2 = o r, + 1 . tan-^ « ; 371 A=- 3 a' 1 tJ 777 :7rz 7 ^TT, + + tan 'uL 6(l+iJ-)-^ 8.(1+^5-)" 2.8 l+t?' 2.8 ' from >? = CO , that is from u = co , to ^ = to V = 0. Area B,C A ^"-^1.-.'^='^^; 16 2 32 therefore whole area = 4 . B,CA IVx^l, ~ iraR 8 ' 3 a- - W «2 _ je 8 a .S (a' - hy - TT ; 8 ah A A 2 372 THE LENUTHS OF CURVES. 76. To find the lengths of curves, or to integrate d^ / dy' , ,. , ^ = V 1 + :7^,, vvhen p =f{v). ax dx~ Ex. ]. Find the length of an arc (measured from the vertex) of the common parabola. _?/ = 4 ma?; dy 2 m dy' 4 m- m • • , ~ i •'• —J Tj = ., — dx y ax~ y~ x "\/ X 4 m Jv dor - •»• .1' Jx ■\/ X m tn r X + ~ r - r X + m / 2 / 2 *^a V tT^ + mx " \/x~ + mx \/ x' + mx = 'V X' + mx + — h. 1. (x + — + \/ ^v' + mx) 4- C, 0... 4-- h. 1. - + C. 9 V 2 Since *• = 0, when x = 0; /-T, w? /'2ti? 4- m 4- 2 V •^' + ^"^\ .-. A' = V ■*"" + *"'^" 4- — h. 1. . 2 V '" / Ex. 2. Find when curves included under the general equation y = nx" are rectifiable. dy m '^n!' ~-= -ax " ; dx n / m^ d' 2 m— 2 n .-. fi = \ \/ 1 \ ~ .x^* ; wliich is integrablc. LEN'GTHS OF CURVES. 373 (l) When IS an integer = 7', w 1 >n 1 2r + 1 or 1 = — , or — = — + 1 = n 2 7* n 2 7' 2r 71 (2) When + ^ = an integer = q, ^ ^ 2777 -2n ^ '' ^ m 1 m 2q or 1 = , or n 2q - 1 n 2q - 1 Let 7-= 1, 2, 3, &c. 7=1, 2, 3, &c. ; m 3 5 7 7n 2 4- 6 •.— = -, -, -, &c. and — = -, -, -, Sjc. n 236 71 13'5 7/i 3 (3) Let — = - , or the curve be the semi-cubical parabola ; 3 . dy 3a , \/ x , . /- 2 7/ = a.i'2, and — = — x-'i = ^^, by putting \/c= — ; ^ dc77 2 ^c ''3a « = /" \/ 1 + - = — ^ . /r \/.r + c = — ^ - (a: + c)^ 4- C. ^' vc " v c ^ 1 2 But if 5 = 0, ,r' = ; .-. C = 7^ - c5 ; 1 2 (4) Find the length of the cycloid. = v^ dy /2a — x dx X dii' 2a — X 2 a ... 1+/^=^! + = — ; dx^ X X 374 LENGTHS OF CURVES. •■• «= / V — = \/2«. [—7-== 2\/2a.v + C, and C = 0, since s - when .v = ; therefore s = 2\/2ax = iw'ice. the chord of tlie arc of the generating circle, corresponding to the arc of the cycloid. Hence the cycloid is rectifiable. And if .V = 2o, s = 2 y/^ci' = 4«, or the length of the semi-cycloid = twice the diameter of the circle. (0) Find the length of the arc of an ellipse. ^> / . ij - - \/ a' - = — . ;, , &c. ; .-. also / — ; = — ; therefore elliptic quadrant TTffl . 1 ., 1.3 l-3'.5 g . a series which is rapidly convergent when e is a small fraction. (6) The length of the elliptic quadrant may be found by circular functions. For since x is never > a, Let .■r = acos0; .-. « = -Va' - a'cos- ^ &sin 0. a ds ^ I dec- dtr / 2 • ■' a — r' — ^ — = \/ — - + -^ = V rt'^ sin- G + &' cos- ^^^"^ dd ^ d&^ ' dO = a y/l - ^ coi^ 9 = a ? 1 - 1 e- cos^e - ^- e' cos^ 9 - ^-^ e" cos" 9 - &c. ^ ^ 2 2.4. 2.4.6 ' which must be integrated from 9 = 0, to 9 = - . 2 Now fe cos'^G = + sin a.cos^''-^ + (2w - l) ./o cos^"--0. sin^0 sinOcos2"-^0 2n-l . „_,_ = + — . /cos~" ~ y, 2 w 2 ?i "^ 37() LENGTHS OF CURVES. and sin cos-""' = 0, when = 0, and = -; o calling je cos^" = P.^„ 2n - I 2w 2w - 3 2w — 2 P., = 1 . P, = i = 1 - from f^ = to = - ; (2w - 1) .(2n -3) 3.1 TT '" ~ 27^.(27^-2) 4.2 *2' /ec„s'e = l.^; /,cos'0 = l^.f , 1.3.5 IT ■' 2.4.62 _7r« 1 3 1.3 ^ 1.3^5 6 1.3^5^7 ,_ . '' *'" 2 '^'~2^''' ~2^?^ 2^4^6=^'^ 2^4-^6^8'^^ ^'^' (7) Find the length of a hyperbolic arc. h / d\f b X V = - V x'^ - «-, -^ = - . — ; a doc a ^ x"" - a^ ds ^ / dy- ^ /(b' + a') .1- - a' . /e'x' - a' d,r rfa'" a^ {x - « ) A- - a and as .v is to be taken from .v = a to w = x ; therefore ^ must be taken between ^ = 1, and ;y = x ; But a f \/^-^—l=ae iz LENGTHS OF CURVES. 377 e'z~ J _ ^ 1 1.1 1 1 . 1 . .'J 1 ^r^-l'\ '^ {ezY 2.4>'{ezy 2.4.6 ■ (e;^)« 1 . 1 . .S . 5 1 ■1 whence, after multiplying every term of the expansion, it appears that every term except the first depends upon the integration of / 7 , when m is odd. -^z Z'" VZ' - 1 . ^ r 1 1 ^yz^ - 1 111 -2 r 1 Now / -. = . h . / > - z"' ^/z~ - 1 m-l z'"-' m-l J, -"'-2 ^z' - 1 and vanishes both when .^ = 1, and z = zt ; til 1 ' tU — \ ~ 1 7n -2 r 1 from ;? = 1 | Jz i%75^ ~ '«' - 1 ^'- ^'"-V^'-l ' to ^ = CO J ' ^ z \/ z^ — 1 TT But / — = sec"^^ = - from r = 1 to r = co ; Xi A/ A* /: I -i!r 9 O ' an z^ y/^ - 1 . r ^ r 1 ii_^ -. V/77^^^"''^^V^^^i ^•'^■~' 1 ^ C ^ 1 . 3 . a TT ^"^^^ J77F~1 ^ ^ ■ ^'.rV^^^ ^ 2~^- 2 &;c. = &c. 37^ LENGTHS OF CURVES. , z ira 1 1.1 1 ,-. .s = ae . / —. • ] -g • - + -7— • -, ' /v2_ 1 2 ^~ e 2^4 «■* 1.1.3^ 1 1 . 