Aw, Paes +4 ee ° ge Br, ae pre as A chef Pee . ‘ Mes ws * Sad 2 - - Mas d 3 v a IY 2 ar ; ¢ f te > pe : 4 i ae . ; + i & * & : ‘ ‘- * F ‘ fh t ‘ s Y Ke Cae A $ ¢ f a : » ane ae 5 4 . : } Ry a ‘ ee hy ey 7 s f 5 e r hai ; 1‘ . : aes sights ate oF Bui nos © ndusteial ie ILLINOIS. Return this book on or before the Latest Date stamped below. University of Illinois Library L161—H-+41 Hoyat tio. N ia 2O VT - ROBINSON'S /MATHEMATIOAL SERLES. A NEW TREATISE ON THE ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL Peat CU 1. US: EDITED BY 1 eA QUINBYp A.M, LED: ARRWETERRY | PHILOSOPHY, UNIVERSITY OF ROCHESTER. a / ." . : oF al < rIVPr | ae 3 / 1 KK are” PROFESSOR OF MATHEMA NEW YORK: a IVISON, PHINNEY, BLAKEMAN, & CO., of 47 & 49 GREENE STREET. PHILADELPHIA: J. B. LIPPINCOTT & CO. CHICAGO: S. C. GRIGGS & Co. 1868. w tt cof n ' cP 7! Cen Sie et > ey ae et cart > sa 4 ts eh Sets ' . Gey ; ta gt bi sayy ‘Ke Ne aus MiNi Wwa whe s bid 2 DANIEL Ww. FISH, AM, In the Clerk’s Office of the District Court of the United States for the } of New York. “35 (i e n * /~ ‘ - ’ ~ ~ “ d «a . 4 a - “ as be . f 4 « # \ ~~ - « . ’ / yer oe i . : MM : “4 ‘ 7s pe ~ tm ew P ¢ : * t ‘ ag cogs - is ee ee a . im ri a iy 7 a pia t x \ tes VATHEMATIC® Linpsre THe design in preparing this treatise on the Differential and In- tegral Calculus has been, not so much to produce a work that should cover the whole ground of this extensive and rapidly extending branch of mathematics, as to produce one that should be complete within the limits assigned it, and adapted to the wants of students in the higher schools and colleges of this country. Many of the subjects are much more fully discussed in this volume than in other elementary trea- _tises; while many are entirely omitted here which are generally included in such works, though they are not essential to, and are rarely embraced in, the college course in this or in other countries. The necessity devolved on the author, either to be limited in the num- ber and full in the treatment of the subjects selected, or full in the number of subjects, and limited in their discussion. The former choice was taken, keeping in view the logical and progressive develop- ment of the principles. This will account for the omission, among other subjects, of the integration of differential equations of the different orders, and of the ‘* Calculus of Variations,” the latter of which, when fully treated, would make a volume equal to the present in size. 4 PREFACE. It will be found, however, that the time usually given to this study will render it impossible to take, in course, all the subjects herein treated. The following are what may be left out in the class-room without serious breaks in continuity : — DIFFERENTIAL CaxcuLus. — Part First.— Section V., from Article 68 to the end of the Section. The whole of Section VII. Section X., from Article 110 to the end of the Section. The whole of Sec- tion XII. Section XIV., from Article 139 to the end of the Section. DIFFERENTIAL CatcuLus. — Part Second. — The whole of Section II. Section IV., from Article 177 to Article 181; from Article 188 to the end of the Section. INTEGRAL CaucuLus. — From Example 4, Section IV., to the end of the Section. The whole of Sections IX. and X. It will be observed that the fundamental proposition of the Differ- ential Calculus is based on the doctrine of limits; and that of the Integral Calculus, on that of the summation of an infinite series of infinitely small terms. The author adopts these methods merely on logical grounds, but ventures the opinion that these, and what are called the infinitesimal methods, are based on the same metaphysical principles. | THE AUTHOR. NOVEMBER, 1867. DIFFERENTIAL CALCULUS==-=—- pe ASE Ly 2 FE, SECTION I. PAGE. GENERAL PRINCIPLES AND DEFINITIONS . . : . : ° Sn a SECTION: Tk DIFFERENTIAL CO-EFFICIENTS OF EXPLICIT FUNCTIONS OF A SINGLE VARIABLE, 23 eo. CT LON EL. DIFFERENTIAL CO-EFFICIENTS OF INVERSE FUNCTIONS, FUNCTIONS OF FUNC- TIONS, AND COMPLEX FUNCTIONS OF A SINGLE VARIABLE . : 2 Z . 41 SECTION: Ty: SUCCESSIVE DIFFERENTIAL CO-EFFICIENTS ., : ; 2 r 7 ‘ ‘ BO 5.0 TE TO NV. RELATIONS BETWEEN REAL FUNCTIONS OF A SINGLE VARIABLE AND THEIR DIF- FERENTIAL CO-EFFICIENTS.—TAYLOR’S AND MACLAURIN’S THEOREMS ., - 69 SECTION - VI. EXPANSION OF FUNCTIONS ,. « i Be ON Whe - APPLICATION OF SOME OF THE PRECEDING SERIES TO TRIGONOMETRICAL AND LOGARITHMIC EXPRESSIONS . : i a eeat A ’ e vs : . ae EY, S ECcoLOone VIL: DIFFERENTIATION OF EXPLICIT FUNCTIONS OF TWO OR MORE INDEPENDENT VARIABLES, OF FUNCTIONS OF FUNCTIONS, AND OF IMPLICIT FUNCTIONS OF SEVERAL VARIABLES A - - : : - F - 7 J - . BEA lr: SECTION IX. SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE INDEPENDENT VARIABLES, AND OF IMPLICIT FUNCTIONS. ° ° . . ° . - . 133 SECTION X. INVESTIGATION OF THE TRUE VALUE OF EXPRESSIONS WHICH PRESENT THEM- SELVES UNDER FORMS OF INDETERMINATION , ° - ' ‘ pe A Paty, SECTION XI. DETERMINATION OF THE MAXIMA AND MINIMA VALUES OF FUNCTIONS OF ONE VARIABLE . . s e s e . . . . s e . . ° e 176 SECTION XIL EXPANSION OF FUNCTIONS OF TWO OR MORE INDEPENDENT VARIABLES, AND . ° ° . ’ ‘ « 90 INVESTIGATION OF THE MAXIMA AND MINIMA OF SUCH FUNCTIONS. ‘é - 188 ee OLE = ee CHANGE OF INDEPENDENT VARIABLES IN DIFFERENTIATION , = d e2it PS lk Od fad Se. @ ELIMINATION OF CONSTANTS AND ARBITRARY FUNCTIONS BY DIFFERENTIATION, 226 5 6 CONTENTS. PART Ty GEOMETRICAL APPLICATIONS. SECTION IL. Lp a TANGENTS, NORMALS, SUB-TANGENTS, AND SUB-NORMALS TO PLANE CURVES , 238 ~ Sie GLO No ASYMPTOTES OF PLANE CURVES.—SINGULAR POINTS. —CONCAYVITY AND CON- VEXITY e . . e e e e . es e e e e . . . 249 SECTION III. POLAR CO-ORDINATES. — DIFFERENTIAL CO-EFFICIENTS OF THE ARCS AND AREAS OF PLANE CURVES.—OF SOLIDS AND SURFACES OF REVOLUTION , 5 - 266 SECTION IV. DIFFERENT ORDERS OF CONTACT OF PLANE CURVES. — OSCULATORY CURVES.— OSCULATORY CIRCLE. — RADIUS OF CURVATURE. — EVOLUTES ANDINVOLUTES, 281 , INTEGRAL CALCULUS. SHCTEON SRE MEANING OF INTEGRATION. — NOTATION. — DEFINITE AND INDEFINITE INTE- GRALS.— DIRECT INTEGRATION OF EXPLICIT FUNCTIONS OF A SINGLE VARIA- BLE.— INTEGRATION OF A SUM.— INTEGRATION BY PARTS.—BY SUBSTITUTION, 315 On Ons tan OB ae 8 INTEGRATION OF RATIONAL FRACTIONS BY DECOMPOSITION INTO PARTIAL FRACTIONS . : 2 ' ° F ° F : ° * . . F “ . 343 SE.C TION ei FORMULZ FOR THE INTEGRATION OF BINOMIAL DIFFERENTIALS BY SUCCESSIVE REDUCTION . . ° . . . F ° z . * r A - » 360 SECTION oly; GEOMETRIC SIGNIFICATION AND PROPERTIES OF DEFINITE INTEGRALS. — AN- OTHER DEMONSTRATION OF TAYLOR’S THEOREM.— DEFINITE INTEGRALS IN WHICH ONE OF THE LIMITS BECOMES INFINITE.— DEFINITE INTEGRALS IN WHICH THE FUNCTION UNDER THE SIGN / BECOMES INFINITE. — DEFINITE INTEGRALS THAT BECOME INDETERMINATE, — INTEGRATION BY SERIES . O74 SEC TIGR. GEOMETRICAL APPLICATIONS. QUADRATURE OF PLANE CURVES REFERRED TO RECTILINEAR CO-ORDINATES. — QUADRATURE OF PLANE CURVES REFERRED TO POLAR CO-ORDINATES . - 392 SECTION. VI. RECTIFICATION OF PLANE CURVES . ° ° ° . ‘ . ‘ ° R . 405 SECTION VIL. DOUBLE INTEGRATION.— TRIPLE INTEGRATION . e « . e e F « td SECTION © VII. QUADRATURE OF CURVED SURFACES. — CUBATURE OF SOLIDS . ‘ : ‘ » 417 SECTION IX. DIFFERENTIATION AND INTEGRATION UNDER THE SIGN “,.— EULERIAN INTE- GRALS. — DETERMINATION OF DEFINITE INTEGRALS BY DIFFERENTIATION, AND BY INTEGRATION UNDER THE SIGN /. ° . . . . . . . 429 SECTION X. ELLIPTIC FUNCTIONS : F ; x : 4 F ‘ ‘ ; A ° ° » 465 DIFFERENTIAL CALCULUS. LEAS CONE 1S NGA See Shed ee oi ed Ae SECTION I. GENERAL PRINCIPLES AND DEFINITIONS. 1. In the branch of mathematics of which it is now pro- posed to treat, we have to deal with two classes of quantities, —constants and variables: constants, which undergo no change of value in the investigations in which they are involved; variables, which may pass through all values within limits that may be restricted or indefinite. Variables are usually represented by the final letters of the Roman alphabet; and constants, by the first letters of this, and sometimes, also, of the Greek alphabet. 2. When variable quantities are so connected, that, one or more -of them being given, the values of the others become fixed, the latter are said to be functions of the former, which are called the independent variables, or simply the variables. The functions are also called dependent variables. Thus, in the equation y — ax’? + bee, y is a function of x, and in this case becomes not only fixed, 7 8 DIFFERENTIAL CALCULUS. but known, so soon as a value is assigned to x. So also, in the equation y — ax + bx +cz +d, y is a function of the two variables # and z, and is known in value when values are given to x and z. 3. An Huplicit Function is one in which the depend- ent variable is given directly in terms of those which are re- garded as independent. In the examples given above, y is an explicit function of # in the first, and-of « and z in the second. In general reasoning, when we are not concerned with the particular form of the function, explicit functions are denoted by the symbols y = Lia), y=f(x), y= 9,4), &e. 4. An Implicit Function is one in which the relation between the function and the independent variable or variables is expressed by an equation that has not been resolved in respect to the function. Thus ax + by +ce=—0, ax” + bry + cy* + dx + ey +f=0, x? —az?+ cyxz +d =), are equations which require solution to render the variable, taken as dependent, an explicit function of the independent variables. Such functions are also designated by the symbols ta, y) =O; oe; Ua ae 5. Functions are also classified, in reference to their com- position, into semple or compound, according as they are the result of one or of several operations performed on the varia- bles. They are algebraic, when, in the construction of the GENERAL, PRINCIPLES AND DEFINITIONS. 9 function, the only operations to which the variables are sub- jected are those of addition, subtraction, multiplication, divis- ion, involution denoted by constant exponents, and evolution denoted by constant indices; transcendental, when, in the composition of the function, the variables have been subjected to other operations, combined or not with those regarded as algebraic. mee ey log. ae, yy = Sin. a, 4 = smn! a,* * are examples of transcendental functions, and are exponential, logarithmic, or circular, depending on the mode in which the variable enters the functions. 6. A function may be continuous or discontinuous. It is continuous, when, by causing the variable to pass gradually from any value to another separated from the first by a finite interval through all the: intermediate states of value, the func- tion will itself pass gradually through all the values interme- diate to those corresponding to the extreme values of the vari- able; and when, besides, the law of dependence of the function upon the variable does not change abruptly in the interval. y —f'(x) is continuous, if, by giving to « the infinitely small increment h = aa, y receives the infinitely small increment Ay=—F(x+ax)— F(x). When the law of the function is such that these conditions are not satisfied, the function is discontinuous. “7 The Limit of a Function is the value towards which it converges, and from which it finally differs by less than any assignable value, when the variable upon which it depends itself converges towards some fixed value. * Read arc whose sine is x, and frequently written arc (sin. = x). The notations cos.—' x, tan.—'z, &c., have like significations. 2 10 DIFFERENTIAL CALCULUS. It is of the highest: importance that we should have a clear conception of the nature of limits as above defined, as this conception is at the foundation of the differential calculus as developed in the following pages. The following examples will illustrate the meaning of limit, and give distinct notions on the subject to those who have not already formed them : — 1st, In the geometrical series $4+4+4+4éc., the sum S of the first m terms is given by the formula ga80—r) 40 0) 1l—r 1—} Fk and it is obvious, that, as 2 increases, (4$)” decreases; and, when n becomes greater than any assignable quantity, (4)” becomes less than any assignable quantity. In the language of the definition, as converges towards infinity, S converges towards unity. Hence the limit of the sum of this series, when v is indefinitely increased, is 1. od, The ratio of an are of a circle to its sine has unity for its limit when the arc converges to zero; that is, limit sin. & | 1. For it is plain in the first place, that, for sensible x values of «, the sine is less than the arc. And again: since the triangle formed by the radius, the tangent, and the secant, has for its measure 4 #. tan. x, while the corresponding sector is ‘measured by 4 &. x, it follows that the are 2 is less than tan. a. Therefore sin. x a sin. x sin. & x x x tan. x sin. & Hence we conclude, that, for sensible values of the are a, a GENERAL PRINCIPLES AND DEFINITIONS. Teh is always included between two ratios, both of which have unity for their limit. It must, then, have the same limit; and _ -sima we have lim. tl I a Again : sin. x . e Hb; SIn. @ COs. x tan. # a a cos. ©, and 46; x 4h Penn 2 ; t ‘ lim poaiees LI cos. 3 but lim. cos; a ==. 1, } eg tan. therefore the limit of sari must also be unity; 7.e., the limit of the ratio of an arc to its tangent is unity. Cor. The limiting ratio of the arc to its sine, and of the arc to its tangent, each being unity, it follows, that, when the arc is infinitesimal, the arc and its sine, and the arc and its tan- gent, may be regarded as equal. 3d, For another example, let us take y = , and trace x 1l+2 out the series of values which y assumes when positive values are given to x Beginning with x—0,we havey—0. By whi 1 Lae division, the value of y takes the form 1— ——-; from which it ctl is seen, that, as x increases, the subtractive part of the value of y decreases, and y itself increases ; and, as # approaches + oo, y approaches its limit 1: but, for all finite positive values of x, the values of y are less than 1. The difference between y and this limit can be made as small as we please by giving to x a value sufficiently great. Thus, if we wish to make this 1 ] ——— = difference less than 1,000,000" we make x = 1,000,000 In this same example, let us now give to x negative values, and observe the changes in the value of y as x increases nega- 12 DIFFERENTIAL CALCULUS. tively from 0, and approaches — oo. Replace x by —4#, then x ane. y fia ei Waa ale TG Beer and let us consider the values of y answering to values of ¢ between the limits 0 and 1. Beginning with ¢ = 0, we have y —O0: for all other values of ¢ between these limits, the denominator of y being negative, y is itself negative. As ¢ increases, ¥y Increases numerically ; and, when ¢ differs from unity by less than any assignable quantity, y is greater numerically than any assignable quantity; that is, —o is the limit of y for ¢ = 1, which answers to x——1. This is equivalent to saying that y has then no finite limit. ; : t When ¢ passes 1, the denominator of the fraction —-— be- comes positive, and y changes from negative to positive. In this case, y passes abruptly from — o to + o while ¢ is pass- ing through the value 1. The value of y may now be put ye : under the form y = ———. For all finite values of ¢ greater i halbees t than-+-1, y is greater than 1: but y decreases as ¢ increases; and finally, when ¢ becomes greater than any assignable quan- tity, y will differ from its limit, unity, by less than any assign- able quantity. Trigonometry furnishes a case of limit similar to that of this example, when ¢ passes through the value unity. As an arc increases continuously from 0, its tangent also increases continuously, but. more rapidly than the arc; and, as the arc approaches 90°, the tangent approaches its indefinite limit +o. When the arc passes through the value 90°, the tangent changes suddenly from an indefinitely great positive to an indefinitely great negative quantity. GENERAL PRINCIPLES AND DET TOaS. 3 P 8. The exact meaning of the word «limit » will be under, stood from what precedes; but it is well to tall attention to \, abbreviations of expression frequently sed in this/ connéc. ’ tion. In finding the limit of" when « Nore 7 | out limit, it would be said, we ae POPS N ae limit = 1 when «= 0; but it must be borne in mind that ats a } cannot reach this limit so long as a has any value. And, if we actually make Re aine 2 : x = 0, the ratio has no meaning ; in fact, ceases to exist. Ai 4 It is true, that if « be not supposed to vanish, but simply to differ from 0 by less than any assignable quantity, that is, if x becomes infinitesimal, the ratio retains its significance, and its value will differ from its limit unity by less than any assign- able quantity. In this case, the language is an abbreviation for this or its equivalent: “As a is diminished, the ratio sin. & converges x towards unity, and can be made to differ from it by as small a quantity as we please by taking « sufficiently near zero.” And, in all similar cases, the language is to be interpreted in the same way. In other cases of limits, the inconsistency just pointed out does not present itself. Any finite value of y, in the example “ LY tiem ey that answers to an assumed and finite value of a, x may be taken as a limit of y; and it would be strictly correct to say a 1 limit of ——— = — when zw = 1. z+ i 2 This corresponds to the definition of limit given in Art. T. y 4 y 14 DIFFERENTIAL CALCULUS. 9. Rules for the evaluation of functions, which, for particu- lar values of the variable, assume the indeterminate forms 0 : ; 0” oe 2 ,0 x wo, 0°+ 1%, will be established in a subsequent section: but itis necessary for our purposes to consider in this 1 *place the function y = (1 + a), and to find its limiting value when x = 0; the function then taking the indeterminate form 1°. The variable « may converge towards its assigned limit zero through either positive or negative values. Let us first is cee suppose x to be positive, and represent it by the fraction —;. m then, as x diminishes, m increases ; and, when « becomes a very small quantity, m becomes a very great quantity. If m be an entire positive number, we have, by the Binomial Formula, (+2) =(145 ny) =1+14 a Go m? Re gl it SP m m—-l1 m—2 1 B ey 3 mit &., a development which will contain m + 1 terms. Dividing both numerator and denominator of each term by the power of m that enters the denominator, we find E z| m 1 td 1 9 (+2) e € + . =245(1 5) +33(1—<) (12) ei 1 2 33 +3537—z) (1—=) (1—S) + og Under the hypothesis that m is a positive whole number, the 2 33 . o,@ expressions 1 — —, 1 — —, 1 — -, &c., will each be positive, eat m m GENERAL PRINCIPLES AND DEFINITIONS. 15 oN \n and less than unity. Therefore (1 +- =) = 2 + some posi- (a>: Again: the development will be increased in value both by tive quantity; that is Le neglecting the subtractive terms —, —, m oo? and also by m m replacing each of the denominators 2, 3, 4, &c., by the least denominator 2; that is, the true value of the development is less than 2 plus the series Digits fia cae ORS Ten an as But this series cannot exceed 1, however far continued: 1 2: 1\” therefore (1 + x) 2 (1 +=) is always included between m the limits 2 and 3. n ee if — = mis a fractional number, it will be found between ae two consecutive whole numbers m andn=m-+1. Lets and t be two positive proper fractions, whose sum is always equal to 1, and make LK — ° m m + s =n —t, whence ) 1 and the expression (14 ca will be included between 16 DIFFERENTIAL CALCULUS. Now, as 2 decreases indefinitely, m and increase to infin- kof Ty? 1 ity, and the two quantities € +5) and € a =} both con- verge to the same limit, which, as was proved above, is includ- ed between 2 and 3; while the exponents 1+ =, 1 Bd, n converge to the limit 1. ? 1 It follows, therefore, that the two expressions € ot =) m Dyes ee € 4. a1 have the same limit, and that this limit is the same 1\™ as that of € a. =) when m, regarded as a positive whole number, is indefinitely increased. Finally, if x is negative, and either entire or fractional, make 1 : oe ani : so that, for all values of x numerically greater Z than 1, z must be negative and included between the limits 0 and 1; but, for values of x less than 1 (and it is with these alone that we are now concerned), z must be positive, and increase to infinity as # decreases to zero. Making this substitution for x, we have eae 1 SAE eae 2+] z+l (i+e)*=(-775) Sn z+1 1X6 i++ Hence, when « approaches its limit zero through negative 1 values, the limit of the expression (1 + 2)? ee i Aw) ark _ € + =) is the same as when this limit is reached Z by causing x to decrease through positive values. GENERAL PRINCIPLES AND DEFINITIONS. iy. To find what this limit is, resume the equation (adfase BHO DO-2 Hasta) 5) 5) + eee and suppose m to be wee. then , 1 Lae I 1 sed a Teer} & im. ( $2)F = 45 Hy eR 4 eB ee 2 — 2:0000000 ... fe — ‘5000000... 2 1 es — ‘1666666... 2:3 1 BT et te, — -0416666... pay wae 1 pvericadl a — -0083333 .. 9.3.4. 3 ' — 0013888 ... 2.3.4.5.6 1 pee eee ee O01 984). - 2.3.4.5.6. / eee pa , - = 0000248... ... 2-4.4.5.6.1.8 1 se — > Dike: 9.3.4.5.6.7.8.9 Ne Sum = 2°7182818 . .. ,anumber that is incommensurable with unity. It is the base of the Napierian system of logarithms, and will be denoted by e: 3 18 DIFFERENTIAL. CALCULUS. Therefore 1* (1+ 2)?_, = 27182818 =e. The symbol 7 will be hereafter used to designate Napierian logarithms, and Z will denote other logarithms. 1 10. Taking the Napierian logarithm of (1+ a#)2, we have | Bp ie I iL @) = Clie eee and, passing to the limit a making « = 0, we have ax 1 x Uil+a)jeslea. In any other system of logarithms, we should have L(-e)f =— Lan) = 249, and at the limit Dili dae wee cee 1 lim. Te ee Cate lim. L(t ee)? ee Le = ja? a being the base of the system characterized by Z, and ob- serving that, since the logarithms of the same number in two systems are as the moduli of those systems, we have Te: -le 3s: Me), ov ers si ee eee Lat late Hes, or ot tas AE 1 and therefore Le = MI = ia? the modulus of the system of al which a is the base. 11. Since lim. (l + 2)" =e when @ is decreased without 1 + * The notation (1 4 x) x indicates that the value of the expression correspond- 2= ; ing to x =O is taken. GENERAL PRINCIPLES AND DEFINITIONS. 19 1 limit, we can from this deduce the limit of (1 + az)? in which a is any constant quantity. Thus, Le Ae (1+ az)? = («a + az)™ | : Now, as 2 diminishes without limit, az will also diminish without limit; and therefore 1 lim. (1 + az)e =e; l Zz lim. (1 + az) — oo 12. In any system of logarithms, 1 e) 1 Lz) = = LU +2); Ll 1 hy lim. SOT) tim. £0. + 2)* = Lies and, if the logarithm be taken in the Napierian System, lim. Ae) mnt aed z 13. Resuming the equation phe 93 0 Ee L(+ 2) = oes and making 1 + z= a’, whence (taking logarithms in the sys- tem of which a is the base), v= Z(1+ z) and z=a’— 1; therefore . ) Reifel, Pa kets = } ( = ) a’ — 1 ’ or, by taking the reciprocals, 1 : a°’—1 Lli+s2) vw Now, as z diminishes without limit, so also will v, and they will reach the limit zero together ; therefore 20 DIFFERENTIAL CALCULUS. 1 Soh, Oa ——, = Im. ; L(+2) v 1 putin: ose ae L(1i+ 2); Le Oh gt —— 1 1 lim. = — =lawhenv = 0. vy Le Suppose a = e”, whence m = la; and therefore lim. cas ee en 4 , 14, To define some of: the terms, and explain the meaning of some of the symbols, employed in the calculus, let us take the explicit function of a single variable ; y =f (2), and give to x an increment denoted by ax; y will receive the corresponding increment ay =af(2) =sf(e + a2) —/(2), and therefore ay _ fle +a) —f(2) Aw AX . 0 . ° When ax = 0, the ratio 47 takes the form o; yet it has in AX fact a determinate value, which is generally some other func- tion of x, and expresses, as will be seen presently, the tangent of the angle that a straight line, tangent to the curve of which y =f («) 1s the equation, makes with the axis of the variable x. This limiting value of the ratio of the increment of the variable to the corresponding increment of the function is called the differential co-efficient, or derivative of the func- tion, and is represented by the notations y', f'(2), aa lim, lim, #8 Aw Ax GENERAL PRINCIPLES AND DEFINITIONS. 2% It is to be observed that the characters a, d, are not factors, but symbols of abbreviation ; the former signifying increment, difference, or change in value, without reference to amount ; while the latter is restricted to particular increments called differentials, having such values, that the ratio of the differ- ential of the variable to that of the function is equal to the differential co-efficient or derivative of the function. The differentials are usually regarded as infinitely small. ’ Sleep se! ‘ Ane . 15. Before the ratio == reaches its limit /“(x), it must differ from it by some quantity which is a function of aw, and which vanishes when Av = 0. We may therefore write, Ay x + Ax)—f(a ’ eo PEPIN apa) +7 AX and, by clearing of fractions, ay = f(x + sa) —f(a) =f’ (aw) aw + pace. ‘Dg - AY ‘ eats From this it is seen, that, as the ratio me approaches its limit Jj’ (x), y must approach zero; and when Az, and consequently Ay, becomes infinitely small, 7 must also become infinitely small, and should therefore be neglected in comparison with the finite quantity f’(x~). We shall then have Ay dy —=/f'(x) = oe Aja (ae) A — dy. AL It is therefore true, that, when the differential of a function is infinitely small, it is sensibly equal to the increment of the function. These considerations are of importance, and are made by many authors the basis of the definition of the differential of a function; viz., “The differential of a function of a sin- 2? DIFFERENTIAL CALCULUS. gle variable is the first term in the development of the dif- ference between the primitive state of the function and the new state which arises from giving to the variable an incre- ment called the differential of the variable; the development being arranged according to the ascending powers of the in- crement.” 16. The definition of the differential of a function follows from that of the differential co-efficient. It is the product of the differential of the independent variable by the differential co-efficient of the function. The object of the differential calculus is to explain the modes of passing from all known functions to their differential co-efficients, and the application of the properties of such co- efficients and the corresponding differentials to the investiga- tion of various questions in pure and applied mathematics. The operation of deriving from functions their differential co-efficients is called differentiation. SECTION II. DIFFERENTIAL CO-EFFICIENTS OF EXPLICIT FUNCTIONS OF A SINGLE VARIABLE. 17. Iv will be convenient, before proceeding to establish rules for finding the differential co-efficients of the different kinds of explicit functions of a single variable, to investigate certain principles which are applicable to all forms. Constants connected with functions by the signs plus or minus disappear in the process of differentiation. The increments of the function and the variable will be char- acterized by the symbol A when they are written in the first members of equations ; but the labor of making the transforma- tions sometimes required in the second members will be les- sened by representing the increment of the variable by the single letter h, which will, of course, be equal to az. Let y =/(x) +c, and give to the variable in this equation the increment A; then yt sy=f(e+h)+e; therefore Ay=f(«+h)—/f(x), Ay f(@+h)—f(x). AD h Passing to the limit by making ax =A =—0, a AT dy lim, —~ ——~ — f/f’ wt fie oa de 1); ‘and dy — f(x ao. 23 2A DIFFERENTIAL CALCULUS. This differential co-efficient 1s manifestly the same as that which would have been found had there been no constant united to f(a) by the sign plus or minus. As, from their very nature, constants admit of no change of value, c¢ has the same value in the new that it had in the primitive state of the func- tion, and must therefore disappear in the subtraction by which the increment of the function is obtained. dy rox, aL y=—a+a, reach di) =a dy AUX. ee a SS eee 18. lf a function of a variable be multiplied or divided by a constant, the differential co-efficient will also be multiplied or divided by the constant. Let y= (x), then ade of (2 +h), = of (x +h) — f (x) =c (f(@+h) —f(a)), (rie+® —F(2)), An Passing to the limit, a lim. Ae iota = and dy = of (x) da. Again: let y ==), i . eels then y= -(f(@+h)—/(2)), ay 1 S(@+%)—s (x); Ax h hae ee oY 8 =f" (x), AL DIFFERENTIAL CO-EFFICIENTS. 25 and dy = f(a) dx. d Ex. 1 y=actb, =a, dy = ada xv ae 1 Ex. 2. = -— —-=- wet : Seren, e Ditas oy ay “a 19. The differential co-efficient of the algebraic sum of sev: eral functions of the same variable is the algebraic sum of the differential co-efficients of the separate functions. Let y=f(e@)tG(v)ty(w)t..., then ytay=/(e+h)+G(eth)ty(eth)+... ay =f(@ +h) — f(x) &(9(@ +h) — 9(e)) + (w(@+h) — (a) + eee Ste N ie) ret is 9) Ax h h whence, passing to the limits, and using the previous notation, = ee ee 5 aed SP (Ee) Mirae, ; Ax L and dy —f'(x)dx+ g’(x)dxtw'(x)dr+t... = (f(x) £ 9! (@) ew! (2) + ae .) dx Ex. 1. y—ax—ber+e ix, 2. y = af(xz) + br/— 1 g(a) a — af’ (x) + br/— 1 g’ (x) dy = (af" (x) + br/— 1g’ (x)) dx. 26 DIFFERENTIAL CALCULUS., 20. The differential co-efficient of the product of two functions of the same variable is the sum of the products obtained by multiplying each function by the differential co- efficient of the other. ) Let y¥ =S(&) X4(z), then y+ay=/(e@ +h) x p(w +h) ay =f(% +h) X g(a +h) —f(a) X (a) =(S(@ +h) —f(@))9(@ +h) + (9@ +2) — (a) Sm); eth xe h) — g(x 44 fet D—SO) ig 1» 4 eos Passing to the ey by? making Aas fies ‘ we see that lim. DMT: © ale) = f'(x), lim. p(x +h) =qg(a) = o’(x)shenge lim: 24 =p : x (2) + 9a) x f(2). Dividing this equation by y=/f(a) X g(x), member by member, we have dy dx f'(@) , 9’ (2) yf (a), uo (a) 21. The rule just demonstrated for finding the differential co-efficient of the product of two functions of the same varia- ble may be extended to the product of any number of functions. Let y =f(x) X p(@) X w(z), and make E(x) =g(x) X w(x); then Y = fl DEX OH a), and oY — 7" (@) x Fle) + F(a) x fle); but ul = F" (x) = q(x) X w(x) + w’ (x) X g(e). DIFFERENTIAL. CO-EFFICIENTS. 21 Substituting, in the value of i for F(x) and F’(x), their values, we have d 1p =H (2) w(x) Sf (@) FF (@) v (2) 9! (2) +S (2) 9 (x) v’ (22). This process has been carried far enough to discover the law, that the differential co-efficient of the product of any num- ber of functions of the same variable is the algebraic sum of the products found by multiplying the differential co-efficient of each function by the product of all the other functions. Ex. 1. y=(a+ bx) (¢ — ax) mx ee = b(c —ax) mx — a(a+ bx) mz + m(a + bx) (c —azx) c m( ac + (2be — 2a*—3abz)ar) a= (ac + (2b¢ — 2a? — Sabar)ar) de. 22. The differential co-efficient of the quotient of two functions of the same variable is equal to the divisor multi- plied by the differential co-efficient of the dividend minus the dividend multiplied by the differential co-efficient of the divisor, the result divided by the square of the divisor. teeliftx Tia +h) biti g@y O88 YT gw FR) fle +h) fS@) OT gah) g(a) Bihet+ h) p(@) —g(@ +h) fe). g(a +h) g(a). = (fe +h) —A@)) 9(@)—(9(@ +4) —9(@)) Aa) g(x +h) p(x) therefore = Kx Oe fle | (a) — $e) pa) p(x+h) g(x) 28 DIFFERENTIAL CALCULUS.: Passing to the limit, by making aw =h=0, tim, 2% — a _ f'(2) oe) = 9 @) fle), BCs Dy (( yx) This result may also be obtained thus: =75) N&)=Y 9(%); therefore, by Art. 20, J (%) = y' 9(%) + (wYy; yf (2). 2 SO) therefore y! = ae da p(x) (9 a 2) = f'(£) p(&) —F (a) 9! (a). (9(2)) Bx. 1. ye an Le dy _(b+ax)b—(a+bx)a_ b6?—a@ das (b+ ax) = 6+ ax) i) eee hie (b+ ax)? 23. The rules which have been thus far demonstrated in this section are independent of the form of the functions char- acterized by the symbols f, g, w, &c.; and it has been assumed that the differential co-efficients of these component functions of the compound functions considered can be obtained in all cases. Before showing that this assumption is correct, by the actual differentiation of all known forms of simple functions, ~~ : d. it is proper to make a few observations on the symbol oe , used da to denote the differential co-efficient of the function y of the variable a. DIFFERENTIAL CO-EFFICIENTS. 29 In the doctrine of limits, dy represents the limit of the ratio di eo and it must be borne in mind, that, at this limit, aw, and . Ax consequently a y, become zero. There would be, therefore, an inconsistency in viewing dx and dy as the representatives of the terms of a ratio that have vanished, until it be proved that the ratio itself does not also vanish. If the ratio remains, although its terms disappear, then dx and dy may be taken as indeter- minates, having for their ratio the final ratio of the vanishing quantities. This is the view to be taken of the differentials dx and dy, according to definition, Art. 14, and which justifies us in regarding these differentials as the terms of a fraction in < 24. Analytical geometry furnishes instructive illustrations of the meaning of differential co-efficients, as was intimated in Art. 14, and suggests many useful applications that can be made of the doctrine of limits. Whatever may be the nature of the function y = f(a), every value of x that will give a real value for y will be the abscissa of a point of a curve of which y is the corresponding ordi- nate; and, if the assumed value of x gives several real values for y, x will be the abscissa of a like number of points of the curve, having for their ordi- nates the several values of y. The curve is, therefore, the geometrical representa- tive of the relation between « and y in the equation y In the figure, suppose SP’P to be the curve represented by the equation y =f(z), 30 DIFFERENTIAL CALCULUS. - and let PI be a value of y corresponding to the assumed value OM for x; then give to @ the increment MM’=h, y will receive the increment P’Q = ay, and we have P’Q=ay=f (ath) —f (x) = P/M’ — PM: sy _ Se+W—fe)_ PQ ADS h Veda expresses the trigonometrical tangent of the angle P’PQ, PQ PQ | wuich is the tangent of the angle that the secant line or chord PP’ makes with the axis of the variable z. Now, it is evident, that, as h = MM diminishes, the point P’ moves along the curve towards P, and the secant line approaches coincidence with the tangent line 7’7”; and finally, when / vanishes, the coinci- dence of the points, and of the secant with the tangent line is complete. The tangent line to the curve at the point P is then the limiting position of all secant lines which have P for one ay +s the lite dx ing value of the tangents of the angles that such secant lines of the points in which they cut the curve, and make with the axis of x. 25. The fraction 47 always represents the ratio of. the Ax assumed change in the value of the variable to the corre- sponding change in the value of the function. These changes, when small, are properly called increments ; and it is evident that their ratio is the measure of the rate of the increase of the function to that of the variable: but it will be seen, that, for functions in general, this rate of increase will vary both with the initial value of the variable and the value of its increment aw. If, therefore, the value of the increment Ay were left arbitrary, the value of. the fraction = would be equally so. But the conception of the limiting value of the DIFFERENTIAL CO-EFFICIENTS. Sal, ratio removes all uncertainty, and suggests to the mind what the rate of change in the value of the function is, in the imme- diate vicinity of its value for any assumed value of the varia- ble. dy In the case of the curve, in the last article, the limit + dc does not depend upon the increment Ax, nor upon the form of the curve at finite distances from the point whose co-ordi- nates are (x,7), but depends only upon its shape and direction within insensible distances from that point. 26. Let us apply these remarks to the equation y =4/ 2 pat, which is the equation of a parabola referred to its axis, and the tangent line through the vertex as the co-ordinate axes. Giy- ing to & an increment, Y TAY =N 2% (x +h) AY =+/2p (Va +h — VJ) mi /Bp ae )= hp ViFhtye) Vethtvye (7 TA ea Dc Wx thtr/x therefore lim. 4.7 — dy A ig Ne em a Aa da 2/n V2pe” y From analytical geometry, we know that P is the natural tangent of the angle that a tangent line to the parabola, at the point whose ordinate is y, makes with the axis of the curve. 27. The differential co-efficient of a function, which is a power of the variable denoted by any constant exponent, 1s ‘the exponent multiplied by the variable with its original exponent, less one. “~ Bae DIFFERENTIAL CALCULUS. Let te ey then yt ay=(a@+h)" syatetir—araan (14h ah Had fae x | U hate Ax h | x Make “atl 24? and { 1 i: the : p= ect y (+3) = a Put also (lt) lee ee Making these substitutions in the expression for re gat becomes A ics pete Ax t Both ¢ and z diminish with hf, and reach the limit zero simultaneously with it. Taking the Napierian logarithms of both members of the equation (1 + 4)” =1-+ 2 we have ni (1+ t)=1(1 +2) oe f(1+ z) ee ae ey tits Biba ary... Chere and east both have unity for a their limit; hence qe | , aD (1 ae das Mel CL Anat a ee l(l+a2)é = maser io) > z But, since n is a constant, 4° 2 lim, 2s, 4p nt ae as | lim. -= 7 n DIFFERENTIAL CO-EFFICIENTS. 33 __ dy _ ae 28. The rule of the last article may also be demonstrated 5 A A Zz therefore lim. ~! — lim. #2”! [= ner, x as follows : — x cS, nu Let, as before, y = x”, then ytay=(x+h)” xt h\n Ay (@+h)*—ar _ i x y= Poa oe i Nol Sen gaan aaa core aa ‘Now, whether m be a whole number or a fraction, positive or negative, it may be represented by He Z in which p, g,and s are positive whole numbers. ath Make ——— = %.*.h=a2(z — 1), x A’ A. and oe, pe ; Ax z—1° : A a J as eee therefore lim, 2% = lim. @™4 ito AX z—1 Ash converges towards its limit zero, converges towards the limit unity, and # and z reach their respective limits simul- r-4 taneously: we have then to find the limit of ar Liga as Pane armen ge t x P-4 Make wu =z‘; whencez* = u?4,z=— u'‘,and the limit of u is unity also. Making these substitutions, we have p-4 ee uP | a? 148 eee Ue st), Gt wt 1 ee t(ut—1)— we t(u?— 1) Dividing both numerator and denominator of this last frac- tion by wu — 1, it becomes iil eee te 1), wie buf. FI) a 34 DIFFERENTIAL CALCULUS. and, at the limit where w = 1, this reduces to Psy LAY hence lim. —~- = lim. 7”? —————. = _ =- = —— a” AX g— 1 “de 8 29. The rule proved in Arts. 27 and 28 is general; but, when the exponent n is a positive whole number, the demon- stration below is more simple than either of those given. Let y= a; Xa, K 2.4. @,, 1m which wa ecee ames n? tions of the variable ~; then, Art. 21, * / HX Wy XK 2. yy XO, +a X LyX. . eee to n terms. - / duc ? Now suppose “4, = 2 = a@, =. 1) = 2 buenas er teem v',y==...== x’; and each term in the value of y’ becomes dy fs Hence yi = nx, thy Under this supposition in reference to n, we may develop (a +h)” by the Binomial Formula, and thus get an expression for the ratio =f of which the limit can be readily obtained. Thus y =a", Ay=(x+h)” — x” et | —nxzth+t+n Ree xe”? h? +. &e. AY — ng +n id +; L mh + &c., ‘ Ax a in which all the terms in the second member after the first term contain i as a factor ; hence him: 4 se oe = nav", DIFFERENTIAL CO-EFFICIENTS. 35 "ate xl. y=at be’, dg = 30% d 2 Ex. 2. y Se Or, a = — = oe ore ae Ex. 3. eee oT d + 2 de 2h (x* + a*) — 22° (x?