1 . 3- . 5- 1 + . 1 . h &C. > . 2-.4^ 6 e^ 2=.4^6-.8 e' ' Now the equation to the asymptote is 7 = — ; leno-th of asymptote = 'V .v + — — = .r \/ 7, — = ex = aez. But ae / = ae\/%'~ - 1 -aez from .? = 1 to ;i: = x • -- \/ z^ - 1 If therefore ^ be the length of the asymptote, and H the length of an infinite hyperbolic arc, TTff , , 1 1.11 1 . 1 . 3^ 1 -. + e ii- . 4 e'^ 2^ . 4- . 6 e' 1 . 1 . 3- . .---,2 1 2-. 4-. 6^8 e' ' (8) Find the length of an arc of the logarithmic curve. . dif Here y = «', and -^ = Aa^ = A .y, dx , (Zs dy dw dy Ay r Vl +AUf r Ay r 1 •y Ay 'Jy\/\ + A~f -^yAyy/^+^'f a/i +A'y' 1 , , Ay ^^ = + - h. 1. . + C, A A I +^1 + A'y- \/i + A' 1 and 6' = if // = 1 ; ••• C = 7 h. 1 A ' ' ] +\/-i + A''' LENGTHS OF CURVES. 379 .s>^ -J^l + ^-Z/--\/l +.J^ + h.l. 7^ 1- (y) Find the length of an arc of the Lenniiscata. r- = d~ cos 2 0, and s = / \/ 1 + r" . o . . dd ., / r' d9 /- de Now - r = a" sin 20 . -— = rr \/ 1 . — - = v'^ a' - r' . -— ; dr a* dr dr rdO - V a dr y/ a » _ 4.1 dd' a^ -. 1 + r- dr a — r = /—.■ =- = ff . / — ,- , if r= a.^ -- \\/l — .«- Vl + 2f"v r 1 , ^ \ .3 . \ .'1.5 ^ , . J. y/i-z^ ^ 2 2.4. 2.4.6 ^ + 1 . o' . \ .3.5 Ijct the integral be required from 9 = 45° to = 0; i. c. from r -- to r = a, or from ;i- = to itf = 1 ; «" J TT / — , . -^ = — . — , and / -^ — . , J.^l-;^- 2.4 2 J.^i-:^^ 2.4.6 2 •■• -^ = — • ) 1 : + ; : r, + —, — z~;^> — z - &c. - . 2 ' 2' 2' . 4- 2' . 4-^ . (r 9.- . h~ . (r . 8~ ' The whole lenirth of the lemniscata = 4s = the circumference of a circle rad = a multiplied into the series between the brackets. 380 VOLUMES OF SOLIDS. THE VOLUMES AND SURFACES OF SOLIDS OF REVOLUTION. 77- To find the volumes and surfaces of solids, or to integrate the functions dV , .dS / dy' -— = Try', and -- = ^tt// V 1 + ^ ax d.T aw Ex. (l) To find the content of a cone with a circular base. Let a = altitude b = radius of base. Then if the vertex be the origin and the altitude the axis of J from X = — r/, to x = + a 4 = - 7r 3 h-a. If the solid content of the oblate spheroid, which is formed by revolution round the minor axis be required ; take the minor axis for the axis of ,r, and the major axis for that of y. 382 VOLUMES OF SOLIDS. Then in equation (l) put y for a; and ,v for y, we have ¥ a- or 0' from ,v = — b, to j? = + />, 4 .-. solid = - TTft'b; 3 therefore prolate spheroid : oblate spheroid :: b~a : a~h :: b : a. Coil. Hence sphere on major axis : prolate spheroid A A, ■? 7 2 2 i2 :: -7r« : -irb n :: a : o, 3 3 an d sphere on minor axis : oblate spheroid :: /r : (r. (5) Content of the solid generated by the conchoid round the axis of a\ ,vy = (a + y) v^/)- - y', dV dV dw ^ dx dy dx dy ' dy dw a y- -^ (a + y) ab^ + y' and — - = -w b- y - dy r \/h^ - y' //• \/l>' - if ' -^r \\/lf — y~ y/b'^ — y and f 1 =-f ^//>=* - rf + 2 f„y Vfr - v'' Jy V h^ - y^ VOLUMES OF SOLIDS. 3B3 and F = 0, w\\e\\y = h; .-. = C - tt < «&--'/ ; .•. C= ^ ^^ ; I 2j 2 V = Tr'ab' o Let y = . 0; TT |ff// Sin _,z/ Vb^'-y- (f+2lr)}. ■. whole volume = 1 = -1?4} (6) Find the content of the solid generated by the revolution of the cissoid roiuid its asymptote, AB = 2a, BM = A , MQ = y. Now NQ~ ^^' BN or X' = (2a - y)' y .'. solid = 7r/ri/' = Try'Kv - 2 tt/,, //.!■. Rut .v^-y- = y (2a - yY; .-. coy = y/y . (2 a - y)' ; f^(a- y) \/2 ay - y- + a f,^ \/2 ay - y' M B <'\ii (2 ay -y"')"^ + a X circular area, ver-sin — y : ,384 VOI,UMES OF SOLIDS. .-. solid = TT {(2az/ - i/)^-f {2 ay - f)^ — 2a . circular area, ver-sin = y + C] {2ay-y~)i . = TT 2r/7r X circular area ver-sin = y + C. 3 ^ Let y = 2a ; .'. solid = 0, and therefore 0= - 2 3.5 , . ,y — ' — a^ \/2ay — i/ h or ver-sin ^ - , 2.3 2.3 a from 2/ = to y = 2a ; . - 2 « • T ' V^ay-y •^A/2a'"-'"2 r = ► " 3 (8) Find the solid generated by the revolution of the cycloid round its axis. dV If V be the volume, — — = ttw^, and V=7r f.y^'^Tr lyKv- 2 f^xy.—\, , dy ^/2ax -x" . and — = (equation irom vertex) ; dx X '•' i^3Gy—^2J^y\/2ax-a)'. X But if = ver-sin ^ - , a y = a{0 + sin 0), \/2ax - x"^ = a sin 0, dx and d7 = a (l - cos 0) ; •'• t^ = sm ; d V .'. l^y y/2ax-x"~ = a' fe sin^ 0.(0 + sin 0) = a^ /e (0 sin'^ + sin=^ 0). But fe 0. sin^ = /(sin^ 0) - fe fe (sin^ 0), sin 20 and /e sin- = | /^ (l - cos 20) = - - , , !0 sin2 0\ 02 cos 20 and / = _ + 4 J e V2 4/4 8 Bb 386 VOLUMES OF SOLIDS. _ r- sin 20 0^ COS 26) fO sin^ 9 = 2 4 4 8 e^ esm29 COS20 tt' from = 4 4 8 " 4 ' to 9 = ::i J r • 3 /I ^^"^ cos and /sin^ = ;; -| cos 3 = I from 9 = to 9 = IT, and y^cT? = (Tra)^ .2a from .i? = to ct? = 2«, or y = to y = 7ra ; (9) To find the volume of a conical figure, the base of which is bounded by any given curve. From A draw AD perpendicular to ^ the base, and = a. In AD take AN = x, N being a point in a section he, parallel and similar to the, base BC Let A = area of the base, S = area of section be ; S _ hN^ _ AN^ _ x^ ■ ~A ''bD^'IlIT ~ "'' a' S=A-, a'' and - — = S = A .— dx a^ .-. F= -/,.T?2 = ^^ + C, and C = 0; ABC^ Aa^ A. a so" = base X ^ of the altitude. VOLUMES OF SOLIDS. 387 Cor. This proposition is manifestly true for a pyramid of any base. (10) To find the content of a Groin, a solid of which the sections parallel to the base are squares, and those perpen- dicular bounded by a given curve. Let the given curve AD be a quadrant. therefore generating area = (2y)' = 4y" ; .-. — = 4«- = 4(2aa? - cT- ); dx I ^^\ .-. F = 4 I a<2?^ I , and from x — to .v = a, To find the surface : generating surface = perimeter of square = 8y ; a dS ds /— -— = 8 V ^— = S\/ a^ - OCT . dx dx ^y d'^ — ,xr . S = ^ax = % a^. = 8a; And similarly may the content and surface be found, whatever be the curve APD. Also, if the base be any other figure, of which the area is a function of y as a circle, a parabola, a triangle, &c. and APB be a curve of which the equation is y=f{x), the surface and solid content may be found. (11) Find the volume of the solid gene- rated by the revolution of a parabolic area round its ordinate. AM = a?, BN = Xi , AB = a, MP = %, NP=y,, BC = h; B B2 388 VOLUMES OF SOLIDS. dV J = T^yC = TT a OB I = IT {a-cc)~= 7r(a--— ) V 4-m/ 4-ma — y^\ ^ tt and 4 m ; (4m) dV _dV _ TT dxi dy ;, \b' -2b^y^ + y*\; (4 m)- •■• ^ "= t:;^ {^'y ~ "^r' ^ \ \ ^^""^ y = o to y = t, X J 2bY y'] :; {b*y + — ' (4m)- \ S 5 7r b' (4m)' f>-l+Si TT // 8 (4 m)- 15 But b^ = 4-ma ; .-. .-. F=— 7rff-6. 15 1 «=* (4 m)- 6^' (15) Find the volume of the \x solid generated by the circle BQP s which revolves about an axis ■A ANx, m Its own plane. Let AO = b, OB = a, MQ = y, OM = X. Then surface generated by QP = 7r(NP' - NQ') = 7r\(b + yf-(b- yf] = ^^rby; dV TTfl- .'. = 4fTrby; .-. F = 47r6 Ljy = 47rO -, dx ' 2 or solid = ^TT^ba^. d 9 ds Surface = 27r. L{NP + NQ) . — = 47r6. T-— = 47r6.7rff dx dw = A-TT^bn SURFACES OF SOLIDS. 389 (13) The surface of a sphere. J ; , ^y a - cr y — \/ 'iax—x , and ^— = ^ — f^* y/^ax-.tr dy^ (a - xy a^ d 1+-T^=1 + .2 o„^ ,.,2 dix^ 2 aw - ar 2ax - w y . / dy~ . a r ^ Surface = Stt/,?/ V 1 + ;7^ = Stt/,?/. - = 2irj^a = ^-nax + C Surface = 0, if a; = ; .-.0 = 0; .-. surface of a segment = Stt^.t ; .-. surface of sphere = 27ra.2« = 47ra. (14) Convex surface of a paraboloid. , dy 2m V' = \>mx, — = — ; dx y dy^ 4fmr 4m^ m x + m dx y imx x x sur rface =• /^27rw \/ 1 + -—^ = 4 7rv w . Lvx \/ — dx = 4!'ir\/ m jx\/ X -It m = 4 7r\/m| {x + m)^ + C, = 47r\/m| W7.2 4- C ; face = . \ (x + w)2 - m^l . (15) Find the surface generated by the revolution of the cycloid round its base. dy \/'i.ay--y- ^ / d^ \/'iay Here -f = ^— ^ ; .-. V l + :A = -'■> dx y dx y dS _dS dx _ ^ / df dx ' dy dx dy ' dx^ dy / — y v'^ay-y' 390 SURFACES OF SOLIDS. y ^^<\/9,a — y = 2 TT \/2 a\-2y \/2 a - y + 2 J^ \/ 2 a - y\ = 27r'\/2a {- 'iys/ 9.a -y (2o - 2/)^|, from ^ = to y = 2a\ 4 . 32 . •. surface generated by sevni-cycloid = 27r.- (2a)^ = — ttk'^. (l6) Find the same when round the axis. Measuring irom the vertex, — = 'v " • dx CD Surface = 2 7r /"v-r- = 27r. . - „ SURFACES OF SOLIDS. 391 S^^irb or e 1 5- = -/"V^l -^'' if ^= — > ex a 1 y* '^ Sf' sin * ^ + Vi -«■--/. a/i- »' = -| sin ^ ;^ + -VI -%'•■, 2 5 = ^.jsin-("^ ea' + from a;' = — a to a? = + a. 2'7rba . , / ry Surface = . |sin-^e + eV 1 -^ \ = 277(1^] -\/l - e^ sin~^e + 1-«1- Let e = 0, or spheroid become a sphere ; .*. sin 'e = 1, and surface = 27ra^{l + l} = 47ra^ (18) To find the surface of an oblate spheroid. BM^x^ CN=x MP^y, NP = y. dS _ ds dS ds or -- = 27r<2?.--; dy dy dS ds ^ /a^ - e^x •. -r- = ^irx -— = 27ra? V — : r dx dx w — x'^ 392 SURFACES OF SOLIDS. du /o - e^ Let — =1 X SJ — d cc ff - OL a?' Make \/ff^ - a^ = -z% .