+ a’) 2x 22° 2a? a Fa wa (a tary 30. The differential co-efficient of a function of the form y = a", a being a constant, is the function multiplied’ by the Napierian logarithm of the constant. Let y = a”, then —y fay = att = ata" Ay = a7 a* — a* — a* (a* — 1), and Passing to limit, A a h hes’ Wee Se 1 im, a pg eae waa L AX dic h But, by Art. 13, ' a" oe 1 lim. —la therefore lim, 2 ae Ce £OL If y =a", then y =(a°)*, and ot ht Ta any 1: _— ebtten?) - — jlitee”) d (b a ont) 9%exre ( bea”) (tL AL * The letter d thus written before an expression indicates differentiation. Thus, d(atbx? d if u=a-+bexr?, then peers) is equivalent to as dz dc 36 DIFFERENTIAL CALCULUS, n O. et Roe on hier? y= ere ey —er 4 — nee, > dx dic Ex. 3... y= e*¥=14 ¢-*¥>1 hs Agar (e*vr1 e*¥=1) Fix: 43 Y= ee me — eter. 31. The differential co-efficient of a function which is the logarithm of the variable taken in any system is the modulus of the system, divided by the variable. Let Tema bo: thenytay=L(x+h), Ay=L(et+ h)— La = L2 4", Whence Ay L = x : Kove ehoah Make h = «xz: therefore ay _LO +2) _1E0+e Ax we x z But h and z reach the limit zero simultaneously; and, by Art. f mea ee) for 2= 0, is equal 10 Tea 4 Ho the: limit: 0 . la = M, the modulus of the system: therefore iin ee Aa” ds x ; ys wed. Hence, if (poy ts ire Hx1 meme fi vicltes ay | 6 ee : aiie . Y ae, a= FOb ) ie ey | : dy Ex. 2. Reese == @ + Dalam dx DIFFERENTIAL CO-EFFICIENTS. 37 22. The differential co-efficient of the sine of an are is equal to the cosine of the arc. Let y= sin. x, then yt+ay=sin. (x +h) Ay =sin. (x +h) — sin. a == 2 COB, (2 + 3) sin. f . (Eq. 16, Plane Trig.) ey sin. — ; TI fi See = 4 1érerore coe D COs. (2 L ,) ‘ 2 But, when / 1s diminished indefinitely, the limit of oy) sin. — 2 , hy a ==) (Art. .(), and lim. cos, (2+ 3) <= CORO. 2 therefore lim. “4 = dy aOR. a. Ag 10s 33. The differential co-efficient of the cosine of an arc is equal to minus the sine of the arc. Let if =itOs, Wunen ytay=cos. (x+h) Ay = cos. (a +h) — cos. x ane) oa h — —¥gin. = sin. & x . ( at . aed Mh ade Synth ee Ay 2 sn f i aa pe sin. (2 a 3 pees /t sin. - 1 At the limit ah i Te sit: (2 = be 5) en BIT Be 2 38 DIFFERENTIAL CALCULUS. therefore chee ths, fey dy =o lim. A ane 34. The differential co-efficient of the tangent of an arc is =~ Sines equal to 1 divided by the square of the cosine of the arc. Let y = tan. x, then y+ay =tan. (x +h) Ay = tan. (%-+h) —tan. « sin. (w+ h) sin. & cos.(7-+h) cos. x sin. (% + h) cos. x — cos. (% + h) sin. « cos. (x +h) cos. x be ~~ sin.(a+th— a) cos. (a + h) cos. - (Eq. 8 Plane Trig.) A sin. h ~ cos. (a+ h) cos. x therefore Ay sin.h 1 Aw . Ah. cosi(ac- fh) coat at the limit nee aan nA ae " : h cos.(«#-+h)cos.« cos.’ x di 1 hence lite tee - see Ax « dae Tecos.6a 35. The differential co-efficient of the cotangent of an arc is equal to minus | divided by the square of the sine of the arc. Let y= cot. x, then y +ay = cot. (a +h). Proceeding with this as in the case of the tangent, we should find DIFFERENTIAL CO-EFFICIENTS. 39 26. The differential co-efficient of the secant of an arc. is equal to the sine of the arc divided by the square of its cosine. Let y = sec. x, then y+ Ay =sec. (x +h) Ay = sec. (x +h) — sec. x io 1 _ 1 ___ cos. x — cos. (w + h) — cos.(@-+h) cos.@ cos. @ cos. (a +h) ae : sin. (2+ 5) a | dy __dydz__ 4 dey. de”. deer da de 7 DIFFERENTIAL CO-EFFICIENTS. 45 By placing for z, in Hq. 1, its value from Eq. 2, we find : dy __ y = 4x" — 62 — 5, whence a 8x — 6; the same result as was found by the first process. 43. Differential co-efficients of the inverse circular func- tions. : Ist, Differential co-efficient of y = sin.—'z. Since y = sin.-'w, «= sin.y; and therefore, by Art. 41, dx pe as and therefore, by Art. 41, dy epee Lie pry it, ] dx COs. ¥ ai W/ 1 — x? 2d, Differential co-efficient of y = cos.‘ a. Here y= cos.-!x gives x =cos.y: therefore, Art. 33, dd. : hl Saeco de =< _ siny= e/a and therefore, by Art. 41, dy _ 1 i th, 3p SN Gn ie eee It would be superfluous to point out the necessity for the | sions =, =, before the differential co-efficients in this and the preceding case. 3d, Differential co-efficient of y = tan.“! a. From y = tan. ‘a, we have « =tan.y: therefore dx . fh = EET: sec” y= 1-4 tan.2y (Art. 34) ; dy se 1 ; and 3 Rea oT a ancaie (Art. 41). Whence ame ae 46 DIFFERENTIAL. CALCULUS. 4th, Differential co-efficient of y¥ = cot. x. From y = cot. a”, we have # = cot. y: therefore LE ari Daa ae == = cosec.?.y — — (1) cote Art. 35 Cy ee har | ( cot.”y) (Art. 35). dy Seg at 1 : 5th, Differential co-efficient of y = sec.—! a. From y = sec.—'x, we have x = sec. y: therefore da = 50-9 & 207%, Gin dy cosy a ecOs..o) 1 and = = a 5 (Art. 4d), dx sin.y sec.2y sin. y 1 But sec. y = —-—,, hence cos. y = ie = 1 ; and COS. ¥ Set. Wy taae tei oa gine Vax? —1 1—sin?y = 5a sin. y = ps Se aati therefore 6th, Differential co-efficient of y = cosec.—! zg, We shall merely indicate the steps. Gl») 4.0 COB I is od ae TOO eerey., Cay ean a cosec. y cot. y (Art. 37): dy 1 1 1 ——— <= —— » Sm. Y = =; dc cosec. y cot. ¥ cosec.y | & pa d 1 e0t.. y= + A/a? — 13-42; fo ee xMV x? — Tth, Differential co-efficient of y = vers.—! a Taking « for the function, we have em vers.y = 1 — cosy; DIFFERENTIAL CO-EFFICIENTS. 47 dx OG sa, tk ; —— 8 therefore dpe =sin.y (Art. 38), and eT a) (Art. 41): ae Bees. : z sin. Y . pl cos.” ¥ Ale (1 — vers. y)? 1 1 =—t a /1—(1— 2) / 20 — a2?’ 1 ay >. te he hae yc nae 44, The principle demonstrated in Art. 41 has greatly simplified the investigation of the formulas expressing the differential co-efficients of the inverse trigonometrical func- tions. They may, however, be determined directly, without the aid of this principle. We will illustrate the manner in which this may be done by a single example : — Let y= sins) g, then ytAy=sin.!(«+h); and Ay =sin.~!(« +h) — sin.-'a. The second member of this last equation is the difference of two arcs whose respective sines are «+ anda; and this difference is, by trigonometry (Plane Trig., Eq. 8), equal to an are having for its sine the sine of the first arc multiplied by the cosine of the second, minus the cosine of the first multiplied by the sine of the second. Expressing the cosines of these ares in terms of their sines, we have Ay =sin.—“' (x +h) — sin. = sin ((2+h)/T— 2? — a/ [1 — (a +h)']): Ay _ sin.— a ((e@+h)/T=% — © —x¢re/[1— (x +h)? 1) e . A Mo h 48 DIFFERENTIAL CALCULUS. “Make g=(ath)/1l—a’—a/[1—(«+h)]: Ay __sin.'z sin. —'2 z therefor — : ees AX h Z h Now z and / diminish together, and become zero simultane- —l, ously. At the limit, a —1. To find what ; becomes at the same time, multiply and divide the expression for z by (wth) /l— a? + x/ (1 — (x + h)*); then ate (7+ hy? (1 — x?) — a? (1 —(x#+ i)?) Her) (2 +h)V1— a? 4+ e/[1 — (a+ ny") Nek 2x2 +h (@ +-A)/1 — ow? +2 v(1 —(e#+ h)*) Pass to the limit by making 2 = 0, and we have he@Ji_e? Vi—a dy paren daz /J1 — a 45. Differential co-efficients of functions of the form y = #° in which ¢ and s are both functions of the same variable a. Taking the Napierian logarithms of both members of the equation y — f*, we have ly=sit. By Art. 42, the differ ential co-efficient with respect to x of ly is Oly dy. au. One Do ds a, au ds dene Cae ds dx dt. -0iae at inet Oe and, by Arts. 20, 42, that of s/é is lt Now, since the equation ly = slt is true for all values of a, DIFFERENTIAL CO-EFFICIENTS. 49 the differential co-efficients of its two members must also bs equal: therefore Ldy _ 1,48 s dt. yada da ' ¢ dx’ dy cise 8 UE whence TY (" ee we : 7m) OTN ear, Parle =t! ee sti = t! (uate ae) 46. From an examination of the particular cases treated in Arts. 19, 20, 45, we deduce this general rule for finding the differential co-efficient of any compound function: Differen- tiate each component function in succession, treating the others as constant, and take the algebraic sum of the results. Rules have now been given for the differentiation of all known forms of algebraic, logarithmic, exponential,and circular functions of asingle variable ; and we have seen, that, in gener- al, the differential co-efficients of these functions are themselves functions of the same variable. 47. The following exercises are given that the preceding rules may be impressed, and that the students may become expert in their application, and familiar with the forms of the differential co-efficients of simple and complex functions :— 1. y= an i — Baa? (Art. 28). 2. y = abx*® — cx? a = 3abx? — 2cx (Arts. 19, 28). 2 3. eam ee rs =e dy _2ax(b—«x*)? + bax’ (b—-ax’)? As (b—«x*)® _ 2ax(b + 2x7) 1 (O ears): ; (Arts. 21, 22). 50 DIFFERENTIAL, CALCULUS. 4. fetes Ve + 3V/ete= (2+ (w ot oF Put 2s=2+(e2-+c)t; theny = 2%, dy dy dz and ae (Art. 42). dy 1 1 But = = rn] dz 1 2 3(a@+c8+1, and Se he Bae 6 hee sree de Fe 6 BON ON Sees therefore oY so fig ee Y 2(e+(a+e)*) 3@+e)h 82/(a cyl ~ 6 +a be XY (e+e) 5. ys=l(atvVitea?). Makez =x +WV71+a?: then — Vert pa sah? oY = at But epee arevias (Artogin dz % gti Pi ee | Rtg dic Has ates 1 oe 1 oA Tat ee therefore dy — de e+J/t+a? Vita? Vita The utility of substituting a single symbol to represent a complicated expression before differentiation is exemplified in this and the preceding examples. Oftentimes the labor, both mental and mechanical, of the mathematician is greatly abridged by the adoption of suitable artifices. DIFFERENTIAL CO-EFFICIENTS. 51 Veet o MRL nominator by the numerator, then y =1(1 + 2a? — 2a /1+2°)- Put pee Oe _9¢-4/ 1 + 2? Whence y = lz, Multiply both numerator and de- Oey b dy _dydz, ae dx dz dx’ dy 1 1 but es —— a dz, --% 1+. I? dar/1 +e? em yp spor oa" and BSE dx at(1 +a)’ 64. From the ee nmelowmn sin. ne ; : 2 2 sin.e+ sin. 2~7-+ ... + sin. nx = ——_______, prove, by differentiating both members, that cos. 2 + 2 cos. 2x te 3cos.3%-+ ... +ncos.naxis equal to \ 2 cae aot te 5 (ain 2a n+1 2) sin. — s, sin. 2 2 sin? x yy} DIFFERENTIAL CO-EFFICIENTS. 59 65. Admitting* that sin. sin (2 4-2)in.(“* +2) are sin,("=2n +a) sin. mx a 9 pena mts in which m is a positive whole number, prove that cot. x -+ cot.{— +a)+.... cot. m-+- x |—=mcot.me. m mm * As the equations assumed in this and the preceding example are not usually given in treatises on elementary trigonometry, they will be demonstrated in the Key to this work. SECTION IV. SUCCESSIVE DIFFERENTIAL CO-EFFICIENTS. 48, Tue differential co-efficient of a function, f(x), of a single variable, being in general another function, /’(2), of the same variable, we may subject this new function to the rules by which /’(x) was derived from f(a), and thus obtain the second derivative, or differential co-efficient, of the original function. The second differential co-efficient will, in turn, give rise to a third, and so on; and we thus arrive at the successive derivatives, or differential co-efficients, of a function. The notation by which these successive differential co-effi- cients are indicated will be best explained by an example: — ° Let us take y =a"; then = ty hee a Pe TS es 1st diff. co-efficient. d*y WW —2 d dj 1 —~=y"”a=n(n—1)a** . . 2° diff. co-efficient. da? any sof OP ans eee ee m diff. co-efficient. da™ These are the first, second, . . . m'” differential co-efficients of the functiony—=/(x). It is sometimes convenient to de- note these by writing the function itself with as many dashes as there have been differentiations performed: thus /’(a), S(x),. .. f(x), are. the first, second, . . . mm ™oditeren 60 DIFFERENTIAL CO-EFFICIENTS. 61 tial co-efficients of f(x), and have the same signification as DP osc YO. In the example just given, it is evident, that, if m be a positive integer, the m™ differential co-efficient will be inde- pendent of x, that is, a constant, when m=; and that the function will not have a differential co-efficient of a higher order than the nm“. In other cases and forms of function, there will be no limit to the number of differentiations that may be performed. d*y d*®y dy The symbols Fictia ad cued ee RR Cae ae ivalent t siseey ae ies and equivalent to Py seer tl are read second, third, ... m‘" differential co-efficient of y regarded as a function of «; and are to be viewed as wholes, and not as fractions, having d’y, d*y,. ..d™y, for their numerator, and dx’, dx’, ... dx”, for their denominators: nor must the indices 2, 3, . . . m, be considered as exponents of powers, but as denoting the number of times the function has been differentiated. 49. Successive differential co-efficients of the product of two functions of the same variable. Leibnitz’ Theorem. Take w= yz, in which y and z are functions of x; then, by Art. 20, we have du dz d eC eae and, differentiating both members of this equation with respect to x, we have au dz, dydz dy dz d*y det 9 det" de de * de de * de® ae dy dz d*y =U eit ae det de® 62 DIFFERENTIAL CALCULUS. In like manner, we should find du d*% dy d?z dz d*7 d*y dist = 4 gs T° ae deat? de dar © * dina” and d*u d*z dy d*z d’y dz dz d*y d‘y dct = Y dat * de dat tet da? 1 * de da Fat This has been carried far enough to enable us to discover inferentially the laws which govern the numerical co-efficients, and the indices of differentiation in the expressions for Mu du dtu dx?’ dx*’ dxt co-efficients and exponents in the Binomial Formula; for, in These laws are the same as those for the respect to differentiation, y may be regarded as y®, and mas 2), | To prove these laws to be general, let us assume them to hold when » is the index of differentiation. Then du = dz dy dvtas. m—1 dyd™ ts da 4 gt de damit” 9” de® da™ ne) (n= 2) 00. (n= 7-21) cae ar 2.3; seks eds das’ dar’ (n—1)(n—2)... (n—r) dttly dr—@tbgz fs 9.3... r(r-+1) datt! de»—@FD d"y tee. lene Differentiating both members of this equation with respect to x, reducing, and arranging the result, we find d™+ly ang dy d”x dgett Y ggrat "Toe due Nee Qin s(n pion) dehy dre | dat > DR (r+1) dart} dum * eae dantl DIFFERENTIAL CO-EFFICIENTS. 63 Now, the laws of the co-efficients and indices in this devel- opment are the same as those assumed: to be true in that from which it was immediately derived; but, by actual opera- tion, we know them to hold when n = 4: they therefore hold when n= 5; and so on: hence they are universal. As an example of the above, take wu = e“*y ; then, observing Npax ae ae. We find a" u Re Or: n—1 son n—?2 a é (ary + nar $ a 4+ a ey lees as) Now, by examining the expression within the i se that we discover, that if (« 1. eS y be developed by the Binomial Theorem, treating the symbol if as a quantity, and (s.)¥ i da d\2 (te) ’ we get that factor of the development of fa bence OP ee Oil 6S Y ye in an. dat dae (a+a) 9 is a convenient and abridged form of writing the n™ differen- dy wy d"y d (Cz =) be then replaced by. op er are hae oe d tial co-efficient of the function wu = e*"y. 50. If n be a positive whole number, we may prove that d"u sd” uv aia ( a) ta ae ( me "de® ds® — da®1\" dar 1.2 dae2\" dai) d"v — do. do... +(—1rus"... (1). For let y = uv, in which both w and v are functions of « ; then, differentiating with respect to x, we have 6 9) alam) hee 71 amas da da" da +” da’ 64 DIFFERENTIAL CALCULUS. yee _ au dv, dx da dx’ and the theorem holds when n = 1. whence Now, differentiate both members of Eq. 1; then Cot naeaty | alae a” ( Uda?!) Ge dx®. ane) 7 aged ae n(n —1)d"—} d?z a yy aC a , fs dv Se Rare TA ae) ie: If the theorem holds for y = wv when the index n has some assigned value, it will also hold for x ae when 7 has the same dx value. Changing, in Kq. 1, v into ad we have dv d™u _ d” CS dv\ 9 d?v dx dx” Clare da) "dat" da? Se at Oe = ( i) 1.2. de—?\" Ge d®tly — &e.+...+(—1)"u aaagh AY Subtracting Eq. 8 from 2, member from member, and redu- cing, we find d’tiy d®tlay d” Lie OE mea CLS are 1) Toa) = 9 + (n+ 1) — a(u To) ee thy mre ces 1)"4)u antl” Hence, if the theorem is true i any assigned value of 2, it is true when the index isn+1. It is true when n=1; it DIFFERENTIAL CO-EFFICIENTS. 65 is therefore true when n = 2; and so on; that is, it is univer- sally true. EXAMPLES. ia dy 1 al) 1 ee! eo dat a Orem ye 1.2.8 da? a? det a dry aa ae eae Sama ae 2. = An. 2, = = OOR) d= SiN. (2 + 5) a nm Scere are oe = COS. ch Ray ire ( +>} Yy | , 1 3. Y= C0s.2, 7 = — sin. & = cos. & Ts }; / — e ; = n P) 4. if == COS. 22, — Oe COR. (aa + ) 5. 4 = tan.¢-+ sec. @, dy 1 sin.x 1-+sin.& 1 dz cosa cos2x% cos?a 1—sin.2’ d*y COS. & da? (1 —sin. x)? 9 66 DIFFERENTIAL CALCULUS. | 3 sin. « — sin. 32% el OP ase Ko n ; e dl” aN n —4 = jain (« + I eee sin, (30 +3) es a Yai eta i ee DEAT ibe 8 Tract oll a8 he a | d” 1.2.3...(n—1 9 Yi ger} in; “is —- - ) x ay 4a? = 2 2 ard gta SP EN ast Bete AHO) fiom 6 fae ehea a da! (ae SEE ere dty —x ah y= Or ORL, dnt = — 4 cos. x2; hence di ait fy = _1l-«z d”y no Le Loy > 0 if ieee F(a, + Ax) = F(x) <0 if F’ (x) <0. 54. Suppose that the two functions F(x), f(x), are real, and that they, as well as their differential co-efficients, are continuous between the limits, answering to the values a, and x, +h of the variable; suppose also, that, between these limits, J’ (z) does not undergo a change of sign; that is, for the in- termediate values of a, f(x) must constantly be either an | increasing or a decreasing function: then the ratio of the differences B(a,-h) — F(a,), f (ei +h) -—f(*1), will be equal to that of the derivatives I’ (x), 7’ (x), when in these x has some value between a, and 2, +h; that is, if 0, be a proper fraction, we shall have E'(¢%,;th)—F(a,) F’ (#,+6,h) J (#1 +h) —f (#1) (2, + Oh) To prove this, let 4 be the least and B the greatest algebraic F(a) J (2) tween x,and x, +h; then the two differences, values that the fraction can have for values of x be- ¢ DIFFERENTIAL CO-EFFICIENTS. val must have opposite’signs for any of these values of 2; and the same will be true for En" (x) — Af’ (x), FY (x) — BS’ (a), because, by hypothesis, 7’ (a) has an invariable sign between its limiting values. But these last expressions are the differ- ential co-efficients of the two functions E(x) — Af (x), # (x) — BY (a). One of these functions, therefore (Art. 52), must be constantly increasing, and the other constantly decreasing, while the values of x are limited by x, and x, + h. | | If, then, the value answering to x, be subtracted from that answering tox,-+h for the one and the other, we have the two expressions, F(a, +h)— F(a) —A(f (+h) —F(m)), F(a, +h) — F(a) — B(f(@ +h) — f(x); one of which must be positive, and the other negative. Wherefore it follows, that, if both be divided by f(x, +A) — f(x), the quotients F(ey+h)—F (am) _ J (41+ 4) —f(#1) Bia +h) — # aA S(t +h) —f(#1) me ite ey Leh 4) that's, F(a, +h) = Fa) than A, and less than B, and is therefore comprised between have opposite signs ; is greater these greatest and least values of sae ae a But F” (x) and /’ (a) being continuous, while x passes by insensible gradations from x, to x, +h, the ratio ee) must pass through all values in- J’ (2) > 72 DIFFERENTIAL CALCULUS. termediate to its greatest and least values. Hence there must be some value of «x between v,anda,+ Ah that will render the ratio of the differential co-efficients equal to the ratio of the differences of the functions. id Let 6, be a variable proper fraction: then, from what pre- cedes, a value may be assigned it, that, agreeably to our enun- — ciation, will cause it to satisfy the equation E(x, +h) — #1) _ E(x, + O,h) S(tith)—f(m) f(t + he 565. It has been assumed in what precedes that /’ (a) re- tains the same sign between the initial and final values of x ; but the proposition is true when the assumption is made with reference to £’ (x), instead of f’(x). For, if #” (x) does not change its sign, by the same course of reasoning we can prove that S(a +h) —S(%1) ff (1+ 4) h) F(@,--h) — F(a) F (@- 0h): whence F(a,-+h) — F(a) F’(@,+0:h) F(@:+h)—f(a) ~ fi (a+ Oh) 56. From the theorem established in Art. 54, we deduce the following consequences :— Ist, If F(x) and f(x) both become zero for the particular value « = x,, then F(a, +h) F(a, +04) Fler+h) F(a Fahy 2d, If the differential co-efficients up to the (n — 1) ™ order of both F(x) and f(x) vanish for « = a, those of the second being constantly positive or constantly negative between the limits corresponding to x—=a,,«—wa,+h, while the fune- tions themselves do not vanish for this particular value of 2, DIFFERENTIAL CO-EFFICIENTS. ia then, from what has just been proved, we shall have the fol- lowing relations : — FE’ (a, + Oh) we EM” (x; +O, h) af (+ 9, 4) JU (@1 + 92h) FY (a+ 0gh) F(a -+ 05h) fl (@1 + O,h) fl (a1 + Gh) Fe-y (xy + Gey h) tis Fe (x, + 0, h) : foamy (xy ar SS h) Re 7? (ay +6, h) $ therefore F(e; +h) — F(a) F (a, +0,h) Feb) —f (a) ~ F (e+ Oak) Since, in the reasoning, no condition has been imposed on 6,, except that it be a proper fraction, we may omit the subscript mn, and thus have (ae, +h) —F(a2,)_ &™ (w+ 6h) (2) (ti +h) —f(%1) fh? (#1 + Oh) If the functions reduce to zero, with their derivatives for = @,, we have F(x,+h) F™ (a, + oh) (c) Fath) Ff (a+o) Making the further supposition that 7, = 0, then Lh) « F@ (6h), F(h) — f(Gh) but, because this is true for any value of h, « may be written for h; and thus Pi (at) (002) (d) J (a) f (G2) # 38d, The conditions relative to f(x) that have been im- 10 74 - DIFFERENTIAL CALCULUS. posed in the preceding propositions are satisfied when f(a) = (x —«a,)”": whence J" (%) = 1 (@ — 2)? J (x2) = n(n —1) (x — a)? fOr? (2) = n(n— 1)... 2(e — ay FOE) 1.2380, nee eee Here f(x) and its successive derivatives, up to the (n —1)", vanish for « = «,; and since, in Kq. b, the denominator of the first member, SJ (#1 +h) —f(%1) = (41 — & A)” — (@, — @))" =H", we have Fh) —Fe@)y= oe 2.3. en FS When n = 1, this gives F(x2,+h) — F(x,) =hF’ (x, + Oh). If F'(x,) =0 as well as f(x,) = 0, then he Ft) > o55 Making 2,—0, and then writing « for in the preceding EL (x2; + Oh). equations, they become E(x) — #(0) = aaa (62) F(x) — F' (0) = xf” (6x) F(2)=75 en - F (622). 57, The equation B(x) _ Pe F(a) — f(a) (Kq. d, Art. 56), ex- pressed in words, enunciates the following theorem; viz.: If there be two functions, /’(x), f(x), which, with their differen- tial co-efficients, are continuous, and which, with these differ- DIFFERENTIAL CO-EFFICIENTS. ex ential co-efficients up to the (n — 1)* order inclusively, vanish for «= 0; and if, further, the first » differential co-efficients of one of these functions are constantly of the same sign for values of the variable between zero and another assigned value; then the ratio of the functions will be equal to that of the n™ differential co-efficients, when, in the latter, some inter- mediate value is given to the variable. The importance of this theorem warrants us in giving it an independent demonstration. Let F(x) and f(x) be two functions which vanish for « = 0; and suppose, first, that the differential co-efficient f’ (a) of the second does not vanish for this value of the variable, and that it retains constantly the same sign between «= 0 and «=A, which requires that f(x) be continually increasing, or continu- ally decreasing, between these limits (Art. 51), and therefore constantly positive or constantly negative, since f(x)=0 when x = 0; and let 4 denote the least and B the greatest of Bee) for values of «a be- J (&) tween zero and f: then the two quantities, gas A fe) he F’ (a) ean B, will have opposite signs; and, since /’ (2) does not change sign, the same will be true of the differences, E(x) —Af’ (a), #° (%) — Bf’ (a): the values assumed by the ratio but these last are the differential co-efficients of the two functions, i'(x)— Af (x), F(x) — Bf(a), one of which (Art. 51) must therefore be constantly in- creasing, and the other constantly decreasing; that is, since ‘both F(a) and f(x) vanish for « = 0,one must be constantly 76 DIFFERENTIAL CALCULUS. positive and the other constantly negative between the limits answering tox=0,x=h. Therefore, because f(z) is of in- variable sign, F(x) —Af(#)_ F(a), Fle) — Bf@) _ F(a) J (2) Sha) Sas nes J (2) are of opposite signs: whence it follows that the ratio of the — B, functions 1s comprised between the least and the greatest values of the ratio of the differential co-efficients. But, if the variable be made to pass by insensible degrees from 0 to A, Ga! which is by hypothesis continuous, must pass tT (x) bi yP } Pp through all values intermediate to 4 and B. If then 6 denote the ratio a proper fraction, it will admit of a value such that the equation Hide). Et! (0) T(") ff! (92) If the differential co-efficients. of both functions, from the will be satisfied. 1st to the (n—1)™ orders inclusively, vanish for «= 0, by reasoning upon them as we have upon the functions, we have F020) OE ORG ea ay _ ONG, fi(0,") f" (09%) f"(032) 9 a whence TFC an ORT Oe} f(x) f (6x) 58. Itis to be observed that the only conditions upon which the equations F(x, +h) — F(a)=hF" (x, + 6h), F'(x)— F'(0) = ak" (6x), F(@) = py5— F (6), aero. depend, are, that 2’ (x), and its differential co-efficients up to the order involved in the equations, should be continuous between the assigned limits of the variable. DIFFERENTIAL CO-EFFICIENTS. ba 59. From the equation (2, +h) —F(x,) =AF’(x,+ 6h) of Art. 56, it may be shown, that, if the differential co-effi- cient with respect to x of any expression is zero for all values of x, such expression is independent of «: for, if I’ (a) is zero for all values of x, the above equation becomes F(a,+h)— F(x,;)=0; or, F(x#, +h) = F(x); that is, the function does not vary with, and is therefore in- dependent of, x. It is plain, that, if the differential co-efficient is not equal to zero, the expression will vary with x Hence those expressions only are independent of a variable for which the differential co-efficients with respect to that variable are zero for all values of the variable. And further: if two func- tions have the same differential co-efficient with respect to any variable, such functions can differ only by a constant; for the differential co-efficient of the function which is the differ- ence of these functions is zero by hypothesis: therefore, by what precedes, this difference must be independent of x; that is, constant. 60. Suppose F(x) to be real and continuous; then, by means of the foregoing principles, we may find the develop- ment of this function arranged according to the ascending positive powers of &. For we have, Art. 56, F(x) — F(0) = ak” (6x) = «F’ (0) + fax by making F! (62) = F’ (0) +f; whence F(x) — F (0) —aF’ (0) = Ra: from which it is seen that #,x is a quantity that reduces to zero when # is zero; and the same is true of F(a) — F’ (0), which is its first derivative with respect to x. Its second deriv- 78 DIFFERENTIAL CALCULUS. ative is /'” (x). Wherefore, by the article already referred to, on? F(ec)—F(0)—aF’(0)=Rye= 3 F'” (0x). Making F"(60) = F"(0) + Ry, then, as before, oe? oe? 2 and it is evident that 2, a is a quantity, which, with its first and second derivatives, E(x) — °° (0) — el" (0), 2” (x) — 2” (0), vanishes with «, and that its third derivative is #'’”(a): there- fore we have of a” // —2 a 4/1 F(x) — £(0) — «n; whence Dene ear. fy < (Ga DIFFERENTIAL CO-EFFICIENTS. 83 But, if x is finite, (<,) will be zero when vis infinite: hence n oe ay when nN —=0O; and the same is true of A” ait Denar Therefore it follows, that if F™ (0x), F™ («+ 6h), are finite, the products ac” | F (62), _ F (022), hn 2d i SEA Ga) TAs posse. ape CE tle Meza A. will diminish without limit as m is made to increase without limit; and we can, in such cases, employ Maclaurin’s Formula for the development of /’(x), and that of Taylor for the develop- ment of /’(«-+-h), into series arranged according to the ascend- | ing powers of « for the first, and of either a or h for the sécond. 64. Maclaurin’s Theorem, when applicable, may be stated as follows: The first term of the development of F(a) is - what the. function becomes when « = 0; the second term is x multiplied by what the first differential co-efficient of the function becomes when x = 0; the third term is the second power of « divided by 1 X 2, and this quotient multiplied by what the second differential co-efficient of the function becomes when x=0; and the (n+ 1)”, or general term, is the n” power of x divided by the product of the natural numbers from 1 to n inclusive, and this quotient multiplied by what the n” differential co-efficient of the function becomes when a 0. This theorem is of very general application for the expan- sion of functions of single variables, examples of which will be shortly given; but it is by no means universal: for 1 ee eae COL. 0, If — az , 84 DIFFERENTIAL CALCULUS. are functions which become infinite when «2 =0; and hence the first term in Maclaurin’s Formula would be infinite, while the function for other values of x would be finite. There are other functions, such as y = ac? for which, though the func- tions themselves remain finite for « = 0, their first, or some of the following differential co-efficients, become infinite for this value of the variable; and, in such cases also, the for- ‘mula would fail to give the development of the functions. 65. Taylor's Theorem may be enunciated as follows: When a function (x +h) of the algebraic sum of two varia- bles can be developed into a series arranged according to the ascending powers of either taken as the leading variable, the first term is what the function becomes when this variable is made equal to zero; the second term is the first power of the leading variable multiplied, by the first differential co-efficient of the first term taken with respect to the other variable; the third term is the second power of the leading variable divided by 1 X 2, and this quotient multiplied by the second differential co-efficient of the first term; and the (n + 1)”, or general term, is the n” power of the leading variable divided by the product of the natural numbers from 1 to n inclusive, and this quotient multiplied by the n™” differential co-efficient of the first term. 66. In Taylor’s Formula, the co-efficients of the different powers of the leading variable are functions of the other variable. When one or more of these functions are such, that, for a particular value of the second variable, they become in- finite, the formula fails to give the development of the origi- nal function for that value of the second variable; for then the function ceases to depend on the second variable, and is a DIFFERENTIAL CO-EFFICIENTS. 85 function of the first variable alone, and will not necessarily be infinite for the assigned value of the second variable. For example, if we have P(x) =V/ x —a, then F(a thy=r/(«—ath). When «=a, F(x) =0, and the first and all the higher differential co-efficients of #'(x) become infinite for this partic- ular value of x; while, for this value, F(a +h) = Wh. It will be observed that there is a marked difference be- tween the failing cases for Maclaurin’s and Taylor’s Theorems. When Maclaurin’s fails for one value of the variable (x = 0), it fails for all; whereas Taylor’s may fail for one value of the second variable, but give the true development of the function for all other values. 67. If a function becomes infinite for a finite value of the variable, its differential co-efficient will be infinite at the same time. In the case of an algebraic function, this follows from the fact that such function can become infinite for a finite value of the variable, only when it is in the form of a fraction whose denominator reduces to zero. But the denominator of a fraction never disappears in the process of differentiation : hence, if the function has a vanishing denominator, so will its differential co-efficient. In the case of transcendental func- tions, it is only by the examination of the different forms that the truth of this proposition can be established. Thus, in the logarithmic function y=Ix, y becomes infinite for «=—0; za _1 is also infinite for this value of x; and for the expo- ce. 2 1 nential function y = a”, which, if a>1, becomes infinite when 86 - DIFFERENTIAL CALCULUS. BAA IhT) la 1 sores x —0,the differential co-efficient is mee which is infinite when « = 0. The circular functions tan. x, cot.x, sec. x, cosec. 2, which may become infinite for finite values of x, when expressed in terms of sin. 2, cos. x, are fractional forms to which the reason- ing in reference to algebraic functions applies. If a function becomes infinite for an infinite value of the variable, it does not follow that the differential co-efficient becomes infinite at the same time. Thus, in the example y = lx Whe : and y is infinite when ’ dx Dre spite ce! = 0 for this value of a. dx 68. It was remarked in Art. 62, that, unless #'(#) and F(x +h) are such that F(0) +2F" (0) +75 F"(0)+..-, ; 52 F(a) + LF! (a) +75 F” (x) + es give rise to converging series, the formulas of Maclaurin and Taylor will not serve for the expansion of these functions. A series in general is a succession of quantities any one of which is derived, according to a fixed law, from one or more of those which precede it. If wo, %1,&.,U3,...%,, are such quantities called the terms of the series, then we have Sn Uo tht Uy $ Ug +. s -Un 4 for the sum of the first m terms. When this sum approaches indefinitely a finite limit S, as » continually increases, the series is said to be converging, and the limit in question is called the sum of the series; but, if the sum S, does not thus DIFFERENTIAL CO-EFFICIENTS. 87 approach any fixed limit as n increases indefinitely, the series is said to be diverging, and has no sum. The geometrical series 2 n a, ar, ar°,... ar, having ar” for its general term, has for its sum 1—r” a ar” Pelee rr et. 7") = eh a Coren aa ) l—r 1—r 1-r It is evident that, as 2 increases, this sum converges towards a the fixed limit i if 7 is less than 1; and that, on the con- trary, as ” increases, the sum also increases indefinitely if r is greater than 1. We are assured of the convergence of the series Uo, Uy, Uy + © s Un, when, as 7 increases, the sum po ty 4 yp i yy converges to a fixed limit S, and when, at the same time, the differences Bee Dn — te; Data Bn = Ua t Uni aS T | vanish when 7 is made infinite. The limits assigned this work do not permit an investiga- tion of the rules by which, in many cases, the convergence or divergence of a series may be ascertained. 69. Admitting that #(x) can be expanded into a series arranged according to the ascending integral powers of 2, Maclaurin’s Theorem may be demonstrated as follows : — Assume F(ez)=A,+4,07+ A,w?+...4+A,0? in which A,, 4,, A,..., do not contain x, and the exponents 88 DIFFERENTIAL CALCULUS. a,b,c..., are written in the order of their magnitude, a being the least; then, by successive differentiation, we have E(x) = aA, x9 + bDAje""} -. .". 4 EF" (x) =a(a—1) A;x** + 0(b—1) A,w? 7+. +p(p—1) 4,2? F'(x¢) = a(a —1)(a — 2) Ayx*— 14 3(b ee atest a +... 4+7(p—1)(p—2) AgmP=?. : The assumed and all the following equations, being true for all values of x, make x = 0; then, since £(0), #”(0), #”(0)..., would in general reduce neither to 0 nor too, we should have A,=£(0), a=1, 4,;=F"(0), b=2, EEO) B(0) Wig ee 123° oF F(x) =F (0) + 2F"(0) += bg which is identical with the formula of Art. 62. «0. Taylor’s Theorem also admits of the following simple demonstration when the function /'(#-+h) can be expanded into a series arranged according to the ascending integral powers of one of the variables with co-efficients which are | functions of the other variable only. Assume F(e+th)=f(x) +fi(~7) ht +f(xe)h?+...+f,(a) h?, and differentiate with respect to x, and also with respect to h; then GF@+M _ GO) Ki) ja 4 Kl) Val) py eM MMe dn" oa dF (a +h) To Vi (4) BO Hf (ae) RO ow es fn (@) AP. DIFFERENTIAL CO-EFFICIENTS. 89 But /(x-+h) involves hf in precisely the same way that it it does x; and, if we place x +h = y, we have (Art. 42) dF(a+h)_aF(y) dy_ aFy), , dx Cyeye. / Oy : dF(x+h) dF (y) dy dF (y) aoa = les dh dy dh dy dF(a+h) dF(a+h hence ee ) ae eee) that is, these differential co-efficients are equal for all values of # and h, which can only be the case when they are identi- cally the same. This requires that eee) spi - HO), e= 8, Alo)——, A) dis also, by making h=0 in the assumed development, we find f(@) =F (2); whence ipa 8) ae Lt! (ar), fy (8) os eae: therefore F(e+h) = F(a) +hF (0) +75 F(a) +. +75 a P” @). 12 SECTION VI. EXPANSION OF FUNCTIONS. V1. THE application of the formulas, demonstrated in the preceding section for the expansion of functions, gives rise to many important series, some of which we shall now deduce. 1. If F(x): (14.2), then Fi(eysm(l- 2), F(a) =m(m—1) (1+ 2)", Fe-v (xz) =m(m—1)...(m—n+2)(1 + a)r—"tt FO (2) =m(m—1)...(m—n+1)(14+ 2)"; therefore (0) =1, £’(0) =m, F'”(0) =m(m—1)..., F°-) (0) =m (m—1)...(m—n+2); and hence, by Art. 60, : (1-+ay=1+me+ mary (m—1)...(m—n+2) .(n—1) : —n+1)a” bee, 1) 058 goes (Lop eee When ~ is less than unity, the last term in this development will diminish as m increases; and, by making 1 sufficiently great, the series 1+ mem ot 4m UE “ty 90 EXPANSION OF FUNCTIONS. 9] will approximate more and more nearly the true value of (1+ a)” the greater the number of terms taken. 2. Let (ge) ——.e7. Then Bee) = eee emer ee eg i) (ot) Ae ee eee 0 (0) = OY (0); F (6a) =e 0%: Pct eres c* therefore é Sway gt oak ihe xe” vx Bitiao, Seer el) To a ee Making in this « = 1, we have = gb ge a ee sala Sieg on Low pages a series that may be used for finding the approximate value of e. 38. Let F(x) =sin.2; then a y—= COS. @ —- STi, (2 + 5 d sin. (2 oe 2) F" (x) = See" — cos. (2 a 5 sim (2 ote i) Qn d sin. (2 = a - y} oe Qn ae 37 FE (x) Pa a pe —= COS. (2 + e) sano USO (2 + a) | ey — Sin; (2 ao 7) Bete On (Oyo 1 FY (0) —- 0, Fh" (0) = — Te Therefore F-) (0) = sin. dese as and we have 92 DIFFERENTIAL CALCULUS. ; oe ac® sin. & = ®— 754 +79 Re ea erg Mego of - n—d TL28.c(n = 1) we” ; nn Ta 28 es (00-475 4, Let F(a) = cos. « ; then ' I" (x) = — sin. & = cos 2 -E 5) FY" (a) = cos a ap zat fu (x) prem 16). (2+3) ae (x) — cos. (2+ 3) LEO) = eae) = 07h 0) — 1, F (0) 0% ih pa” rain ) c= COR. s mn: ac? act h Pia | ee ae Seem oy ake van oe ont n—1 vf AM at nn T4508 eae G ay =) By Art. 63, it will be observed that the last terms in Exs. 2,8, and 4, diminish as 7 is increased, and finally vanish when nm becomes infinite. 5. Let #(x)=/1(1+ 2); then / aes 1 /1 — 1 V1 oe 1.2 eke ea i (2) — eee (7) (aaa f Bae) (ar) = es rt Oe Gaia): hence E(0) =.0, 2! (a) = 1, 2 (00) I ee Bo (0) = ( — 1)*- 11.2.8... (n— 1); and thererane 2 3 4 es a (—)* be a nm ° (1+ 62)" EXPANSION OF FUNCTIONS. 93 An examination of the last term of this expansion shows, that, when x does not exceed unity, this term necessarily de- creases as ” Increases, and vanishes when n becomes infinite. And, since the factor aoe ; under this hypothesis cannot exceed unity, the sum of the series, up to the n™ term in- . . 1 elusive, cannot differ from the true sum by more than—; and . n hence, by increasing 7 sufficiently, this difference can be made as small as we please. Changing the sign of x, we have a 2 bistemte.. ee a ganl (= 1 yes a” 1) (1— Ga)" 6. Let | Pe) —— tar, ee then Hee ey = E 5=5( cee Ite 2\l—aV/—-1 1ltae/—l 1 ee gore =3(a-ev=1)> 4c +av=1y") P(n) =5 x ee sere AY AT 8 aes ies 1 avi Soatt a I TE I a a ie tes 1 gees! ee =V=1x35(a—ev=1)*—(- 1a ev=1)") L(g) Be UN ult 2 shal iis 1) ee yi) 1 (L+2V—1) Ae SP itor ite: ee ea) oA) ea) (eV) (— yee 2)" 94 DIFFERENTIAL CALCULUS. therefore ah 12.3.6: ( be PO =v) Whence it follows, that, if n is an even number, #"™ (0) = 0; but, if m is uneven, then n—1 1.2.3. .(m —1) . FM (0) = (= 1) 5 2 EY 103) (na + /—1 /—1 Hence we have 3 5 n—1 gel te a ee po taney aso Acres teas med eee (1—oar/—1) "(1+ 1)". n Qr/—1 The final term in this development is not in a convenient form, as it stands, to decide whether the series is converging or diverging; but by referring to Ex.