• — X dz y/aJ'-w' dx e du du dz d% J dx dz dx dx a 2 C + ^ ' •. u= - ej^ \/ & + %^ = \z \/(? + ^ + c^ h. 1. {z 4 y/ z- + c^) ^ . The integral must be taken from x = Q to .t? = a ; .'. % must be taken from z = a to s? = 0. = - . < a v c^ 2 I «* = - + a'^ + (? h. 1. « + va" + c~' !■ •. surface = 9.8 = Sttc . lay/ & + a^ + c^ h. 1. > = 27re<- a / 0-«')u 1 /!+« = 27ra2.|l + ^^ ^h 2e •••(^:)1- Let e = 0, or let spheroid become a sphere. Then, since — h.l. f 1 = 1 when e = 0, 2e VI - e] the surface = 47ra". CHAPTER VII. DIFFERENTIAL EQUATIONS. 78. In the integrations which have been performed in the preceding Chapters, the differential coefficient, has been either given as a function of one of the variables, or else in such terms of the two, that by a very evident process, it has been reduced to a function of one only. We now proceed to integrate differentials, when the differential coefficients and the variables w and y are mingled together. 79. Differential equations are divided into classes, de- pendent upon the order and degree of the differential coeffi- cient. Thus an equation involving, dy d^y d^y d"y dw da;"^ dw^ dx"^ is called a differential equation of the w"^ order and of the first degree, while one containing d_l^ Idyy^ idyV'^ ^^ (dy_V dx ' \dxj ' \dxj ' \dxj is said to be of the first order, and of the n^^ degree : and finally an equation in which are to be found the n^^ powers of the differential coefficients, and the m"^ differential coefficient is named an equation of the w*^ order and the n^^ degree. We shall confine ourselves to the more simple classes, beginning with that in which the first power of the first dif- ferential coefficient is alone found. 394 DIFFERENTIAL EQUATIONS. Differential equations of the first order and the first degree. 80. These are included under the formula dw where M and N may be any functions of x and y, we shall begin with homogeneous equations. dy 81. Let M + N — = 0, be a homogeneous equation, in which the sum of the indices of y and oc together, is the same in every term. Make y = wz ; .-. — = z + ai — — . dx dx Divide by N and the equation becomes, M dy M dz M N But "t; must be of no dimensions, and will be a function of y . M - or ^: let .-. — ^f{z)\ dz ( , dx 1 xdz z +f{z) \CJ JzZ+f{z) the right hand side of the equation may be integrated by the ordinary rules. We put X =yz, or y = .vz, as may be most convenient, for the solution is more easily effected, when we substitute for that differential coefficient which involves the fewest terms. DIFFERENTIAL EQUATIONS. 395 We may here remark that the notation fXclx, and /^X, both of which will be met with, mean the same thing. dy Ex. 1. Let .t + y = (a- - v) -j- • dy dz Here make y = xx; .-. -— = z + x -—; da; dx dz ^ + y 1 + « ... z + Of -— = = . dx X — y \ — % dz I + z^ .•. X dx I — z dx 1 — z 1 « xdz I + z' I + z~ 1 + z~ •. log [-] = tan-^ z - \og\/l + ^; .-. log ( - V 1 t- ^- , or log = tan ' - 07 Ex. 2. Find the curve in which the subtangent is equal to the sum of the abscissa and ordinate. ^, dx . . Here y — = x + y ; and let x = yz ; dy dx dz X -\- y .-. -— = z + y —- = = ^ + 1 ; dy • dy y ■ '^''=1; ...log (??!=. = ? ydz \c ) y Ex. S. Find the curve in which the subnormal = y - x, dy dy x •^ dx '' dx y dz I z - I Let y = xz ; .'. z + x -—= 1 = ; dx z z 396 DIFFERENTIAL EQUATIONS. clx — % ,vdz z^ — z + l"* Ex. 4. Find the curve in which the distance from the origin to a point in the curve equals the subtangent. y /J fp Here AP = NT, or Vy' + x" = y — . dy -., , dz \/ x^ + ?/ / Make cc = y%% .-. z + y . — = — = v 1 + « . dy y dy I /, 7> _ ydz ^y\JrZ~-z c^ {x ^ 's/ ,11^ + y")] y' Ex. 5. {y CG - y/ y) = \/y . — . Make x = yz. dy Ex. 6. X y = y/ XT + y~. Make y = xz. dx Then xr = c~ + 2cy. 82. The equation (a +hx + cy)dx+(ai + hxX + Ciy)dy=0 can be rendered homogeneous by making V = a + bx + cy, and z = ai + b^x + Ciy ; .'. dv = bdx + cdy, dz = b^dx + c^dy ; .•. c,rft) — cdz = (bci — b\c)dx, bdz — b^dr = (Ac, — b^c)dy ., DIFFERENTIAL EQUATIONS. 397 whence by substitution the equation becomes v(cidv - cdz) + z{bdz - b^dv) = 0, or (vci — b^z)dv + (bz — cv)dz = 0, which is a homogeneous equation. CoR. This method is inapplicable when bc^=b^c-, but since then c, = — - , the equation becomes a, + b]X + bi — \dy = 0, i. e. {a + b.v + cy) dw + \a^ + — {bx + cy) idy = 0, an equation in which the variables may be separated by , . dss - cdy making bx + cy = z ; .