-18, p. 67, making a = 1, and observing that there 6 = 5 — tan.'x, we have 2 —— Fo (a) Oe (= 1)*71 19. Chena (n ’ 1) sin. er _- ntanmte \: (1 + a”) y therefore 3 5 tan“ e =a — = +5 —& 4+... (— eee ie fe sin. (s — ntan.! ® (12%)! ae This form of the final term shows, that, if 2 is less than unity, the numerical value of the term may be made as small as we please by giving to 7 a value sufficiently great. The above form for /"” (x) might have been used for find- ing all the differential co-efficients of tan. a as readily as that specially deduced for that purpose. EXPANSION OF FUNCTIONS. 95 The following is a more simple process for getting the expansion of tan.~! # : — Assume tana =— A+ Be+ Cx? + Dx?+ ke. (1), and differentiate both members with respect to x; then 1 2 eee wet Ae (2); but by division, or by the Binomial Theorem, aes —xvtoat—a>te*§—&+ ke. (3). The second members of (2) and (3) must be identical: hence, equating the co-efficients of like powers of x, we have 1 Je femty ie Gia. 0; eat I ies ATA ee and, since the assumed development must be true for all values of x, make x = 0 in (1), and we find 4=—0: therefore 7 — Wo ee 7 tan.-!'2%—a—- 3 et feat —— sin, ' 27, assume sin. ‘w= 4+ Bet Cx? + Dx? +... (1), and differentiate both members; then 1 ; OO) a Si ee te ae la ace). but, by the Binomial Theorem, we find 1 1 1.3 letra Trial tar tag t+agaee t+ (3). The co-efficients of the like powers of w in the second members of (2) and (3) must be equal: hence i Loa) B-1,0=0,D= 55, #= eae boa | and, by making x = 0 in @) we get A = 0: therefore Tends) Val oso ce a es ney | x a raat 545 12467 96 DIFFERENTIAL CALCULUS. 8. Let y=e* "= and assume y=A,+4,24+4,07?+...+4,0"+... (1). Differentiate twice; then oy 2A4,12.34,¢-+. + (n—1)n4, 0" 7 (3). But dy asin T zh et ee daz Ji — 2? ary — prsin te a? | See da? Le (1 — a)? and hence (1—a?)"%— a7 Faary... (4). 8b dy d*y Substitute in (4) the values of —*, —“, taken from (2) and hae Gaee (3), and we have 24,+ 2.34,0 +3.44,07+--+1(n—1)n4,0"74+.. — (24,0”°+2.34,2°+3.44,2t+--+(n—1)nA,x"+..) —( 4,*%4+24,0°+34,0'+4A,v'+. +. i a? A,+a’?d,x-+a?A,x? Be a tet dye" Equating the co-efficients of the same powers of x in the _ two members of this equation, we find eas and generally 2 2 Bows. 4 2 Uh on Yael ind Areal SIN (n—1)n EXPANSION. OF FUNCTIONS. 97 If then A, and 4, be found, formula 5 will give all the following co-efficients in terms of these two. . - —l . A, is what e**" * becomes when «= 0: hence 4, = 1. ~ becomes whena = 0: Jt And A, is Snakes — e@ sin) —___ dx hence 4,= NS Be 10\ 1 13. ty=(“ wae z) yr show for what values of « Taylor’s Theorem fails to give the development. It fails for 7 —c; 1st term is then infinite. It fails for «=a; 2d differential co-efficient is then infinite. SECTION VIL. APPLICATION OF SOME OF THE PRECEDING SERIES TO TRIGO- NOMETRICAL AND LOGARITHMIC EXPRESSIONS. 72. Leva and b represent any two real quantities what- ever; thena+b*/— 1 will be the most general symbol for quantity, since, by giving to aand b suitable values, it may be made to embrace every conceivable quantity, real or imaginary. The two expressions, a + bW — 1,a—bW — 1, which dif. fer only in the signs of their second terms, are said to be conju- gate; and their product, (a + bW — 1)(a —bW —1)=a?+2?, is always real and positive. The numerical value of the square root of a+ b? is:the modulus of either of the conjugate ex- pressions. Denote this modulus by rv; then it may be shown that the expression a + b*/ — 1 can be put under the form | 7 (cos. 6 + / — 1sin. G). Horie, @=7 cos. 0, 6 —rsin. 6: b ; 2 Ha tan. 0 =~, r? (cos.? 6 + sin? 6) =r? =a’?+ 6’, r=Va? + b?, Now, if we suppose the arc of a circle to start from—Z, and to increase by continuous degrees to + 5 passing through zero, the tangent will at the same time increase by continuous degrees, and pass through all possible values between — a and +-2». Among these values of the tangent, there must.be one that will satisfy the equation tan. 6 = ie and the arc an- 99 .100 DIFFERENTIAL CALCULUS. swering to this tangent will be that whose sine and cosine will satisfy the equations a —=~rcos.0, b=rsin.0, and therefore render 7 (cos. 0 + / — 1 sin. 0) the equivalent of a+ b7/— 1. 73. Let us resume the series (Art. 71, Exs. 2, 3, 4). 2 3 e=ltoetoat+ypogt F a8 a? Bl, B= 2 — Tog to 34 5 a? oot Ey seers Gee EN ak MMS ek ee bes I soa y 1277234 Gh: and in (1) write «WV — 1, —a~/— 1, for a successively; then x? eirn/ —] a" eVE1 Fay eecenee SOP lbey: i737 ae ei a/—] mae heap gag ch = C08: acta ae as is seen by comparing this result with the second members of (2) and (3). Also e-*¥-1 =1—avW—1— 1.2 L255 1.2.3.4 PV Aas, | poll. ie che = 00s. &— 7 — 1 sin. @: therefore cos.a+%—Isin.2=e7%—-!.-- (4) cos. ¢ — 4/— 1 sin.@ = e7*~~1s =. (5), also cos. ¥ + 4/— 1 sin, y =e") ae (6); multiplying (4) by (6) (cos. a -E A/T sin, x) (cos. ¥ + /— 1 sin. Tie er+yy=1 = cos. (a + y) + Y— Lsin. (a + y). Effecting the multiplication in the first member, and then equating the real part in one member with the real part in the TRIGONOMETRICAL EXPRESSIONS. LOE? other, and the imaginary part in the one with the imaginary part in the other, we find cos. (x + y) = COS. & COs. y — SIN. XSiN. ¥ sin. (w + y) = sin. «cos. y + sin. ¥ Cos. x, Again: (cos.x-+4/ — 1 sin.) (cos.y + — Isin.y) (cos.24- — Isin.z) = ettyts...)Y-1— 98. (at ytet-.)+v —Isin.(etytet-.), from which, by making «x =y—=z=.---, we have (cos. @ + 4/ —1 sin. a)” = cos. max + / — 1 sin. mx, and generally (cos. g + 4/— 1sin. a) ™— cos. ma + / — sin. mz, which is known as De Moivre’s Formula. Hence the multiplication of expressions of the form of cos. 2 + 4/ —1sin.ax, and therefore of all imaginary expres- sions, is thus reduced to an addition, and the raising to powers to a multiplication. V4, Dividing formula (4) of the preceding article by (5) of the same, member by member, we have a cos.a+7—Lsin.2 sol-b Man, ee —1 cos. ee a I sin. x 1 — MrT, 1 tan. nee whence, by taking the Napierian logarithms of both members, 9a /—1=!1 (1 + /— 1 tan. a) af (1 —/— I tan. 2). Expanding the terms in the second member by Ex. 5, Art. 71, Sow tate tanta 9n/—1=r*/—1tan.2+ Day —V-1- aes “ a SE EA ey Rag ee pass eae x —(- S/o tan.! x Ss tare? x a rea ore te ot : ») ‘102 DIFFERENTIAL CALCULUS. Equating the imaginary parts in the two. members of this equation, and then dividing through by 2/ — 1, we have tan’a , tan®a . tan.te ome Bat. their a series that may be used for the calculation of z, and which 2— tan. x — agrees with the formula in Ex. 6, Art. 71. 4". To find the expansion of cos.” in terms of the cosines of multiples of a. Make ee" Fl ay? then er, ) ey ee eame/—1 — iby Yt From formulas 4, 5, Art. 73, we find 7 = = 1 2.cos. # = e*~—! + e7#v-1 — y + oe : 1 24 —1sin.2@ = et! —. (Te also, from De Moivre’s Theorem, we deduce 1 press 1 2cos.ma = y™ + yn 9/—1 sin. ma = y™ — re Because eowe=y +i, 2" conn (y +=) a but Va. yr it. Ae 1 ‘ n n—2 (+5) BY CUS Solan ee De tate 1 1: Ringer Fi 3 pat pat oe ay pat uy n—2 1 =a" boat (vee = +, iar lee tal: by combining terms at equal distances from the extremes: hence TRIGONOMETRICAL: EXPRESSIONS. 103 n — i | CORN T= | COB: Mee +n cos. (n — 2)x n— cos, (n—4)xe4+.-. ) (b). Since there are n + 1 terms in series (a), when 7 is even, the number of terms is odd, and the middle term, that is, n(n —1)(n—2)... (942) +1) ? pete fe 2 9 will be independent of y, and consequently of #; but, when n +n is odd, n + 1 is even, and there is no middle term in series (a), and therefore no term independent of x. In the first case, there will be within the ( ) in formula (0), besides the term n sy» that does not depend on a, 5 terms, containing as factors the first cos. nx, the second cos.(m —2)x; and so on to the last, which will have cos. 2 for a factor. In the second case, that is, when 7 is odd, there is no term within the ( ) in formula (b) that does not involve x; but the iz x : terms will then have for factors, severally, cos. nx, cos. (m — 2) xa. .., cos. 3a, cos. x. eae eas 26 = i (cos. 4a + 4 cos. 2% + 8) , 1 Ex. 2. cos. et — 3 (cos 5a + 5 cos. 8a + 10 cos. *) : 76. To find the expansion of sin.”x in terms of the sines of multiples of a. By formulas 4 and 5 of Art. 73, we have, employing the notation of the last article, 2/—Isin. 2 = et¥—! ey aie ere) deny a 104 DIFFERENTIAL CALCULUS. aes Rea Nes eae se therefore 2” (Vf — L)* sin. = 2" (= 1)7 ee (y be. -) n—1 att ee n—2 ms = ny +n 9 y™*— &+- ae p —1 1 Fs 1 al RGM Mp G eer ge 1 1 i n ee if n na n—-2 _ poe | n> 1S ey =(y een) a) n(y pec: mi) n(n — 1) n—4 n—2 1 is 130. (y ie 7) a he (a). An examination of tlus series shows, that, when 7 is even, the second terms within the ( ) are all plus; and, when v is odd, they are all minus. In the first case, the expansion of sin." will involve only the cosines of multiples of #; and, in the second case, it will involve only the sines of these multi- ples. n The factor (—1)? in the first member will be positive and real when 7 is any one of the alternate even numbers:begin- ning with 0; that is, when m is 0 or 4 or 8 or 12, &.; and negative and real when v7 is one of the alternate even num- bers beginning with 2. In like manner, (—1)? will be imagi- nary and positive when m is any one of the alternate odd numbers beginning with 1; and it will be imaginary and nega- tive when 7 is any other odd number. Let & represent any positive whole number, zero included; then the different series of values above indicated for n will be embraced in the four forms, 4k, 4h + 2; 44 +1, 44 +3. It would be of no advantage to make formula (a) conform to each of these cases by special notation, as it can be easily applied, as it now stands, to the examples falling under it. TRIGONOMETRICAL EXPRESSIONS. 105 Ex. 1. Expand sin.’a in terms of the sines of the mul- tiples of x. 22 (a/ 21) sina (y' — 3 — 3 (y - y" y = 2/— 1 sin. 3x — 6 —1 sin. 2: 1 Aap sin32a2 = — rr (sin. 3a — 3 sin. @). Ex. 2. Expand sin.‘z in terms of the cosines of the mul- tiples of x. 7 oe 1 1 24(4/—1)‘sin.tc = (y'+ 7) —A4 (v"+ =) +12 = 2 cos. 4% — 8 cos. 2a + 12: hse 52 (cos 4x — 4 cos. 2x + D) Ex. 3. Expand sin.x in terms of the sines of the mul- tiples of x. 2? (4/—1) sinc — (y— = — 5 Cis 7) + 10 (y _ ) = 2/—I1 sin. 52 —10/ — 1 sin. 3u + 20 /— 1 sin. a: ; Ie : sin.“ = ry (sin, 5x2 — 5d sin. 3x2 +10 sin. =) Ex. 4. Expand sin.*a in terms of the cosines of the mul- tiples of a. | —\_ . 1 1 1 Day — 1)*sin* a — (+ _ — 6 (y+ a) +15 (y+ — 20 y° Y y° = 2. cos. 6% — 12 cos. 4x + 30 cos. 2x — 20: ee — sal 08 6x2 — 6 cos.4%-+ 15 cos. 2% — 10), 77. To find the different n™ roots of unity. Let x represent the general value of the n™ root of unity; then, by the definition of the root of a number, «”=1, or 14 106 DIFFERENTIAL CALCULUS. 2” —1= 0; and the object of the investigation is to find all of the values of x that will satisfy the equation x” —1=0. By De Moivre’s Theorem, Art. 73, we have (cos. y + /—1 sin. y)”= cos. my + /—1 sin. my; an equation which holds, whether m is entire or fractional, positive or negative. Now, if k be any whole number, 2ka — will be an exact number of circumferences to the radius unity, and cos. (y + 2kx) = cos.y, sin. (y+ 2k) = sin. y: therefore (cos. y + /—1sin.y)” = (cos. (y + 2kx) + /—1 sin. (y + 2kn)) 1 ee Make m = 73 then (cos. y + “/—1 sin. y)” a (cos. (y + ka) + of —1 sin. (y + 2k) 2h pesae 2k Yee re emrs i n == COS. In this last equation, make y = 0: whence, as cos. 2ka= 1, and sin. 2kz = 0, we have 2kn 1 2k ue (1)" = cos. —~ + /—1 sin. n n 1 But, from the equation «”*—1=—0, we get x=(1): hence we conclude that the different values of x, or the roots of the equation x” —1= 0, are the values that may be assumed by 2h att 2h cos, -— + /—1 sin. — n n by assigning different values to k. Since k may be any whole number, take for it successively 0, 1, 2, &c.; then, 1 when ‘amet US 1" = cos. 02 '4/ 1 sins Oe 1 2 ee 2 when Derek 1. 08: — + 4/—1 sin. ~ ee TRIGONOMETRICAL EXPRESSIONS. 107 L 4 basa 4 when ae 1" = cos, = + /— I sin. ‘3 and so on, continuing the substitutions for & until the arc reaches such a value as to cause the expression 2hr — . kx cos. —— +k 4/—1 sin. ae to reproduce the roots it has already given. When n is an even number, this will be the case for k = 5; for, . n z Tiny, at eT eo if Bohs iB Reet tafaley at 4/—1 gin. nu; n .: er, if Liaary 1? cos. 2 AW 1 gin hs ST n u n+ 2 eee ie if a rl, 1 COs: gt + 4/ —1 gin. nt; — 2 pee 2 but Cos. “ git / —] sin. — nm v 2 een’, 2 _) SaeaY te iat aN a mt: 11) therefore the two roots corresponding to k= 5 + 1 are the same as those corresponding to k = 5 —1. So, also, those ob- tained by substituting 5 + 2 for k are equal to those obtained n 2 tions for k after the value 5 would merely reproduce the roots by substituting 2 for k, and so on: whence all substitu- already found. Again: when 7 is an odd number, the substitutions for & must be continued until k = al ; for, 108 DIFFERENTIAL CALCULUS. 1 n RT ie eon od Coeds ma ra if Lan eet Pemee ep Paine my but cos. one 4A OT eine tee Ve = és a n n n =e 4/ — Lsin. te n+] n hence the substitutions of ~ aa na 2 ait : for k give the same roots. So, also, it may be shown Re the substitutions of Lees Pande 2 for k would give the same roots. Therefore we should merely reproduce the roots already found, if we substituted values for k greater than k= m—1. 2 When n is even, k =0andk =5 give, the first the root +1, and the second the root —1; and the intermediate values of k give each two roots. When n is odd, k=0 —1 2 inclusively, give each two roots. In either case, the expres- gives the root 1; and all the other values of k, up to i Qhrn Qn sion cos. —— ++ 4/— | sin. —— can assume vn different values, | n and no more. Hence it follows that the equation x” — 1= 0 has n different roots, and can have no more. By the aid of the foregoing principles, the roots of the equation «” — 1 = 0 may be expressed under the form of ex- ponentials. Since, by Eqs. 4,5, Art. 73, we have Soe JES cos.2Ea/—1sin. e— ert” ’ TRIGONOMETRICAL EXPRESSIONS. 109 the successive values taken by the expression hn a. : Qlen cos. — + 7 — | sin. — n n may be represented in order by eet AL arial € ,e ” a ts #5. € ; when ” is even, and by Pye ot 8 yo, tS el € ay rT taal stacotge Curti® ) when is odd; the first term in each series of roots being unity, but the last term in the first series is minus 1, since it is equal to cos.7 + 4% — lsin.w—=—1. Both series of roots are the terms of a geometrical progression, the first term cl sei +0/—1 : . ° . = 1, and of which the ratio is e of which is e~ Ex. 1. What are the three cube-roots of unity? They are the roots of the equation «*— 1=0. Here n = 38, and the proper values for & in the expres- sion Cos. ass aA/ — 1’sin, a are (0 and 1: hence the first gives qa)" — cos.0 + /— 1 sin.0 et 1 =1. * The second gives 5 gas 1 ae Q7 Bags. LET, (1) = Bosaeeet A/S sin. =e Ex. 2. Find the roots of 7° —1=0. Here n = 6, and the proper values for & are 0, 1, 2, 3. (1)8 —cos.0 + V — 1 sin. 0 = coger heey , It c2e ee mu ae (1)8 = cos. g + VY — 1 Sling Sac a bee eae ae) re ga (1) = cos. Sf /—1 sin. Baga +rr/—1 (lye = cos. 4/1 sin. x = e+ Key. 110 DIFFERENTIAL CALCULUS. For each root of the equation «” — 1 = 0, there is a binomi- al factor of the first degree with respect to # in the first member of the equation. Since k=0 gives but one root, unity, there will be but one corresponding factor z—1: k=1 ~ gives two roots, and the corresponding factors are | Zee 2 ne! 2 — (cos. Zid isin.) © — (cos. St — 7 Tain, 2) n n n n which by multiplication will produce the quadratic factor x” — 2x cos. a +1. In like manner, each pair of simple factors may be reduced to a quadratic factor. If is even, the last factor is « + 1, which may be combined with the first factor « — 1, producing the quadratic factor «7 —1. Hence, when 7 is even, we have xn — I =(2" = 1) (2 — 28008, + 1) (2 — 2x roma 1) n n.* att) and, when n is odd, a —1=(e—1(2 — 2 eos. = +1) (22 — 22008, +1) ‘chanel 1} Tex, he (CL) = 1)(2* — 2x cos. + 1). Ex. 2. («#*—1) = ts — 1) (2 — 2x cos. a 7 (2° — 2x cos. ae 1) 78. The solution of the equation «” + 1 = 0, and the reso- lution of its first member into factors. Nn : (2° — 2 Gos. Nn ; (2° — 2a cos: Resume the equation ] (008. y Tain, y)* = 000. LO 5 5/ TRIGONOMETRICAL EXPRESSIONS. 111 of Art. 77, and make y=; then, since cos.2 = W— 1, and sin. z = 0, this eS becomes 1 (==) 1)\=='co tee I 1 sin, Bea, But, from «” + 1=—0, we have x = (— 1)*; hence the roots of the equation 2” + 1=0 are the values of which the expres. sion Cos. ae + / —1sin. a ce Ya will admit for ad- missible values of k. But k may fe any whole number in- cluding zero. Therefore take for k successively the values (leer then, 1 fork=0, (— 1)"=cos. § +7 — 1 sin. ; n n 1 fone ty (—— L)* = cos. 2 4 I sin. a min aie ena for k=5— in ie = cos. When 1 is even, substitutions for k greater than 5 —1 will only reproduce preceding values for (— 1); for, if k= 5 then (— 1)" a cos. (x i) aay Tid (x + 7 a=s7 COS, (« — ‘\ev=a sin. («— n n which is the same pair of roots as that given by the substitu- tion of os 1 fork. In like manner, it may be shown, that, if n ; *S5 +1, the pair of roots would be the same as that for b= 52; and so on, 14194 DIFFERENTIAL CALCULUS. ‘When n is odd, the substitutions for k must be continued stil eae if k=" Z 3, (— 1) = 008. "ae =I sin.” fe : n—t1 r ‘ Vey = ,(—1)"=co. 2+ 1sin.z= — 1, Now, for the next value of k, that is, 4 = are Ee) | jee 1 (—1)*=cos. 1 * i ea 1 sin. Tee 2 n—2 ere == 008: te / — 7 sind ee n and therefore this substitution for k gives the same pair of Stee roots as is given fork = am °, and the higher values of k merely cause preceding pairs of roots to recur. | Hence, whether 2 be even or odd, there will be n, and only n, differ- ‘ : ent values for (— 1)”; and the equation x” + 1 =0 has n, and only n, different roots. These roots can be put under the form of exponentials, as in the case of the roots of #” —1=0. Ex. 1. What are the roots of «'+1=0? Here n = 4; and the formula iL a ad (—1)*= cos, Tht VI i ir n e mt So i eee gives, for k =0, (— 1)» = cos, 7 + VW — 1 sin. 73 for ieeoek, fect Na EE om = Isin, ve TRIGONOMETRICAL EXPRESSIONS. 113 Ex. 2. What are the roots of «° +1—0? Here n = 5; and the formula gives, 1 fork—=0, (—1)" = cos. 7 + Vv —Tsin. 5 om , ay Mites, (—— 1) cos. 2 + 4/—1 sin. -? 1 for k = 2, (eee Conte A 20 sin. wie) 1. For each root of the equation x” + 1 =0, there is a corre- sponding binomial factor of the first degree with respect to x in the first member of the equation. When v is even, all the roots enter the equation by conjugate pairs, and the factors of the first member, answering to the simple roots of each pair, may be compounded into a rational quadratic factor, and we should have o +1—(a4— 20 cos.= +1) (2 — 22 008.5% 4-1). - (2 = 22008." Sa +1), tv When 7 is odd, there will be rational quadratic factors for IO eww |. ..,up to k= inclusively; but, for k= n —I1 2 case, we should have ot + 1=(2*—22008,% +1) (2-22 cos, 1): “se , there is only the simple factor x +1; so that, in this n—2 (22 — 2x 008, x+1)(e +1), The solution of the equations x” —a—0, «"+a= 0, and: the resolution of their first members into simple and quadratic 15 114 DIFFERENTIAL CALCULUS. factors, may be at once effected by the formulas in this and the preceding articles: for these equations give respectively Le o de 2—a"(1)", «—(a)"(—1)*, in both of which na is the numerical value of the oe root of a; and this, multiplied by the different values of (1), will give the roots of «”—a=0; and, multiplied by the values of (— 1), will give the roots of «”+a=0. 79. The determination of a general expression for the log- arithm of a number positive or negative. In any system of logarithms, the logarithm of 1 is 0, and the logarithm of 0 is — if the base is greater than unity, and + o if the base is less than unity; while the logarithm of oo is + 0c or— o, according as the base is greater or less than unity. It thus appears, that, whatever be the system, all pos- sible positive numbers between 0 and will embrace for their logarithms all possible numbers between — oand +. The logarithms of negative numbers, if they admit of expres-— sion, must therefore fall in the class of imaginary quantities. In the equation cos.% + 4/—1sin.a=e*~—! (Hg. 4, Art. 73), write « + 2kz for x, k being any whole number; shen cos. (a + Qk) + 4/—1 sin. (a + Qhkw) = e@ +27) Y=1, Por 2.0, this pives.-1= ¢#*"—- for «= 7; this gives —1—=e@t+07~—-1, Taking the Napierian logarithms of both members of these equations, we have L(1):== Qh A/T, (1) = (2k + 1) Sn LOGARITHMIC EXPRESSIONS. Ls These are the general expressions for the Napierian logarithms of 1 and —1: and, since k may be any whole number, it fol- lows that both +1 and —1 have an infinite number of log- arithms; but all of them, except that of +1, corresponding to k = 0, will be imaginary. From this it may be shown, that any positive or negative number, in whatever system, has an indefinite number of logarithms. | ; For, first, suppose y to be any positive number, and « its arithmetical logarithm taken in the Napierian system ; then Tg em*—et*x1=e*~x euavy—i net ertuny—1 . ae y= = x + 2ha Mel which is the general Rote ees of y, and will ad- mit of an unlimited number of values. Denoting the arith- metical logarithm by /(y), we have ip Uy) 2k —1.. > (Mm). Again: let Ly denote the general logarithm of y, taken in the system of which a is the base, L(y) denoting the arith- metical logarithm; then, since we pass from Napierian to any other logarithms by multiplying the former by the modulus of the system to which we pass, multiply Eq. m by _ the a modulus of the system characterized by LZ, which gives 1 1 Apa oa ly X yee L(y) X ia + ig 2h —13 or, 2ht 4/1 eat a 5 (2). a From Eqs. m, n, we conclude that the arithmetical logarithm of a positive number taken in any system is the value of the general logarithm corresponding to k = 0. 116 DIFFERENTIAL CALCULUS. Now, suppose y to be negative ; then —y = — 1 X y, and xX YO ore bb, paw Leen e@k+lav—1 a ettQk+l)rV—1 . l(— y) =v@+ (2+ 1)aV/—1... (p), _&-+(2kh+ lav—1 (q). la also L(—y) Eqs. p, q, are the general expressions of the logarithms of a negative number, and show that such a number has an unlim- ited number of logarithms, all of which are imaginary. From the equation 1(/—1) = (2k+1)aW —1, we get pio el gel (2k+1)/ —1 This and the preceding remarkable results developed in this section must be interpreted with reference to the symbols and the character of the quantities with which we are dealing. It must be remembered that e and w are the representatives of arithmetical series, and that the formulas have meaning, and can be regarded as expressing true relations, only when the rules for combining imaginary quantities with each other and with real quantities are strictly observed. SECTION VUI. DIFFERENTIATION OF EXPLICIT FUNCTIONS OF TWO OR MORE IN- DEPENDENT VARIABLES, OF FUNCTIONS OF FUNCTIONS, AND OF IMPLICIT FUNCTIONS OF SEVERAL VARIABLES. 80. WHEN several variables are involyed in an equation, any one of them may be selected as the function or dependent variable; the others being regarded as independent. If the value of the function is directly expressed in terms of the va- riables, we have an explicit function of several independent variables; but, when the function and the variables are in- volved in an unresolved equation, we have an implicit function. Let wu = F(a, y) be an explicit function of the two independ- ent variables, x, y, and give to these variables the increments, Ax, Ay, whereby wu receives the increment Aw expressed by the equation au=F(a+ a0, y+ ay) — F(a,y) = F(x +ax,y) — F(2,y) +F(x+au,ytay)—Ma+tan,y)...(a). The partial derivative, or differential co-efficient, of a function with respect to one of the variables involved in the function, is that which comes from attributing an increment to that va- riable alone. The partial derivative, or differential co-efficient, of u=F(2,y), taken with respect to x, is denoted by (x,y), or ae In like manner, /”)(a, y), or 7 denotes the partial dif ferential co-efficient taken with respect to y; and F’., (a, y), or 117 118 DIFFERENTIAL CALCULUS. jae is the partial differential co-efficient taken with respect dady’ to x of the partial differential co-efficient taken with respect to ¥. Now, if 7), 7, 73, are quantities which vanish with Aq, Ay, then, by Art. 15, we have the following : — F(x+aa,y) — F(a, y) = F(a, y) sa +7, a, F(a+aa,y+ay)—F(a+aa, y)=F, (e@+an,y) sy + ray, Fi(a+ aa, y) —F, (x,y) = Fy (@, Y) 4a + 1, Aa 5 from which last we get Ei(x+ aa, y) = Fy (x, y) + Fi, (a, y) da + 1; Aa. By substituting these values in Kq. a, it becomes AU ce? yyrsc+ F(x, yay tr sctryay + Ff TED y) AXAY + Tr, AxAY; or, Si oe aa +- Tay +r ae + rg Ay wi pea tea breawsy wo The increment aw : a function of two independent varia- — ‘bles is, therefore, like that of a function of a single variable, composed of two parts; the one, of Aw + oe Ay, of the first degree with respect to the increments Ax, ay, and in which the co-efficients of these increments do not vanish with the increments. The other part is made up of terms which are either of a higher degree than the first with respect to Ag, Ay, or they are terms in which the co-efficients 71, 72, 73, of the first powers of Ax, Ay, vanish with these increments. 3 From what precedes, we pass by what seems to be a natu- ral extension of our definition, Art. 16, of the differential of a DIFFERENTIATION OF EXPLICIT FUNCTIONS. 119 function of a single variable, to that of a function of two varia- bles. If we write du, dx, dy,.for Au, Ax, Ay, respectively, in Kq. 6, neglecting at the same time all the terms in the second member after the second term, we ae a 1 4 acmemmyee 5, ae 1 mu: ly (c). Here du in the first ange ie the total differential of du du u, and is different from the dw in ) i, In this, as in former .,. du du ae cases of differentiation, eeay? are to be regarded as the limits of the ratios of the increments of the variables to the corre- sponding increments of the function; the distinction being, that now in each of these ratios the increment of the function is partial, and refers to the variable whose increment is the du du denominator of the ratio. We must treat ie as wholes, and not as fractions having dw for the numerators, and dz, dy, for the denominators. . It is true that du, dx, dy,in these differ- ential co-efficients, may be regarded as quantities rather than as the traces of quantities which have vanished, by assigning them such relative values, generally infinitely small, that their ratio shall always be equal to the differential co-efficients. In du We this case, =- dx would reduce to du; but this is the partial rae differential of w taken with respect to x, and should be written d du. So likewise Zy dy should be written d,w. To indicate du du that davai are partial differential co-efficients, they are some- du\ /du times enclosed in ( ); thus (z ), eS From Kq. ¢, we conclude that the total differential of a func- AO DIFFERENTIAL CALCULUS. tion of two independent variables is the sum of the partial differentials taken with respect to each of the variables sep- arately. 81. To find the differential of u= F(x, y,), a function of the three independent variables a, y, z, denote as before, by 1,7, 73.-., quantities that vanish with aw, Ay, Az; then Au= F(a@-+ av, yt ay, 2+ Az) — F(a, y, 2) = (a+ Aa, y,%) — F(a,y,2) + F(a+ aa, y+ ay, 2) —F (a+ Ax,y,2) +f (a+ aan, y+ Ay, 2+ 2) —F(x+au, y+tay, 2)... (d). But, Art. 16, F(a+ax,y,2) — F(a, y, 2) = F(a, y, 2) Av +7, Aa, F(a+Aax,y+ ay,2z)— F(x + aa, y, 2) ) = Fi(a-+ ac, y, 2) ay + may. SN Te ANSE ae =F} (x +An,y tay, %) Az + 7; Az. —F(x+an,y+ ay, 2) Also, from same article, Fi(x + au, y, 2) — F, (a, y, 2) = Fy (2, y, %) 4a 4 Bae; and therefore F(a + Ax, y, 2) = Fy (ax, y, 2) + Fey (2, y, 2) Ae + 1 Aw. So, likewise, Fi(t tax, ytay, 2) =F) (x4 +a, y, 2) . + Ey, (2 AL, Y, BAYA MSAY, and Fi(a+taa, y, 2) = Fy (a, y, 2) + Fy, (a, y, 2 Ae+ re Ax. Making these substitutions in Hq.d, and denoting the co- efficients of the terms containing the products of Aa, Ay, Az, by each other, by m,, m2, m3, we have AUu=F (x,y, z)Ac+ Fy (a, y, z)Ay + F, (a, y, 2) dz trArtrnAytrAztm, ATAY+M,AXTAZ+M3AYAzB; DIFFERENTIATION OF EXPLICIT FUNCTIONS. 121 or, bu ae Seay tones trae + ray + 1,42 +m, ALAY + M,ALAZ+ MAY Az. From this, by the same considerations that led us to the expression for the total differential of a function of two inde- pendent variables, we conclude that ins oe ee a5 aay + oe de which may be written du =d,u + d,u + du. The course to be followed for a function of four or a greater number of independent variables, and the results at which we should arrive, are obvious. The total differential of a function of any number of independent variables is therefore equal to the sum of the partial differentials of the function taken with respect to each of the variables separately. 82. In Art. 42, a rule was given for the differentiation of a function of an explicit function of a single variable. It is now proposed to treat this subject more generally. Let u= F'(y, z) be a function of the variables y, z, which are themselves functions of a third variable x, and given by the equations y= q(x),z=—w(x). If x be increased by ag, u, y, and z will take corresponding increments, which denote by Au, Ay, Az; then Au=Miy+Ay,2+Az)—F(y, 2) =F(y+ay, z) —F(y, 2) + Fy + Ay, 2+42)— Fly + ay, 2). Dividing through by 4, and in the second member multiplying and dividing the first two terms by ay,and the second two by Az, Au (y+ 4y, 2)—F'y, 2) Ay a AY Ax i et ot) UY A AS AZ AX 16 oe DIFFERENTIAL CALCULUS. Passing to the limit by making Aw = 0, and remembering that Ay and Az vanish with aa, the first member becomes ~ pe the first term of the second member becomes la dy . dx | dy dx see clearly what the second term of the second member be- comes, suppose, first, that Ay vanishes; then this term reduces to Ey, % + 42) — Fy, 2) AZ, y AZ Ax’ L(y, % + Az) + Fy, 2) and it is evident that, now, the factor a is the ratio of the increment Az to the corresponding incre- ment of the function: hence, at the limit, this factor becomes ae and the second term ge of ; and therefore we have ae” dz da du dudy | du dz dx” dy da ' dz da’ and alu du du 17 dx uU zy dy + Te dz In general, if w= F(y,2,u,V...), Y, %, U, Ue functions of the same variable x, we should have dw dwdy , dwdz , dw du de — dy dah ds dein de do aa dw dw kN) BE dy‘ yr di pee aie ae dy) ete Prue tial co-efficients and partial differentials of the function w; - (a) dw = de (6). du SLADE : dy, hs dz, are the partial differen- while a and dw in the first members of these equations 2 are the total differential co-efficient and total differential FUNCTIONS OF FUNCTIONS. 123 of the function: hence we may enunciate the following the- orem; viz., the differential co-efficient of a function of any number of variables, all of which are functions of the same in- dependent variable, is the algebraic sum of the results obtained by multiplying the partial differential co-efficient of the func- tion taken with respect to each dependent variable by the dif- ferential co-efficient of such variable taken with respect to the independent variable. This is the meaning of Hq. a; and Eq. 0 admits of a like interpretation. If in the function, w= ’(y, 2), we suppose, for a particular case, that y and z in terms of w are given by the equations : oe 1 a Eee and the sec- ond term in the second member of the equation, du __dudy , du dz ao nets’ 02.000) fete x; then dz— dx du dz of u— f(y, 2) = f(y, x) taken with respect to w. This would reduce to —, which is the partial differential co-efficient cae dus must be in some way distinguished from dg 1D the first mem- x ber of the equation, which is the total differential co-efficient of the function. This is usually done, in cases where the two kinds of differential co-efficients are likely to be confounded, by enclosing the partial differential co-efficients in a paren- thesis. Thus the above equation should then be written du /du\dy , /du dz \dy) dx \de) 83. It may happen that some of the subordinate functions are themselves functions of the others, and thus complicate the example; but the principle just demonstrated is easily extended to such cases.. For example : — 124 DIFFERENTIAL CALCULUS let b SP e, 0,2), Uf Uy es ae y= 9 (2); Z=w («); from which, by making the proper substitutions, w could be made an explicit function of x, and thus the differential co-effi- cient of w with respect to x be found. But this result may be reached without making these substitutions. Differentiating each of these equations with respect to a, we have du /du\ dy du dz du\ dv du to (ay) ae + as )ae* (te) ae * ate) dv _ /dvu\ dy dvu\ dz dvu\ dan a) dx hs dat Ce y, in which we distinguish partial from total differential co-effi- cients by enclosing the former in parentheses. By substituting dv dy in the first of these differential equations the values of rele yes = derived from the others, we get, finally, ul (3 “9 («) + (4 ee @) +(){ Gro xarro+@) (a) Ex. uw=yteta2y, ¥y = COS. &, Ar, du Reeth mafia st Din Vee 3 lh} 2 aria AL Pera + azy, dys ZO oe Ip ee. Fe IMPLICIT FUNCTIONS OF VARIABLES. 13s therefore eee = — (2y + 2”) sin. @ + (32? + 2zy) e* = — (2cos. x + e*) sin. w + (3e” + 2e* cos. x) e* — 38e** — e* (sin. x — 2cos. x) — sin. 2x; a result identical with that obtained by first substituting in u the values of y and z, and differentiating the explicit function, u — cos.” x + e* + e” cos. a. 84. When the relation between the variables is expressed by an unresolved equation, any one of the variables may be assumed as a function of the others regarded as independent. It is often inconvenient, or even impossible, to solve the equa- tion with reference to the variable taken as the function, and thus convert it into an explicit function to which preceding rules for differentiation are applicable ; and hence the necessity for investigating special methods for the differentiation of this class of functions. ~ Consider, first, a function of a single variable, which, in its most general form, may be written w= F(x, y)=0. LHither y may be taken as a function of ~, or x as a function of y. It generalizes our result to leave the selection of the independent variable undetermined. Let Aa, Ay, be the simultaneous incre- ments of «and y. ‘The increased variables «+ aa, y+ ay, are subject to the law of the function /’(x,y) = 0, and hence must satisfy the equation, F(a + sa, y+ ay) =0: Au= F(x + Aa,y + sy) — F(a, y) =0. Treating f(x -+ ax, y+ Ay) — F(a, y) as was done in the case of a function of two independent variables in the last therefore article, we have AU 0 Se sy PT, T's; oe quantities that hs be Ax, AY. ap tmey tramway. (a); 126 DIFFERENTIAL CALCULUS. Now, by whichever of the increments we divide through, and then pass to the limit, by making that increment zero, it is manifest, that since, from the mutual dependence of x and y, Aw and Ay become zero together, all the terms in the second member of the above equation will vanish except the first two. Dividing through by Aw, and passing to the limit, we have day dus) Vikyon dw sad dat dy! se de Ty dex : du dy ai whence ee di dy Dividing through by Ay, and passing to the limit, we get du dudx’ dw’, , dx _dy ay: dy dy sae hike dy du e dic In Kq. a, writing du, dx, dy, for Au, Av, Ay, and omitting | all the terms in the second member after the first two, it be- comes du du d mae rae 5 See ae, . Aas. dic +- a dy = 0 85. Ifu=F(x,y,2...)=0 be a function of any number of variables, one among them may be taken as a function of all the others regarded as independent. Were the equation. solved with reference to the variable selected.as dependent, we should then have to deal with an explicit function of several independent variables, —a function which has no total differen- tial co-efficients, such as there are in the case of explicit func- tions of a single variable; and we are, therefore, concerned only IMPLICIT FUNCTIONS OF VARIABLES. 127 with the total differentials of the function, and with its partial differential co-efficients of the different orders. Suppose z to be the dependent variable, and that the value of z, in terms of the other variables, is z=/(z,y...): then Ceri, | (Ls Yon «yee j emi and, considered with reference to x alone, wu is a function of a, and of a function of a function of z But, by the law of the function F(x, y,2...), w must be zero for all values of the inde- pendent Oe atlcs hence its partial differential co-efficients, taken with respect to these variables, must be zero. Denote by (a) the partial differential co-efficient of wu taken with respect to x, and, through «, with respect to z; and, by st a the partial differential co-efficient taken with re- spect to # and z separately: then, by Art. 82, du\ du , dudz _ Ge) = at a dx et Similarly, by adopting a like notation with reference to Yy,5,t..., we have du. du dudaz Gq)=qtaa=? © du\ du dudz_ 0 a Sabie ds as ee Kqs. a,b, c..., will give the partial differential co-efficients of z with respect to the variables severally. Thus, from (a), we have du du detie nOe dai pyre ay te Fa? and, from (0), Fe, ees 128 DIFFERENTIAL CALCULUS. Multiplying Eqs. a, b,c... through by dz, dy, ds ... respec- tively, and adding the results, observing that du dz du dz du dz du da dan ds dy! + dds Ne a we have du du du du det bi Mahe Fd cia - + 50s e(m). From Eq. m, we may find the total differential of any one of the variables regarded as a function of all the others; du du du thus ir a ay egy age foarte: Roe ne AE BEETS Se du dic 1 bp. A eS u= ay + Ba? — ah? = 0, Mh tion. WOE Mae | alice ae therefore cy + Bex <0. Hae. From the given equation, we get y= ova — 27, an ex- plicit function of y; and, by differentiation, we obtain directly dy i bx b? x de’) gA/ai=onige ae Bx. 2. u-—y> +2* — sary = 0, “t= 30” — Say, 7 =P — Bae, dy) 2%" = OY) ty ae des o® — aw oy? — aa’ a result that it would be difficult to verify, as was done in Lixeals 86. When we have given the two implicit functions, was la. yes.) 0, uf (%,Y, See IMPLICIT FUNCTIONS OF VARIABLES. 129 of the same variables, we should have at the same time du = 0, dv = 0, from which can be determined the differentials of any two of the variables considered as implicit functions of all the others; and, in general, if the relation between the m variables, xv, y, z...,18 expressed by the m equations, u—0,v—0,w=0..., we should have at the same time the m differential equations, itera iathoe 0 O10 ma Ost. and, by means of these, could determine the differentials of m variables regarded as functions of all the others. If the number of variables exceeds only by 1 the number of equations expressing the relations between them, one of . the variables alone can be independent; and we may find the differential co-efficients of all the others regarded as functions of this single variable. Let us have n equations, tigteea hy (Opie Boa 6) a= 0; Merny (an Yeti « bjoa— Oy it eect Liye (2D) ned. «== 0, between the n + 1 variables a, y,z...¢. Differentiating all of these equations with respect to x taken as the independent variable, we have Cee OU, dy. du; dz du, dt eed) de de da at de” ewe. dy du, dz du, dt eee Gs de haa Gee au oy du, dz du, dé Pd. dak? neds. dx There are n of these differential equations involving the n 17 : 130 DIFFERENTIAL CALCULUS. required quantities, dy be i: ag which may therefore be dx’ dx da’ determined. 87%. When the variables enter the function in certain combinations, the results of differentiation take special forms, and peculiar relations exist between the partial differential co-efficients, depending on the manner in which the variables are combined. We shall first consider the case of homogene- ous functions. A function is said to be homogeneous when all the terms entering under the functional symbol are of the same degree with reference to the variables. Thus E(x, y, %) = ax® + by? + ca? + 2eyz is a homogeneous function of 2 dimensions, and Y is a homogeneous function of 0 dimensions. A property of such function is, that, if all the variables are multiplied by the same quantity, we obtain for the result the original function multiplied by this quantity raised to a’ power whose exponent is the number denoting the dimensions of the function. Therefore, if F(x, y,z...) is a homogeneous func- tion of a dimensions, and ¢ denotes a new and independent vari- able, we have EE tec, ty tz,.3. 1) = EO Put $0. Ub, ty UB; te EU, 0, .. 3) ECG oy 2 eee and differentiate both members of this equation with respect tot: the result is, dFdu dF dv . dF dw ——— —— = — eee taht — (Tf oe : du dt cae PEED Be dt zt ING, Y, 2 ve a)e IMPLICIT FUNCTIONS OF VARIABLES. T3d But a as Oe therefore dF dk ak * a dy dw Now, since ¢ is entirely arbitrary, make ¢ =1; then Cie onlin Glen lug) ex du dx’ dv dy * pee arige CL Ys Riess c)s Mae 7,20 — %..,..,, and whence we have cet lag tae ee Se ey pares). The first member of this equation is the sum of the. products obtained by multiplying the partial differential co-efficients of the function, each by the variable to which it relates; and the second member is the primitive function multiplied by the number denoting the degree of the function. If the function is of 0 degree, dF dF dF Be dy ee da te = eT Ex. I. F(a,y,2) = ax? + by?’ + cz? + 2eyz 4+ Afzx + 2gay, ae Zax + 2fz + 2qy, re 2by + 2¢e2 + 29x, dix dy dF _ 2cz + ey + 2fa, and a = 2: therefore (2ax + 2fz + 2gy)a , + (2by + 2e2 + 29x) y + = 2(ax? + by? + cz? 4+ Qeyz + Bea + (2ca + 2ey + 2far)z + 2gxy), an identical equation. 132 DIFFERENTIAL CALCULUS. edi Titre exe ay LG ee? dey’ dy y? a= 0: therefore dk LE 2) Sy: <2 ae ae dy yy yy ‘ 88. Let us next take the case of the function of the alge- braic sum of several variables, x,y,z... If the function be u=F(eatkytz+---)) and we put ctytet..-=#, it becomes wu = F(t). Now, if the original function be differentiated with respect to x, y,%... separately, we shall have, by reason of the equa- © tion u = F(t), dF edFdi dF _ dF dt dF dFdt dx dt dx’ dy” dt dy’ dz” dt dz But the equation x t+ytzt---=¢ gives dt dt dt a] aS te ee tS SS +6 apt da ! dy a dz = therefore dF dF dF dg ay 7 dee that is, the partial differential co-efficients of the function are numerically equal. du du x. lo we (oy Tian a i na + y)r-}, Db. Te (x—y)", oH — =e du du __ sf Hox, '°3, u=IV a ty, da dy 2a ty) du du 1 Ex. 4. uxlvxz—y, da dy a—yy SECTION IX. SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE IN- DEPENDENT VARIABLES, AND OF IMPLICIT FUNCTIONS. 89. In Sect. IV., rules were investigated for the succes- sive differentiation of explicit functions of a single variable. We now pass to the successive differentiation of functions of many variables, all of which, at first, will be supposed inde- pendent of each other. By Art. 81, the total differential of a. function of several variables is the algebraic sum of its partial differentials; and it is evident that each partial differential co-efficient is, in general, a function of these variables, which may be again differentiated with respect to a part of the variables, or with respect to the whole of them. These operations give rise to what are called partial and total differentials, and differential co-efficients of the different orders. 90. lfu=F (a, y, 2...) be a function of the independent variables x, y,z..., then du, d’u, d*u...d"u..., standing by themselves, will denote the first, second, third... total differentials of the function. = is the first partial differential wx co-efficient of uw taken with respect to w; and is OG, Ol atk: is the corresponding partial differential. us e pee 188 is dx\dx/ dx? the second partial differential co-efficient with respect to x; 138 134 DIFFERENTIAL CALCULUS. 2 and the differential corresponding to it is st da? OTe x 2 ay at pees is the second partial differential co-efficient, dy\dx/ dydx taken, first with respect to x, and then with respect to ¥; dydx, or d,d,u. 2 and the differential answering to it is oo dydx 3 a Tt) a a ef is the third partial differential co-efficient dy \dx’ dydx” of the function obtained by differentiating twice with respect to a, and then once with respect to y; and to this we have on yen ib ‘ 2 E the corresponding differential ARE dyda*, or di.d,u, which may also be denoted by d%:,u. In like manner, the notations dtu d‘u hh a : ld indicat datdyde’ da*dyde dx*dydz, d®.d,d,u, di2,,u, would indicate four differentiations: one with respect to z, one with respect to y, and two with respect tox. From what precedes, the significa- QUMtnr+P---y UMrr+P---.y dx™dy"dz?...’ damdy” dz... dd" d...u, dm? u, will be readily understood. ane ee tion of the notations da™ dy" dz? , The remarks in Art. 81, in reference to partial differential co-efficients of the first order, are equally applicable to those of the higher orders. Keeping in view the principles there laid down, there will be no risk of confounding any order of partial differential of the function with the total differential of the same order. Thus, in oe the d?u is always. associ- dady’ ated with dxdy written below ae and in this way the con- struction of the expression indicates both the character of the differential co-efficient, and the variables with reference to which it is taken. It is often convenient to attach to the symbol of the func- FUNCTIONS OF TWO OR MORE VARIABLES. 135 _ tion the characters by which are denoted the order of differ- entiation, and the variables involved in the operation. Thus, Fy (%,y,%---), Bry (2, y, +++), Ha, (Gy, %-.-), have re- tively the same significance as atlas Les as Ree 5 dx’ dxdy’ da*dy’ above explained. 91. -Before proceeding farther, we must prove, that, in whatever order in respect to the variables the differentiation of a function of many independent variables is effected, the result is always the same: that is, if w= F(a, y,z...) 1s to be differentiated m times with respect to x, and times with respect to y, the result is the same, whether we perform the m «-differentiations, and then the n y-differentiations, or re- verse the order of differentiation in respect to w and y; or perform first a part of the m z-differentiations, then a part of the » y-differentiations; and so on until the whole of the m and n differentiations are effected. This principle may be demonstrated as follows: Take the function w= f(x, y,%...) of the independent variables ©, Y,%... Suppose, in the first instance, x to be variable, and all the other variables constant, give x the increment h, and develop by the formula of Art. 61; then, in the result, suppose y to be variable, and all the other variables, including z, to be constant, give y the increment k, and develop the terms by the same formula. The final result will be the same as that we should have reached by giving the increments to a and y simultaneously. Changing x into 2 + h, then, Art. 61, DGG yeaa.) d! (00,Y, @..<-) + hl, (2, y, 2...) ie +5 Fa (e+ ah, RCAS Pa ea | (1); 136 DIFFERENTIAL CALCULUS. in which #';(@-+ 0h, y,2...) isa function 2, h,y,z..., which remains finite when h = 0. If in (1) we change y into y +4, the first member becomes F(e«thy+hk,z...); and the terms in the second member become respectively Fic,ythkh,z...)=F (a, y,2...) + ki Gaps) hae a + a Lie (4 Y + 92k, &---), AF, (c,y kh, 4...) SAE (2, y, 2.. +) ani eee hk? mr ap SO TTES (v, y+ Osh, 2...) 2 F'(@+hythz...)=” = FY, (@ + 1h, Y, a) Ra ip += Fi, (©+ 6h, y+ Ok, 2...). Making these substitutions in Eq. 1, we find (0, Y, & 0s.) PRET yy aes) + hE) (x, y, 2...) | a +AKE,, (x, Y,%.. +) + 7 Fa (@ + Oh, Y &-:) Bahl 1h) Key ata eae hk? "I we ee (x +6, h, Pniey ire F If we begin by giving y its increment, we shall have the equation Pay +kh2...) =F (x,y, 2...) + kF, (x, ys eee) k? a hy Fy (Usa, 2 ey and in this, giving to # its increment h, and developing the terms as was done above, we have FUNCTIONS OF TWO-OR MORE VARIABLES. 137 [ (a, y, 4...) AP, (x, y, %---) +E, (a, y, 2. +) AE js (By Beek ke) $5 P(e + Oh, Y, %+-) kee 2 BFiaxth y+k,2z...)= + 5 Fe (2, y + Oh, 2...) kh? mt / +-5 Fya(e+o TRU icra) 4+ Be Fyn, (@+ Oh, y + Oh, %...). It is to be observed, that, in the several preceding equa- tions, the factors Ff), (x + 6,h, y,%..-), Fia(®, y+ 9:k, 2°..), &c., of terms in the second members, remain finite when / and k, separately or together, become zero. Equating these two values of F(a+h,y +h, z...), sup- pressing terms common to the two members of the resulting equation, and dividing through by Ak, we have wr k yr FY (0, 4,2. +E P(e y+Okye...) | i we fey 30 (2+ Oh, y + Ak, oH ai h rr jig (2, Y, 2) Fee hs (w+ Wh, y,%...) a k we | +5 Fi (@+ Ov hy + Ob, 2...) This equation must be true, whatever the values of / and k. Make h=0,k = 0; then , Frey (%) Y, % +++) = Eye (®Y, #) (2). The first member of this equation is the second partial dif ferential co-efficient of the function obtained by differenti- ating, first with respect to x, and then with respect to y; the second member is the second partial differential co-efficient 18 138 DIFFERENTIAL CALCULUS. which comes from differentiating, first with respect to y, and then with respect to x It is therefore immaterial in what order the differentiations are performed. This theorem being demonstrated for derivatives and dif- ferentials of the second order, it can easily be extended to ) those of any order. Suppose we start with Whether z this be differentiated, first with respect to x, and then with respect to y, or we invert the order of differentiation, the re- sult is the same by what has been proved. So that dui d*u dadydz ~ dydadz But the order of differentiation with respect to z, and either x or y, may also be inverted ; and therefore CLP Us TN LA ed ee dxdydz dydadz~ dadzdy and generally, for the function v= F(a,y,z...), (me rnr+p U Ar tmtrPy qdm@rp+n U da™dy"da? — dy*da™dz? — da™dz?dy” re Exe]: pray AI & du any du 4x y dx (a? + y?)? dy (oa au _8ay(x?—y?) du. sage) dyda (x* + y?)) " dady (a? + y?)F Ex. 2. tan abe; EER AG) Ce! Tee dx ay?’ dy wy’ azu ih a? — y? REY. v2 —y? dyda (x+y?) dady~ (Fy? , FUNCTIONS OF TWO OR MOR [7 ABET 139 a able: is zero. Now, since the he len Sea\ function of the Slama Ea ealee ge Gigs réspeet to vat end the corre- yf co-efficient by the differential of the variable to which it re- lates, it follows that, in subjecting such differential of the function to further differentiation, we may set aside the differ- ential of the variable as a constant factor, and operate on the differential co-efficient alone ; restoring in our final result the constant factors set aside: thus, if w= f(x, y,z), in which x,y, and z are independent, then etal, (ant, Bae == a dx, d,d,u = dad, F(x, y, 2) = da Fy, (x, y, z)dy gene Lady a dxdy, Mu dady dd ,d,d,u = daxdyd, Fy, (x,y, %) = dadyF')’ (x,y, )dz deu ey mas, ONT he ea pe Y; 2)dady di dadydz dadydz, and, generally, a2 dy Uy te = Fenn sm (0, Yy 2) ce dy de? QUMrrr+Py m nN Any DP = quam dy"da da™dy"dz?. 93. By Art. 81, the first total differential of the function u = F(a, y,2), of the variables a, y, z, is du =F de + oi y soil seks (eer Taking the total eer pa each of the partial differential du du du co-efficients rE Nae 7,7 We have 140 DIFFERENTIAL CALCULUS. de = mete Ut oak Ca) ats ap 204 4G) aa? eg te and therefore fo ae dat 42 2 tag + gi 7, dad _ a 4, te (2) Proceeding with (2) in the same manner nee we did with (1), we should get the third total differential of the function ; and so on. For the function w= F'(a,y) of the two independent vari- ables a and y, the successive total differentials will be et ge d?u 2 dt = 5 de” pee i edy + Gat - au fee ad*u au = x? d Ba 5 a: dy? nda sh y ail cay au .,. Bae i du AMA SY, lw = ——~ dy” Outs Vek pies, VO Ten Br _ n(n—1) d”u da”—*qy? + tee tS 1.2. da" —*0y? nm(nm—1)... (n —(n —2)) d™u ss 1.2..-(n—1) anag FUNCTIONS OF FUNCTIONS. 141 the law of the co-efficients being the same as that in the de- velopment of (1 + x)”. | dix. 1, Meco e du = yzdx +- xady + xydz, d?u = 2 (adydz-- ydadz + zdady), d?u = 6dadydz. Ex. 2. w= (a? + y?)?, mea end Se Te OE Ova BUA : da (w+ty?)? dix? (x? + y2)3 dic3 (x? + y2)8 du __ y Oe ee ge ant ey ire dy (e+yrft dy? (oye dy (wt yt meer Ce Byler ye) dy (a? +y) dutdy (a? +y?)! du ¢(2y? — a”). dady? (a? + y?)? 1ST own, (~ 3xy? de® + 3y (2x? — y?) du*dy + 3a (2y? — a”) dady? — Byotdy) Bee. (+yi Ex. 3 ip ee Ee ed Tpke ght S ti gd erate du — bet + by dl? — brewtly dy ” dy? d*u Sees ax-+by. a. abe : Mu = (a*dx? + 2abdxdy + b? dy?) erty = (adx + bdy)e +, 94. If, in the function s= Fu, v, w), u, v, and ware functions of the independent variables wx, y, and z, we have a 142 DIFFERENTIAL CALCULUS. case of a function of functions of independent variables ; and the first total differential of s is ds ds ds _ —e —d 4 ty, : ds FE ate ves z (1) But since u, v, and w are all functions of a, y, and z, the par- tial differential of s, regarded as a function of a, is (Art. 82), ds ds du ds dv ds dw deo” du dete} do ee ds ds du ds dv ds dw ] dy = — —dy+—— — — dy; Be ay! andy 2) du dae ar ds ds du ds dv ds dw aes ayes — — daz, eae ign dude nde The total differential of u is du du = — de r+ yt eS and, for the total differentials of v and w, we have like expres- ds ds sions: therefore, by substituting these values of — cE dx, ay dy, Y d. =, de in (1), and uniting terms, we have ds th cP du ae , +4 Tae dw. A second ait Ae ee give d’s d’s ds 2 of——— 2 w Gist me ie sa dv Sale! ~ dw +24 duh dudv dudw + 2 abe dudw He — ae dvudw 2 Br ae bak as ds 4. ; ie ao” and from this we pass to d’s, and so on. The general rule is, then, to differentiate as if uw, v, w, were independent variables, and substitute in the results the values’ HOMOGENEOUS FUNCTIONS. 143. of du, dv, dw; d?u, d?v, d?w..., derived from the equations giving w, v, w, in terms of the independent variables a, y, z. If uae + by + ca +d, v=a’et by +c2+d’, wale + b/y + 0/% + d”, are the expressions for wv, v,w, in terms of a, y, z, these func- tions of the first degree, with respect to the independent vari- ables, are said to be linear. In this case, we should have eee 0, 0) de = 0; dee. s+ and the successive differentials of s = F'(u,v,w) would then have the form of the successive differentials of a function of three independent variables: thus ds d’s ds d*s Ligh. 2 2 2 d's— — du sre ra + a, dw aude" 7 dudv ey ae dudw + 2 leah dudw. Ex. 1. s=fF (u,v), u=—axc+by+e, v= ale-Lbly +e, n __ a*s ds n—l d's vy", eee ae du” ae — iq du du-+.. a dU". Ex. 2. s=f'(u)F(v), u=ax+ by+c, v=a'a + byte’, ds = EF" (u) F'(v) du + Fu) £” (v) dy, C3 ae E’(v) du® 4+- 2F"(u) “ v) dudv + a ee ee de = Fw (u) Flo) oe Le nPo- D(u) Fi(e) oie ee + nf" (u) F°—” (v) dudu”—! + F(w) ae ee 95. If the function s = F(a, y,z) of the independent varia- bles, x, y, 2,18 homogeneous, and of a dimensions, then, by Art. 87, dF dF. dF da Yay tae It may be shown that similar relations exist between the func- Peat ao? ) = tion and its differential co-efficients of the higher orders. 144 DIFFERENTIAL CALCULUS. Since the function is homogeneous, if we change a, y, and z into tx, ty, tz, and make u = tz, v = ty, w = tz, we have Pu, 0; w) = t F(x, 9,2). Differentiating this twice with respect to ¢, observing in the d /du\ d /dvw\ d /dw second differentiation that — ai ( i) sila aia) are each zero, we find, dF du .dFdv . dF dw du dt ' dv dt ' dw dt CKdu? PE dv dFdw* | au? at S det dS dw dt d?F’ du dv d?F du dw eS nog By =< Ley | §% ‘ ste dudv dé dé dudw dt dt | a(a—-1) 7 F(a, y,2) ae at?) F(a, Y; a). , a?F dv dw AsO aon But Sy = 2, ate Y; e — 2; and, if¢=1, the second partial differential co-efficients of the function with respect to u, v, w, become the second partial differential co-efficients with respect to x, y, %, respectively ; and hence the last of the above equa- tions becomes 2H 2 2 PE yi 0 ad? ar a’ Dine pe eh Te pene dady ady potas dadz Tea dydz = a(a—1)t**F (x,y,z). By a third differentiation, we should get a pst Wie let ain of ha: DPF nis = a(a-1)(a—2)t? F(x, y, 2). +32? Y itayaae xy Ga iapas DIFFERENTIATION OF IMPLICIT FUNCTIONS. 145 Example. F(x, y, 2) = ax? + by? + cz? + 2ery + 2fxz + 2gyz, 3 eae: iG ar "pn Ga = Ga = % ar fa ak Og va ay ee dady ” dads 1 dydz sath oe F dar ae be pace F y eee ae gst. aed ak d?F 2 Bie iy eae ene dada ged? dydz 2 (aa? + by? + cz? + eay + 2fxa + 2gyz) = 2K (a, y, 2). 96. To express the successive differential co-efficients of implicit functions, take the function wu = F(a, y) = 0, in which y is implicitly a function of w; then, by Art. 84, du, du dy _ iad dy da The first member i. i is another function of w and y, which =—0 (1). denote by v; whence v=0. Differentiating v= 0 as we did u — 0, we have dv dy _, : qe ASE dy da Bee) | 2 2 oa dv du, du dy , dudy dx dx? ' dady dx ' dy dx?’ dv du , du dy me dy dady ' dy? du These values of on 7 substituted in (2), give os 9 Wu dy d?u (dy du d?y _ dni’ ” dedy dot dy? = (ae) as): 2 From (1) and (3), we find the values of gh ath Kqs. 1 and 3 are called differential or derived equations of the first and second orders respectively; and, with reference to them, u — F(x, y) = 90 is the primitive equation. 19 146 DIFFERENTIAL. CALCULUS. : | at od The above process is somewhat simplified by putting ages then du | du PM py eng LAE aa ae (1’) dv. dtu au du (dp aE ese : dz da? us dady © Ay alae dv .d'u du du /dp ot dy dady + ap? © wa) These values in (2) give du du (dp du dite ip eae (2 8(6) Ze ee du du fy dp\ | - det dady? + Gp? to But (2 ) me = te (Art. 82); and hence au au du, , du dp “he — 1p? + 1 = OF - 2 Rea de dy da (9), d d. du ad? 2 d’u dy Y\ at oe ce Te eal dady et ae dy’ ag | T dy da? d. We call attention to the notations Ge ee ee by re- marking that p is generally a function of «and y; and that dx/’ \d tion, the first with respect to xv, and the second with respect to y: whereas ed is the differential co-efficient of » with re- spect to x, p being considered, as it is, a function both of « G ) & ) are the partial differential co-efficients of this func- and of a function of a function of x. Thus suppose, that, by solving the primitive equation (x,y) = 0, we find y= f(z), then DIFFERENTIATION OF IMPLICIT FUNCTIONS. 147 d p= (2), =f"(2). Suppose also, that, without solving the primitive equation, we find p=9(2,¥) =9(2,f(a)); then a ee (=o ens (F) =a au). But, by Art. 82, d / ' dl 1 = 92(% Y) +99 (2, Y) = S'(a):.. Q). This points out the necessity of distinguishing, in certain cases, partial differential co-efficients, such as those of p in Kq. a, by the parenthesis, or some other mark, that, in the course of an investigation, they may not be mistaken for others, as 2 in Kq. 0, of the same form, but having a differ- ent significance. The value of oy deduced from Eqs. 1 and 8, or from I’ da? and 3’, 1s d?u /du? d*u. du du. d*u /du2 dy at je) * dxdy dx dy ' dy? a ; TE ah du\3 Mi) The expressions for the higher orders of differential co-effi- cients of implicit functions are so complicated, and so little used, that it is unnecessary to proceed farther with this divi- | sion of the subject; but we will conclude it by giving the differential equation of the third order of the implicit func- tion y of the variable x, given by the equation wu = F(x, y) = 0. This differential equation is 148 DIFFERENTIAL CALCULUS. d*u d*u dy d'u /dy\? d'u/dy dix? ide dx*dy dx eS dacdy’ cs 195 dy? ete d’y , dudy\dy' du dy _ 3 ed Lis a (ser dy? es dx? ' dy dx’ Sy 97. Suppose we have given the two simultaneous equa- tions ie shi Ge a! Pt) =m) tall b bp U = fat, 20 ae It is theoretically possible, by combining these equations, to eliminate either variable, and get an equation expressing the relation between the other two from which the successive dif ferential co-efficients of one of these regarded as a function of the other might be obtained. But without effecting this elimination, not always practicable, we may proceed as fol- lows : — Suppose «x to be the independent variable, and differentiate (1) with Ne to x; then (Art. 85) du dy , dudz wt 95 de ae ae = Oe In like manner, from (2), du dy , duda _ a Pal dy dx ' dz dz Se: From (3) and (4), we find du dv __ dv du dx dudv dvdu dy dz dy dz and dv du du dv dz da dy dx dy Be Sy ax dv dus du dv DIFFERENTIATION OF IMPLICIT FUNCTIONS. 149 The first members of (3) and (4) are functions of a, y, 2; and, by differentiating them with ae to x, we have d*u d’u dy d’u dz = Ge d’u dy dz dx? 1 dady dx ee? dadz da +o dy? vox dydz dx dx dz du d*y dud?z i +o Ae ae apa dada 7 ee and dv 9 d°v dy dy dz aaa ay dy dz dic? dicdy aes dade EQh oie da dydz dx da d’ Gal du d*y | dvudz Pb a Fare, enh See) FA nae Ban rea: es tla Oe From (7) and (8), by substituting in them the values of BS ee in (5) and (6), we may deduce the values of cd ie and oe They may also be found directly by differentiating (5) and (6). 98. For an application of the methods of successive differen- tiation, suppose we have the single relation wu = F(a, y,z) = 0 between the three variables x, y,z; then z may be consid- ered as an implicit function of the two independent variables a. It is required to find the first and second orders of the par- tial differential co-efficients of z with respect to # and y without solving the equation u= F(a, y,z) = 0. The first partial derived equation with respect to @ is (Art. 85), du dz Bate Bis teen and that with vt to y 18 du dz 150 DIFFERENTIAL CALCULUS. du du du dx’ dy’ dz’ of uw, taken on the supposition that the variable a, y, or z, to in which are the partial differential co-efficients which they separately relate, alone varies. Kqs. 1 and 2 will give oe 7 Differentiating (1) with re- spect to x, and (2) with respect to y, and also either (1) with respect to y, or (2) with respect to x, we get du di aN ia fda\t s duad2 ee ? Uega.dig's dat i Th a5 ape le (3), du Mu dz dz\2 du dz dy? it? dydz at ast sen da ia (4), du Ouida: d'u-dz . d’u de dzaaiaiuaeee dxdy ' dedx dy 7 didy da T aa dy di T a dady and from the five Eqs. 1, 2, 8, 4, and 5, we can deduce Bemeeiee 02" On mies da’ dy’ da®’ dy?’ dzdy d*y Ex. 1. Given y* + x«* — 3axy = 0, to find the value of meee =0 (5); The first differential equation is and the second, a? (y? — ax) (i) 2a e+ ay (OE) +2x=0 (2). Substituting in (2) the value of taken from (1), we find, after a little reduction, 2 (y* — ax)? a — 2a(ay — a*)(y*? — ax) + 2y(ay — x*)P + 2x(y? — ax)? =0: whence ay __ 2a(ay — x?) (y? — ax) — 2y(ay — x")? — 2a(y? — ax)? dz? (y? — ax)? 72 DIFFERENTIATION OF IMPLICIT FUNCTIONS. 151 and this, after performing the operations indicated in the nu- merator of the second member, and reducing by the given equation, becomes dty - da* xy Oe (yy? — aa)" Ex. 2. Given 6?c?a? 4+ a?@y? + a’?b?2? — a’?b?c? =0, to find I Sell SR ae da” dy” dady’ az c?(a?2? +c?x7) dz c?(b?2? + c*y”) i an aiet ae ae bt? . ieee Cry eee oe 0" 2" SECTION X. INVESTIGATION OF THE TRUE VALUE OF EXPRESSIONS WHICH PRE- SENT THEMSELVES UNDER FORMS OF INDETERMINATION. 99. Ir sometimes happens that the expressions under con- sideration assume, for particular values of the variable or . . > 0 variables involved, some one of the forms ~, +2, 0x0, ie @) | 0 0°, 0°, + 1°, o —o, called forms of indetermination, though he value of the expressions may be determinate. Our object now is to establish the rules by which may be found the true value of an expression which reduces to any one of these forms. 100. Of the Form - This form can only result from a fraction in the numerator and denominator of which there is a common factor, which factor becomes zero for the particular values of the variable or variables which reduce the expression P (x —a)”™ Q («@—a)”’ may or may not be functions of «; but, if they are, they do to 4 -Thus take the fraction in which P and Q not contain the factor « —a, and therefore do not become zero when «=a. If in this fraction, as it stands, we make « =a, it takes the form . but if, before giving «x this value, the t Hi sae fraction be written — (#—a)”~", 1t1s seen that the true value ~ of the fraction for c=a is 0 if m >n,o if m < n, and A if m=n. This suggests the following rule for the evaluation 152 INDETERMINATE FORMS. 153 of expressions which take this form; viz., discover, if possible, the factors common to the numerator and denominator of the fraction, and divide them out. What the result reduces to by giving the variables their assigned values is the true value of the expression. vetoar—3e—3 0 = - Btens he cane Geo Ee ie ear e g When # /'3 x + x? — 84 — 3 ee er) 2 te e* — 20? —34+6 (w?—3)(a—2) aw—2 1473 — for «= V3. Js, ee Many cases of the form : may be treated as follows :— Example. but i Take the fraction Pre ein which becomes D when «=a. ah? 0 (a? — a")! Make «x =a-+h; then Sense 7): a e ace i = =.0. when (a? + 2ah+h?—a*)s h3s(Qa+h)s (2a+h)s h = 0, which corresponds to # = a. oe 0 Also the fraction Ve— Va is Vea O = — for g — a; mak- Ja — a iG ing «—a-+h, —Wath—Va+vh _ Vath —(va—Vh) / 2ah bh +h? / Qah + h? Multiplying numerator and denominator of this by Va+th + (Va —wWh), we find 2*/ah 1 (Qah + 12)! (a+ ht + (Wa —Vh)) ~ / 2a" for h = 0, after dividing out the common factor h?. The examples already given have been solved by common algebraic transformations; but most of the cases which present 20 154 - DIFFERENTIAL CALCULUS. themselves can be more easily solved by means of the differ- ential calculus. 101. Suppose the fraction to be Fe aot and that both F(a) and f(x), as also their successive differential co-efficients up to the (n— 1) order inclusively, vanish for «=a; then it has been proved (Art. 56) that Fiat hy Pease on ie Path) f(a oh)” and consequently, by making 2 = 0, we have Ba) _ F(a) AY GSRE@S Hence, to obtain the true value of the vanishing fraction Hm) when « =a, form the successive differential co-efficients J(&) of both terms of the given fraction until one is found, whether of numerator or denominator, that does not vanish for z=a; and take the value, when «=a, of the fraction whose terms are respectively the differential co-efficients, of the order of that thus found, of the corresponding terms of the given fraction. If one of these differential co-efficients vanishes, the value of the fraction will be 0 or «, according as it is that of the numerator or of the denominator; and it will be finite if the first of the differential co-efficients that do not vanish is of the same order in the two terms of the fraction. Tix ; OT rue 2 a). sin. & 0) d(x) = e* —e-*, P(x) = e* + e-*, /(¢) ae Bate ee Ce Gann Cray oe aa % INDETERMINATE FORMS. BS Ex. 2. 2 — sin. x its b.— cos. 2 an sin. a aie tion? oe eae Ore fin Ex. 3. ( Ue ) =(5) nih e—1/,—1 LY 1 102. Form — If the two functions /'(x), f(x), become £2) infinite for « = a, the fraction Fiz) reduces to - But io ii this case the fraction may be put under the form LE), which, F(a) for x = a, becomes zs and may therefore be treated by the preceding rule. ‘Thus : f(a) . Fa) f(a) (4)) ~ (F@) pray | Meo Ka ( fla)) 7). @ (roy | whence ah — 7 mi and the true value of the ratio aa = F(a) J’ (4) Tf all the differential co-efficients of both terms of the fraction is the value of become infinite up to the (7 —1)™ order inclusively, then Bea) Se F(a)’ and the true value of a ratio, that, for a particular value of the variable, takes the form =, is the value of the ratio of the dif ferential co-efficients of the order of that first found, whether 156 DIFFERENTIAL CALCULUS. of numerator or denominator, which does not become infinite for the assigned value of the variable. Example. For «= 0, l l = . 9 e ee O-: sm? ae. 2 6in, cede cosec.x 0 wcos.v cos.e—a@sin.x 103. The rules which have been given for finding the true value of ratios which take the form ; or — are applicable for infinite as well as for finite values of the variable. This fol- lows from the fact, that the reasoning by which these rules were established requires only that the value attributed to a, causing the fraction to assume the one or the other of the above forms, should be the same in both terms, but does not involve any supposition in regard to the magnitude of this value. The rule depending on differentiation may be demonstrated directly when the form of indetermination comes from the hypothesis ee Represent the terms of the fraction by /'(x), /(x), as before, and suppose, that, for « = 0, we have either (x) =0, f(x) =0, OF 2? (2 )— 00, () = cor; tthen; putting | for &, F(a) _* () scales a ee BQ" TOA but, by rules already given, [70)| _[s"Q)] = INDETERMINATE FORMS. 157 hence ae f F(2) _|73)| 2 | 2 (ae 3 Oe || orf ee Ex. 1. For x= we have, whena > 1, a*_atla_ Tig as Wine ee ly Len 7. —"oo.., fe 1 _ 9 x ala Ex. 3. Whenw=—o, and n is the integer which immedi ately follows a, a* _a(a—1) (a 2)...(@ —n+1)_ 4 oF erage 104. Form 0x. Let F(x), f(x), be two functions of a, one of which becomes 0, and the other infinity, for a particular value attributed to x. Forx=a, suppose f(x) =0, f(x) =a. The product may be put under the forms u= F(x) x f(x) = ae Ae as ; the last two of which, for the assigned value F(z) (2) ‘ of x, take respectively the forms 0” =, and can therefore be treated by the preceding rules. Exel, w= 1(2—5) x tan, 57 = OXen when «=a. a 2a But 1(2 )xt PER Reset? a 2 1 1X Gots == tan. 2 ‘3 158 DIFFERENTIAL. CALCULUS. Geo (2-2) pple and - eles i _4 ote ’ ti J 5 [ Ue elo | ET: a2 = [ 2a. rt=a ie oe or cla = —0XKo, lar las aie: aclac = eth and eae = ee 0. Ex. 3. «™”(lxv)"=0xXo for «=0, when the exponents m and are positive. Make «= - then 2” (Ie)\"—(— 1). This, by? Bxo3;* oe Art. 103, is zero when ¥ =o, which answers to « = 0. 105. Forms 0)’, ~°, +1”. In the explicit function y= (Fwy of the variable a, suppose that f(x), f(x), are such, that, for the particular value | x =a, y assumes any one of the above forms; then, to deduce . a rule for the evaluation of y, we proceed thus : — Take the Napierian logarithms of both members of the equation y= (F (cs and we have ly = f(a) UF (2) =). J (2) Now, since, to have one of the proposed forms, #'(x),. for the assigned value of ~, must take one of the values 0, 00, or 1, lF'(x) will become either — «, + 0, or 0, and a will take J («) one or the other of the forms o and may therefore be INDETERMINATE. FORMS. 159 used for calculating the true value of ly, from which we pass to that of the function itself. Ex. 1. «* forz=0 becomes 0°. In this case, BGO a | Ge which, for « =0, is equal to s 4 acall = by ae Oe Geert 1 ; a a T=0 1 Tx) 2. @2,— 0" Bee Ti 00. Here i) ss oe and this, when 7 =o, = bile 0: SiS: x J (2) site == .0 a Seal ] fie 3. Cees woek ws Is BEE (a ke #0) "es ee Ranh whens 13 fx bya —Il: Sea ime 106. Form «~—~«. If the functions F(x), f(x), of x, both become infinite when a =a, then, for this value, E(x) —f(x#)= 0 —o. To deduce a rule for the Aggie) of a that take this form, make f(x) = F(a le \ = F(a)’ ; then the value of x that causes f(x), f(x), to become infinite, must reduce /}(@), f(x), to zero; and, if a be this value of x, we have pee bel Dee of @) — Fy (a), 2 2 F(a) —f(a)= ———. : ~ F(a) f(a) F(a\A(a) = and the case is thus made to fall under the rule of a 101. 160 DIFFERENTIAL CALCULUS. n exe; Sec. « — tan. 72 = 0 — oo when = 95, 1 sin.a 1—sin.a 50C.-@ —tan v7 = : — ‘ COsS.% CcoOs.ax COS. @ Sein we cos. x and (pe ee ice fh r =O. cos. a. /*=5 sm, oy 1 2 Hix. 2. ——-~ =0o—o whenx =I, lc -~ la dV 2\ ae a ee lee ny Na eS ) = 107. It may happen that. not only do F(a), f(x), in the ratio aay vanish for the assigned value of the variable, but also all their successive differential co-efficients, however far the differentiation be carried. For suppose F(a) = a5 which becomes 0 for « = 0 if a and n are positive, anda >1; then 1 nla.a E(x) = reas Wide wil. ST faa nm+1 EY" (2) =nlaca sz (see Gee) Making « = I these differential co-efficients become z nlaz™ +} Ee (avers PL nla (nlaaro+ —(n+1) vt) EF" (x) a aie a a (8 Ie It is needless to carry the differentiation further to see that each differential co-efficient will contain a factor of the form m in which a, m, and are positive, anda >1. This factor an? takes the form 2 for z=; and if we apply to it the method INDETERMINATE FORMS. 161 for finding the true value of such expressions by differentia- tion, differentiating p times, p being the whole number next above m, z will disappear from the numerator of the ratio of the differential co-efficients of the order p, and this ratio would be of the form aot in which & is a constant, and (2) a function of z, that becomes infinite when z=. Therefore all the differential co-efficients of #’(a) vanish when x = 0, 1 which answers to z =o. Hence,if f(x)—b «*, the terms E(«) | J(%)? for «= 0,if a, b, n, g, are positive, and a@ and 0 are each of the ratio and all their differential co-efficients, vanish ereater than 1. The true value of this ratio cannot then be found by the method of differentiation. ae When n = gq, the ratio becomes () «” the true value of which, for z=0, is 0, ifa>b, anda ifa< 6. 108. The solution of cases of indetermination is often facilitated by transforming the example so as to make it take a form of indetermination different from that under which it presents itself. Thus tae | ae 0 = — becomes — when xv'= 0; hy x W) but. 1 e # 1 1 — =—; = ~—— when «= 0; a Tes 0 xX & and the true value of the given expression is 1 divided by. 1 the true value of ze when x= 0. 21 162 DIFFERENTIAL CALCULUS. Again: if (x) becomes infinite when «=o, then (Art. 102) FC) =(F"@)) \ x r= ve But (Art. 56) Fe tM = FQ) ae and it is evident, that as x increases, and finally becomes infi- nite, the second member of this equation converges towards and finally becomes F”(x): hence Eoye =e Bi it es or by making h = 1, as we may, since hf is arbitrary, (= ~) = (Fw to Nie F(z)) If, now, the value of (F(a))=, when «=o, is required, we have (#2) =e? (1); a true equation, as may be seen by taking the logarithms of both members: therefore (#12) = eed (2); and the proposition is thus Bee to the evaluation of iF LF (2) ae x) e * , or rather nes when c=o. INDETERMINATE FORMS. 163 By what is proved above, (=e) = Cxc seh F(z) = (: a ae (3). But, from Eq. 1, we have 1 i(sH(0) = ae ; therefore, by Kq. 3, (7): = era i Let this be applied to the determination of the true value of wo 1 “ z= when @=oo. Dees Now, by what has just been proved, the required value is that of eee 2... fet 1\" 1\2 < 1.2...(@+1) ira = 7 y=(+]) for c=o. But (Art. 9) (1 fe a et? 109. Thus far, the discussion of the indeterminate forms has been confined to functions of a single variable. A few cases will now be considered in which these forms present themselves in functions of more than one variable. We re- mark, that a function of two variables may assume the form * either when a particular value is attributed to but one of the variables, or when both variables have particular values given them. An example of the first case is b(a —a) y(x* —a*) + (@— a) C— which, for <= a, reduces to "4 whatever be the value of y; 164. DIFFERENTIAL CALCULUS. but by dividing out the common factor 2 — a, and then making 2% =a, we have z= z—. 2ay An example of the second is _ e(% —4) a, any DY which takes the form : for «=a, y =), and, for these values of the variables, is really indeterminate. For let p denote the ay G y— bi p is an arbitrary quantity, and z is therefore indeterminate. ratio then z= oe ; and, since # and y are independent, 110. To investigate a rule for the evaluation of the inde- terminate forms of functions of two or more variables, take the function w= F(a, y), « and y being independent, and suppose the function to be finite and continuous for all values of x and y betweenz =a, x«=at+h,y=b,y=b-+k; and further, that all the partial differential co-efficients of the function, up to (n — 1)™ inclusively, vanish for «=a, y=06; but that those of the n™ order neither vanish nor become infinite for these values of x and y. For the time, denote by ht, kt, the increments of a and b; so that the function, when the values of « and y with their respective increments are substituted, is /’(a-+ ht, b+ kt), which becomes F'(a+h,b+k) by making #=1: then de- noting F(a -+ ht, b + kt), which is a function of ¢, by /(¢), we have Fiatht,b+kht)=f(t) (1); and, making in this ¢=— 0, F(a, b)=f(0) (2). If f(z) is finite and continuous for all values of ¢, from ¢ = 0 INDETERMINATE FORMS. 165 up to ¢= any assigned value, ¢ 7; and if, in addition, all the differential co-efficients of f(t), up to the (n— 1)™ inclusively, vanish for ¢= 0, while that of the n™ order is neither zero nor infinite for = 0; then (Art. 56) Fe Oe 8) (3) To simplify the application of this equation to our purposes, -make w’=a-+t ht, y’=b+kt; whence os h, aes =k, and S(t) = B(x! 9’): i dx dk dy’ dk us = ee ds’ ai dy! a dat” +5 ay Making ¢=—0, observing that then x7’—=a, y’=6), and that what paged! become, will be identically the same as what da’? dy’’ ee iy become when «=a, y = b, and denoting these dif ferential co-efficients for this value of ¢ by ) eae we y have P= (Ga) b+ (Gy YF If cat 29 , both vanish, then /” (0) = 0, and we must pro- da ), \dy /, | ceed to the 2d differential co-efficient of /(¢), which is OTe. " Rial Ody? T dyn 3 and, in this making ¢ = 0, we have, by adopting a notation in harmony with that in the expression for /” (0), 4/ VEN ak Cae Dire 70) = (Fe) +2 (soa) tet (Gps BY 166 DIFFERENTIAL CALCULUS. 2 2 7 and in this also (Gr) ... are what sans become when Oa hg ua? . x—a,y—b. If all the partial differential co-efficients of the second order vanish for ¢ = 0, then /”(0) =0; and we must pass to the 3d differential co-efficient of /(¢), and in this make t#=0. We should thus find GF iy a dF 3 hk +3 jie it =) a é ay) e Gee) aay r= (5 and so on; the expression for /”(¢), all up to that pene for ¢ = 0, being ft) = d"F ri ad a ae a \An Biss i a h=—1k n—1 aa) : cp ix. (ey ) - ame err (x5 das dy" Fine hk ; ad" Rae eR +(+) the laws governing the co-efficients and exponents being ob- viously the same as in the Binomial Formula. Now, since f(x,y), F(x’, y’), differ only by having x and y in the one replaced respectively by x’ and y’ in the other, it follows that any partial differential co-efficient of F(a, y) will be the same function of # and y that the corresponding partial differential co-efficient of L(x’, y’) is of x’ and y’; and hence the hypothesis that renders « =a’, y=y’, will, at the same time, cause these differential co-efficients to be equal. There- fore make «=a2’=a+t+ht, y=y’=b+kt, and we may write OE dx a. dy Ret le ot ee poodle | 1” Tedy*— Tay : ae INDETERMINATE FORMS. 167 all of the several orders of partial differential co-efficients of F(x, y), up to and exclusive of the n™, vanishing for «=a, y = b, that is, for t= 0 in /f’(t)... f-» (t); but all of those of the m™ order not vanishing. Then, writing #¢ for ¢ in Kq. 4, and substituting in Eq. 3, we have F(a + ht, b+ kt) — F(a, b) and if, in this equation, we make ¢= 1; then F(a+h,b-+k) — F(a, b) 1 /d*F oF | 123. an ae eg evar aCe 8 ad cipnog Se; dady mis eer atle +- dy a eee (5), y=b-+ gk which enunciates a theorem relating to a function of two inde- pendent variables analogous to that demonstrated in Art. 56 for a function of a single variable. In Eq. 5, suppose both a and 6 to be zero, and then change h and k into y and x, as we may do, since / and & are not only independent of each other, but may have any values, and we have F(a,y) —£(0,0) = 1 anh, a deh Se =2(e—a), 7 = 2(y—B); and therefore atid 7 0F df: da dy dy dx becomes a(y —b)—y(%—a)=0: b whence ey Se, ye ao Substituting this value of y in oh we have w* — 2ax+ a? chen z+ b?—c?