-. d,v ~ 7 ; dz — cdy ( biZ\ .-. (a + z) + («! + — - lrf7/ = 0; .-. (a .+ z)dz — (ca + cz — a J) - b^z)dy = 0; dy (a + z) (a + z) dz ca - a^b + (c - bi)z a + (^z where a = ca — a^b and (i = c - b^, the integral of which may be readily found. 83. To integrate the linear equation, (so called since the first power of y is alone involved). ^+Py=Q, ax in which P and Q are functions of x. Since -— (yef'^) = — - e^-^ + e-^^ ^ . Py dx dx r^-+ 398 DIFFERENTIAL EQUATIONS. It is obvious that if we multiply both sides of the equa- tion by ef'^, the left hand side will be a complete dif- ferential, and the right hand a function of x alone ; both sides may therefore be integrated. Multiply therefore by e^'^. ax .-. yef'P = C + Jef'P.Q; dy Ex. 1. Let v y = ax"^- dx Here P = 1, /,P = .r ; .-. e^^ = e% Q = aar^ ; .-. ye" = C + aje" . .-r^ = C + ae^' {x^ - 3x^ + 6x - 6\ ; .-. y = Ce'" + a \x^ - S.x^ + 6x - 6] . Ex. 2. (l + x^) -^ - yx = a; dx dy X a or — -y dx 1 + a^ 1 + x^ a? , „ , 1 r T, 1 HereP=--^-;; /;P = log-^.= ; 6^^^ = ^^^^ 1 r 1 1 /•I aa? = a / -} c^ ! = I- + c ; ^r (1 + a?-)i y^l + ,^2 -•. y — ax -v r V 1 -I- tr^. DIFFERENTIAL EQUATIONS. 399 84. The equation y"'"^ — + Py'" = Qy" may be reduced to the preceding form, in the following manner. Divide by y". ... ym-n-X _y_ ^ pym-n ^ Q , dy dz Let ?/"-" = (m- n) %, .-. t/'"-"-* -^ = -— . .'. — + (m - n) Px = Q ; dw which is of the required form. d 1) ft 1) TYl Ex. V = — - . (WheweWs Dynamics, p. 182.) ds s s~ ^ „ dv dz Let v^ = 2z; .-. v—— = — ; ds ds dz 2hz m ds s s^ HereP = -— ; .-. /, P = -2Alog (..) = log^ ; ■•■ef'''=^r s s s .-. zs ^ = - m jgS ' ' = c + 2A + 1 (2^+ 1)5 Integration of exact differentials. The method of finding a factor which will render a function infegrable. 85. The equation Mdx + Ndy = is not always the result of the differentiation of f{a)y) =c: for after the dif- ferentiation its terms may have been divided by some common factor, or the equation may have arisen from the elimination of an arbitrary constant between the primitive equation and its derivative. 400 DIFFERENTIAL EQUATIONS. But whenever Mdw + Ndy = is the complete differential of a function of two variables, the condition = — is docdy dydx die du fulfilled, or since M = —- and iv = — - ; dof dy dM d-u dN dy dacdy dw Hence as it is necessary that every equation Mdx+Ndy=0 which is a complete differential should fulfil this condition, we have conversely a method by which we may find whether any equation is or is not a complete differential ; and since then — = M, and -— = JV, we can by integrating these partial daj dy differential equations, find the integral. dzt 86. For since M = — , J/ is the partial differential co- rf ct; efficient of w, with regard to x, considering x alone to vary, and its integral will give all the terms in which x is to be found : let the integration be performed. Then Here instead of adding a constant C, we have put F, for as y has been supposed not to vary, the constant will include those terms of the original equation, which are functions of y alone. Next to determine Y: differentiate with regard to y\ du _df^M dV dy dy dy But— =iV, .•.-— = AT- _ ; dy dy dy dlM^ u =/^-^((--^)^- DIBFERENTIAL EQUATIONS. 401 87. Since Y ought to be a function of y only, a-- dy should be independent of x. To prove this, let y + hyhe put for y in j^. M, when we have f,{M + -— Sy + kc.) = f,M + Sy f ~z- ^ kc. ay J.V dv h is removed from beneath the sign of integration since j^ refers to the variation of x only. df,M _ rdM dy /r dy Hence = / — — ; J.r dv dM^ Jy \ J.T dy j dY_ rdM dy Jx dy = 0; Now differentiate with regard to x ; d'Y _dN dM dxdy dx dy „ r I ^. dLM\ . , or F= / \N J contains y only. Jy \ dy } We may remark that had the partial differential coefficient A^ or — been first integrated, the same result would be dy obtained ; and in the application of the theory, that differential must be chosen which appears most likely to facilitate the solution of the equation. 2d-27 2xdy Ex. 1. Let du = Co 402 DIFFERENTIAL EQUATIONS. Here M = ; N = - 2x dM s/ x^ -if y y x^ - y^ 2y dN -2 ( -y^ 2y u dy {x'-y')V dx y \{x'-f)^j {x' - f)V = f,M+ Y=2f,y/x~-y'+ Y = 2\og {x + ^^^') + V, du -2y dY -2x = + dx •■ X' 3dx^ •*• -dy, I - + 1 yf SdXi < = Sa^dx^ ; .