=0, 2 or Be ore ons a whence ee ac =a Vea By differentiation, we get from Eqs. 1 and 2 du 2bc? gas fi (c? — (x — a))t Observing that the upper sign in the value of « answers to ee. au a Cage the upper sign in the value of — and the lower sign in the one to the lower sign in the other, we discover that ac CRW ay ED d*u ' : renders —— negative; hence this value of # makes w a maxt- da? mum: but, when a — dl? is substituted for a, ae. be- ac Vai +e? dae 184 DIFFERENTIAL CALCULUS. comes positive, and w is therefore a minimum for this value OL as. 119. When we have n variables connected by n—1 equations, by processes of elimination, we may reduce the n — 1 equations to a single equation involving but two of the variables, and thus bring the investigation of the maxima and minima of the variable which is taken as dependent to the case treated in Art. 117. But, in general, it will be found easier to operate as follows: — | Suppose that the four variables, x, y, z, uw, are connected by the three equations, Si(@; Y; %, U) = 0, fo(@, Y, %, U) =, f's(%, Y, %, U) = 9; and that the maximum or minimum value of w is required, x being the independent variable. Differentiating with respect to x, we have df, 4 df, dy | df, dz | df, du __ dx. dyide i da da dita d df, d df, dz | df, du a Hf et - das F = de a d, fy dy c= Of, 07 ie Of, OF Tete ETE aot ae =?) One of the conditions for a maximum or minimum for u being Be = 0, introducing this in Kqs. 1, they become Of hy AY Sais eee das! sly das anda: | dj, _ afi dy df, dz __ dz tidy deigs de =. (2). df, df, dy , df, dz _ dat dy dzT de dz~"} 0 | dy dz The equation which results from the elimination of —% dx’ da’ MAXIMA AND MINIMA. 185 from Eqs. 2, together with the three given equations, which we will denote by f, = 0, 4, = 0, f, = 0, will determine values of x, y,%,and u. To decide whether any or all of these values, or rather systems of values, make uw a maximum or minimum, d?u dic”’ when the values of the variables are put init. By differen- we must ordinarily pass to and find what sign it takes tiating Eqs. 1, the resulting equations and Eqs. 1 will give a 120. Before concluding the subject of the maxima and minima of implicit functions, we will briefly refer to the limi- tations made at the beginning of Art. 117. Resuming the equation du Ce ee SW, roma cr dy we remark, that the necessary condition for a maximum or 1 dy on Tyr minimum value of y is, that 7, change its sign, which it can da do orly when it passes through the values 0 or 0. Now, du ne dy becomes zero when — = 0, -— being finite; or when dx dx dy du... ; ee et. ‘i oe, sa being finite. Again: “2 becomes infinite when dy dic dix d d ae : du 5 aut ee 0, o* remaining finite; or when — = ow, — being finite. dy x da dy It therefore appears that the methods heretofore given for determining the maxima and minima of implicit functions are quite incomplete, as they omit the discussion of several cases that may give rise to these states of value. Most of the functions with which we have to deal are those 24 186 DIFFERENTIAL CALCULUS. whose maxima and minima are indicated by a change in the sien of the first derivative when it passes through zero. It often happens that the conditions of the problem to be inves- tigated enable us to decide some of the questions relating to maxima and minima, which, if referred to general rules, would require great labor. EXAMPLES. : ie © rt jet eae Whenw=-,wuisa = — 9 lta—2 ee minimum. x When « =1, wu isa max. 2. et 2° F A + & “« e=—Il, wis a min. 3. u=e*-+ 2cos.e¢+e—-* Whenx=0, u=—4, a max. 1 4, Tie A max. when 7 =e. 5. Divide the number a into two parts, such that the pro- duct of the m™ power of one part and the n™ power of the other part shall be a maximum. i) ma m+n mem +n a m+n their product, mn” ( ) when m+n mand nare evennumbers. The prod- The parts are Ans. uct may also have two mimimum states. 6. Find a number such, that, when divided by its Napierian logarithm, the quotient shall be a minimum. The function to be operated with is . Ans, © =e. MAXIMA AND MINIMA. 187 } n T. w=sin. x(1-+ cos. x). A max. when «= A x — , — A max. when x = cos. &. aay 9. Find the number of equal parts into which a given number a must be divided, that the continued product of these parts may be a maximum. Pe part must be e, the number of Ans. parts “ , and the product (e)¢. 10. Of all the triangles standing on a given base, and hav- ing equal perimeters, which has the greatest area? Denote the base by 6, and the perimeter by 2p, and one of the two unknown sides by «. ; (ein es 2p —b Ne triangle 2 is isosceles. 11. Of all the squares inscribed in a given square, which is the least ? Ans. That having the vertices of its angles at the middle of the sides of the given square. 12. Inscribe the greatest rectangle in a given semt-ellipse. Let the equation of the ellipse be ay? b2 a? — a? b?. Nie fee sides 00/2, Top and its area is ab. 13. Given the whole surface of a cylinder, required its form when its volume is a maximum. Represent the whole surface by 27a’. rani of the base Ln aXx1s ig Ans. : Me Ma volume — ae SECTION XII. EXPANSION OF FUNCTIONS OF TWO OR MORE INDEPENDENT YARI- ABLES, AND INVESTIGATION OF THE MAXIMA AND MINIMA OF SUCH FUNCTIONS. 121. Let it be required to develop, and arrange according to the ascending powers of 4 and k, the function F(a +h, y +h), when F(a, y), and all its partial derivatives up to those of the n™ order inclusive, are finite and continuous for all values of x and y included between the values w anda th, y and y +k; h and k themselves being finite. For the time, replace h and k by At and kt respectively ; so that, when it is desired, we may pass back to h and & by mak- ingt=1. Then F(a+h, y +k) becomes F(x + ht, y+ kt). — Now, by hypothesis, x, y, h, k, and ¢, are all independent of each other; and, considered with reference to ¢ alone, we may write Fla+ht,y +h) =/() (1), F(a, y) =/f(0) (2). For all values of ¢ between the limits ¢ = 0 and ¢=1, it is evident that /(¢) and its derivatives, up to those of the n™ or- der inclusive, satisfy the conditions above imposed on F(a, y) and its derivatives. Hence, for such values, we have, by Maclaurin’s Theorem, F(t) =/(9) +(0) 5 +770) Aen cea ALO) pay ti) pay 188 EXPANSION OF FUNCTIONS. 189 Deducing the values of /(0), f’(0), f”(0)...f(#), by the method pursued in Art. 110, except that now @ and y are not replaced by the particular values a and 8, and substituting them and that of f(t) in Eq. 8, we have | dF dk Fle + M, y+M) = Fley) +4 (Gh +G, 8) Y /@F 9 OF ae wae ag da a) i n/a kr” a ak d?f 2 2 3 cos (a 7 Bard: Bi, aa dady Gees ied =H) 4- oR a d if dF pn-1y dF to (F' 195 dx" ¢ rae ike Dias ao (4). The ‘notations « —x-+ 6ht, y=y-+ Okt, attached to the pa- renthesis of the last term, signify that in the derivatives Gen i d°k ye da™ du—"dy?"* dy”? x is replaced by «+ 6ht, and y by y + Okt. In (4), make ¢ = 1, and we have 4 dk dk Fethyth=Fyt+ abt 7k ar 1? d? iF aa a 3 (qr ee Boat ay? > k*) 1 (iF oP ar iF a eit DS 3 2 : a2 1 3 + rage +3 sage BE+ 8 Tal + Sok) a. ee, F OF, Bees Pee n h?- =) . + 1.2.. al et mn, ao" Tas dy" ; ae T ay si er) y=yt Ok which is the development sought. 190 DIFFERENTIAL CALCULUS. If, in Eq. 5, we make x= 0 and y= 0, and then in the re- sult write x for h, and y for k, we find F(a, y) = F(0,0) + Cre yi Gy )s i) Pare a*f | ar ae 1 9 2 T 19 ae de os (aedy) pi + (aa ‘)e ou y =0 pH!) ae . : OF: ae 1 n n—1 123 (as a" as ean Ya y= oy y=9Y af. 3 (ae eb ¥ =6Y which is the formula for the development of a function of two independent variables into a series arranged econ to the ascending powers of the variables. The extension of formulas (5) and (6) of this article to func- tions of more than two variables is easily made. For the expansion of M(a +h, y +k, 2+7...), we should find Fixth, ytkh, 2+7...)= F(a, y, 2...), ar, , oF dni h®— 1k : Se aah oP east (1); EXPANSION OF- FUNCTIONS. 191 and if, in this, we first make a, y,z..., severally equal to zero, and then in the result write 2, y,2..., for h,k,7..., respec- tively, we have dF dk ak Flay...) = Fle yt. +(e 2 +(G (Glee 0 1 ar ; ar \ a2 FF: : eet Gert »(@F y (aay, rt d 1! Een of LN, eg RE (=) +(a5 saat he grt 8 e oe eas Yate ts Busy, ( )3 y=0Y z=62 a formula for the development of a function of any number of independent variables, and in which the notation (_), signifies that the variables. entering the expression within the paren- theses are made zero. In formulas from (5) to (8) inclusive, of this article, the last terms are remainders expressing the dif ference between the value of the sum of the preceding terms of the development and the value of the function. When the form of the function under consideration, and the values at- tributed to the variables, are such, that, as m increases without 192 DIFFERENTIAL CALCULUS. limit, the remainder decreases without limit, then, by taking n sufficiently great, the remainders may be neglected. 122, Maxima and minima of functions of two or more ip- dependent variables. A function F(z, y, 2...) of several independent variables is @ maximum, when, being real, it is, for certain values of the variables, greater than J’(a+h, y +h, 2+17...); the increments h, k,i..., being very small, and taken with all pos- sible combinations of signs. On the contrary, the function is a minimum, when, under the same conditions, it is less than Pia th, ythk,2z+7...). Let us consider first the func- tion J"(x, y) of the two independent variables x and y, and endeavor to deduce from the conditions of these definitions, the criteria of a maximum or minimum of this function. tesuming Hq. 5, Art. 121, stopping in the second member at those terms which involve the third order of the partial deriv- atives of the function, and transposing f(a, y) to the first member, we have dF dF Fieth ytk)— F(a, y)= RECURS : ae ee hi avd ke? roa? 1 andy aye WY fo Ba pend d*k dr (1), j 3 Dee By ad Hore: Sane ey Rei dy mM ae 7 met the last term of which we fe denote by &. Now, if F(a, y) is a maximum, the first member of (1) is negative ; and therefore its second member must be negative also, and this whether hf and & be both positive or both nega- tive, or either one be positive and the other negative; and whatever be the values of f and k, provided only that they be very small. If F(a, y) is a minimum, the second member of MAXIMA AND MINIMA. 193 (1) must be positive under the same conditions and limitations in respect to the signs and values of h and k. But, when f and & are taken sufficiently small, the sign of the pie member of (1) will be the same as that of Be Be tei h which must therefore be permanent and nega- tive if G y) is @ maximum, and permanent and positive if F(x,y)isaminimum. It is plain, however, that the sign of oi a aif will change by changing the signs of f and k. To make ne sign of the second member of (1) invariable, whether positive or oe we must have say me A rae AE and, since / and & are entirely independent of each other, this requires that dF dF oe = 0, and iam (2). Let x=a, y= 6, be values of x and y derived from these dad’ dF dik. dx*’ dxdy’ dy?’ respectively become when these values of « and y are sub- stituted in (1); then (1) becomes equations, and denote by 4, B, C, 2,, what F(a+h, b+kh)—F(a,b) = a5 (Ah? + 2B + Ok?) + R, (8). When the values of # and & are very small, and only such values are admissible, the sign of the second member of (3) will be the same as that of Ah* +- 2Bhk + Ch?, which may be put under the form Bh Ais (+2 at 4) 25 194 DIFFERENTIAL CALCULUS. The sign of this will be invariable, and the same as that of A, if the roots of the equation h? Bh are imaginary ; : being treated as the unknown quantity. Solving this equation, we find h —BtvVvB— AC, ke A ‘ from which we conclude that the conditions for imaginary roots are, that 4 and C have the same sign, and that the prod- uct AC be greater than B’. In recapitulation, we say, that if «= a, ¥ = 6b, make F(a, y) a maximum then for these values of x and y we must have dk dF dx he dy =a ea ae both negative Magee Aah 8 , Cra a? k\? ee dx? dy? Ga) ; If c=a, y —), make F(x, y) a minimum, the conditions are 2 a hed MK oh The existence of real roots for Eq. 4 indicates that we may the same, except that then must both be positive. give such signs and values to f and & as to cause the expres- sion Ah? + 2Bhk 4- Ck* to vanish, and, in so doing, change its sign, which is incompatible with the existence of a maximum or a minimum state of f(a, y). 123. There remains to be examined the case in which AC— B*?=0. When this condition presents itself, there may also be a maximum or minimum value of the function. MAXIMA AND MINIMA. 195 By the theory of the composition of equations, or by in- spection, the expression Ah? + 2Bhk + Ck*® may be written Ie? h 2 alu 5 (45 +2) Je oN t aren ye hence, when AC — B? = 0, this becomes q(45 +B) , the h ; é sign of which, except when (4 7" —- B) vanishes, is the same as that of 4; and F(a, b) is a maximum if 4 is a negative, eta ~ h : and a minimtim if 4 is positive. Should (4; + 2) vanish, as it does when ae pps cannot tell, without further inquiry, that M(a+h,b+k) — F(a, b) does not undergo a change of sign. To decide this, let Z, M, N, P, represent the I P ove Hy Baek: ) will always be positive, and the sign of (b) will be the same as that of 4. Putting this quantity un- der the form At B— AQ’. BU AC (Av AR) G Rema, Fes Fy) we see, that, to make it negative for all values of ¢, it is neces- sary and sufficient to have A” — AB< 0, ie., AB—A”>0 (c). @4 8 = AC"? < (BY — AC) (A"* AB) (d). When conditions (¢) and (d) are satisfied by the values of x,y, #, deduced from Kqs. 2, (a, b,c) is a maximum or a mini- mum; for then the sign of expression (0), and consequently that of the second member of Eq. 3, is permanent, and the same as that of d. (a, b,c) is therefore a maximum, if .4 is negative; and a minimum, if A is positive. 200 DIFFERENTIAL CALCULUS. Hence the conditions necessary for the existence of a maxi- mum or minimum of F(a, y, z) are, that the values of a, y, z, derived from the equations dP, ak _ 9 CF _ 0, da ie ij rre, eae should make O73 2k dad? dx? dy? Se dy ) > 0, that is, positive (c’); and Pd Fore ieee dxdy dxdz da? ae ar a CF) (OCF Cr ir di) (Saas) ~ da* dz* ) (say) ~ dx? dy? er A necessary consequence of conditions (c’) and (d’) is, that Bd LO ae es oF GE |) fai SO: (ae dx? ie oe * da? da? —( Fa Cr PF ar and hence da? dy? di must all have the same sign, which is negative when F(a, y,%) is a maximum, and positive when F(a, y, 2) 1s a minimum. 176. If we have a function F(a, y, 2...) of m independent variables, the first condition for the existence of a maximum or minimum would be Git ak es eee |, whence, because h, k,i..., are pia of each other, dk A Bh on he a 0 ra age = e dx ’ dy ’ dz wee Kqs. 1 determine values x = a, y =b, 2 =c..., which may or may not produce a maximum or minimum state of the func- MAXIMA AND MINIMA. 201 tion. To decide this question, we should have to examine the term 1/@F_, @F., @PF dF | Sy ee a aay SIE i dare seks mites ne era Tapa, eG ) in the expression for F(athytk, z+...) —F(a, y,2...). If, for these values of x, y, z..., the sign of this term is per- manent, and negative for all admissible values, and all the com- binations of the signs of h, k,7..., the function is a maximum: if the sign is permanent and positive, the function is a mini- mum. It would be found, that, to insure either of these states meee. Oh dei of the function, aie dak sign, negative for a maximum, positive fora minimum. But -++, must all have the same the investigation of all the conditions to be satisfied in this general case, in order that the function may be a maximum or minimum, is too complicated to find a place in an elementary work. 127. Maxima and minima of a function of several variables some of which are dependent on the others. Let it be required to find the maxima and minima values of the function u= F(x, y, 2...) of the m variables a, y, 2..., which are connected by the n equations | i EQUAL ee at Bete Wen Fl; Y; 2.) —0 | By eliminating from uw, n of the m variables, by means of the n given equations, «would become an explicit function of m—n independent variables, and its maximum or minimum could 26 202 DIFFERENTIAL CALCULUS. then be found by the method just explained; but this elimi- nation may be avoided, and the determination of the maxima and minima of w greatly simplified, by the process which fol- lows : — Suppose the variables 2, y, z..., to receive the respective increments h, k,7..., by virtue oft which the function passes from a given state of value to another in the immediate vicinity of this: then, if the given state be either a maximum or a minimum, we must have LE dF , Ba): yf a ee dz v fe MEAN or ( ). The partial derivatives of the first members of each of Kas. 1, taken with respect to each of the variables, are separate- ly equal to zero. Taking these equations in succession, multi- plying each partial derivative by the increment of that varia- ble to which the derivative relates, and placing the sum of the results equal to zero, we have apie Aare ee Ving Hep His 20 ny if, dfy . *. ip a i a te ME es? iy Ae ian 4 There an a ea (nx) of Eqs. 3, these with (2) make n + 1 equations of the first degree in respect to the m quantities h,k,v...: hence, by the combination of these equations, we may eliminate n of these quantities, and arrive at a final equa- tion involving the remaining m — n quantities, and also of the first degree with respect to them. To facilitate this elimina- tion, let My, Hy...,4,, denote undetermined quantities; and MAXIMA AND MINIMA. 203 multiply the first of Eqs. 3 by »,, the second by p,..., the nm by u,; add the results to (2), and arrange with reference toh, k,1...: we thus get df dfs Ta 7, ate By at a a rel tela ead a a df, df, df, +(a, +8 oa gp tir ol aols ze | dF df, df, eG fy tb Ho Foe a 2 + ° . . e a true equation when F(x, y, 2...) is susceptible of a maxi- 0 (4); mum or minimum, whatever be the values of p,, M2. ..-, Ma: Place the co-efficients of n of the quantities h, k, 7..., in Hg. 4, equal to zero: the m equations thus obtained will deter- mine ft), /)...,“,. By substituting these values of (1), fg.-+) Mn; in (4), 2 of the quantities h,k,7..., will vanish from that equa- tion; and, if the co-efficients of the m—n of these quantities remaining in the equation be placed equal to zero, we have, including the n given equations, Re a, ital irae te eh oy Uf 8s) 0, m equations from which to determine values a, b, c..., for the m quantities x, y,z..., respectively. This is equivalent to equating the co-efficient of each of the m quantities h, k,7..., in (4), to zero; and these m equations, together with the n given equations, will make m+n equations, by means of which we may eliminate the n indeterminates py, ,..., M,, and find the m quantities x, y, z It remains to be ascertained whether the sign of the expres- sion for (a+h,b+kh,c+7...)—F(a,b,c...) is invariable; and, if so, whether it be positive, which answers to a minimum; 204 DIFFERENTIAL. CALCULUS. or negative, which answers to a maximum. Theoretically, this examination is very complicated; but, for most cases in which this method is applicable, the form of the function en- ables us to decide at once.which of the two states, if either, the function admits. When m — n=1, or there is only one more variable than there are equations connecting them, the case discussed in this article reduces to that of an expression which is implicitly a function of a single variable. 128. In the case in which it is required to determine the maxima or minima of a function, the several variables of which are connected by but one equation, the process may be still further simplified. Let u= F(x, y,2...) be the function, and J(&, Y, %---)=0 (1), the equation expressing the relation between the variables x, y,%...: then, by the reasoning employed in the last article, we have eht+ 5, bp Se 9 ay e hae t, ke t, i+. (3). Multiplying a by Re eee vu, subtracting the result from (2), and arranging with reference to h, k,7..., we have a | hea aE a: dB) ape Ae Kquating to zero the co-efficients of the several quantities h, k, v..., we should have, with the given equation, m +1 equa- tions, by means of which we can eliminate », and determine the m quantities. But from the co-efficients of h,k,7..., in (4), placed equal to zero, we find EXAMPLES OF EXPANSION. 205 dF dF dF eS _ dy: anes Ga a aie da: dy dz that is, the ratio of the co-efficients of h, in Eqs. 2 and 3, is the same as that of the co-efficients of k, of 7... . This rela- tion will be found to facilitate the determination of maxima and minima. : The examples which follow are arranged in the order of the articles in this section under which they fall. EXAMPLES. 1. If F(a, y) = «?(a+y)’, find the expansion of | (w@thy(aty +k} in the ascending powers of / and k. (2?(a-+-y)'+20(a-+y)h-+ 30%(a-ty)%h +(aty)h-+6e(a-+y)%hle-+3a%(a-+y)e? +3(a+y)?h?k + 6a(a+ y)hk? + x?k? + 3(a + y)h?h? + 2xhk® + h?k?. 2. If F(a, y, 2) = ax® + by? + cz? + Lexy + 2guz + 2 fy2, find the expansion of F(a+h,y+hk,z-+7). an? + by? + ca* + Lexy + 2gxz + 2fyz + 2(aa + gz + ey )h+ 2(by + fa + ex)ke + 2(cz+fy + ga)i+ (ah? + 0k? + ci?) ee 2(fki + ghi + ehk). (wh) (aty+k)= Pathy+h2to= 3. Expand 2 (atarg tari: )(bo tot +? AEE -) in the ascending powers of x and y. 206 DIFFERENTIAL CALCULUS. a ee 2 (a+ a, i tars + \(b4 64 +b + +) =A by + box + aydyy 1 de 1.2 (A,b)x? + 2a,b, xy + a,b,y”) ‘| + 1.2.3 (a,b) x° + 3a,b,x°y + 3a,b, xy? + aobzy’) foe 4, Find what values, if any, of w and y, will render the func- tion F(x, y) = wy + xy* — axy a maximum or minimum. From the equations ae carl WY a = 0, we get four systems dy of values, viz. 31) mayb emer 1 i | a | 3 ) ? ) } y=0 Hot a yas 3 none of the first three of which satisfy the condition ee 2 oC ee ae males oi tyare ) (Art. 123), dady and must therefore be rejected. The fourth system reduces Be se F cok Poe ae tite q he is inequality to 9° ny which is true, and at the same 74 2 2 : time makes both Es and ee positive: hence the values on dy? z=%, y =o make the function a minimum, and this mini- a’ mum 1s — 57° 5. Determine the values of « and y that will make Eo y) — en F(a Boa a maximum or minimum. EXAMPLES.— MAXIMA AND MINIMA. 207 bam |) dF dy dx oo -{) fl emed ea—-t1 at ae} a0 b which we will examine in succession. Pr ar a The first system gives Tx? Say Hiatal dears hence the values.« = 0, y = 0, make the function a minimum. = 0, give the three systems of values With the second system, we have al eT SI fee Ls ae dx? = 2(a — be", as — — 4be7!, ee hence the existence of a maximum or minimum depends on the ; i at relative values of a and 6b. Ifbis greater than a, —,, =~’ ain aia have the same sign, which is negative, and the function is a max- 2 2 imum; but, if b be less thana, ee at have opposite signs, and the second system of values of x and y make the function neither a maximum nor a minimum. For the third system, we find 2 2 2 = — 4ae, at = 2(b — a)e™", ia Sa fe from which we conclude that = +1, y =0, will make the ' function a maximum when a>b; but, when a< J, it has neither a maximum nor a minimum. 6. The equations of two planes, referred to rectangular co- ordinate axes, are Ji(%, Y, %) = Av + By + C2 — D=0 (1), PA, Y, 2) = Ale + Bly +C'e—D'=0 (2). It is required to find the shortest distance from the origin of co-ordinates to the line of intersection of the planes. Let F(a,y, 2) =a? fy? +22 (3) represent the square of the distance from the origin to the 2.08 DIFFERENTIAL CALCULUS. point of which 2, y, z, are the co-ordinates: then, if a, y, 2, are the same in the three Eqs. 1, 2, 3, the concrete question is reduced to the abstract one of finding the values of a, y, 2; which, when subject to the conditions of Eqs. 2 and 3, will render f(x, y, %) @ minimum. | By Art. 128, we have 20 + p,A + p,A’=0 2y +4,B+¢,B/=0 ¢ (4). 22 + yO + pC’ = 0 Multiplying the first of Eqs. 4 by A, the second by B, and the third by C, adding the results, then, by (1), we have (A?-+B? 4 Ou, + (AA + BB+ OO), +2D=0 (5). In like manner, (A + BV+ CO”), + (AA + BB’ 4+ CO) pu, 4+ 2D'=0 (6). From Eqs. 5 and 6, we get the values of »,, u,; and Eqs. 4, when these values of ,, u., are substituted in them, will de- termine x, y,z. Multiplying the first of (4) by a, the second by y, and the third by z, adding results, and reducing by Eqs. 1, 2, and 3, we have 2F (a, y, 2) + Dey +D/p,= 0; from which we get F(x, y, 2). In this case, it is unnecessary . to examine the sign of F(x-+h, y+hk, c+1) — F(a, y; 2), when the values of a, y, z, are substituted; for we know from the conditions of the geometrical question that the function has a minimum. 7. Required the values of a, y, 2; that will render the func- tion Ue — aes Fee a maximum, the variables being subject to the condition yv=ax+byt+ecz—k=—0. EXAMPLES.—MAXIMA AND MINIMA. 209 We find Ci BAF os | fy isd) Ute wae cas! taal y ra U, dy y dz” 2 ’ dv dv 7 i ; ome dy, de therefore (Art. 128) VA) 8 GINS al aby Cz I ; p k q k r k Se - nae Oe ce aera LA oo patacreman per prea peer 9 bp +g+r Ra rN! fn ef These values of x, y, z, make the function a maximum. For we find en de ct” dy? y dy ye dz? du __ Clase Goad. . Oe GL os au redu ee : i a dd | a8 and these, because ‘= 0, for the values of dx ’ dy dz x, ¥, %, become ere i AO Qe int r 0 de tm aa all of which are negative, —a necessary condition for a maxi- du du dey mum; and, by getting the partial derivatives diedly dadz’ dydz we see that the other conditions (Art. 126) to insure this state of the function are also satisfied. By making a= 1, b=1, c=1, the above becomes the solu- tion of the problem for dividing the number k into three such parts, that the product of the p power of the first, the g power of the second, and the r power of the third, shall be a maxi- mum. 8. Inscribe in a sphere the greatest parallelopipedon. If a be the radius of the a the parallelo- Ans. ; pipedon is a cube having —; ai ~ for its edge. 27 210 DIFFERENTIAL CALCULUS. 9. Determine a point within a triangle, from which, if lines be drawn to the vertices of the angles, the sum of their squares shall be a maximum. The point is the intersection of the lines Ans. 4 drawn from the vertices of the angles to l the centres of the opposite sides. A function F(a, y,...) of two or more variables may be of such form, that it admits of a maximum or minimum for values dF df of the variables which make —_, ——,... indeterminate or in- dx’ dy finite. There are no general rules applicable to such cases; but each one must be specially examined. 10. What values of x and y will make wu = ax? + (x? + dby?)8 a minimum ? LE Sy yall 2 : du _ 2 y dx 3 (a? + by*)8 1) (a? + by?)8 For «= 0, y = 0, these differential co-efficients take the form 3 but their true values are infinity ; for, if we make y = mz, they become eae dx 1 du 2 m af (1 + m2b)® dy 3 a8 (1 + m2b)3- Hence for « —0, and therefore for y= 0, at the same time, =2an +5 we have du Te O, dy =—=,00); For «= 0, y = 0, we have uw = 0; and no real values of # and y can make w negative. Hence w is a minimum for «=0, Uieoe Us SECTION XII. CHANGE OF INDEPENDENT VARIABLES IN DIFFERENTIATION. 129. It is often required in investigations to change dif- ferential expressions, obtained under the supposition that cer- tain variables were independent, into their equivalents when such variables are themselves functions of others. Suppose that, having given y=/(z), c= q/(z), it is required to express the successive derivatives of y, taken as a function x, in terms of those of w and y taken with respect to z. We have found (Art. 42) dy dy dz Gai dz ds’ and (Art. 41) dy Cau lor ty. dy dz JOD EER GG We tase dz dz dy dy gy ad’ da d dz dz eecS dat = da de — da du dz ‘AT *) dz dz d*ydx dx dy dz? dz daz? dz dz = a ae . ao d iy dx da dy _@da dedi da 1 0 5a de dee: (e) i 211 yl DIFFERENTIAL CALCULUS. So also dy dx d’x dy ey) dda? de og) oe dx? dx da\* =) d’ydx d’a dy _ addz? dz dz? dz dz rae. Vda ®t TS de (i) Ge dx da a) (=)- a(T) 7 (at da d? x a dz? dz dz* dz/\dz dz / dz* \dz2 dz dz* dz/dz 7. ap. uy. fda\®, 0b): cn a, : (a axes O°2 ) a wa (ae de .d*m Z) dz? de dz> dz/dz “dz? \dz* de daz? daz -, dax\* ‘ dz 4, 5 In the same manner, we may find es , ee .. Substitut- nay Lata dz’ dz’ dz? ing in these the values of -++, found from y =f(2%), c= 9(2), we have the values of the successive derivatives of y with respect to a, in terms of those of « and y with respect to %. 130. Having y =/(x), to change the independent variable dy d*y from x to y in the expressions for — daa! jn : ee ees eee CHANGE OF INDEPENDENT VARIABLE. 213 1 Since aa da (Art. 41), Meh dydads vir dy dy d*x da mo ey dy et Jay? Ae nee dy a) Similarly, d*a ete dey Hi dy a a ae raf 3 (= a\? dy? 7 ae (es ) hee: ul ie dx iy) d®a da d?a\? — — dy* dy a) 1 ‘ ay a4 In like manner, we may find the expressions for dpe? a3 a These formulas may also be found from those in the preceding article, by making z = y; whence dy _ ae a 6 diy _ 4 ea Ce Ocoee Dieter’ C7 mer a2" ATS: EPR ae ci dake dy By the introduction of these values in the formulas of Art. 129, they will be found to agree with those just established. 131. Having given iar Oy GL), and also eecus. 0, 77 sin.§ (2), 214 DIFFERENTIAL CALCULUS. it is evident that we may eliminate x and y from these equa- tions, and get a direct relation between r and 6; and thus r becomes a function of 6. dy a? : It is required to express the values of ae a" -» derived Ce ae: from Kq.-1, in terms of —- SPU By Arts. 41, 42, we have dy dy dydo dy1 _ do da di dx dé'da™ (da do dé sin. 6 ia + 7 cos. 6 = 8 from Eqs. 2 cos.6 ie 7” sin. 0 also ; sin. 05 abr COs sin poo dy da a TES ole "dé dx? da +e on dx cos. 0. — resin. 0 . cos. 7, — r sin. 0 Performing the indicated differentiation, we find for the nu- merator of the result RTA IP Meade ie ot eee sin. AT acm COs. Pee: sin. Ji eos 35 ee ) d*y Py woe dr —( cos. 6 aan eater me PE sin. 0 Tse : dr d?r hich cme § which reduces to p29 (a) - 7 Jp?! ‘ . ae dr and the denominator, remembering that — —cos.6é— —rsin.6, dé do ae ee ee COS. ee fe ASB . do is CHANGE OF INDEPENDENT VARIABLE. y Mies ; dr\? d?*r eds 42(2) ml Se Ey" da? ew, Ae dr ; : 3 . a) These formulas are used in the applications of the differen- Hence tial calculus to geometry, where a change of reference is made from rectilinear to polar co-ordinates. 132. Suppose that we have the expressions for aie ae dic - dy found from the equation vu = F(x, y); but that the variables «and y are connected with two other variables, 7 and 6, by the equations x = f(r, 6), y= F,(r, 6): then we may conceive x and y to be eliminated from these three equations, and w to be a function of r and 6. Required the equivalents of the expressions for na oe in terms of the derivatives of das ody | x, y, and w, with respect to r and 6. By Art. 82, we have du __dudx , du dy ) dr ~ dx dr ' dy dr | (1); du _dudx _ du dy idee a dy do. and from these two equations the values of an pes can be | Cone a1) found. When the equations expressing the relations between Ueeemoware 7 — I(x, 7), 0 = Fi (x, y);, mstead of those given above, then du _dudr , du da ) dz dr dz + di dx | du _ du dr, dud | dy dr dy ' dé dy | (2). 216 DIFFERENTIAL CALCULUS. If the variables x, y,7, 6, are connected by the unresolved equations F\(z, y, r,6)=0, F(x, y, r, 6) = 90, we proceed thus : — By Art. 82, al cca Oe diy dy _ db} de (de dy dia dF, dl, ds diy dy am Gokia da dy ao} in which it must be remembered that (a) (=) are par- tial derivatives of /,, #,, with respect to 6. Differentiating /,, F,, with respect to 7, we get two similar equations involving oe ; and the four equations thus ob- . » 04 dy “dawrdy mene | tained will determine a?) Hat ieee hich must be sub- stituted in formulas (1) or (2), Art. 131. The following example will illustrate the manner of using the above formulas : — Given u=/(x, y), «= 7 cos. 0, y= 7 sin: 0, 1b Ieee du du du -— du to express in terms of 7, 0, a Mey’ dr? das We have dx dy Ap 6, q9 == 7 COBO; dx dy a i] aw POs cp cos.4, ae sin. 0 Hence, by formulas (1), du ; A ee ee + sin. 6 = CHANGE OF INDEPENDENT VARIABLE. 217 du Cine roe du | whence Tp = 008 OG — Bin. 05, | r (a). rey jes Ue ee | ere digeany d0) To make formulas (2) applicable to this example, we first deduce, from the equations x = r cos. 6, y =7 sin. 6, the values ef r and @ in terms of wand y. We find — Jar + y?, 6.== tan.>} e whence dr_«a dr_y ddO_ _y @_&, dx” r’ dy. r’ der?’ dy r?- and, by means of these, Eqs. 2 become du _ «du y du dze™ r dr’. r* do du _ydu, « du dy rdr' r® do . . e Hb; aoerelations. « — 7 cos. 0, y = 7-sin. 0, give cos. 0=-, A (b). sin. 0=% , by means of which we can pass from formulas (0) to (a), or the opposite. 133. Attention is here called to the necessity of attaching their precise signification to the symbols pemarivds dy i dd} dio du dy dz’ dy’ dr’ dr’ dx’ dy’ do’ do’ which occur in formulas (1) and (2), Art. 131. It must be borne in mind that these denote partial differ- ential co-efficients, and that those referring to the same varia- ai da bles, such as —_, , have not to each other the relation of dx dr to a which are derived from the equation f(x, y) = 0. With reference to these last, we know that one is the recipro- 28 ca 218 DIFFERENTIAL CALCULUS. cal of the other, or that their product is 1; but this is not true for ue x ae . The consideration of the meaning of the dr term “ differential co-efficient,” and the difference between the equations connecting the variables in the two cases, will re- move all difficulty. In getting formulas (1), x and y were given as explicit functions of the independent variables r and 6; and a change in either r or @ will produce changes in both # and y. Hence, in the operation of finding oe r, c, and y vary, while 0 remains constant. In formulas (2), 7 and @ were given as explicit functions of « and y; and a change in the value of either « or y will produce changes in both v and 9; and hence Port! the increment attributed to « in getting “” causes 7 and 0 dx also to vary, while y remains constant. ‘That is, in formulas | fan, zs supposes 7, x, and y to vary together, 0 being constant; while, in formulas (2), - supposes a, 7, and @ to vary, while y remains constant. Thus it appears. that these two partial de- rivatives are obtained on different suppositions in respect to the variables which receive increments, and those which remain constant. In the example just given for formulas (1), we have = = cos.0; and, for formulas (2), i. = cos. 6; and the product Th a inate fae 154, Having u= F(a, y, 2), and three equations express- ing the relations between a, y, z, and three other variables du du, du-:: dz’ dis da?” terms of the different co-efiicients of w with respect to 7, 0, w. vr, 0, w, it is required to find the values of CHANGE OF INDEPENDENT VARIABLE. 219 By Art. 82, du __duwda , du dy du dr dx dd dx 2 dw dec ' dr dx du dudod .dudw . dudr Aiiee 1): Se deay » dw dy © dr dy (1) du _dudd, dudw , du dr da @ dat dy dat dr a The three equations connecting a, y, z, 7, 0, w, will enable ag do) - da dr dw ee. dy’ Hae yen) Wee and Eqs. 1, when these values are substituted in them, give us the expressions us to determine sought. du du du da’ dr’ dw’ du du du expressed in terms of eas zi OF We may find these By solving Eqs. 1, we can also find the values of values from the equations Gia duds Jjduidy > du. dz do dx do dy do ' dz do du. dudx dudy du dz ee ieee Midi si deductions”: gp du de du dy du dz | dr daz dr" dy dr‘ dz dr 135. Let the relations between the variables a, y, z, 6, w, 7, be , een. OCOs., Y—rsin. Osin. yw, 2=rcos.d (1’). From these we find ee r cos. 8 cos dy r cos. 6 sin dz r sin. 0 Eat wie ul Ln = ° e ) - = cas | oe do Y Y a9 peed da : ‘ d : dz dw = — rsin.@ sin. yw, oe =r sin. 0 cos. w, Gib aU, dx 3 d , . dz ae = sin. 0 cos. y, oe = sin. 7 sin. y, mE ee COS eda IP() DIFFERENTIAL CALCULUS. and formulas (2), Art. 134, by the substitution of these values, become oS = 7 cos.6 cos. = + r7cos.6 sin. w a = sind ] “ = —rsin.ésin.w os +7 sin. 6 cos. p is (a). ae = sin.@ cos. wy ue + sin. sin. w 7 + cos. 6 = From Eqs. a maybe found the required values of se i ey, du du du do’ dw’ dr° Again: squaring Kqs. 1’, adding results, and taking square in terms of root, we have r= /(x*-+ y*+ 27). Adding the squares of first and second of these equations, we find r? sin’6=a2*+y’; whence 7 sin. 6 = 4/(x? + y’), sin. 6 =i V(e+y"): and from this, and the last of Eqs. 1’, we find n/ (x? + y*) 2 1 972 ony pias 6 tan NY v4 & Dividing the second of Eqs. 1’ by the first, we have GAT. 4) oe a w= tan. Hence we have Apt nee r= a/(x? + y? + 27), 6 = tan! Vee ye tan 18 (2/). From those by differentiation, we have Ae ee dr : drt ine io oe 6 cos. W, ae a = sin. 0 Sinia 7s Ga cos. 4, dl z ee hy cos. 8 cos. w eres Ere re y] ye A /c8) 1 97? r d9 z y __ cos. # sin. w y- PLP pe Very? CHANGE OF INDEPENDENT VARIABLE. 220 Coe 4/(e? + y*) sin. 8 dz wt ttt gt oe ay Cl dye cos.w dw 7, de — wtty? resin. dy a+y? rsin.0’ dz — By the substitution of these values in formulas (1), Art. 134, . we have du cos.dcos.w du sin. w du . du dz r dé rsin.é6 dw ye i dus cos.@sin.w du — cos.w du : du a 2 fli UB Le bad ee yee b). dy r da + r sin. 0 dy Bart Oe dr ) du —s sin. 0 du du dz re da Rin dr dl The values bids’ to agree with those given directly by formulas (0). given by formulas (a), will be found EXAMPLES. 1. Transform area (ayN Ay) eae + (Ge) — Som ho) into its equivalent when neither x nor y is independent, but both are functions of a third variable z. Substitute for ote and dy their values given in Art. 129 das” dx ; and we have dy dx dx dy : dy dz? dz ee og is Leet ALANA Pom A mee da? ie) acs and, multiplying through by ( | pe kad, Cae eee. dy zs dx 0 32 ” de? da da® da +(4) -5 i(Z)= ee 222, DIFFERENTIAL CALCULUS. If we make «=a, this reduces to the given equation. Making y = 2, (2) becomes eae 2/ Oa Soa —1=0 (8). dy Hg) . Equation (3) is the equivalent of (1) when the independent variable is changed from @ to y. 2. Change 3 a* (ae) eae Xx into its equivalent when both w and y are functions of a third variable z. c(ay\? | (de\" 4, (Py de _ dew dy eae) dz + dz “13 dz? dz \, da? da wie If y = 2, the above becomes sy a 2 ee ee ne } dy in which y is the independent variable. 3. Eliminate « between bay DO = + +y=0,anda2?=46 (1), and find what the differential equation is when @ is the inde- pendent variable, and also when y is the independent variable. First suppose both w and y to be functions of a third varia- ble, 2; then the differential equation becomes (Art. 129) ay dx d*x dr 1 ch da dx 8 ite u(g) ta(q) =9 @ dz? dz dz? dz ' « dz \dz dx dx d6. dy bea da do da Gp ee dix _4 46 d?x 1 d LIN _1d*6 04 (Z) ee iy det od hh ° 92 a From 2? = 46, we have «= 26? CHANGE OF INDEPENDENT VARIABLE. yap hs: ; a oe by their values, we have _1 46 d*y 1 4*6 dy dé? dy da Meade. ded!” tat ao iG) =9 (9); which does not contain w Making y =z, (3) becomes d?0 do\? da\3 eG.) -9(3) pet and if, instead, 0 =z, we have In (2) replacing x dy mtg ty=0 4. Given the relation «=e, to change the independent variable, in the differential expression x” | By Art. 42, we have een Of a" y\ da (| jdty MAY, z(? —ieraG a el Gale a dA” at from « to s. — n ood ir Ge tte ey hes dic” Bis dart} ad d"y dy Ge Pay See | ee n — grr . Al am 0 a” dig +1 or, writing the first member in an abbreviated form, ad A/G Me BRE ee ae ae Making » = 1, this gives From «= e', we get - —=e'=2; also we have dy dy dx a dy . Hagieada Us 9). dx hence (2) becomes 224 DIFFERENTIAL CALCULUS. When n = 2 in formula (1), then and, putting in this the value of x? oe from (3), d*y d d dy Oe on 1 ee -— i. ieee dic? G i) G YS The law governing the construction of these equations is ob- vious; and we may write, generally, ah d d d d d. The meaning of the operations denoted in the second member d d of formula (4) is, that if the expressions ae v ah be combined by the rules for multiplication, the result will dy represent, in terms of indicated differentiations on ds? the value Daas ) d”y n of aw oe . 5. If we have p= — — , and the relations 2—=rcos.6, y—rsin.6, find the equivalent for p when a change of independent variable is made from «& to 6, and also from x to 7. When @ is the independent variable, CHANGE OF INDEPENDENT VARIABLE. 225 and, when 7 is independent, ech d’6 aS) | | Q S Ss 29 SECTION XIV. ELIMINATION OF CONSTANTS AND ARBITRARY FUNCTIONS BY DIFFERENTIATION. 