-. dyi + Sa^yi'^dx^ = 3Xi~'^dxi, which, as has been shewn, is integrable. 92. To integrate the differential equation of the first order and of any degree. Let m\pm"\Qm-\!.c.^V^O be the \dxj \dxj \dx) dy equation ; let it be solved with regard to — ; and let Ci X dy ^1, X2, ^3, &c. be the values of — , thus found; then each of the equations p = X ^, p = X.,, p = X-^, &c. when integrated will satisfy the proposed equation, as also will the equation formed of the product of all these integrals. But as the original equation arises from the elimination of a single constant, raised to the n'*" power, and since each simple integral introduces a constant, the solution will appear to contain n arbitrary constants, and therefore to be more general than that from which it has been derived. But if we consider that the constants are quite arbitrary, we may, by giving all values to them, make each equal to that particular constant which belonged to the primitive equation, and thus the result will be of the required form. dy' „ dy , dy Ex. 1. Let -— = o^ ; .-.--= a, and -— = - a; dx dx dx .-. y = ax + c, and y = - ax + c', both of which satisfy the equation. Their product {y - ax - c) (y + ax - c') = will also satisfy it. 412 DIFFERENTIAL EQUATIONS. For differentiating we obtain l~ -a] (y + ax - c') +{—- + a] (y - ax - c) = 0, \a£0 J \a,v I an d making successively y = ax + c, and y ^ - ax + c', we (^y dy , ^ get the results -^ = a; -— = - a ; as we ought. ^ dx dx The integral {y — aw - c) (y + ax - c) - contains two arbitrary constants, and appears to be more general than those of the other equations which involve but one con- stant; but we must remember that each factor ought to be separately considered, and that we obtain no other lines but those which would result from an integral including one constant only, of which constant this equation is also susceptible. This equation may be obtained by observing that if dy we refer to the original equation we have — = ± a ; and integrating y-c= ^ax, and squaring both sides, {y-cy=a^ar^. This equation gives two lines, inclined at different di- rections to the axis of x, but both cutting the axis of y in the same point ; and by giving to (c) different values, we may have groups of such lines in pairs. And the integral of (y - ax + c) (y + ax - c) gives the same result, except that each factor only represents lines inclined in the same direction ; but by giving to c and c all possible values, and taking care to collect together those straight lines in which c and c are equal, we shall find the solutions comprised in the equation (y - cf = a^x^, which is limited to the single constant c. dy" /— Ex.2. Let^ = aa?, or p = ^^y/ a^'-> dx .-. — = \/ax. and ^ = - y/ax; dx dx DIFFERENTIAL EaUATIONS. 413 '. y = - \/ a x^ + c, and y = -- y/ a x' 2 /— 3 2 /— a y = - V a x^ + c, and y = -- y/ a x^ + c', 3 3 4 each of which is comprised in {y - cy = -ax^. Ex. 3. Find the curve when s = ax -\- by. ds . / dy'~ dy Here -— = V l+-^ = a + 6 — . ctj; aa^' do? . ^ dy . . . . ^ dy And •.• — IS obviously constant, let -— = m; dx " dx .•. y = mx + c, the equation to a straight line ; y-c cV = m, an \ X J \ X J 93. When the equation only involves x and p, and the equation is easily solved with regard to x, we can in- tegrate thus : Since x = f(p) = P, and -— = p; dx whence t/ is a function of jo, and therefore of x. Ex. 1 . Let X + ap = b v i + p^ ; .-. ?/= - ap^ + bp\/l + p- - fp(-ap + by/l + p~) The elimination of p will give y in terms of x. 1 /l—x Ex.2. Let (l4-«^) .27=1 ; .'. x = -,and«=\/ — --. 1 +p'^ X .'. y = px - / = px — tan"^ p + C = V^.r-.'P--tan-i V -^+ C. 414 DIFFERENTIAL EQUATIONS. Ex. 3. Let v ^= V 1 + -7^, ' ax dx' .-. 'px = \/\ 4- /J ; .•. 't? = -; .-. y =: px - JpX = px - / '^p P r ^ r P = px - / . - / . ^ppy/l+p^ -^py/l+p^ = px-\og I - ] -a/i +p^ \1 + V 1 +pV ••• ^ = ^"S I ^^^ j = ^°^ I c 1 94. When the differential equation contains, y, x and p, and is homogeneous with respect to y, the variables can be separated by making y = zx; for then x will disappear, and we shall have ^=f(p)- dz But ■.■ y = xz; .-. p - z = x -— ; ^ dx 1 dx 1 se dz p — z dz Idx dp f {j)) X dp p-fip) p-f(p)' r ^P And X being found a function of ;>; y = p^- - j^^ ^ ^ y may be determined in terms of jo and therefore in terms of x. Ex. Let y -px = x 's/l -\- p~. Make y = a%; .-. z -p = 's/l + p'^ ; z 1 dz .-. z^ - 2zp = i^ ^^' ^^ "" o ~ ~ ~ ^ "^ ''^ J~ ' DIFFERENTIAL EQUATIONS. d% z 1 _ 1 sf^ + 1 dx 2 2% 2 z doe 2z 415 ,vdz 1 + «""' •. log — = loff = log .-. we have dx dw dw = (a? + P') — ; .-. — = 0, or .v + P' =^ 0. dx dx d 7) If we make -— = ; « = c ; .-. y = ex + c'. dx This equation appears to have two arbitrary constants ; but if we put c for p in the original equation, and C for P, C being what P becomes when c is substituted for p, we shall have y = cx+C; .'. C = c', and the equation has but one arbitrary constant. This is the general solution of the dif- ferential equation. Again from x + P' = 0, a value of p will be obtained which is a function of x or y, and does not introduce into 416 DIFFERENTIAL EQUATIONS. the original equation the constant by the elimination of which the differential equation was formed, such a solution of the equation is called a singular or particular solution. The particular value may, however, be derived from the general solution, by making c to vary ; and as y = ex + C is the equation to a straight line, we evidently see that the par- ticular solution gives the equation to the curve which is the locus of the intersections of the straight lines denoted by the general solution. Ex. 1. y — p,v = a\/l + p^ ; dy dp ap dp doc dx y/i ^ p'^ rfiT ' { ap ] ^ dp {x + —-=^=\ = 0, and 3^ = 0; .'. p = c, and y = c,T + a \/l + c"^ which is the general solution. - ap a^ 1 +p- \/ a^ - oc^ 1 But cf = — > ; ••• -; = — 5—; •'• = ~ y/\^f x^ p" 00 p cV / - ap -a a^ a' a^ - x' 's/ d~ - x~ \/a? - x^ \/d^ - x^ = - \/d- - x^ ; ■'- y^ + x^ = or- which is the solution to the following problem. "Find the curve, in which each of the perpendiculars drawn from a given point upon the tangent, is equal to a given line :" and we find (see Art. 178, Diff. Calculus); that it is the curve which is formed by the intersections of the line defined by y = cx + a VI + c^ DIFFKRENTIAL EQUATIONS. 417 Ex.2. Let y =pa? + -(l +p"); •'• y^ = ^a(a + = Ex. 3. Integrate — ^ = ^ + m dx^ * d.r^ D D 2 420 DIFFERENTIAL EQUATIONS. Let-— =p; .-. -- =g + mp'; dx ax doo dp g + mp^ ' . dy p and —- = dp g + mpr both of which are integrable, and a relation between y and x may be found. d^y ,^ d y^ Ex. 4. Integrate — -^ = F + m -— :; . dx^ dx- Make^=p; ■•. ^ = ^; a = 0; [ ax] du whence u + Pu + Q + — - = 0, aw an equation of the first degree and order ; but which is seldom integrable when P and Q are functions of x. 101. Let P and Q be constant, and let P = ^ ; Q = B. du ^ J r, .-. — + u~ + Au + B = 0; dx du ^ or - — I- (w - a) (^^ - o) = ; dx an equation which is satisfied by making u— a and u = h., ^ r II M.T + c' ^ ax and y = ef^' = e'"^'" = c,e' 2C , either of these values when substituted for y will satisfy the conditions of the differential equation, but the complete solution, which must comprise two constants is y = c.e'"' + C2e*% for by substitution we find that their value also satisfies the condition required. Cor. 1. If the roots of the equation u^ + Au + B be impossible, then a = a + /3\/-l, and 6 = a - /3'\/ - 1 ; = e"'' \ (ci + Cg) cos (ix + (ci -Cg) v- 1 sin (ix\. Make Ci + 0^ = A sin ^, (cj - Cg) v- 1 = A cos ^ ; .-. // = y/e"-^ {sin ^ cos (ix + cos ^ sin (ix\ = ^c"'' sin {(ix + ^). 422 DIFFERENTIAL EttUATIONS. Cor. 2. Let the roots be equal ; or a = b. Then y = e"'^ (cj + c^) = c^e"'' which has but one constant. To find the second constant. Suppose b = a + h; .-. y = c,e'"" + c^e"'"' + ''■'' = e"' jci + c,e*'j = e''' {ci + c^ + c^hx + - — ~ + &c. | ; make Ci + Cg = c', c.^h = c", and /t = ; .-. ?/ = e"' (c' + c"a-0. 102. The equation d^V ^dy dw dx is seldom integrable when P and Q are functions of ,v ; it can however be solved when P = ; and Q = For make a + 6.^ = e'"" ; rf;;r 1 d?/ dy dz dy 1 (/,77 a + bx^ dx dz dx dz a + bx d^y d'^y dz 1 dy b dx" dz' dx a + bx dz (a + bx)- d~y_^dy\ 1 '^dz^ dz) (a + bx)' whence by substitution, and multiplying by {a + bx^, d^z ^ , ,dy which may be integrated by the preceding methods. DIFFERENTIAL EQUATIONS. 423 103. To integrate the general equation -^ + A- — 4 + 5— -4 +&C. + L2/ = 0; where A, B, C, &c. L, are constant. dy d'y „ Let y = e'"'' ; .