136. WueEn an equation is given in the form F(x, y)=c (4), the constant c will disappear on the first differentiation, and the successive differential equations derived from (1) will be identical with those derived from F(a,y)=0 (2). Though an equation may not be given under the form of (1), it often happens that one or more of its constants may be made to disappear by successive differentiation alone. Let (y—b)'+(w—a?P—r?=0 (8), and differentiate this equation twice, taking x as the independ- ent variable. We find (yb) +e-a=0 (4), yn T+) +1=0 (6); and thus the two constants a and 7 of (3) have vanished in the two differentiations which lead to (5). A third differen- tiation gives dty , ody dey _ Y= Nima + > ae gat = 9 A 226 ELIMINATION OF CONSTANTS AND FUNCTIONS. 227 From Kqs. 3, 4, 5, and 6, we get eve holy. (ela (Oy) Go bay dl ba) Oa dy d*y dy dx dx" _8r%(a—a). ee Lee, Sep ote Ss and, by eliminating y — b between (5) and (6), we get dy\? dy dy /d?y ae re) +1} 55 95) Tene Between (3) and (4), we may eliminate any one of three con- stants a, b, r; and, by taking these constants in succession, we should have for our results three differential equations of the first order, each containing two of the constants. By a proper combination of (3), (4), and (5), we can arrive at two differen- tial equations of the second order, each containing but one of the constants of the primitive equation; and between (3), (4), (5), and (6), we can eliminate all three of the constants, ob- taining for the result a single differential equation of the third order. It thus appears, that, by differentiation and elimination, Hq. 3 will give rise, 1st, To three differential equations of ‘the first order, each involving two of the constants a,b,r; 2d, To two differential equations of the second order, each involving but one of these constants; 3d, To one differential equation of the third order, from which all of the constants have vanished. By means of Eqs. 3; 4, 5, the values of a, b, r, may be ex- pressed in terms of x, y, and the derivatives of y of the first and second orders. Denoting these derivatives by y’, y’”, we find +1 ACESS PONCHO OF 228 DIFFERENTIAL CALCULUS. 137. In general, if we have an equation between a and y, and n arbitrary constants, and we differentiate this equation m times successively, we shall have, with the primitive equa- tion, m+ 1 equations, between which we can eliminate m constants. This will lead to a differential equation of the m™ order, in which there will be but n — m™ of the constants; and, as the constants eliminated may be selected at pleasure, it is evident that as many equations of the order m may be formed, each containing »—m constants, as we can form combinations of n things taken m in a set, which is expressed by n(n—1)(n— 2)...(n—m+1) OES Ar ] When the original equation is differentiated n times, we should have altogether n + 1 equations, between which the n constants can be eliminated; and, as the resulting equation would involve the n™ differential co-efficient of y taken with respect to x, it is said to be of the n™ order. The order of the highest differential co-efficient entering any of the equa- tions at which we arrive,by the steps above indicated, deter- mines the order of the differential equation. It is worthy of remark, that if any one of the differential equations of the m™ order, obtained by eliminating between the first m derived equations, and the primitive equation, m of the constants entering the latter, be differentiated n — m times in succession, then this equation of the m™ order, and its nm —m derived equations, would enable us to eliminate the re- maining constants; and the final equation at which we should arrive would be the same as that obtained by effecting the elimination between the primitive equation and its n succes- sive derived equations. ELIMINATION OF CONSTANTS AND FUNCTIONS. 229 We et sa To illustrate, take the equation a = x — Fae and differ- entiate with respect to x. We should find, after reduction, yl" (yl +1) —8y'y” = 0, which agrees with Eq. 7. The theory of the elimination of constants by differentiation is sufficiently simple, and needs but little explanation. It is referred to here for the reason that a knowledge of the forma- tion of differential equations assists in understanding the more difficult and highly important operation of passing back from such equations to those from which it may be presumed that they have been derived. 138. Functions known and arbitrary may also be elimi- nated by differentiation. Let y=asin.2; then VT Bs RA So ANY par Sere. In dy? - ° . a 2 me oe a a 0 ° an equation which no longer contains the known function sin. &. Again: suppose s=9(0) in which w and y are independ- ent, and g denotes a function of the ratio of these variables, the form of which is not given, and is therefore called an arbi- trary function. a PS SU OL’ salen. eee HP Ong HPO, (¢), diz dt x ea oy! ES Ee ee a a0 Ogg =~ 9 O 230 DIFFERENTIAL CALCULUS. This last equation is true, whatever may be the form of the x : x Ret function oy denoted by ; it may be z =1(5), 2 — Sie ae or z—=e,: and for each of these cases the differential equation subsists. Take the more general case, w = » (v), in which wand v are known functions of the independent variables x and y, and of the dependent variable z, and g(v) an arbitrary function of v. Differentiating «= g(v) first with respect to # and z, and then dz Rae with respect to y and z, and, for brevity, making - eh eH) Ege we shall have du du ; dv dv hk Pig ee © +7) du iy eee a dv dv yaaa ta cae Dividing these equations member by meniber, we have du du dv dv det? da dat? a; du du — dv dv dy srl dz dy + as Clearing of fractions, and making Pp dudv du dv dudv dudv , dudv dudy ” dy da dz dy “a de de de! a we find that the partial differential co-efficients of the first order are connected by the equation Ppt Gea and this equation is in no wise dependent upon the form of the function characterized by ; in other words, this function has been eliminated. 139. Suppose c and c, to be two known functions of a, y, Z, ELIMINATION OF CONSTANTS AND FUNCTIONS. 231 expressed by c=/(a, y, %), . =/i(%, y, 2); and that, in the equation F (x,y, 2, 9(e), 9(%1)) =9 (1), 9, 1, denote arbitrary functions. Let us see if it be possible to pass from (1) to a differential equation which shall not con- tain g(c), g,(¢,), or their derivatives. The equations df di Ee es a 2 Fp ay 8 ash ba (i hel buat Ta? cE ae yar (3), that we get by differentiating (1), will contain the unknown functions g’ (c), 9:/(¢,), 9” (¢), gy’ (e1), which, with g(c), o,(¢;), make six quantities to be eliminated between Eq. 1 and the five equations of groups (2) and (3), which are generally in- sufficient. Passing to the equations Cr a ad? a | Gia deidy "day yt SD we introduce two additional arbitrary functions g’”(c), g/(¢), and only these two. We shall now have ten equations, viz. Hq. 1, and those of groups (2), (3), (4), and but eight arbitrary functions to eliminate: hence the elimination can be effected, and we may have two resulting differential equations of the third order. We have said, that, in the case supposed above, it is gener- ally impossible to effect the desired eliminations without pass- ing to Kqs. 4. It may happen, however, that the forms of the functions f(x, y, z), f\(x, y, %), are such that Eqs. 1, 2, 3, will prove sufficient. ger: DIFFERENTIAL CALCULUS. Suppose z—g(e«+ay)+ q,(*«—ay); then dz ny (et ay) + oi (z— ay); dz / / Pas (x +ay)—ag’(x— ay), d*z HW “/ qn? (7 + ay) + g,/(x — ay), d?z Gp © V2 Oy) Oot re From the last two of these equations, we find d?z As hed =, = a" -_-=. dy? dic” 140. Suppose that we have two functions, F (a, Te ORLY mi(c)-++) wae Fi (x,y,% C; (ley mi(¢)-+) ao Ui in which ¢ is an implicit function of x, y, z, and g(c), g,(¢).-., are arbitrary functions of c. Itis proposed by successive dif- ferentiations to eliminate c and the arbitrary functions. To accomplish this, z and c must be considered as functions of the independent variables x, y; then, having differentiated the given equations a number of times successively with respect to 2, and also with respect to y, we must eliminate the quan- tities des de. d7¢ id? cimga eG ° de’ dy’ dx?’ dady Vaya (1); g(c), p’(c), p”(c)..-, gic), gi’(e), i’(e).-- (2); between the given and the differential equations. Let m denote the number of arbitrary functions P(e), Pr (C), g.(C).--, and m any positive integer; then, if we stop with partial ELIMINATION OF CONSTANTS AND FUNCTIONS. 233 derivatives of cand of g(c), g,(c), of the n™ order, the num- ber of terms in series (1) will be expressed by eee tee The number of the arbitrary functions G(), P/(C) ++) P™(C), il), Fr (C) +++) PIMC) ve sy will be equal to m(n+ 1). Again: since each of the given equations will give rise to two derived equations of the first order, three of the second, four of the third, and so on, the number of given and derived equations together will be equal to (n+1)(n+ 2). Hence to be able, in the general case, to eliminate ¢ and its arbitrary functions, and their derivatives up to the n™ order, we must have (n +1) (n+ 2) 2 (n+1)(n+2)> + (m-+1)m,or5-+1>m. This condition will be satisfied if m = 2m — 1, which will give 2m(2m + 1) equations between which to eliminate 4m? + m quantities. The number of equations exceeds the number of quantities to be eliminated by m: hence there will be, in gen- eral, m resulting differential equations. When the proposed equations contain but one arbitrary func- tion,g(c),of c, they become EF’ ( a, Y, %, ©, 7 (c)) = 9, F(a, Y, 4, ¢; 9(c)) = 0, each of which gives two partial derived equations of the first order; and we shall thus have, including the given equations, six equations between the quantities dz dc dc a fi XH, Y, %, p= ae! t= Gy! C, ape dy’ p(c), Q (c), the elimination of the last five of which will lead to a single 30 234 DIFFERENTIAL CALCULUS. partial differential equation of the first order between the variables a, y, z, of which x and y are independent. If there are but two arbitrary functions g(c), ,(c) of c, we should find that the given equations F (a, Y, %,C, PC), 7) = 0, F(a, Y, %, C, P(e), ni(e)) = 0, with their partial derived equations of the first order, making in all twelve equations, would involve twelve quantities to be eliminated ; viz., i UG. d6y U2 CrtO.C mae ? dx’ dy’ dx dady’ dy” (Cc), 9(c), (Cc), pile), Pile), 91 (Ce): hence the elimination cannot be effected, except in special cases. Passing to the partial derived equations of the third order, we should then have in all twenty equations, with eighteen quantities to be eliminated; viz., the twelve above given, and O70 02c acorns Ree dx” da?dy’ dedy” dy” g’’(c), #1 (¢); additional: and we may therefore have for our results two par- tial differential equations of the third order between a, Yi i the latter being the dependent variable. In certain cases, it is unnecessary to make as many differen- tiations as have been indicated to enable us to effect the de- sired eliminations. Suppose, for example, that the given equations contain but three arbitrary functions, (c), ,(c), g2(c): in this case, m=3, 2m—1=5; and, to effect the eliminations, it would be generally necessary to form the de- rived equations of the fifth order, and we should have for our results three partial differential equations of the fifth order between a, y, z. But if the arbitrary functions are so related ELIMINATION OF CONSTANTS AND FUNCTIONS. 235 that g,(c) = g’(c), g2(c) = 9” (c), the proposed equations be- come F\ «, Y, %, C, p(c), p’(C), p’”(c) = 0, F,{ x,y, 2, ¢, 9(c), 9’(c), »”(c) | = 0; and these, with their derived equations of the first and second orders, make twelve equations, involving the eleven quan- tities COmOCa G (076d "¢ C, da’ dy’ dx” dady’ dy” p(c), p’(c), ep” (Cc), gp” (e), (Cc); and the elimination will lead to a single partial differential equation of the second order. If the value c be found, as it may be, theoretically at least, from one, say the second, of the equations F\a, Y, 2, C, p(C), gi(¢) { = 0, Fe, Y, %, C, p(C), g1(¢) | = 0, and this value be substituted in the first, we should have for our result an equation of the form Fh x, y, 2, (a, y, 2), ¥i(a, y, 2)} =0, which is evidently equivalent to the two proposed equations. By Art. 139, we shall generally be unable to eliminate the two arbitrary functions w, w,, with this equivalent equation and its derived equations of the first and second orders; but it would be necessary to pass to the third derived equations to effect the elimination. EXAMPLES. 1. Eliminate the constant a from the equation Wo? 4/1 y? = a (2 — y). a/ ey dy Ans. Se et 2 /1— 2 236 DIFFERENTIAL CALCULUS. 2. Eliminate c from the equation ee 4? Cas dy Pi bel foes ory Pi Ans. ¥ 2xy Rime 0. 3. Eliminate the functions e” and cos. x from y —e*cos.x = 0. d*y ody fs Ans. Dot Ie tha 4. From y = asin.x + bcos. eliminate the functions sin. a, COS. &. a Ans. a + Y= 0. 5. If y=ce™'*, prove that dy 6. Ify = be” cos. (nx +c), show that d (1a) 74 — =O: d*y dy 2 ond: ogre Srrice 2a". + (4 +n*)y=0. T. From the equation y = oie ee eliminate the exponen- tial functions. Ans. y+ 1 0: 8. From z= q(e*sin.y) eliminate the arbitrary function characterized by q. Ans, sin. ia — cos. eee 0 Y lye Y de 9. From -e ee, ie he —~-—1=0 eliminate the constants a; 0,0. 2 Ist Ans. wa bt (7) - ae, =a d dz\? ag 2d Ans. yz aity(g)—a= ELIMINATION OF CONSTANTS AND FUNCTIONS. 237 Poser rom wu — af (2) + g(axy) eliminate the functions : ig (2), g (xy). sue Eliminate the functions from u—f(e+y) + ryg(2—y). dey d?u d?u d?u iM ( 2u Pale 2a da? dy?/ oo dc! dy dxdy* dy® a«+y 12. From xr we? —y s=r(t) ea 8) xv eliminate the arbitrary functions /, g, w. d*z az dz GE (rae dz. 2 ee fF |, es gy ae, a ee —_ — == Vi Ans. (x carat) - z+ (: oo AG aed dy) 0 2 2 13. From the equations phe ee Be pa = w(x, y), elimi- nate the variable z ; i.e., change the independent variable from % tO &. d d w(x, Y) te p(a, Y) it Ans. 2g (x 7] ) OG hve? ee ‘ da dvy : da? 14. Eliminate the arbitrary functions from e+ hs() d*z d* x , a2 dz dz Ans. © ag Tt 2axy Sear Sree nay ee ae DIFFERENTIAL CALCULUS. ea PA Rie SiC Oo iNees GEOMETRICAL APPLICATIONS. SECTION I. TANGENTS, NORMALS, SUB-TANGENTS, AND SUB-NORMALS TO PLANE CURVES. 141. The tangent line to a curve at a given point is the limiting position of a secant line passing through that point, or it is what the secant line becomes when another of its points of intersection with the curve unites with the given point. It is now proposed to find the form of the equation of tangent lines to plane curves. Let y =/(x) be the equa- tion of the curve RPQ, and take on this curve any point, as P, of which the co-ordi- nates, referred to the rectan- ‘Ss’ x gular co-ordinate axes Oz, Oy, are x andy. This point will be briefly designated as point (x,y). Give to a, taken as the independent variable, the in- crement Aa, y will receive a corresponding increment Ay, and 238 TANGENTS AND NORMALS. 239 a+ Ax, y+ Ay, are the co-ordinates of a second point, Q, on the curve; then, if x,, y,, denote the general or running co-ordinates of a straight line passing through P and Q, the equation of this line will be eta — 2 22 ele — é (x4 wy x), or A Vi Y =<" (@—2). Now, conceive the point Q gradually to approach the point P, oa = will, at the same time, gradually approach its limit 4 ye and finally become equal to this limit when Q unites with P; but then the secant line becomes the tangent line. Hence the equation of the tangent line is d Wy = 32 (a1 — 2), or yi —Y = Y' (2 — &), dy dx makes with the axis of abscissa. Calling this angle 7, we in which is the tangent of the angle that the tangent line have Dy ae (are ans 9) tan em Ua cot "= ay qe da 1 1 i Sete — COS. T Vity? J (% 3? 1+/(— da dy ; y! dic ; ae a en gIn. T Vity? aa} 1+/(— da 142. The normal line to a curve at any point is the straight line passing through the point at right angles to the tangent line at that point. 240 DIFFERENTIAL CALCULUS. Since the normal and tangent lines at a given point are per- pendicular to each other, denoting the angle that the former makes with the axis of « by », we have hw laste hacen Lo) Se tang: yf = yr : ny dc and, if x,, ¥,, represent the general co-ordinates of the nor- mal line, its equation is 1 —¥= (=), Ory — Y= — Fea). y’ dy Cor. When the equation of the curve is in the form F(x, y) = 9, or the ordinate y is an implicit function of the aP abscissa, we have (Art. 84) oY — st hence the equation dy of the tangent line becomes (ea) + = 9, and that of the normal (v7, — )- (ars y= 0. 145. To find the equation of the tangent line passing through a given point out of the curve represented by the equation (a, y) = 0, we should make a, y,, in the equation of the tangent, equal to the co-ordinates of the given point. Then, since the point of tangency is common to curve and tangent, the co-ordinates of this point must satisfy both the equation of the curve and the equation of the tangent: hence these two equations will determine x and y, the co-ordinates of the point or points of tangency. In the same way, we may find the equation of a normal line passing through a point not in the curve. TANGENTS AND NORMALS. 241 Now, if we have two curves, of which the equations are F (x,y) =0, F (x,y) —c =0, respectively, the equations of the tangent and of the normal to the first curve will be iden- tically the same as those of the corresponding lines to the second (Art. 142, cor.). Hence, if for given values of a, y;, and any assumed value of ¢, the values of x and y be deduced from the equations dil’ dk Py) 0 — 0, Sordi re BA biome py a such values will be the co-ordinates of the points of tangency of the tangent line drawn through the point (7,,¥,). In like manner, the combination of the equations (x,y) —c=9, (a, — =) (*1—Y) = 0, will determine the points of intersection with the curves of the normal lines drawn from the point (x, 7,). Since the equation dk (2-2) +(yi—y) = 0 is independent of c, it will represent a line which is the geo- metrical locus of the points of tangency of the tangent lines drawn from the point (#,,¥,), with all the curves which, by ascribing different values to c, can be represented by the equation F(a, y) —c=0. So also (x) — a (2 yo =0 is the equation of the geometrical locus of the intersections of the normal lines drawn through the point (x,, y;) with the same curves. Hence, if these geometrical loci be constructed from their equations, their intersection with the curve answer- — ing to an assigned value of ¢ will be the points common to the curve and tangents, or normals, as the case may be. 31 2492 DIFFERENTIAL CALCULUS. 144. Formulas for the distances called the tangent, the sub-tangent, the normal, and the sub-normal. Def. 1. The tangent referred to either axis of co-ordi- nates is that portion of the tangent line to a curve which is included between the point of tangency and the axis. Def. 2. The sub-tangent is that portion of the axis which is included between the intersection of the tangent line with the axis and the foot of that ordinate to the axis, which is drawn from the point of tangency. Def. 3. The normal is the part of the normal line in- cluded between the point of tangency and the intersection of the normal with the axis. Def. 4. The sub-normal is the part of the axis in- cluded between the normal and the foot of the ordinate of the point of tangency. The relation of sub-normal to normal is the same as that of sub-tangent to tangent. In the figure, let P be the reference to the axis of x, PM being the ordinate of P, Pt is the tangent, J/¢ the sub-tan- gent, PN the normal, and ZN the sub-normal. With refer- ence to the axis of y, the lines of the same name are PT, MT, PN’, and UN’, re- aE ) 2 OY eee 1 ae dx Nowa — an. Pie Migs — tt dy dz or Mt = subtangent = y a : TANGENTS, NORMALS, &c. 243 Again: MN it __ dy WP tan. MPN = tan. Pix = rie dy dx Also Pi=—iM + PM =? (F)+y" ease (Ge) +t: al. 2 Pt = tangent = y Ge) +1, = a ie ere are 2 2 and PN = PM + UN =? + r(Z) seh (2) tig MN = sub-normal = y anh? VPN — normal = y JZ) +1. Grouping these formulas, we have aay ; d Tangent = y sl e +1. Sub-tangent = y 7 d Normal = y JZ ZL) +1. Sub-normal = y 2. x 145. A curve may be given analytically by two equations: of the form y = g(t), «x = w(t), which, by the elimination of ¢ between them, may be reduced to one of the form y= /(z). Without. effecting this elimination, the equation of the tangent line will be dx (Yi —Y) ae (1 — a) Os and that of the normal, d (Hy — y) B+ (a — 2) = . When the co-ordinate axes are oblique, making with each other an angle o, the limit of the ratio ot or fa no longer 244 DIFFERENTIAL CALCULUS. sin. T sin. (wo —T) expresses tan. 7, but In this case, the investi- gation and the form of the equation of the tangent line remain unchanged; but the equation of the normal line becomes eae EXAMPLES. 1. The equation (x, — x)x + (y,— y)y = 0 of the tangent line to the circle can be put under the form which, if « and y are variable, and x, and y¥, constant, is the equation of a circle, the centre of which, having *) : 5 , for its co-ordinates, is the middle point of the line drawn from the point (x,, y,) to the centre of the given circle. The radius of this circle is equal to Va 28 Y, . Now, for assigned values of x,, ¥,, the points of contact with the given circumference of the tangent lines drawn from the point (x,, y¥,) must be in the circumferences of both of the circles; and, since*(1) is in- dependent of r, the circumference of the circle of which it is the equation is the geometrical locus of all the points of con- tact with the given circumference of the tangent lines drawn from the point (#,, ¥,) to the different circles that we get by causing 7 to vary in the equation x? + y? = r?, 2. The general equation of lines of the second order (seuie sections) 1S u— Ay* + Bry + Cx? + Dy 4+ Fx t+ F=0: TANGENTS, NORMALS, $c. 245 du du in By + 2Cx +L, Be eee and the equation of the tangent is (x; —2) (By +2C24+E)+(y —y)(Bu+2dy+D)=0, which the given equation reduces to w,(By +2Cx+ IL’) + y;(Be+2Ay+D)+Dy+ Lx 4+2F= 0. 3. The logarithmic curve is that which has y= 7 li for its equation. For it we have dy = ant and the equations dx. «la of its tangent and normal lines are ala(y; —y) — (%,— #%) = 0, alae, —@) + (y,—y) = 9. The sub-tangent on the axis of y is expressed by ot = = and is therefore constant, and equal to the modulus of the system of logarithms. 4. The logarithmic spiral is a curve having m2 2 Bee ty? x } of tan 12 — —= WW/ a? + y' + y? — UR, for its equation; whence dx ety? dy «“+y, py wy de e—y’ and the equations of the tangent and of the normal are (%, —a)(@+y)+ (Yi—Yy)(y¥—#) =), (,— 2%) (y—#)—(y¥.—y) (@+y) =0. When @,, 7, are considered constant, and x, y, are made to vary, these last equations represent two circles, the circumfer- ences of which cut the spiral in the points of contact of the tangents to the spiral which are drawn from the point (x,, y;). 5. Denoting the tangent by T, sub-tangent by 7,, normal 246 | DIFFERENTIAL CALCULUS. by N, and sub-normal by N,, determine these lines for the fol- lowing curves : — First, The circle: 2? + y=r’. 7? ge ; WV = 7, Ne eee T= +7 (r? — 28, {Bytes 2 Second, The ellipse or hyperbola: a = - = side Cre a0” at )2 r=\ (Samazg)te-S nas(tx 2), 2 4 2 wai Gri)eeeh N = +a, a)\a Third, The parabola: y? = 2pa. T= 2424 ay Die 2a, N=p? (p+ 20), N, = The sub-normal in the parabola is constant, and equal to the semi-parameter; the sub-tangent is double the abscissa of the point of tangency. Fourth, The logarithmic curve: #« = i ly. alent 2a Ji 221 LN: 221 = (Gq) +2 =—, Nahe (p+ : N, = lae*™. In this curve, the sub-tangent on the axis of x is constant, and equal to the modulus of the system of logarithms. 146. The Cycloid is a curve which is generated by a point in the circumference of a circle, while the circle is rolled on a line tangent to its circumference, and kept con- stantly in the same plane. Suppose the circle of which C is the centre, and which is tangent to the line Ox at the point 0, to roll on this line from THE CYCLOID. 247 O towards x. While the point of contact is passing from O to NV, the radius CO, which, at the origin of the motion, was perpendicular to Ox, will turn about the centre of the circle through the angle VC’P ; and the generating point will move from O to P, describing the arc OP of the cycloid. To find the equation of this curve, take Ox, Oy, for the co-ordinate axes. Let r= CO= radius of the generating circle; o— NC’P the variable angle; and «= OR, y = PR, the co-ordinates of the point P: then £2£=0R=ON—RN= are PN— PQ =reo—r sin.o =r (w— sin. 0), y= PR=CN— C'Q=r—r cos. 0 =r (1 — cos. @). From y =r (1 — cos. w), we have r—y r—y 1, ~<———_, cos.@ = ——, sin. o = + vary — y’, co. = cos. ~! ee and these values of , sin. w, substituted in the equation x—=r(o— sin. w), give cr (cos.— =") =F V/ 2ry —y’, which is the equation of the cycloid. The minus sign before the radical must be used for points in the are OPO which is described while the points in the semi-circumference OLA are brought successively in contact with the line Ox; and the plus sign must be used for points in the arc O/B. The point O’ is called the vertex of the cycloid, or rather the vertex of the branch OO’B ; since, by continuing the motion of the gen- 248 DIFFERENTIAL CALCULUS. erating circle on the indefinite line Ox, we should have an unlimited number of curves in all respects equal to OO'B. From’the equation of the cycloid, we get == | ie ees YS dy Ore yee ae ee Hence the equation of the tangent line at any point is n-y= { (0; =o and of the normal, oo (x; — @). If, in this last equation, we make y, = 0, we find %,—e#=Vy(2r—y) = rsin.o = RN. Substituting the values of al a ; for tangent, sub-tangent, normal, and sub-normal (Art. 144), we in the general formulas, have for the cycloid 2r y Pay | tay |e N= V2ry, N= Vy Ora which last agrees with what was found above; and from which we conclude, that, if supplementary chords be drawn through the extremities of the vertical diameter of the generating cir- cle in any of its positions and the corresponding point of the cycloid, the lower of these chords will be the normal, and the upper the tangent, to the cycloid at that point. SECTION IL. ASYMPTOTES OF PLANE CURVES.—SINGULAR POINTS. —CONCAVITY AND CONVEXITY. 147. Wuen a plane curve is such, that, as the point of tangency of a tangent line is moved to a greater and greater distance from the origin, the tangent line continually ap- proaches coincidence with a certain fixed line, but cannot be made actually to coincide with it until at least one of the co-ordinates of the point of tangency is made infinite, such fixed line is said to be an asymptoée to the curve. Hence we may define the asymptote of a curve to be the lmiting position of a tangent line when the point of tangency moves to an infinite distance from the origin of co-ordinates. To establish rules for finding the asymptotes of curves, re- sume the general equation of a tangent line d Piesypias a (1 — x) (Art. 141), and find from it the expressions for the distances from the origin at which the tangent intersects the co-ordinate axes. These are, dz f= x — y — = distance on axis ofx (1), dy dy as ee ea = distance on axis of y (2). Now, there may be two cases in which asymptotes will ex- ist: 1st, Both x—y 7 and y — x dy may remain finite for the di 32 249 250 DIFFERENTIAL CALCULUS. values x= 0, y=o. 2d, One of these expressions may re- main finite while the other becomes infinite. If the expression for the distance on the axis of « is finite while that for the’ distance on the axis of y is infinite, the asymptote is parallel to the axis of y; and it is parallel to the axis of « when the distance on the axis of y is finite, and that on the axis of a is infinite. | Ex. 1. The equation of the parabola is d da 2x y= Spe; .°. me a Spr aa oe and, for these values, expressions (1) and (2) for z=, y=, are both infinite. The parabola, therefore, has no asymptote. Ex, 2, The equation of the hyperbola is bb) a*y? — ba? = — a*b?, ory= at — WV/ x? — a’, a da bx dz x? — q* a? ert odo 0 os a dy which reduces to 0 for v=o: y —2& a will also become zero x dy b when «=o, and dn becomes -— 5 Hence the hyperbola has x two asymptotes passing through its centre, and making equal angles with the transverse axis on opposite sides. Ex. 3. The exponential curve: d eee min, = = a*la, da 1 1 a,” aa i rn y—o = a? —aa*la = for « = 00, but = 0 forge and, for pte, °Y = orig a da ASYMPTOTES OF PLANE CURVES. 251 Hence the axis of x is an asymptote to the curve, and ap- proaches the curve without limit on the side of x negative. In this reasoning, we have supposed a>1. Ifa<1, the axis of x is still an asymptote; but, in this case, the curve ap- proaches the axis on the side of x positive. 148. An asymptote to a curve may be defined as the line which the curve continually approaches, but which it can never meet. An investigation, based on this definition, may be given that differs somewhat from the preceding. Let y= ax+ 6 be the equation of a straight line, and y = ax+ 6+ v the equation of a curve, v being a function of a and y, which vanishes when # and-y are made infinite, or, at least, when one of these variables is made infinite; then the straight line is an asymptote to the curve. Tor the formu- la for the perpendicular distance from the point (a, y) to the straight line is Bees Y Var +1 point of the curve. Hence when v vanishes, as it does by 2 hen the point i Se wy eh e poin Is a V/ a? +1 hypothesis, for one or both of the values cz =x, y=o, the straight line is an asymptote to the curve. From the equation y= ax + 6+, we have ogy -|- Pre ry x whence ois the limit of a when w and y are increased without limit. In general, for these values of # and y, takes the dy ee rao 1 ae form 2; but its true value is ah oe = So, also, 8 is the limit of y — ax, and @ is the limit of ae therefore, in general, £ is et: dy fy—— «2. the limit of y In” When the value of & and # thus determined are substituted 752, DIFFERENTIAL CALCULUS. in the equation y= ax-+, it becomes the equation of an asymptote to the curve. 149. When two. curves are so related that the difference of the ordinates answering to the same abscissa converges towards zero as the abscissa is increased without limit, or the difference of the abscissa answering to the same ordinate converges towards zero as the ordinate is increased without limit, either curve is said to be an asymptote to the other. Suppose we have a curve, the equation of which may be made to take the form y=axr"+a,x"—'+.-- 1,0. Sp U0): then the curve represented by y =ax" +a,x"—'4+.---+a,_,e+a, (2) will be an asymptote to the first curve. So also is that represented by y= an fae + fa, eta,F2 ©), and y =ax" + ayx"-'+--- +a,_\2 +4, +2424 (4). It is obvious, also, that of the curves represented by Kas. 1,2, 3..., any one is an asymptote to all the others. Example. Find the asymptotes, rectilinear and curvilinear, of the curve represented by x*— xy? + ay?—0, or ya | xa The value of y may be put under the form y = = a{ 1 — Ae x and, expanding this by the ae Theorem, we have yoto(it i+ Stet) ©) SINGULAR POINTS. 253 which expresses the true relation between 2 and y for points of the curve far removed from the origin; for then : is less than 1, and the series 1 + 5, + eee +.» converges to a fixed finite limit. Whence we conclude that the curve has two recti- linear asymptotes represented by the equation y= = (2 + 5) and an unlimited number of curves, having for their equa- tions a 3a? a 3a? — ies Fae Sos et oo a eos Y Pei tot.) é o(l+s toc ) which are asymptotes to it and to each other. 150. Singular points of curves are those points which offer some peculiarities inherent in the nature of the curve, and independent of the position of the co-ordinate axes. First, Conjugate or isolated points are those the co-ordinates of which satisfy the equation of the curve, but which have no contiguous points in the curve. Ex. 1. x?-+ y?=0 can be satisfied only forxa«—0, y =0, and represents therefore but a single point; i.e., the origin of co-ordinates. Pee a (a — @”). Thisis satisfied by x= 0,7 = 0, and therefore the origin belongs to the curve: but there are no points consecutive to it; for values of « between the limits x= +a, «= —a,make y imaginary. Hence the origin is an isolated point. Ex. 3. ay?— «x + bx? = 0. Second, Points d’arrét are those at which the curves suddenly stop. | Ex. 1. y=alx. Herex=0, y —0, satisfy the equation ; 254 DIFFERENTIAL CALCULUS. but negative values of x make y imaginary. The origin is therefore a point d’arrét. ix 2a ez. If « be indefi- nitely great, and positive or nega- tive, y approaches the limit 1; but, if x be indefinitely small, and posi- tive, y approaches the limit 0; while, for negative and very small values of x, y approaches +o. The curve will be composed of two branches, as rep- resented in the figure, and will have for the common asymptote to these the parallel to the axis of x at the distance 1. Third, Points saillant are those at which two branches of a curve unite and stop, but do not have a common tangent at that point. Example. y= From this we find If x be positive, and be dimin- ished without limit, both y and a ultimately become zero; but if « be negative, and be numerically A v diminished without limit, we have ultimately y= 0, os — 1, -Hence the origin is a point of the curve at which two branches unite having different tangents; one branch having the axis of x for its tangent, and the other a line inclined to the axis of # at an angle of 45°. SINGULAR POINTS. 256 Fourth, Points de rebroussement, or cusps, are points at which two branches of a curve meet a common tan- gent, and stop at that point. The cusp is of the /irst species if the two branches lie on opposite sides of the tangent, and of the second species if the branches lie on the same side of the tangent. Fifth, Multiple points are points at which two or more branches of a curve meet, but do not all stop, or at which at least three branches meet and stop. Ex. 1. y?=«x?(1 — x”) represents a curve of two branches which cross at the origin, at which the equations of the tan- gents arey=— 2%, y= «x. Ex. 2..The equation y? = x*(1— 2’) is that of a curve composed of two branches which meet at the origin, and have the axis of x fora common tangent. The origin is a multiple point. Sixth, A point of inflexion is one at which the curve and its tangent at that point cross each other. 151, We will now establish the analytical conditions by which the existence and nature of singular points in a curve, if it have any, may be generally recognized; omitting, for the present, the case in which the first differential co-efficient of the ordinate of the curve becomes infinite. If a curve has either a conjugate point, a point d’arrét, a point saillant, or a cusp of the first or second species, we may pass through this point an indefinite number of straight lines, such that, in the vicinity of this point, there is not on one side of any one of these lines for the last three kinds of points just named, or on either side for that first named, any point belong- ing to the curve under consideration. 256 DIFFERENTIAL CALCULUS. This is illustrated in the adjoining fig- ure,inwhichJ/,Fig.1, is a conjugate point ; M, Fig. 2, is a point darrét; If, Fig. 3, a point saillant; and J, Figs. 4 and 5, are cusps of the first and second species. Now, if, for any one of these cases, two points, P, Q, be taken on one of these lines, one on each side of the point J, and however near to it, these points may be united by a curve which has no point in common with the given curve AB. Consequently, if w —/(x, y) = 0 is the equation of AB, and u is continuous, as is supposed, it cannot change sign, except at zero: but no values of x, y, belonging to PQ, can reduce wu to zero; for, if so, then that point would be common to 4B and PQ. Hence the values of x, y, belonging to points of PQ, make the sign of w constant; while the values of a, y, belong- ing to the point JZ, reduce w to zero. Since, then, the value of w at the point JZ is zero, and has the same sign at P, on one side of this point, that it has at Q on the other, these points being very near WW, wu must bea maximum or minimum at Jf according as the sign of wu at P and @ is negative or positive. In either case, we must have du _ du dy dx dy dx Again: denoting the tangent of the angle that the arbitrary straight line PJ/Q makes with the axis of x by a, the equation of this line, which the co-ordinates of IZ must satisfy, will be SINGULAR POINTS. 257 y =ax-+b: whence ih —a; and, substituting this above, we have But this last equation must hold for an indefinite number of values for a, since the line PIZQ is arbitrary; and therefore we must have du du ean 0, ee 0. The co-ordinates of the four kinds of singular points under consideration should then satisfy, at the same time, the three equations Two of these equations will determine values of x and y to substitute in the third. Ifa set of these values x=2z,, y= y,, verifies the three equations, the corresponding point may be a singular point, but not necessarily so. To ascertain the nature of the point thus determined, let us ce a dy ame seek the value of a which the equation ae gives under the form f The second differential equation, because of the conditions au —0 oe ee Mee = 0, reduces to ete) + 2a ast a! dy? \dx dxdy dx. dz Suppose, also, that, by the solution of w=—/(a, y) = 0, we have found y= (x) for the equation of the branch of the 33 258 DIFFERENTIAL CALCULUS. curve on which the point about which we are inquiring is sit- dy uated. The solution of Eq. (a) with respect to op gives du 4 d?u\® d*u dra iy _ ity \ Gay) 1 d?u rence, dy? I. From the definition of a conjugate point and these equa- tions, we conclude that the point x = x), y = yy, will be con- jugate: first, if the two ordinates Y= Fe +4), Y= P(e — A), are both imaginary; second, if the curve at this point has no tangent, which requires that d’'u\? dw da (iy) Podat dyke unless we have hat) Db on Oe dat =) dady > “dy? Il. The point x=2,, y= yp, will be a point darrét: first, =i) when only one of the ordinates y = F(x, +h), y= F(a,—h), is imaginary ; second if the curve at this point has but one tangent, which will be the case when the co-ordinates of the : « epee point satisfy the equation ae a0. III. The point «= x), y¥ = Yo, will be a point saillant: jirst, if to each of the abscisse sz =a, +h, «=a, —h, there is but one corresponding ordinate, differing but little from y,, or if there are two, and but two ordinates, differing but little from Yo, corresponding to one of these abscissa, and none to the other abscissa; second, if the curve at the point ay, yo, has two tangents, which requires that we have (cam) du d*u SS 0, dady) da* dy? SINGULAR POINTS. 259 IV. The point x, y, will be a cusp, when, the first condi- tion for a point saillant being fulfilled, the two tangents at that point coincide; which cannot be the case unless eu? du du (aedy) ost duit 152. To investigate the conditions for multiple points, let the equation /'(x, y) 0 in rational form represent the curve; then Beene e gS ne) WATE 84), da ' dy dx evi Since at least two branches of a curve pass through a mul- tiple point, two or more tangents may be drawn at that point: d hence ee) for such a point, must have more than one value. dx But, since F(a, y) is supposed rational, oe ae will each ad- xy mit of but one value for the values of x), ¥), which determine : dy the point. Therefore “! cannot have more than one value, dic dk dF mess = — 0, —— dx dy existence of a multiple point. The equation from which to = 0; and these are the conditions for the find the values of dy is dlc ad? re ct ad?’ /dy\? me ay eta.) = © dx dxdy dx. dy* \dx d which will give two real values for A , If, for the values of a, and Yq, CGE\? @Faer (Ger dx? dy? and in this case the multiple point is called a double point. >; 260 DIFFERENTIAL CALCULUS. CF a’r a’F If Soe Se 0, dat dady dy” then Eq. (b) becomes indeterminate, and we must pass to the differential equation of the third order, which, after intro- ve BEA a : ducing the above conditions, 1.e. Nie 0, Si 0 nuts f* aS ad? i’ fdy\?. “ote yag isate ge OY so (2 Tale jaa a" dx* dy dx dady* \dx dy* \dx This cubic equation will give three values for a which, if all real, show that three tangents can be drawn to the curve at the point (a), ¥)): the point is then called a triple povnt. If Eq. (d) becomes indeterminate, we proceed to the differen- tial equation of the fourth order, and thus get an equation of the fourth degree for finding HY ; and,in general, if n branches of a curve unite ina multiple point, the co-ordinates of such point must verify the following equations: LER OLE se Se ae ak... a 0 dx’ dy 7 dx? >" dedy ) ‘G75 thie tO dif ay anes 0: dete dx” dy ine dy" ae and the n™ differential equation of the curve would in general determine 7 real values for th x 153. If a curve has a point of inflexion, the co-ordinates 2 of that point must verify the equation ee coy 8 Suppose the equation of the curve has been put under the form y = f(x); then the difference ay of the ordinates corre- sponding to the abscisse # and a + h is (Art. 61) ite h” Ne h(a) +. 13 B(x) fee ote EF (x + oh). abe vores , —- SINGULAR POINTS. 261 The difference of the ordinates corresponding to the same abscisse of the tangent line at the point (ax, y) is Ay; =AF"(x): hence, denoting Ay — Ay, by 5, we have — iaz E(x) + sdf EM (ae) fe eee ich EL (x + oh) ey 1.2.3 RR ; When # is very small, the first term in the expression for 6 exceeds the sum of all the others; and consequently the sign of 5 for points in the vicinity of the point (x, y) will be con- stantly positive, or constantly negative, according as F(z) is positive or negative: hence, if #”(x) does not vanish, the curve cannot cross the tangent at the point (x, y), and there can be no point of inflexion. If F”(x) vanishes, then the first 3 term in the value of 6 is sas f(x), if F(x) does not vanish -ae v0 at the same time; and the sign of this term will change from positive to negative, or the reverse, as 4 changes from positive to negative. This can only be the case when the curve crosses the tangent at the point (x, y); and this point is therefore a point of inflexion. If £’”(a) = 0, then, by the same course of reasoning, we prove that the co-ordinates of a point of inflex- ion must verify the equation #’”(x)=0, &c. Thus, to find the co-ordinates of a point of inflexion, we seek the roots com- mon to the equations d*y y= F(x), F” (x) =0, or f(a, y) =0, 77 = 0. A system x= 2,, y= ¥Y,, of these roots, will be the co-ordi- nates of such a point, if the first of the derivatives that does not vanish for them is of an odd order. 154. Throughout this investigation of the conditions for singular points, we have supposed /(x), and its derivatives for values of x and y in the vicinity of those corresponding to 262 DIFFERENTIAL CALCULUS. : go the point (,, y,), to be continuous. But, if oe co, We may dx readily determine the nature of the point (x,, y,). Under this hypothesis, the two quantities #'(x, +h), F(a, —h), may. both be real; or one may be real, and the other imaginary. First, If both are real, and both greater or both less than F'(x,), the point (x,, y)) will be a cusp of the first species: if one is greater and the other less than /’(@,), the point will be a point of inflexion. Second, If one of these quantities, say F(x, — h), is real, and the other imaginary, then, if #’(~, — h) has but one value, the point will be a point d’arrét: if (a2, — h) has two values, both of which are greater or both less than /(z,), the point will be a cusp of the second species; but, if one of these values is greater and the other less than /(z,), the point will be simply a limit of the curve. Third, If each, or but one, of the quantities F(a, +h), F(2,—h), has more than two values, the point (x,, y,) will be, in gen- eral, both a multiple point and a point of inflexion. — In conclusion, to obtain the co-ordinates of singular points of curves, we seek the values of x and y that will reduce the : Rs 0 differential co-efficients to zero, to infinity, or to x The na- ture of the point is ascertained by inquiring how many branches of the curve pass through the point, and determin- ing the position of the tangent line or tangent lines corre- sponding to the point. 155, The terms “concave” and “convex” are employed to express the sense or direction in which, starting from a given point, the curve bends with reference to a given line CONCAVITY AND. CONVEXITY. 263 from the tangent at that point. If it bends from the tangent towards the line, it is said to be concave, or to have its con- cavity turned towards the line; but, if the sense in which it . bends from the tangent is from the line, it is said to be convex, or to have its convexity turned towards the line. To find the conditions of the concavity or convexity of a curve towards a given line, take that line for the axis of a, and let P, of which the co-ordinates are 2 and y, be the point at which the curve is to be examined with reference to these properties. Draw the tangent at P: then, from our definition, if at P the curve be con- vex to the axis of 2, the ordinates of the curve for the absciss 2 +h, « —h, must be greater than the corresponding ordinates of the tangent at P; h having any value between some small but finite limit and zero. But, if the curve be concave towards the axis of a, the reverse must be the case. If the equation of the curve is y = F(x), the ordinate cor- responding to the abscissa « + h is yt by=F (x) +hF"(x) + an FY (g) to HM he ig ay, ne Coste eee The equation of the tangent to the curve at the point (a, y) is yj — ¥ = F' (x) (a, —2), or yy = F(x) + 0, LF" (x) —a2 l(a). Observing that x, y, are the co-ordinates of the point of tan- | 264 DIFFERENTIAL CALCULUS. gency, the ordinate of the tangent corresponding to the ab- scissa x + h is . Y, + AY, = F(a) + oF" (x) + AF" (x) — ck" (a) = E(x) + hk’ (x): hence, if 6 denote the difference y + ay—(y,+4y;,), we have h? hn — __ fev eye Be Be iio Patt Umi nen The sign of this difference, when h is very small, is the same F(a + 6h). 72 as that of 3 which has the sign of #”(x) whether h be positive or negative: therefore, if /’”(x) be positive, the curve is convex to the axis of ~; and it is concave if F”(a) be negative. | We have supposed the point of the curve at which its con- vexity or concavity was examined to be above the axis of a, or to have a positive ordinate. Had the point been below the axis, /"”(x) positive would have indicated concavity, and i” (x) negative would have indicated convexity. To include both cases in one enunciation, we say, ‘‘ When a curve at any 2 Yas point is convex to the axis of a is positive at that 2 point ; when it is concave to the axis of a, y — is negative.” 1a Cor. Comparing this article with Art. 153, we conclude, that, when a curve has a point of inflexion, it will be convex to a given line on one side of the point of inflexion, and concave on the other. | EXAMPLES. Find the asymptotes to the curves represented by the fol- lowing equations : — EXAMPLES. 265 fey = "(2a — 2). Ans. y¥ = — oh oe ME y? = (x — a)? («@ —C). Ans. y= x*— $(2a+¢). poe y2— a(x? — y*). ATS yf Saeed, A, (y — 2x) (y*? — x?) —a(y—ax)+ 4a?(x@+ y) =a’. NST) Yi Wy pes ope ee Tm es 3 3 Find and describe the singular points in the curves of which the following are the equations : — 3 x ° e . . e e ede aay There is a point of inflexion at the origin, and also at the point having # = + a4/3 for its abscissa. 6. y(at — b*) = x(x—a)*—ab*. There are two points of inflexion corresponding to the abscisse «=a, «= = T. y3=(e%—a)(x—c). There is a cusp of the first spe- cies at the point of which « = a is the abscissa. 8. «a! — ax?y— aty?+a’?y?=0. There is a conjugate point at the origin. 9. ay? «*?+ bx? =0. There is a conjugate point at the 4b origin, and a point of inflexion at the point having «= z for its abscissa. 84 SECTION Il. POLAR CO-ORDINATES. — DIFFERENTIAL CO-EFFICIENTS OF THE ARCS AND AREAS OF PLANE CURVES. — OF SOLIDS AND SURFACES OF REVOLUTION. 156. Let the pole coincide with the origin of a system of rectangular co-ordinate axes: denote the radius vector by r, and the angle, called vectorial angle, that it makes with the axis of « taken as the initial line, or polar axis, by 0; then the formulas by which an equation expressed in terms of rec- tangular co-ordinates may be transformed into one expressed in terms of polar co-ordinates are x =r cos.6, y=rsin. 0. To express in polar co-ordinates the tangent of the angle that a tangent line to a curve makes with the axis of x, we have, calling this angle 7, tan.c= a and hence (qs. a, x Art. 132) : | ap sin. sae + rcos. 0 tails (Sa ay ees dr i cos. === 7 8insw dé and from this we may readily find the expression for the tan- gent of the angle that the tan- gent line at any point makes with the radius vector of that point. Let MZ be the point, P the pole, ZT the tangent line, and Px the axis of x, from which 6 is estimated ; then 266 POLAR CO-ORDINATES. 267 PMT = MTx — MPT: hence, by the formula for the tangent of the difference of two arcs, in. 0 Hh 0 sin. 6 — Y COS. a0 au Sean Pals) hate yee e feet NS) ° cos Wp ry sin do tan. PUT = St ee ta o( emt ) Le SL tanen TCOS. sas Neo CUR GH Later dr d cos.d— — 7 sin. 0 da This may also be found directly as follows: Take on the curve a second point, Q, the co-ordinates of which are 7 + ar, 6+ a9, and draw JZN perpendicular to PY; then ZN —r sin. 40, and QN=r-+ar—vrcos. ad: hence fon) VOU r sin. AO r+ Ar — rcos. Ad Now let the point Q move towards JZ. The limiting position of the secant QM is the tangent J/7’, and the limit of the angle NQM is the angle PMT. Call this angle 6; then | r sin. Ad r sin. AM tan. 8 = lim. =i lim, r+ Ar — r Cos. Ad ee Pea, 2r sin. 3 + ar r sin. AQ = lim. Ad 2r sin. ? — Ar AO 1 AO sin.” ae sin : in.? — — re sin. AO : : ’ AO The limit of —— — 1, lim. adaae oP ain eae 0, AG 2 Ar dr do ‘lin. —is aenoted by —: therefore tan.B— 7 —- aaa ie y do re 268 DIFFERENTIAL CALCULUS. 15%. To find the polar equations of the tangent and nor- mal lines to a curve, we may assume the equations of these lines referred to rectangular axes (Arts. 141, 142), and change them into their equivalents in polar co-ordinates; or we may proceed thus : — Let r and @ be the co-ordi- nates of the point M7; and r’, 6’, those of a second point, Z, in the tangent line: then from the triangle PLY, making PMR we have whe sin. PLM _ sin. (6 — 6’ +7) 7 STD oh sin. T = sin. (6 — 6’) cot.t + cos. (0 — 0’), or “= 3 au sin. (6 — 0’) +-,cos. (9 — 0’). (CE) observing that cot.7 = iat 2 =i 9 (Art. 106). Eg. 1 may tan. T be written, Poe oy a sin.(@— 0’) (2). 1 1 1dr edd Mak —— pe Soka —— = : STARS ae eC Mtoe tear then 3 1p ae and hence, by dividing both members of (1) by 7, and substituting these val- ues, we find u’ =u cos. (0 — 0’) — Beat sin. (@— 0’) (3). To find the polar equation of the normal at any point of a curve, denote by r and @ the co-ordinates of IZ; and by r’, 6’, those of any point, #, in the normal: then POLAR CO-ORDINATES. 269 . : / A ie ein RM (0 eae :) PR sin. PMR cia ( ; 7 7 = sin. (6’ — 6) tan. t + cos. (6’ — 8) therefore = sin. (0/ — 0) Gas cos.(6’— 6) (4), which may be written d dr () rie Les Vb emer See Drage COR (8 fad iniiea: Adopting a notation like that in the case of the tangent, (4) becomes dp u’ = u cos. (0/ — 6) — ee) sin. (0’— 6) (5). 158. Let P be the polar point to which is referred the curve RMS, and through P draw NT perpendicular to the radius vector PM; then MT being the tangent, and JZN the normal, to the curve at the point J/, the lines MT, PT, MN, and PN, are, re- spectively, the polar tangent, sub- tangent, normal, and sub-normal. To find the formulas for the lengths of these lines, put angle PMT = §, and resume the equation tan. § = ro (Art. 156), r Aces Ak : ‘making oe whence tan. 6 = ets from which we find ie Tees VS ip Vp py Then the triangle Pi re gives cos. Bp = 270 DIFFERENTIAL CALCULUS. MP "Sees — ee ee ee a oo (74 vy ies Ee cos. PMT cos. B Sein cei “ =r Jitr a, ’ PTT = PM iP Ma +o = egiee 74 dr’ PM PM __.. ee ) beh: ire ker, Lito ie 2 pe 2 ig Als cos. PUN sin. PMT Ve =.) +(3) ? dr do | The polar sub-tangent is considered positive when it is on the right, and negative when on the left, of the line PM; the eye being supposed at P, and looking from P toward J. The dr sign of the sub-tangent will then be the same as that of We? that is, positive when 7 is an increasing function of 6, and neg- PN=N, = PMtan. PMN =r cot.g =r’ =r’ = + r ative when r is a decreasing function of 6. 159. An asymptote to a curve referred to polar co-ordi- nates is a tangent line, the polar sub-tangent to which remains finite when the radius vector of the point of tangency becomes infinite. Hence, to find the asymptotes to a polar curve, we must seek the values of 6, which make r infinite while ae remains finite. If m be a value of 6 which satisfies these con- ditions, the asymptote may be constructed by drawing through the pole a line, making, with the initial line, the angle m, and another line at right angles to this through the same point; laying off on the latter, to the right or to the left according as _ do Y ee is positive or negative, the distance represented by yr? ay? and through the extremity of this distance drawing a line par- allel to the first line. The line last drawn will be the asymp- tote. POLAR CO-ORDINATES. ree | Example. In the hyperbolic spiral, so called because of the similarity of its equation r = - or r9 =a, to that of the hy- perbola referred to its asymptotes, we have Boy 08) te coy va 9h i UE alt) mC a eet Oe Hence the sub-tangent is constant, and equal to —a: but Se Oe 6=0 gives r=; whence the line parallel to the polar axis at the distance from it equal to — a is an asymptote to the curve. This curve, beginning at an infinite distance, contin- ually approaches the pole, making an indefinite num- ber of turns around without eo ever reaching it. 160. When the curve in the vicinity of a tangent line at any point, and the pole, lie on the same side of the tangent, the curve at that point is concave to the pole; but, if the curve and the pole lie on opposite sides of the tangent, the curve in the vicinity of the point of tangency is convex to the pole. Let Pp =p be a perpendicular from the pole on the tangent to the curve at the point (6,7): then it is plain, that, if the curve at this point is concave to the pole, p will increase or decrease as r increases or decreases; that is, p is an in- creasing function of 7; and, on the contrary, if the curve is convex to the pole, p is a decreas- ing function of 7. Hence, when the curve is concave to the 272 DIFFERENTIAL CALCULUS. pole, a must be positive ; and, when convex, a must be neg- ative (Art. 51). Cor. At a point of inflexion, the curve with reference to the pole must change from concave to convex, or the reverse. oe ash! : Hence, for a point of inflexion = 0 one > dr We have (Art. 158) : “ r chs Lr ; Siar i I Vries = Jr+® Tair +) art raed c r? Le uh dr\? a pas pT ee aii r+(q) i du «1 dr | (dr ae Make eer then aca Sa eaae (5) —-,r G; 1 du\? 1 dp d*u\ du BE ey ae ay —— es p iat Gi 1 And p*® do (ut 7m) do’ dp do _dp_ ( au do du du P wt oe) dp _dpdu____ 1 dp_p’ Bu Bence dr dudr rdu_ r* a im) 2 _ will generally change Therefore, at a point of inflexion, w + its sign. 161. Differential co-efficient of s the arc of a plane curve. Let: F(a, y) = 0 or Yaya) be the equation of the curve RPS referred to the rectangular axes Ox, Oy; and take, in this curve, any point, P, of which the co-ordinates are x,y. Denote the length of the curve estimated DIFFERENTIAL CO-EFFICIENTS OF ARCS. 273 from a fixed point to the point P by s; then, if x be increased by MM’ = Aa, s is increased by the arc PP’ = As, and it is required to find the limit of the ratio = , or the differential co-efficient of the arc s, regarded as a function of &. The tangent line to the curve at the point P meets the or- dinate P/M’, produced, if necessary, at Y ; and makes, with the axis of x, an angle of which the tangent, sine, and cosine are respectively / y' 1 OS Ey Eg Now, if, within the interval az, the curve is continually con- cave or continually convex to the chord PP’, it is evident thaware sf’ >‘chord PP’, and arc PP’ << PQ + QP". But chord PP’—/ az? + ay?, PQ= soa, OPN pT ieee NV tans OPN = ylAme oe QPimy' sg —Ays hence, substituting in the preceding inequalities, we have As > Waa? + ay?, as YyAx, AU<(y+tAy)Aa: whence y < — n(2y + ay) [1+ fe Be a(2y +9042) + (5 Ly) a(t Nan —n(2) ax dy ae a ——— © + Qry 7 ny At the limit, the second member of each of these inequalities dy? becomes equal to 27y J oo ey : hence TAI Ta at i dy ds Din an aed 1+ (FE) = aay 2 dS = 2nyds. SECTION IV. DIFFERENT ORDERS OF CONTACT OF PLANE CURVES. — OSCULA- TORY CURVES. — OSCULATORY CIRCLE.— RADIUS OF CURVA- TURE. —EVOLUTES, INVOLUTES, AND ENVELOPES. 168. Suppose y = F(x), y=/(x), to be the equations of the two curves RPN, h’PN’, which have a common point P; and let us compare the ordi- nates M’N, M’N’, of these curves corresponding to the same abscissa OM’ —ax-+h, differing but little from the abscissa OM = «x of the point P. We have MN=F(ath) WN=f(x +h): NN = F(x +h) — f(a +h). Developing each term in the value of NN’ by the formula of Art. 61, observing that F(a) =/f(«) by hypothesis, we find mech. df\., i (Br, dif MW = i) + cages — an) + h” (ae ZZ) Rett (a it) da” dx” 1:2 1.2..n+1 dat) dan+! hr? Ree ee (Fe +(x + oh) —f"t?(x + 1h); the last term of which may be written, — 36 281 282 DIFFERENTIAL CALCULUS. n-+1 oan pa (Bete +m) —p4(@+ 02) hrri R 12s # being a quantity that vanishes with 4: hence 2 2 NN’ —} dk df ee ar a‘f dz dx dx* dx? Arr qd? +ift Chass et (ae dx” +} ): dls gd If, in addition to F(x) = f(x), we have Fae es —S the curves x have a common tangent, PZ’ at the point P, and are a to have a contact of the jirst order: and if, at the same time, PP _d' Og? da the contact is of the n™ order if n denotes the highest order 4, the contact is of the second order; and, generally, of the differential co-efficients of the ordinates of the two curves that become equal when in them the co-ordinates of the common point are substituted. 169. When two curves have a contact of the n™ order, no third curve can pass between them in the vicinity of their common point, unless it have, with each of the two curves, a contact of an order at least equal to the n™. For y= F(z), y —f(x), being the equations of two curves, APN, LPN’ (figure of the last article), which have at the point P a contact of the n™ order, let y = g(x) be the equation of a third curve, R’ PN”, passing through P, and having with the first curve a contact of the m™ order, m being less than n. Then, by the preceding article, we should have © dF dg A” i See E(x) = OO eee da™ Pian CONTACT, OSCULATION, §c. 283 nee ym d™+iff duty ee NN 1.2...(m +1) eae dnt pnt qztipt . qe+ i Ree a an . a ee) (); FR and F#, being quantities which vanish with h: hence Galen vm ie" eae ee er ae ae WN (i 2).-.(n + 1) gatip datig dam™+i — qym+ +1) (a), also. cAV4— + £, Since, as 2 converges towards the limit 0, & and &, converge towards the same limit, and reach it at the same time h does, ; ‘ NN and since n > m, it follows that the ratio ——— can be made NN” as small as we please by giving to / a sufficiently small value; that is, when / is a very small quantity, VV’ will be less than NN”, and the curve y= q(x) cannot, in the vicinity of the common point P, pass between the curves y = F(z), y = f(z). It is evident that this reasoning holds when h is negative as well as when it is positive. Cor. When h is sufficiently small, the sign of the expression for VN’ (Eq. 6) will be the same as that of A”+}, and will therefore change with that of 4 if n be even, but remain in- variable if n be odd. Hence, if two curves have a contact of an even order, they will cross each other at the point of con- tact, but not otherwise. 170. Osculatory curves. If the form of the function F(x), and the constants which enter it, are given, the equation y = F(x) represents a curve fully determined in respect to species, magnitude, and position; but if the form of the func- tion only is known, the constants which enter it being arbitrary, the species of curve is all that the equation determines. Thus the equation y = b+ Wr? — (a — a)’, when 7, a, b, are fixed in 284 DIFFERENTIAL CALCULUS. value, represents a circle that is completely known; but if 7, a, and 6 are undetermined, the equation may represent every possible circle lying in the plane of the co-ordinate axes. It is then the equation of the species “ circle.” When a curve of a given species has a higher order of con- tact than any other curve of that species with a given curve, the former is said to be an osculatrix to the latter. Suppose /(x,, Yi, a, b,c...) =0 (1), involving +1 arbi- trary constants, to be the equation of the species of curve that is to be made an osculatrix to the curve of which y= F(z) (2) is the equation. By means of the n+ 1 constants in (1), we can satisfy the n + 1 equations dy dy, dy dy, dry dry, ATI Fy ide) det = dei. ae (3); or, in other words, these equations will determine the values of a, b, ¢..., which, substituted in (1), will make it the equa- tion of a curve having a contact of.the n™ order with the curve represented by (2); and it will be an osculatrix, since, in general, a higher order of contact cannot be imposed. We conclude from the above, that the number denoting the order of contact of an osculatory curve is one less than the number of constants entering the equation of the curve. Example. The form of the equation of a straight line is y—ax-+b; and since this equation contains two constants, a and b, we may so determine them as to cause the line to have a contact of the first order with a given curve at a given point. Suppose y= f(x) to be the equation of the curve, and that «=m, y = n, are the co-ordinates of the point; then the equations to be satisfied are am+b=F(m), a= F’(m), which determine a and 0. OSCULATORY CIRCLE. 285 171. Osculatory circle, or circle of curvature. Assume the co-ordinate axes to be rectangular, and let y = F(a) (1) be the equation of the given curve; then, since (x, —a)?+ (y,;— 6)? =p? (2) is the general equation of the circle, and contains three constants, the osculatory circle will have, with the given curve, a contact of the second order. From (2) we get, by two successive differentiations, da Pie Canty 1 9) a = 14 (H+ n and, because the circle is to be ie circle of curvature, we (3); —0 must have dy dy, dy d’y, Y= 913 dx dx,’ dah Ga oo dy, dy day’ Fibs 7 e—a+ (yb) =0, +(2 ZL) +u-9 G4 =0 (5): These values of 7, substituted in Kqs. 3, give therefore dy dy 4 dy cae Bie) es raaea leer (6) Y cm ody * iy ete idly ; diac? da? By substituting these values of y—b, x —a, for y, —b, x,—a, respectively, in Eq. 2, we find Kqs. 6 will determine the position of the centre; and Kq. 1, the length of the radius of the osculatory circle to the given curve at any point. When the curve at the point of 286 DIFFERENTIAL CALCULUS. osculation is concave to the axis of x, as is the case if y is 2 positive and sa negative, then, to make p positive, we must take the minus sign written before the second member of (7). The first of Eqs. 5 indicates that the centre of the circle is in the normal to the curve at the point of osculation; and from the second of these equations we conclude that y — 6b and 2 da? the circle is always on the concave side of the curve, since’ must have opposite signs, and hence that the centre of y —bis the difference between the ordinate of the point of contact and the ordinate of the centre of the osculatory circle. In general, the contact of an osculatory circle is of the second order, that is, of an even order; and consequently it crosses the curve at the point of contact, except at particular points where the contact is of an order higher than the second. The osculatory circle is often called circle of curvature; and its centre and radius, the centre and radius of curvature. 172. As an application of the formulas of the preceding article, let it be required to find the radius of curvature of a conic section at any point of the curve. If the curve be referred to one of its axes, and to the tangent through its ver- tex, as co-ordinate axes, its equation will be Ba: y? = pe + qa’, which, by two differentiations, gives d x ee dy\* ae ae vat ey me In the last of these, substituting for dy its value taken from the first, we dx have, — RADIUS OF CURVATURE. 287 Ne D + 2pqx + ¢° a? nee y? Se whence Pie SP i. di y>” and for p we have dy?\3 (+8) [ae : p? The numerator of this value of p is the cube of the normal NN’; for from the triangle MNWN’ we have aes dy’ dy? 8 NN a ae et us n= y(1+ 3) and a Therefore the radius of curvature at any point of a conic sec- tion is equal to the cube of the normal at that point divided by the square of the semi-parameter. - The value of p expressed in terms of the constants of the equation, and the abscissa of the point J, is (q+97)2" + 2p + ge +p") p* 173. The equation of the asa line to a curve at the p= point (x, y) being ay Gs AC — &3); the expression for the length of the perpendicular p let fall from the origin of co-ordinates on the tangent is 288 DIFFERENTIAL CALCULUS. whence, by differentiation and reduction, 2 2 2 a! 1+ (4) ee mC oem dp aa” dx dx dx? \ da coe dy\*) 3 {+ (a) 5 dy\ d*y (« TY ze) as Cas roe So hae nae OC. AG +7) (Art. 171). +e) Ss And, if vr be the distance from the origin to the point of tan- gency, dr dy pp 2 aie e — ° mnt a hai To. =o and, substituting this value of x +- y in the expression for a we have dp _1 dr. ; _ i da. a! pdt an tikae dp 174. If «andy are both functions of a third variable, s, then dy d?ydx d*x dy dy ds @y ‘dai ds det ds dx dx’? du? dx\? (1); i a) and these values of a a put in the formula for p (Art. 171), give day* fdy\*)4 UG) +@)3 , Py de dx dy (2). ds? ds ds* ds Supposing s to be the arc of the curve estimated from a fixed RADIUS OF CURVATURE. 289 d dy\? point, we find, from the formula ~ = J 1+ @) of Art. 161, da 1 — , (dae? , (dy? (de? ae eo (sey wt? ae es ds +@) da de ae dys? edea re (3), 1 1 dydx da dy ; dy dz diz dy’ p ds* ds ds® ds ' ds* ds ds? ds From (3), by differentiating, we get dx d?x dyd?y _ : Meegytc us dat Squaring (4) and (5), and adding, we find 1 2a 2 d?y 2 = (ae) + Ge). 2 2 d Eliminating ae abs sa? gr turn, between (4) and (5), observing 8 that da\*? /dy\?__ ta) Ge)? we also find d*y da 1 ds? ds? p ~~ “da dy ds ds 175. To find the expression for the radius of curvature in terms of the polar co-ordinates of the curve, we substitute in the value of p, Art. 171, the values of —%, oy , given in dx’ dx Art. 131, thus getting 37 290 DIFFERENTIAL CALCULUS. dr\? ) 3 aaa) dr\? Giy? 42 (5) ~ ee 1 and, when r =~, we have U Ys fee dr ldu dr 2 /du’ 1 du. desi? do’ do? a, u® do’ and these values, substituted in the above value of p, give _\e+G) | a u? ut So) do” 176. The chord of curvature at any point of a curve is the 8 2 portion of a secant line through that point that is included be- tween the point and the arc of the circle of curvature at the same point. The chord of curvature that, produced if necessary, passes through the pole, is obtained by multiplying 2p by the cosine of the angle included between the radius vector and the normal to the curve at the point; but if 7 is the radius vec- tor, and p the perpendicular let fall from the pole on the tangent to the curve, - is the cosine of the angle included be- tween the radius vector and the normal. But the value of p =~ is readily found to be ee hence the chord of cur- ie) do vature through the pole is equal to du\2 2 2u? + 2 GS ats ie wei. ON se 7 a 2p say 2p Tp iteas mEEy (Arts. 173, 175). RADIUS OF. CURVATURE. 291 177. Denoting by a@ the angle which the tangent to a curve at any point makes with the axis of abscisse, we have tan. & = a oe tan 19; da therefore d°y d°y CGT i da da da? dg dy\? ds (VF ee) ee ' dx 1 since —-- = - : theref . Se ne a ay 1erefore p= - (ae) 5 178. The co-ordinates of the points of the curve at which the radius of curvature is a maximum or a minimum must be found from the equation of the curve and the equation — —0 the latter leading to dy > dy d*y dy\? oe (5) diss dz} i+ (Gt) t= a Differentiating the second of Eqs. 3, Art. 171, we find dy, ay ie dic® + dy, a? dP y en a, 3 - a aa dy, dx, da® x dic (2) da by Eqs. 4 of the same article. Comparing Eqs. dtgy _ aly = os Which da’ Bi proves, that, at the points of maximum or minimum cur- 292 DIFFERENTIAL CALCULUS. vature, the osculatory circle has, with the given curve, a con- tact of the third order. | 179. If a perpendicular be let fall from the origin of co-ordinates on the tangents drawn to the differ- ent points of any curve, as SIS’, the locus of the intersections of the per- pendiculars with the tangents will be a new curve, the properties of which will depend on those of the given curve. Denote the co-ordinates of the new curve by 2, y,; then will the length of the perpendicular p,, from the origin to the tangent drawn to this curve at the point corresponding to the point (x, 7) of the given curve, have for its expression 1 Pi ole ee Y1 i — J te Ge The equation of the tangent to the given curve is d mn ea (vy — 2), wand » being the general co-ordinates. Since the point (a, ¥;) is on this tangent, di Yi — y= (a, — %). The equation of Op is u = hy ; and, because Op is perpendic- 1 da | ular to Mp, = = — ay whence (Yi—y)y¥i = —2(x,— 2), or YY + ee, =a + y?. EVOLUTES OF PLANE CURVES. 293 Differentiating this last with respect to x, we find d d = He by os i ay to = Dy, +n—=+ _ for the distances from the centre to the points H, E’, at which the curve meets the conjugate axis. By two differentiations, we find 1 /u\-3 1 /\-t dy mnt ol Geen 1 /u\-$ 1 /»\-4 /dy\? 1 /v\-3d*y a SY ey Moree tac — ee meets bs m° (“| n* () (=) ee n ) du? whence Since the numerator of this expression is positive, the sign of ad*y . ve tisy ae will be the same as that of the denominator; that is, du and » will have the same sign. The evolute at all its points will therefore be convex towards the axis of x (Art. 155). Moreover, we have I~ [ET nN nr Since this differential co-efficient becomes zero for » = 0, and. w\—-} J dy iG m m (= n infinite for « = 0, we conclude that the axes of the ellipse are tangents to the evolute at the points H, H’, and FE, E’; and that, in consequence of the symmetry of the curve with respect to the axes, these points are cusps. 18%. Radius of curvature, and evolute of the parabola. When referred to the principal vertex as the origin of co- CURVATURE AND EVOLUTE OF PARABOLA. 299 ordinates, the equation of the parabola is y¥? = 2pa, from which we find dy peatdige =p".

F(@,) — Fl@a—1) = ba {f(@n—1) + Po}. Adding these equations member to member, for the first member of the result, we have F'(x,) — F'(x)). The second member is composed of two series, the terms of one being of the form hf(x); and of the other, hp. Denote the sum of the terms of the first series by 3/(x)h, and of the second by Sph; then our result may be written F(2c,) — F(a) = 3f(a)h+3ph (3). If p’, the greatest among the quantities p,, p,...,p,, be sub- stituted for p in the series represented by Sph, we should have Sph

Ay (hy thot +++ + hy) = Ay (L_ — 2p), aAf(a)h BS tex si oan = Ua + A/a? + a). 4, l= Making Va? — a? =¢ ee) and proceed- ing as in Ex. 3, we should find da cm Faas Ot i: [Va? +a’ dx. Integrating by parts, Art. 200, we have [Ve +e da=aV/ax? + a? +2 — | yaaa ae (1). 328 INTEGRAL CALCULUS. 6 cee 2 2 But ii x a aN ie Fe 2). Therefore, by the addition of (1) and (2), a pa dx of[Va? +a dia = an/ a a + a | By Ex. 3, dx a | ara = te Va and hence 2 SJ/2 + @ dz = oF EG +S Ya + opal). [V@ —a’ dex = Sa Fa @ — © ae + aa) The quantity under the radical sign Ua ay in the denominator may be put under the form otra Bort oak) 2 2 =(#+4) +q—%: hence +q-4 Reece; SoUe ar ST ae Making x + £4, and q— Fat, we have dx = dt, and dt Vea In this last, substituting for ¢ and a their values, we have — ,= Ut + Vt? + a?) by Ex. 3. MISCELLANEOUS EXAMPLES. 329 er Oe aa p> p eee tat Cts) +87) a1(0+h4+Va + pe +9) 8. Beit ) aA me dsc. Put «+ P=, iti = a?, then Se pa fy de=JVP FO dt =SIWF EO +o t+ VE Pa), by Ex. 5. Replacing ¢ and a? by their values, we have, finally, Jv epee 9 do= 5 (2+5)Vi +p + 1 : | =e 3( ae (e+$4+ve Fpe+4). ax 9. ee. Letra —i;-thén Jar — 2’? =a? — ¢7, Iam — x? cabanas and dx = — dt: therefore Aga: <== -~(sa— ==; = COS. ne _;a—«x Sey Cs. = = Vors, a he, a 10. lz Put Ges ee? then Lh eee! a" 2ax — a? = fs Betty, waa — a) = eae) e asitad 1 ay t 1—¢ dx =o and therefore adt f dx “ie (1 — Os ee 1 dt tr/ ax — a? — a(1+t) VAL, 2 ia a ae 42 330 INTEGRAL CALCULUS. i Lie Y tere Lie = 1+ gin. ¢ — ~'sin, ~ See a a x fxcos.azde. Assume u=—, v=sin.aw; whence a dv =acos.axdx and [a cos.axdx = f udv: therefore x sin. ax sin. ax fa cos.axada = pind hea da Ped SIG “a COS. Aa a a? __ sin. AL 12, fe sin. aada. Put u > ve; e es ; sin. ax ae COS. Ax fe sin.axdax oa | dx. c c But we have, in like manner, ae" COS. ax poe an a? ay ax [US de= ee + fe emda: Cc hence . * 2. . sin. ax ACOS. AS ox ma: sin. ax e da, fe*sin.axda = eo . C | C c which, by transposition and reduction, becomes e°"(Csin.ax — a COS.ax) e** sin. axdxa = : i i re 13. fe ‘cos, axda. Proceeding with this as with the last example, we should ~ find e°*(e cos.ax + asin. ax) fee cos.aade = * cia dix dx 14. ih as SS a V (q+ pe — 2?) Jia 2 Put g-+4- =a’, wat: >, dt =o MISCELLANEOUS EXAMPLES. 331 Therefore Rare ut hee ald rece ae P a pare ae ee: 2 Real eid S a 42 ar 4 15. fYatpe—ade=f |}q4+%—(2—$) tae 2 ~ Making gto =a', % —f=1, we have IV (a + pe — x?) da Ree —?t? dt = sive 4 = in. mee by Ex. 2, 1 = — (22 — pw/4q 4+ p’ 4 E ihes the saa pee OV a Pr (4q-+2") vipers da 1 dt 16. la ee whence da = — -,, and eas! 1 a era ae renee a Aha Aa/ x =F fe a= av 1 ge f dx dt trae ae : xAa/ x? —-a3 V1 ~ Qt? eet a” —= —-sin. lat = — —sin.-!- But, since ORT pee asia ee sin ~ + cos. sin cos eGo: hence, throwing — may write INTEGRAL CALCULUS. — dx Nee dt 17. loa Make w= 5: : . da = — =, and f dx PER: dt fT 1 es Ss a a es ee a een 2 — pa J pee (t+ =5) d aetat a ax ee x 1 x | wn sae: a+ /a? + a? by including 54 in the constant of integration da 1 1 1 18. ———. = — vee di. vee wal Gees a ks fey ah da = dix 20. Ci — a c+ a 1 eae st pg ere Pls 5 Meta) = - bed If x is less than a, then | f AGS dx ph dx da e—a? 4@—2 Y/\a—a%' ate 14%, => la—x)— 5a +2) = =: prite dic at+2z ang laa tet dix ur da 2 ena (245) 49-4 Suppose g — i to be negative, and make 2 e+f=tq—-t =a’; MISCELLANEOUS EXAMPLES. 90 then, by the last example, ii dx a dt RB jbe, te miaepat+gq Y@—a? a tta i 20 +p — /4q — p? ~ V4q— pp? 2a +p+V/4q—p* 9 “ If q = be positive, then g — apse a’; and das 1 et ie +pxe+q =/aeea= a a : -1_ 20 p V4q —p /4q — p” uP ne mx +n sora 2 2 0 [ee en re a + px +- q a* + pa + g m 2a m dic es yaa 2/7 «+ px+q 2 /¥ «w+ pe + q The integral of the first term is 5 U(x? + px + q), and that of the second is found by Ex. 19. 21. i} es —f2 a als sare S1n. & Make cos.2 =¢; then dices res f 1h oes 1 1! 7) 1, 1+ cos.a cei e))606lCUl ULL ECC cone by Ex. 18: but 1,1-+ cos.c a1 ea) = etm: > 2 1 —cos.a@ >. \1 + cos. a 99. f da =f cos.cde f dsin.a Com ae cos; @ J 1 —sin’a2 334 \TINTEGRAL OALCULOGS* C08. 5 aa aaa wd 1+ singe es 9 2° 1—sin.a 1 ti | cos. 5% — sin. 5 & Seale at ne a7 +5)" da sin.2z + cos.2 a 93. {———— =i A aE da. sin. & COS. x sin. 2 COS. x = f (tan. « + cot. a)da oo tan. 2. = —lcos.x+U sin. =e 24. jouer =f eh oe a sin.” @ cos.? x sin.” x cos.” x = [ (sec? « + cosec.’ w)da — tan: « — cot: a dx a A) scares ee da a(sin2 5 + cos? 5) +b (cost — sin? 5) sec. 2 dex iio Se atb+t(a — b) tan? by observing that roe oC oie sin? 5 + cos. 5c 1; Cos. 2 = cone 5 i sin.? 5? and dividing the numerator and denominator of the result by cos. When a > 8, the last integral may be put under the form MISCELLANEOUS EXAMPLES. 338 x 2 Bens ee ee x | —— tan.— ee LAT) ene 8s, fat aah lato ( ato ») a—b 2 a—b i} dx cee ee Creeigs {eee x atbcos.a/ys, VWa?—b? ” rey vee 5) When a < 6, we have x f , ts d tan. 5 at+tbcos.« b—a b+a, 22 b—a i Vb—atan.=+Vbba J a8 vb o /b—¢ tan. — 5— vba by Ex. 18. ae a+b sin. a . ow 20 a+ 26 sin. = cos. = sie 2 ye dx a (sin? 5 + cos. 5) + 2b sin. 5 C085 x sec? — 50% a(t +- tan. a ae: 5 336 INTEGRAL CALCULUS. d (tan. 5 a ;) a 2, f2 2 ‘iu paleo +(tan.5 +5) Q: 2 ee 2 Best a zc 6b my epieam ea Sa (tan. 5 eal when a > b; but, if a < b, then “% 2 ee f es teh 2 d (tan. +5) at+bsin a Ue ss ( st) oy a Po a? tan.2 + 2 Vb — a a \_ Vb? — @ ton. 1 2 was Le atan. 54+) —/b?—a? =a) OF ate mL |. Bian. | Oe Ne 202. Rationalization and integration of irrational functions. Examples of integration by substitution have already been given: we now proceed to show under what conditions cer-. tain irrational differential expressions may, by proper substi- tutions, be rendered rational, and integrated by the methods previously investigated. Let us assume the form x(a + ban)a dx,in which m, n, p, and q are entire or fractional, positive or negative. © Put alam = st; a= (5 = gu?" 1 nbn I=n (27—a) ™ da: ai IRRATIONAL FUNCTIONS. 33% whence P i oe fara + bx")i da = ty f Pt 8-1(29 a Ye Eee nb * Ey. aaa oe Now, if oat is an integer, the binomial (2?—a)™ —' is rational in form, and may be expanded by the Binomial Formula into a finite number of terms when the exponent 1 ; a : me —1 is positive. Each term of the expansion, being multiplied by z?+%~-!dz, will give rise to a series of monomial differentials which can be immediately integrated. We may also write fom (at ber) dex = fo" (bax-"ide; and, by comparing this with the first case, we conclude that the substitution of z4% for b + ax—” will reduce | Sam*a(d 4 ax-"); dic to an expression that is immediately integrable when m+tl p n qd ; me: ; ar 7 m1 ip. ; 1s a POBrEYS integer; 1.e., when ‘9 ae 1s a negative integer. | . 2M, L . Hence fom (a+ ban)e dx may be rationalized and _ inte- grated when = is a positive integer by substituting z? for we m+ 1 ‘ Fotis a+bx"; and, when __. +~ isa negative integer, by substi- q q tutingy 2? for b+ ax-". It will be shown in a subsequent section (2) that the inte- | tna ae grals may also be found when is a negative integer In . 43 338 INTEGRAL CALCULUS. 1 ree) the first case, and when aoe is a positive integer in n the second. 2 ie fala + ba)’. Hereom=1, n=4, 7 oe 1 Beas = 2,a positive integer. n yl beavar a+ ba =2?: i ee r a day = 7, 2% f x(a 4 bee)? dee =sfe — a)z*dz = 5 fe —aeh)de =7(--9) a+ bu)i ae 5): NTO b? 7 aa . é Ex. 2 f See ;: In this example, (a* +a)? m=8,n=2, 2 oe oe 2 Yana AS ee 2\2 : ada, Pot a?-+ 2? = 2°: 0°. g= (2? — 9’) , and d7=er (2? — a)? x* da = | (2?—a*)dz == — az eS: J ) 3 a. ee (at iets & 2a : 3 Ex. 3 f——. In this case, m = 2, n=4 Boe (a? + 27)’ q pdt ae oe a negative integer. —. (1+ aa “Ede! a? + a)? IRRATIONAL FUNCTIONS. 339 Let Si alee he a (ae eee Coal Tre azdz 8 Gal) GI er Ge a piek.o I : (a? + o:?)* aier tart os TE MEE tae te 3a°(a? +0)” dx p 1 Ex. 4. os Here m= —2, n=2,4+——- x1 +- 2)* q 2 Bute) @=>— 2°: LS ina sees Ea dz—= — ad as (21) (x? — 1) d . | aes ta evar ake? kee cers “Ya@1ta) ~ «1+ —*)" Functions in which the only irrational parts are monomials can always be rationalized and integrated. Thus, suppose it is required to find 1 2 ns 2 ees x) dar 8 : bee we" Putw—t*; .-. da — 6t°dt; and we have +a? —2£ el ere pe, ae —H+%+0—t+e? — a1 6 elt Mate +: ety, 6 Stee ED at Bae #h hs =! re +a +t ze + 2¢ 6¢ + 6 tan.