-. -^ = me'"' ; -^ = w«'e'"% &c. .-. -m" + Jm"-' + Bm"'^^ + Cm^-'^ + &c. + L = 0. Let a, b, c, &c. be the roots of this equation ; then y = e"'', y = «''•'', ?/ = e*^"^; &c. will be particular integrals of the general equation, and the substitution of each in it will satisfy it. Hence the complete integral will be, by the introduction of n constants y = Cie"" + c^e'"' + Cgg'-'^ + &c. CoR. 1. Should any of the roots be equal, as a = b; then for Cje"'' + Cge*', we must put e''^(ci + 02^?); .-. y = e"'^ (ci + CgcT) + Cge*^^'^' + &c. And if three roots be equal, and a be the equal root, we must put for Cje"'' + c.^e'''' + 036''% the term and so on for any number of equal roots. CoR. 2. If pairs of roots be impossible, we must sub- stitute for the impossible exponential functions, the cosines and sines of the circular arcs, to which they are equivalent. Ex. 1. — — • + n^u = 0. dO' 424 DIFFEIIENTIAL EaUATIONS. Let M = e"'^ .-. — = 7«e'"'^; — - = m2e'"% .-, m^e""^ + n^e'"^ = ; .-. rnr + n- = 0, and m = ± ws/- 1 ; = (c' + c") COS nO + (c' - c") V^ sin wO = ^ cos (w ^ + B). If c' + c" = J cos B, and (c - c") \/'^\ = - J sin B. Ex. 2. — - + n-u + a^ = 0. dB Make a^ = n^j^, and u + j3 — w, whence we have d'w and the solution is performed as in the preceding example, d^s ds Ex. 3. -— + 2k — + fs = 0. df dt -^ Make s = e'"' ; .-. ni^ + 2km + f= 0; .-. m= -k^ y/^ s/f- k' = -k^ a y/ - 1 ; .-. s = e-*' (c'e«*^~ + c" e- "'^"') = Jg-*' cos (a# + B). Examples (l) and (2) are useful in Physical Astronomy^ Ex. (3) gives the space a function of the time, when a body moves through the arc of a cycloid, the resistance varying as the velocity. d^y d'^y dy Ex.4. -4-6-4+ 11-/-- 62/ = 0. rf.r* dw^ d,v Let y = e""'; .*. m^ - 6m^ + 11 m - (i = 0, the roots of which are 1, 2, 3; .-. y = Cye^ + c.^e-^ + c-^e^". DIFFERENTIAL EftUATIONS. 425 Ex. 5. Let -4 - 3 —4 + 3 -^ - 1/ = 0. dw^ dx' dw Here if ^ = e""', m? - Sm" + 3m -1 = 0, or (w - l)^ = 0, And .-. y = e" {c^ + 0.^0; + c^ob^). _ d^y dy . Ex.6. — ^^ + 8-^ + lbt/ = 0; .-. y = e-' ■'{€,+ c.,w). dar dx d^y dy Ex. 7. -r^,-'^-r + 34>y = o; •'• y = ^4e^'^cos (5 + 5x). dx~ dw Ex.8, ^-i^ + 1.0. Make *• = e^ .•.— = -; -^ = - -^ , a-j? e " + c^ + dx. d"y Ex. 13. Integrate — - - y = 0. doc" Make y = e""" ; .-. m" -1 = 0, let 1, ai, as, as, a^, &c. a„_,, be the roots of this equation; .-. y = Cie-" + c\e"-'' + Cse"-"'' + &c. + c„e""-'*'. 104. To solve the equation, d^y ^ dy ^ ^ , We shall shew that the solution of this equation may be made to depend upon that of the equation, d,~v dy y + P Y^ + Q// = o (2) dx~ (IX DIFFERENTIAL EQUATIONS. 427 To effect this, we proceed to apply to this equation, a method called by Lagrange, " The Variation of the Para- meters ;" which consists in this, that if y = c'yi + c"y2 be the solution of the equation (2), we may assume it to be that of equation (l), if c' and c" be considered no longer constant but functions of w. Let .•. y = c'yi + c'y.^ be the solution of (l) ; _ dy__ ,d]h n^y^ ^ ^ dx d.v dv * div doo But as we have made but one supposition to determine c and c", we may make another, let therefore dc dc" dy , dy^ ,, dy-^ dx ■ dw ' dw dx dx ' d'^y ,d^y\ n^^lhi dc dyx dc" dy^ dx^ dx^ dx' dx dx dx dx ' whence by substitution in the original equation (l), dc' dyi dc" dy.^ dx dx dx dx which by means of equation (2) is reduced to dx dx dx dx dc" yi dc dx t/a dx dc_lfHh _ y\ dy^\ ^. dx\dx y^ dx) dc whence —-^ is found to be a function of /r, and c' = X. + C\, dx also similarly c"= ^^ + Cg; 428 DIFFERENTIAL EQUATIONS. A similar proof of this proposition applies to equations of a higher order. Ex. 1. Integrate ——; + a~y = cos fix. CI/ Ou The solution of the equation -— - + c^y = is ax'' y = c cos ax + c sin ax ; assume this to be the solution of the proposed equation ; dy , . „ dc dc" . •, — = - c a sm ax -\- c a cos ax + -7- cos ax + — — sm ax dx dx dx = — c'a sin ax + c"a cos ax. , dc' dc" . Since we make — — cosatr+— — sina^ = 0; dx dx (Fy ,2 " 2 • ^^' • ^^" = — ca cos ax — c a smax - a-;— sin ax + a — — cos ax da^ dx dx dc . dc" = - a~y - a — sm a a? + a - — cos ax ; dx dx dc' . dc" ^ .-. — a-r- smaa; + a - — cos a ^^ = cosp**. dx dx „ dc" cos ax dc But -— = - -. — ; dx sin ax dx dc / . cos^ax\ dc / . cos^a^N ^ a — — sm a