— 1b, which becomes the integral in terms of x by replacing #by a, 340 INTEGRAL CALCULUS. The rule to be observed for rationalizing such expressions is to substitute for the quantity under the radical sign a new variable affected with the least common multiple of the indices of the radicals for its exponent. Fractions in which the only radicals are the roots of the same binomial of the first degree may be reduced to the case just treated. For example, required xv* + (ax + b)} dex a + (ax +b)" 1®—b 6t° dt Assume. aextb=?*: .°.%= - , dx = ——, a ak P hye a (axe + by? = t*;(ax-+-b)° = By these substitutions, the expression to be ec be- comes the rational fraction 6 Ale by atti tde a? —b-+ at?’ The general method of integrating rational fractions will be investigated in the next section. EXAMPLES. : 3 + 2a —_ —1 as = sin. ie | n+l IX fortede Se gi (to aoe ; n+ 1 n+] 3. fo sin. 6d = sin. 6 — @ cos. 6. 4, Ss stall... .6". sin. 2x Dom f (1 — cos. 2)'de = 5 —2sin.a MISCELLANEOUS EXAMPLES. 841 a? dc Rea xs . Sma ws ara 1+ cos. x : T. ee: Sin. 0% =U % + Bin. @). . e+ sin. # a 8. 1+ cos.2 adr — ¢ tan. 5 9, J—= AED Us 10. fessin. ma cos. nada e“ asin.(m + Bie — (m-+ n)cos.(m + n)x id. a? + (m+n)? e* asin.(m — n)x — (m — n)cos.(m — n)x ee etna ew Having found the indefinite integral, the definite integral between assigned limits, except in special cases, can be at once determined. eh Cee oem 70,” atk Va? — «0? dx =? U xr/ a? — a” a? for ead Aires , +g sin = ¥(2); a? and w(a)=——» (0) = 0: +. yla) — ¥(0) == 2a 12 3 us 12. J ver. 4 ia eet By making « = a(1 — cos. 6), we find x 2 fver- OY) bs Smee fas sin. ddd a — asin. 0d — a0 cos. 6. The limits 7 and 0 for the transformed integral correspond to the limits 2a and 0 for the given integral. 342 INTEGRAL CALCULUS. 2a 5a” ag = : 13 i aver da 1 da xe on : = -—_lIt a sin. @ + cos. x =F; st © a 3) i i} sin.” eda = (eto na Vatan. © & ; a+ bcos.a ab? ia Vato b een atba? a $ aly (A +- ba?)”. 16. fa/a + ba?dx = ( Bp sm) (a ar”) Doct pally eens 2ar/a — x 17. [Fe ae vee ee 3 + 18. fiat boo") da x 4 = 37, (a + ba") at $Y) (Ot bw) 9 arty Gt be) 2 (a+boy'+a va In effecting this integration, transform by assuming a be? 2". SECTION IL. INTEGRATION OF RATIONAL FRACTIONS BY DECOMPOSITION INTO PARTIAL FRACTIONS. 2038. A RATIONAL fraction is of the form A+ But Ca*?*+.---+ Ma Al Bla Oa? +... Na’ in which the numerator and denominator are entire and alge- braic functions of x; the co-efficients 4, B..., A’, B’..., being constants. Denote the numerator of such a fraction by F(x), and the denominator by f(x). If the degree, with respect to a, of F(x), is not. less than that of /(a~), we may divide F(x) by /(a) until we arrive at a remainder of a degree inferior to that of f(x). Let g(x) be this remainder, and Q the quotient; then F(x a a AE. J(#) f(#) a g(x)dx and Qda + = aes FER pos J As a Qdx can Bae, : found, the integration of the origi- d nal fraction is reduced to the integration of oy in which the degree of g(x) is lower than that of /(x). The integra- tion of a is effected by resolving it into a series of more simple fractions, called partial fractions; and we will now demonstrate the possibility of such resolution in all cases in 343 344 INTEGRAL CALCULUS. which f(x) can be separated into its factors of the first degree in respect to a. F(z) J (2) and that the degree of #’(x) is inferior to that of f(x). If the factor « — a enters /(x) p times, we shall have J (x) = (% — a)? g (2), g(x) denoting the product of the other factors of f(z): whence PO F@) FG) aa) 9a) 204. Suppose the fraction to be in its lowest terms, 7(@) ~ (@—a)?g(@)— (w@ — a) g(a) (% — a)? But F(x) — = g(x) = 0 when x =a, and is therefore divisi- g(a ble by «—a. Let w,(x) be the quotient, then 2 bee ale Ey ne Bs J(") (@— a)?" g(x) © g(a) (@ — a)? Denoting Ae: - by A,, we have g(a F(x) a w1() A, NOMIC er F(a) F(a) has been resolved into two parts, one of which is x Be tatico a wi(2) ipa? In like manner, (oe ay? lpia) w (2) Wy (Z) 4. A, (x7 — a)? l(a) (a —a)?—*g(x) ' (2 —ayPal and so on, until at last we should have may be reduced to Wp —1(2) _ Up (#) ou (w—a)g(x) g(v) ' e@—a RATIONAL FRACTIONS. 345 By the successive substitution of these values of Y1(#) W(x) (@—a)P"y(e)' (@ =a) "H@) of that in which they were deduced, we find in the order the inverse | Pm). wh, (2) A, A, ee tener Gaett tee Proceeding in the same way with the fraction se the 2 F(a) J() completely effected. decomposition of into partial fractions will be at last 205. If the root a is imaginary, and equal toa + pv — 1, then a, = a — BY — 1 is also a root of the equation /(x) = 0. Suppose that all the partial fractions corresponding to the real roots have been determined, and that there remains for resolu- F(x) ( tion into such fractions the fraction y? in which f(x) =0 a 1 gives rise to imaginary roots alone. Suppose, further, that the pair of roots, #—=a+ B/ —1, «=a— pV —1, enters this equation g times. Denote the factor of f,(x) that gives the remaining imaginary roots by g(x); and, to abridge, make Oey — 1, a, — « — b/ — 1: then ae f(a) F(a). F(a) P(x ) (a)? py(x ) 91 (a) 5 Ale) @— ore CEO (@—ayl(@— a, ae i Sie ae’. oie) ( " 2 (2% 0) G,)% yi epeenit 1} ere ay)! 44 346 INTEGRAL .CALCULUS. In like manner, He nN L itiey 91 (a) Vo (eh (eae aa ee) — _ PG) F\(a,) — Fa) ay ee p1(x) eae (4, — @)g1 (41) s (x — a)?—*(@ — ay)"9,(@) fF (a,) — ao a) Sk é (4, — @)g (a) (2). —,a)1~"(e%— ay)2 The last term in the second member of Eq. 2 may be written F(a) — Fi(@) p1(@1) S g1(@) ‘ (a, — a)(% — a)?" (@ — a)? But (a, — a) = — 264/—1, and #i (a1) 2) is derived from fac) 91 (4) (a) by changing the sign of / —1: hence, if ae as eae BT. te then Fila) a34 > RN dana Py (1) F(a) & F(a) = Bae Pi(41) px (@) Therefore (ay 1) F(a) B 91(41) 91 (@) B (a, —a)(e@—ay(@ a," (way @— ayy and the sum of the two partial fractions in the second mem- bers of Eqs. 1 and 2 will become RATIONAL FRACTIONS. 347 B BAA 1 B a ear AL BVITHE (27— 0) BA” sad (2 — ae — 1)? B B (c—a)+A — (a? — Qoe a? + Bt which is rational. It is also seen, that the numerator and de- nominator of the first term in the second member of Eq. 2 is divisible by « —a. Dividing, and denoting the numerator of the result by %,(x), this term may be put under the form X, (x) f Elmina’ <8 (i — von + 02+) "9,(z) hence, by substitution in Eq. 1, we should have F(x) 1, (a) = (ea) +4 F(a) (@— tea pot py "9,(a) 1 Snreanra ye Now, %,(a) is a rational and entire function of x, and the frac- XL (x) (a? — 2ax + a&? + B*)2—! g(x) tion may be treated as was F(x) Si(@)’ | of imaginary roots being of the form By (a) 2, (a) vei iecapeat Ji(@) ~ Oi(%) (2? — 20x a? pet and so on; our result with respect to the assumed pair | M,« +N, | isc® — oe +a? + B? The possibility is thus demonstrated of resolving a fraction, the terms of which are rational functions of a single variable, into a series of rational partial fractions whenever the denom- inator of the given fraction can be separated into factors, 348 _ INTEGRAL. CALCULUS. whether real or imaginary, of the first degree with respect to the variable. The investigation also shows the form of the _ partial fractions answering to the different kinds of factors of the denominator of the given fraction. Thus if f(x), the denominator of , contains the factors F(a) J (x)? i OR Cb (2 ic), (2 —a— pv —1)?, (w—a+pry/—1)?, e—7y—d/—1, w—y + /—1, then her a A By Bis Nexo 16) Fe Go, Co a a (eee a eee ee Mx + N, Mia + N, + a Sen pe + Py + GP pon eee MU, +N, Pe+Q TT ae pop R TT Rep EE The labor of determining the constants 4,, A,...B,...M,, NV,..., for the partial fractions, which, by following the method above indicated, would be very great in many cases, may be diminished by expedients which we will now investigate. The most obvious of these is based on the consideration, that, when the partial fractions are reduced to a common denomina- tor, the numerator of the result is identically equal to the nu- merator of the given fraction. 206. To determine the partial fraction corresponding to the single real factor, « — a, of f(x). Lape (2) J(t) &@—a& g(x) Assume (1); RATIONAL FRACTIONS. 349 in which 4 is a constant, and ae way is the sum of the partial fractions answering to the remaining factors of f(a), arid J(&) = (% — 4) 9(*). From (1) we have F(a) = g(a) + (# —a)w(a) (2), an identical equation. Make a =a, and then eh ei Cea) (a) = Ag(a): . Da: Aaah 9 We also have the identical equation J (x) = (& — a)q(a) ; whence, by differentiation, f' (x) = g(a) + (2 —.a)p’(x), an equation also identical: therefore, making x =a, J’(4) = (2): ent 207. To determine the partial fractions corresponding to the real factor («— a) repeated n times. We now assume Peles Ay oa: A; Levin d as w(x) f(z) (e@—a)* * (wa) —a ~ 4(#) v(x) q(2) factors of f(a) give rise. Multiply both members of (1) by (x — a)", and we have the identical equation, eae (2) g(2) p(X) observing that /(x) = g(x)(a—a)". Denoting the first mem- (1); denoting the sum of the fractions to which the other A, + Ay(w—a)+ +++ +A, (a —a)t (2 — a), 350 INTEGRAL CALCULUS. ber of this equation by z(x), and then, in it and its successive differential equations, making « = a, we have 7(a)' = Aj, x/(a) = A,, 7" (0) = oe ya) — 1.2.0. (0 ee and thus the numerators of the partial fractions are determined. 208. To find the partial fraction corresponding to a single pair of imaginary factors. Suppose « — a — Br/— 1, e—a+ fv — 1, to be the ima- ginary factors of f(x). We then put , F(a) Me +N w(@), f(z)” wv —2ax+a?+6 whence /'(x) = g(x) (Ma + N) + w(x) (x? — 2am + a? + 6?), an identical equation. Make a=atp/—l; then F(a+6¥ —1) = 9(a@+87—1) EAC cc : I) + Nf; or, by making « = a— pv — 1, F(a —BV—=1) =9(@—pV=1){ Mla —BV=1) + MI}. These last equations may be written | A+ BV—1=(C+ DV—1)|M(a+pVv7—1) 4+ FI}, A— BY—1=(0C—DvV—1)| M(a—bY—1)+ I}, in which 4, B, C, and D are known functions of @ and 8. From either of these equations, the values of WM and NV may be found by equating the real part of one member with the real part of the other, and the imaginary part of one member with the imaginary part of the other. | The values of JM and WN may also be found by the method of Art. 206. Thus, for brevity, denote the imaginary factors by « — a, x — a,; then the partial fractions are Faye Rll (i) i ne f(a) &@— a’ f'(a) & a, RATIONAL FRACTIONS. 851 F(a) = sais F(a) my) — . If Fila A+ Br —1, then acai ABN since F(a;) F(a) by changing the sign of /—1: Flay) is derived from —— Fila 7 (a) hence, replacing a and a, by their values a+ bv —1, «— bY —1, the fractions become etn — Meh ae | NS | if er as the sum of which is rene — a) + 2Bs a® — Qo + oc? + B 209. To find the partial fractions corresponding to a pair of imaginary factors which enters the denominator of the given fraction several times. Let « —a —6V —1, «—a+ $V —1, be the imaginary fac- tors, and, to abridge, put a=a-+ b/—1, a, = a— bv —1; then, putting f(x) =} (a — a)(a — a,){"9(a), Atay, Mix + N, M,«+ N, Fe) —Ve—o@—a)f'* [e=ae@=a) ue M2 + N,« EDs eee ay ys (8 G(s) (2) 9(@) which the remaining factors of /(x) give rise. Multiply the att representing by the sum of the partial fractions to first member of this equation by f(x), and the second member -by its equal }(#—a)(x —a,)}*% (a), and we have F(a) = (Ma + N)9(#) + (Myx + N,)(# — a)(x — ay)9(2) + (Myx + N,)}(@ —a)(e —a)}?9(2) + + +} (@ — a)(% — a4){"9(@) (1). 352» INTEGRAL CALCULUS. Now, whether we make a =a, or «=a, all the terms in the second member of this equation, except the first term, vanish. Suppose «=a, then F(a) = (M,a+ ™)9(a) ; and if the real parts in the two members of this last be equated, and also the imaginary parts, we shall have two equations from which to find the values of WJ, and N,. Sub- stitute these values in (1), transpose (,a + N,)g(x), and divide through by (2 — a)(~ —a,), denoting the first member of the resulting equation by F(x); then F(x) = (hx + My)q(a) + (Max +N,)(x —a)(a —a,)9(2) ++ + | (@— a)(@— ay)}*~"g(a) (2). Proceeding with (2) as we did with (1), the values of M, and N, may be found; and, by repeating these operations, all of the constants, 1/,, V,, W,, N,..., will finally be determined. 210. The rational fraction, which may be decomposed into partial fractions by the foregoing methods, being a differential co-efficient, the resulting fractions are also differential co-effi- cients; and the sum of their integrals will be the integral of the given fraction. The differentials corresponding to these partial fractions are of the form Adx (Ma + N)da (e— a)" (a? + px +9) iseger m—1 («—a)™—)!? becomes Al(~ — a) when m =1; and that of the latter, when which The integral of the former is — nm = 1, has been explained in Ex. 20, Art. 201. The integra- tion of the second form, if m be greater than unity, is reserved for the next section. RATIONAL FRACTIONS. 353 EXAMPLES. (3 —2ax)dx F(x) Pie 2 f(a) x+1, «—2. We therefore put 3—2n A A, ee er alma os The factors of the denominator are iE, Substituting — land + 2 successively for x in Ia) = 3 — 2x yA boy) eas 5 i we get 45? A, =— 3° (3—2x)dxe 5 da 1 da eee ee tA ee Bo 2” (3—2x)dx ge Baye be ——— 210 -+1) 3 (a — 2). ie (x? — ae 2)\de (@? e+ lf@ +1) In the denominator of this, the pair of imaginary factors, aa ieee ep t Se 3, e+ 5 + 9 4/ — 3, enters twice; and the real factor e+ 1 once. We put w* — 8a — 2 LPR crates aoe (w+ae+l1y(atl) (@+e+1P?° etoe+l : xv? — 3x —2 —=(M,x«+ M)(x+1) $ Oha + Mart 2+ Weetl) +(e tet Le) (0) Give to x one of the values which reduce x? + x+-1 to zero; then, for this value, (1) becomes x? —32—2=(Myx+ M)(a+1) (2). 45 (2) . Tel 354 INTEGRAL CALCULUS. From #?-+a2+1=0, we have x? =—a—1. Substituting in (2), —4¢—3= Me? + Mct+ Noct+M =M,(—2-1)+Me+Not+™ = — Met Not: whence M,—N,=3, N= — 4, A= — In (1), replacing M, and N, by the values thus found, and transposing, we have e*— 384 —2+(x-+ 4)(e +1) = 2(x? + 4+ 1) = (Myx + N)(x? +e+1)(@+1)+(a?+a+4+1Py(2) (3). Dividing through by x? + «--1, and in the result making x?+oaeti=0, we get 2 = (Mx + N,)( +1): whence Mo — 2, may be found The partial fraction corresponding to aie by the method explained in Art. 206; or thus: After dividing (3) by «? + «+1, replace M, and N, by their values, trans- pose, and again divide by w +a2+1. We find w(a)=2: (x* — 3a— 2)dax (a+ 4)dx_ therefore fst ap Ihe) = @ aI 2ada 2dx ES i ei 3. (Seatac 832, aR filsh ev*— 5u* + 32+ 9 By the method of equal roots, we readily discover that the denominator may be resolved into the factors (w — 3)?, «+1: hence we put 9x? + 9x — 128 A, A, B, gba Lao ay t cS whence 9x? 4+- 9x —128 = 4,(x+1)4+ 4, (x—3)(a+1)+ B(x — 3)’; _ RATIONAL FRACTIONS. 355 from which, by making x=3 and «= —1 successively, we get 4,;— —5, By=—8. If the second member were de- veloped, the co-efficient of x? would be 4,+ B,; equating this with the co-efficient of x? in the first member, we have Meee 9: .*. A,—17; and therefore {ise + 9x — 128)da _ uO ane 5dac Hef Lia: 8da w® —$e?+3a+9 — (7 — gy oh DG ees ae, = ae + 171(a —3) — 8l(a+1). m—l1 : e 211. Integration of ra when m and n are positive integers. If n be an even number, the real roots of «” —1=0 are + 1 and — 1; and the imaginary roots (Art. T7) are given by the expression cos. Wea 1 sin. a , by giving to & in succession the values 1, 2, 3..., i ifr We will denote the arc — = by A, ae - being the fraction to to be resolved into Aa fractions. It has been shown (Art. 206), that, if a be a root of /(x#) = 0, the corresponding m—t\ per i : hence, for the fraction —~ f(a ) sy ’ ar — 1? the partial fraction for the root Nie 1 of the equation x” —1=0 partial fraction is m—1 m Fy oo ee ea ee el and, for the root — 1, the partial na"—* na” n(x —1) so" n(aw + 1) cos. 2k + / —1, sin. 2k0, give the partial fractions fraction is The pair of imaginary roots, 356 INTEGRAL CALCULUS. (cos. 2k0-+ / — 1 sin. 2k9)” n(x — cos. 2k0 — +/ — sin. 2h) 4. _ (cos. 2k6 — /—1 sin. 2k0)™ n(a — cos. 2k0 + 4/—I1sin. 20)? that is, (Art. 73), cos. 2mko +- / — 1 sin. 2mk6 n(a — cos. 2k9 — 4/ —1 sin. 2k0) cos. 2mk9 —4/—1 sin. 2mko n(a — cos. 2k6 + 4/—1 sin. 2k9) __ 2cos. 2mk#(a — cos. 2k4) — 2sin. 2mké sin. 2k0 . i n(x? — 2acos. 2k6 + 1) ; and for each pair of imaginary roots, that is, for each of the values 1, 2,3..., er — 1 of k, there will be a partial fraction of the form of this. Let the symbol S denote the sum of these ; then a ¢ wn Lae Sata —1) +f pate ahs eal cos. 2mk6(a — cos. 2k0) — sin. 2mk6 sin. 2k0 (~ — cos. 2k0)? + sin.? 2k =2Ka— 1) je sPi(e+1) + bs cos. 2mk6l(x? — 2x cos. 2kd +- 1) _,«% — cos. 2k0 sin. 2k0 by observing that the last term under the sign of integration 2 : ed S sin. 2mké tan. can be separated into the two fractions 2 _cos. ee — cos, 2k) ie 2 sin. 2mk9 sin. 2k n~ — 2x cos. 2k9 +1 “Hay Die a ame (2 — cos. 2k9)? + sin? 2k9 m—1 212. Integration of es mand n being positive inte- gers, and n an odd number. EXAMPLES. Se In this case, «* — 1 = 0 has but one real root, +1; and the imaginary roots are the values assumed by the expression COS. eT 1 sin. ais by giving to & in succession the palder lt 2, 3... Hence, by operating as in the ae article, we find am 1 capes we —1) sh ly Cos. miket(a — 2x cos. 2k0 + 1) bag he — l _1 © — COS. 2h9. sin. 2k9 ae = 5 sin. 2nké tan. nN gm =! 213. Integration of ~ ey ~__°" m and n being entire and positive, and n even. Under the supposition, none of the roots of #* + 1=0 are real; and the ee roots are found by giving to k, in the a nee . te/—1 Ane zx, the values expression Cos. ? Oe a: te lin succession. Put 6 for , then the partial fractions corresponding to a pair of these roots will be cos. (24 + 1)4 + /—1 sin. (2k + 1)4 aw — cos.(2k + 1)6 — WV —1sin.(2k + 1)0 cos. (2h + 1)9 — /—1 sin. (2k + 1)9 @ — cos. (2k +1)0 ++/—1 sin. (2k + 1)9' the sum of which is 2 cos. m(2k-+-1)@ } x — cos. (2k-+- 1)9 { — sin. m(2k +- 1)@ sin. (2k +- 1)0 tt | 2 — Cos. (2k 1)0{" +L sin.? (2k-+ 1)0 Hence Tyla 1 1 | eee —~= Scos.m(2k-+ 1)0l | @ — 2xcos.(2k-+1)0-+1} x — cos.(2k + 1)0 sin. (2k + 1)9 +. : * sin.m(2k + 1)#tan.—! 358 INTEGRAL CALCULUS. m—l 0 ated) ? finding the partial fractions corresponding to the roots of In like manner, we integrate when n is odd, by x” +-1=0. In this case, there is one real root, — 1; and the other roots, which are imaginary, are the values assumed by the expression | cos. (2k + 1) = at /— Tain (2a = n— by giving to & the values 1.2.3... successively. We should find m—I — ym f= ety) a es aoe +1)0 a” — 2a cos.(2k +1)0+-1 oe a sin.m(2k + 1)étan.—! % — cos. (2h + 1)6- sin.(2k + 1)0 EXAMPLES. y las ees 3 qin Soe Sate e ep 44) eee cen mars Oa iT tn") f Nee ee 6. [eS = alle yee gee’ Be) a ; /3 | tan.—\(2e — 4/3) — tan.—"(Qa + 4/8) . EXAMPLES. 359 ig J ae. This may be rationalized by putting (1 — a)” Fast etoc? 2°: 2 1 whence da=— Beis (1 — AN es ea (2° + 1)° (1 + 23)° dx 2dz Ca eat rr gu wae 1 _,22—1. htt pe in which, if we replace z by its value ee we have the required integral in terms of a. SECUIO Me FORMULA FOR THE INTEGRATION OF BINOMIAL DIFFERENTIALS BY SUCCESSIVE REDUCTION, 214. Tue integration of differentials of the form x(a + ba”)? dx may be made to depend on that of other expressions of the same form, in which the exponent of the variable without the parenthesis, or the exponent of the parenthesis itself, is less than in the original expression. This is accomplished by the method of integration by parts. We have fen(a + bx")?dx sap agin (a+ bx”)? a—dax = fudv Pat or facia MRL oe by making u = x a pean and therefore am mit % ee m—n—-] (4+ Oras sfx (a+ bx")?dx=«x Aa abl, Le pearie Ma RE Se agent She Sane fe (a-+ ba?) Phd fT). The integration of «”(a-+ ba")? is thus brought to that of a™—"(a-+ bx”)? +!, which last is more simple than the first when m is positive, and greater than n, and when p is negative; for then the numerical value of p + 1 is less than p. But we may find a formula in which the exponent of the variable without the parenthesis is diminished, while that of the parenthesis itself is unchanged. Thus we have the identical equation ™ "(a + bar)? Th = o™—"(a + ba”)? (a + ba) = ax™—"(a + bax")? + ba™a + ba”)?; 360 REDUCTION FORMULA. 361 therefore fen—"(a + bx")? *1dxe = afam—r(a + bx")? da | +b fa™(a + bx")? da. Substituting this value in Eq. 1, we have ber yrr m ba")? dr = EAS fa (a+ bx")?dx= a CBee TS m—n+1 gee % Cea (a + ba”)? dx ea eae y — b fx (a+ bx")?dx; whence, by transposition and reduction, ema tlg + ban)Ptt b(m + np + 1) a(m —n +1) ~ O(m + np + 1) The integration of «(a + bx")? dx is then made to depend fea + ba")? dx = eae + bx")?dx (A). on that of «”~"(a + bx”)?dx; and, by another application of the formula, the integration of this last reduces to that of am— (a + bx")?dx, and so on: hence, if m is positive, and greater than n, and ¢ denote the entire part of the quotient ~ the integral to be determined after a number, 7, of reductions, would be a enn + ba")? dx. If m— im =n — 1, this expression is immediately integra- ble; for Bet Oe") PEE nb(p +1) ’ —21+1, andthe condition ernie + bx")? dx = ( m+1 n but m — in = n — 1 leads to of integrability (Art. 202) is then satisfied. 46 362 INTEGRAL CALCULUS. Formula A cannot be applied when m+ np +1=0; for then its second member takes the form o — oo: but in this m+1 nN case + p is equal to zero, that is, an entire number; and the original expression is therefore immediately integrable. 215. Formula for the reduction of the exponent of the parenthesis. Assume x™(a + bx”)? dx = (a+ bu")?d an = Udve then, integrating by parts, e™tl(a, + ba")? m+ 1 — Bi fem (at bat)? de S411). fora + bx")? dz = In this formula, the exponent of the binomial has been diminished by 1, while that of « without the parenthesis has been increased » units. We may, however, diminish the for- mer without increasing the latter exponent. In formula A, last article, change m intom-+n, and p into p — 1: we thus have m+n n\p—l wha, am that be")? fa +"(a + bx”) We hate aes (m + 1)a m n\p—l/-. ~ tne Ean (a+ bx")? di and this value of fa™+t" (a + bx")?~! dx, substituted in Eq. 1, gives, after reduction, eta + bx”)? np + m-- 1 tet coe fen (a + ba")? dn (1 ae fan(a + bx”)?dx = REDUCTION. FORMULZ. 363 By the repeated application of this formula, the exponent p will be diminished by all the units it contains. This formula will not admit of application when np + m-+1=0; but then the integral fa™(a + ba”)? da can be found at once (Art. 202). By means of formule A and B, the integral fa™(a + ba")?da, when m and » are positive, may be made to depend on the more simple integral fa™—(a-+ bu")?—%dx ; in being the greatest multiple of m less than m, and q the entire part of p. 216. Formula for the reduction of the exponent m when m is negative. From Hq. A, Art. 214, by transposition and division, we find am—ntl(q, + ban)P+! eee + bx”)? dx = a(m —n-+ 1) b(m + mp +1) Fm np salt aeneneist a fa (a + ba”)? da. Changing m — n into — m, this becomes ee ee ba® Pk faa + bx”)? dx = — ae 1) ei bin bc 1 ite eet ) fo-m-+n(a 4 bar)” dex (C). If in denote the greatest multiple of contained in m, then, by 7+ 1 applications of this formula, the integration of a—™(a + bx”)? dx will depend on that of a~™*¢FP"(q 4+ bx")? dx; and, if we have —m+(1+1)n=n—1, Gx" )\2 ei we have era -- ba")? da = (a + ba oy But under this supposition, since hae = — 1, an entire number, the original expression is immediately integrable (Art. 202). 364 INTEGRAL CALCULUS. 217. Formula for the reduction of the exponent p when p is negative. From Formula B (Art. 215), we find e™*l(a + ba”)? anp fora + bx”)? dz = — | i at ae fon(a + bu")? dz; and, if in this we replace p — 1 by — p, it becomes | POM aS Oa rras fara + tory-ran = 2 eee 1 me age fee 1 Jena +- bu")-? Fda {D). By the continued application of this formula, the exponent of the binomial will finally be reduced to a positive proper fraction. When p=—1, it cannot be applied; but then the integration of the given expression may be brought to that of a rational fraction. 218. The preceding formule facilitate the integration of binomial differentials; but it is to be observed that the exam- ples to which they are applicable belong to cases of integra- bility before established (Art. 202), and the results may there- fore be obtained independently. By the application of Formula A, we have 1 (ee 28 0 UT ek on Bae Bes ate mn a and, by making m = 1.3.5... successively, this gives ada : Wit a) x* da aA 2 xd REDUCTION FORMULZ: 365 whence ada Ae orase V1 — a? ‘ a da ac? =a, Vie? (s+ Opa - ada act When m is an odd number, we have the general formula a™ da J 1 — 2? i gam—l (m —1)a™— —3 het) Se = m 45 (m — 2)m a +5 ‘ (vi=a and, when m is an even number, ve i tee (1 )ae Leo Bt (iL) ee ee . Ce om en av T=@ 1.3.5...(m—1).. _, a eiieeeer Uy ky ero Ona 2. {——_. ~= fa eae 8 4g?) 3 de, a +. x)? Comparing this with Formula C, and making 1 ese, te re 6: we find 3 de (at tat) ad aa x)? (m — 1)a? gmt ‘Auta f—"—,; (m—1)a?* «2™—?(a? + x)? 366 INTEGRAL CALCULUS. . Without referring to the formula, this expression may be found as follows : — 2 2 f dix (fete ad(a? +- x*)” My 1 1s x™ (a? 4 2)? rs rae go a 2 2 _(a+2%) +(m+1)f—“~+" _ ae dx ; 7 oe omg? 4 a2)? whence, by transposition and reduction, 1 die a? + a)? d (m + yar f— “2 _, EE) _ 4, f__S _, ere dats Cs fe + a*)* gam +l Aehdee (e 2 NG ii dx See Ce x?) e™(a? + 2)? (m—1l)a?a™! m — 2 dx (m — 1)a? oq? 4 gp) . (Mx+N)dx 219. By means of Formula D, the expression (oT cee ee which occurs in Art. 210, may be integrated by successive reduction. Let a+ ba/—1, «—B/—1, be the roots of the equation 2? +pxe+q—0; then (Mx + N)dax (Mx + N )dax (a? + pe +9)" | (a — a)? + 6?}" _ Mle ade Dy i yo [e—aptey" | 3 Te a By? Putting « — a@=2, we find M(a—a)de ‘ Mee Steep + ep 5 ae Bl 5 1 ib2@.— 1) (ee M tl REDUCTION FORMULZ. 367 Making « —a=y, and Ma + N= WM’, we have fe Meck Mis f MH parry and therefore, by combining these results, we have f (Mea +- NV) dic 2 M 1 | (a — a)? + B?}” 2(n—1) {(w@—a)+ ptr +f" (y? + By "dy. Formula D may now be applied to the term under the sign ‘f in the second member of this last equation; and, by repeated applications, the exponent — n will be reduced to — 1, when the integral will be completely determined. 220. Reduction formule may also be constructed to facili- tate the integration of trigonometrical functions. Let the integral of sin.?a cos.’adzx be required. Make sin.2 = z; then . BK paint cos. a = (1 — 27)", de = (1—2?) “dz, q—l and sin.?xcos.%xda = 2?(1 — 2”) ? dz. Now, if g be an odd number, whether positive or negative, we may always effect the integration of z?(1 — ay ae, what- ever may be the value of ». In like manner, by making cos. = 2, we see that the integration can be effected when p is an odd number, whether positive or negative, whatever be the value of q eee! Formule A and B, we get yP— 1 — 27) 2 q—l UA EN ty 2 Je fe Z*) 2 dz ar 25h fera—ay Fide (a) 368 INTEGRAL CALCULUS. fea a wy? de = Seager) 28 ps (mae | —3 Beene. =f e(1 — 2?) © de | (B’). wach g If p < 0, Formula C gives g—l Pi ue vo = 5 Fan fe? — 2°) 2 d= — AL ld p—l == ; = +PoI—*fa-7411 9!) Fda (0%); and when a Wee 0, by For mula D, we have ) fou ortMyata—e" P| ee 2) F de (D). By the aid of the foregoing formulz, we are enabled to make the integration of sin.?x~cos.%adx depend on that of expres- sions in which the exponents p and q are numerically less than in the original function. From (A’) we have fosin Px cos.2ada = sin.? “'wcos.? Fix Pd 4+ PEA fsine-tecos.tad (1); and from (B’), sin.? tlycos.2—!a Prd +i fsin2 cos.?—"ada. (2). sine cos.!ada — When q is positive, by the application of (2), fsin.?a cos. fada will finally depend on fsin.?adx, or on fsin.?# cos. xda, according as q is even or odd. REDUCTION FORMULZ. 369 By making g = 0, in (1), we have ag et : sin.?—*xcos.« , p—I1e. J sin.2xde = Jsin.? ade; L and thus, if p be a positive integer, and even, fsin.? ada will at last depend on bh: dza—x; andif pbea positive integer, and odd, the final integral to be found is fsin.wda = — cos.” In the second case, that is, when g is odd, we have sin.? tly pri It is therefore always possible to find the integral J sin.? cos. ade = fsin.?adsin.« = of sin.” acos.? ada when p and q are entire and positive. Formule 1 and 2 are inapplicable when p= — q; but in this case fi sin.? a cos.%ada = f tan.? dx = fh tan.?—2a tan2ada = ftan.?~* «(seca — 1)dax = ftan.?—? ad tan. x — ftan.? ada tan? g(x) for all values of x from x=a to x=); then, taking f(x) — g(x) for the differential co-efficient of another function, we should have [\F@ — 92) } de >0; DEFINITE INTEGRALS. 377 since the derivative f(x) — g(a) is constantly positive be- tween the limits a and 0, and the function f'} /() ~- a (2)} de is an increasing function of 2: hence f fo)ae > f 9 @)dx, Also if g,(«) is another function of x, such that g,(x) > /(x), for all values of « between the limits a and b, we should have f f@)dx< 1 pi(x)dx: therefore fo ride > fPfw)dx > f* 9(w)de. When a given differential cannot be integrated, it is desira- ble, and sometimes possible, to find two other integrals be- tween which the required integral, at assigned limits, will be included. EXAMPLE. J 7 sae For values of « between 0 and (1-2)! 1, we have l< 1 1 TTR TE VT ee age cmen yg (Lien (1 —'x?)* |S ewe oA ras a—ayt 46 aa} 0.5< il = 0.5236. bho] dic Ants ——— < sin. (L— x?)" 226. Demonstration of Taylor’s Theorem dependent on the properties of definite integrals. The equation f(e+h) f(a) = ff @+h— oat 48 378 INTEGRAL CALCULUS. is identically true; but successive integration by parts gives [i P@th—Odt=o (@+h—N+f Yr (@+h—ddt, fi tf! (a +h —t)dt = © Pet h —t) S, Seth — t)dt, lies se (w+ h—t)dt= 75-5 f"(w@+h—t) “f agli (eth— tat ‘ n—1 firma th id= Pe thee +f,i Making ¢=/ in these equations, and then adding them — f(a +h — tht. member to member, we have J (@ +h) er wae 1 rats f.nmee If the function to be expanded, and also its differential co- efficients up to the order denoted by n+1, are finite, and continuous between the limits x and w# +h, the residual term 1 h (n +1) as ‘4 oes ifs Sf (7 + h —t)dt may be replaced by i (n+) ume 2, ae (tO and the expansion then agrees with that of Art. 61. DEFINITE INTEGRALS. 379 227. In what precedes, it has been supposed that the limits a and b of the definite integral sf ; J (x)dx were finite, and that the function /(x) was also finite, and continuous between these same limits. It may happen that one of the limits, 8, becomes infinite while the other is finite, the function re- maining finite and continuous. Then the value of the inte- gral is the limit of the value of OL: when 0 is increased without limit. This value may be finite, infinite, or indeter- minate. EXAMPLE 1. fede. 0 For the indefinite integral, we have fede =—e "+0: 2 fi erde=i—S, 0 foetde=1 ee 1 : € FIX. 2: ff, era. The indefinite integral is ferda=er+ a: [ e* dan 67 1) — 0. 0 380 INTEGRAL CALCULUS. Hix. 4. f° cos. ade 0 In this case, fcos.cdx = sin.a + C; and, taking the inte- gral between 0 and the finite limit 0, we have b : J cos.“2dxe = sin.b; 0 but, when 6 becomes infinite, the value of sin.b will be inde- terminate, though confined within the limits 0 and 1. The following investigation will sometimes enable us to decide whether the definite integral i f J (x)dx is finite or infinite forb=co orb=—o. First suppose that 6 is very great, but not infinite, and let ec be a number comprised between a and 6; then (Art. 224) b c b [ fwjde =f" f(a)de + f f(w)de. Since /(x) is finite, [" f(w)de is also finite; and it remains only to examine the value of f f@de when 6 becomes in- finite. fee (2) Ze) g(x) being a function that remains finite for all values of x greater than c. If A denotes the greatest and B the least of the values of g(a) for all values of x greater than c, we shall have A BY b od J F@de ele b A 1 1 oF freee <5 (si- 5) DEFINITE INTEGRALS. 381. Now, when n > 1, the second member of this last inequality A 1 : Cteytte 1: hence, in this case, we for b=oo reduces to know that the integral if “f(a)de has a finite value. When ; n <1, we have ig fi(ajdx.> Bic ae ave [fod Pe we Gi —c}—"), Now, when 1 — n > 0, the second member of this inequality becomes infinite for b =o: hence f f@ae, and therefore f ° f(x)dx is infinite for b= 0 . Te m = 1, then ['serde > B'S = wi; but (2) =—oo whenbU=o: hence f J (a)da =a’. Putting f(x) under the form Bee g(x) being a function that is finite for all values of « between — o and some value less than 0, it may be shown in like manner that {i ° J (a)da is finite if m > 1, and infinite ifn << lorn=1. Thus, if it be possible to put f(x) under the form ue) and the condition imposed on g(a) be satisfied, we can decide whether the integral if : J (x)dx is finite or infinite when one of its limits becomes + 0 or —o. 228. Definite integrals in which the function under the sien of integration becomes infinite between or at the limits. 382 INTEGRAL CALCULUS. The function /(x) may become infinite at one of the limits, b, of the integral if : j(x)dx; in which case the integral is defined as the limit of ff (w)de when @ is decreased with- out limit. In like manner, if /(«) becomes infinite for «=a, 6 aie b ae then 5 J (x)dx is the limit of ff (ade when « is indefi- nitely decreased. Finally, if /(c) =, e being comprised between a and b, we should have b : c—O 3 b f f@de= lim. fi J (x)de + lim. ff (@)am, when @ and £ are decreased without limit. Should there be more than one value of x for which /(x) becomes infinite be- tween the limits a and 0, we learn from what precedes how to define the integral ff (a)dz. 229. It may sometimes be decided whether the integral if : J (a)dx is finite or infinite when /(x) is infinite at one of the limits. Suppose /(b) = oo, and let f(x) = (Ne: ua eae gy (x) being finite for « = b and for all values of x < 0b, and n being Ps), If ¢ be a number comprised between a and 3b, we have b c f f@de= [fade + fo sade. Now f° f(@)de is finite; and hence fF (2)de will be finite or infinite according as if s J/(x)dz is finite or infinite. Denote by A the greatest and by B the least of the values of g(x) for values of w included between c and’. If n < 1, DEFINITE INTEGRALS. 383 we shall have, for such values of x, f(x) < esis ty and (6 — a)”’ therefore fo re@de < f= Geo a}. When @ converges towards 0, the second member of this in- A faa! a Came equality converges towards the finite value hence, in this case, the value of lim. Hi oy J (x)dx, and there- 5 a eye : fore of [ J (x)dzx, is finite. But if n 4 1, the proposed integral is infinite; for, since I(2) >= i= yi we have b- Bdx B 1 1 if “f(a)de ee iis (6 — a) prede TE | Hee me eat and it is evident, that, when « becomes 0, the second member of this inequality becomes infinite: hence, under the supposi- tion, f F(@)ae is infinite. In like manner, when n = 1, we have f(x) > en and therefore van b— 06 cso b—c de * f(x) da > i} a - b ns: 5 ‘ ; But soem becomes infinite when a vanishes: hence f f@)de, and therefore f Sade, is then infinite. EXAMPLE 1. i : ETE — a — 0X — X” P being a function of « that remains finite for all finite 384 INTEGRAL CALCULUS. values of x, and a and b being two positive quantities, we have . | Py fee ey P 1... “Spee ee Nt ee by putting Soa a7) Since the exponent of 6 — « is less than 1, it follows from the rule just established that the proposed integral has a finite value. 1 dz dx Xe ee —=——=* We have 9) s-ccee ee vice We have | 7 . 1a 00. 0 i/o: 0 / 1 — 2 MEN ice: is the differential of the area includ- ae V/1—ax da The expression —- V/1—x ed between the axis of x, and the curve having y¥ = for its equation. This curve has two asymptotes; the one the axis of a, and the other a parallel to the axis of y, and at the distance +1 from it. It is Be from the figure the f ae rep- resents the area bounded by AQ, AB, the curve, and its asymptote BD; and this area, although it extends indefinitely in the direc- tion of the asymptote BD, still has the finite value 2 230. A definite integral may become indeterminate, as is the case for co . if sin. xda == cos. 0 — cos. 0, 0 INTEGRATION BY SERIES. 385 since cos.2, when «& is indefinitely increased, does not con- verge towards any determinate limit. For another example, take J sis ee , In which a and 6 are —a ie 1 : any two positive quantities whatever. Since — becomes in- x finite for the value x= 0, which is comprised between — a and +b, we put ee 3 +o dx. fo gam. fod tim. —; +8 2 a and $ being numbers numerically less than a and 6 respec- tively, and the limits indicated being those answering to Gee tee. “But [i gsta—t, ees 1B x iat ath, eae bts, )+2(5). Therefore tie ee — i) + lim. (5): ON > he : The first term (7) in the value of this integral is deter- minate; but, since the variables @ and £ are entirely inde- pendent of each other, the term lim. (5) does not converge towards any fixed limit, and the integral is therefore indeter- minate. 231. When the integral of Xdz is required, and X can be developed into a converging series, ASUytt,fust-+u,+7, (1), we shall have, after multiplying by da, and integrating between the limits a and 6, iE Xdo= fi wey dar +f wadart + fe, d+ fr r,dx (2). 49 386 INTEGRAL CALCULUS. If series (1) is converging for «=a, « =), and also for all values of x between a and b, we may assume r, < @; a being less than any assigned quantity when n is taken suf- ficiently great. Whence fo rade < fi ada, or f r,dx < a(b—a). Therefore } . y,dx will decrease without limit when 7% is in- a creased without limit; whence the series b Baie b i} wdc + f U,Axe +t... Ete U,ae a a a 7 is converging, and its sum is the expression for J Xdx. The OS es fixed limit 6 may be replaced by the variable x, provided no values of x are admitted which fail to render series (1) con- verging. We should thus have | f° Xe = [i usde + [ude + res + [unde | (3). 232. Formula 3 of the last article still holds true for « = 8, even though the series uw, + u, + u,-+ ---, whichis supposed converging for « < b, becomes diverging for « =), if, at the same time, Series 2 is converging. For, however small the quantity @ may be, we have fo xax lias u, dex lee TA ee ee, + fade, The two members of this equation are continuous functions of x, and are constantly equal; hence their limits fora=0 must be equal, and therefore {FP Xdx= fo mde+ fi ee a INTEGRATION BY SERIES. 387 If the series 2 F(@) =f (0) + 2f'(0) + po S(O) +, to which the development of /(x) by Maclaurin’s Formula gives rise, is converging, we shall have [fl@)dx = 0+ af (0) + £5 10) + yg SO) 45 and, if it is wished that this integral should begin with «= 0, C must be zero; and we then have [Pode =f) + FLO + pag OF EXAMPLE 1. {cS =l1+ <2). By division, or by the Binomial Formula, we have eye's 9 8 eet — eet..-toe eet ae gt ao” a ada —r~— SI cs se é eas Tt+e- 4 Waa vera When 2 is numerically less than 1, positive or negative, the series 1 —v+ x?— @’... is converging, and therefore so also 2 a3 a4 is the series « — 3 + ant artsy between the same limits for >-1 x: hence, when «<4,? ze ge4 Cee ae~ 2 ee ies It may be shown by direct demonstration, that [~ a" dn converges towards 0 as n approaches oo. ° : a4 _ For, if x is positive, we have 5 wows as we”: therefore x ada ae ite eae ae 388 INTEGRAL CALCULUS. n—1 n+ 1 in stopping with any term of the series, the error will be less Now, as 7 increases, approaches 0; and consequently, than the following term, and will be additive or subtractive according as the last term taken is of an even or odd order. If « is negative, and @ denotes.a number greater than z, ig 1 hs Te < ee and therefore 2 w2"dx eiimee 9 L—aw ~(n+1)\(1— a) The limit of the second member of this inequality for n = 00 but less than 1, we have is zero. In this case, the error is always numerically additive. When a = I, the series 1— a+ x?— x@*-+---- is no longer 2 a3 converging; but the series x — a a y —--- is (Art. 231), and will represent the value of /2: hence gnrrl We have gS let eh es n being an odd number, and positive. Integrating, and taking for tan.~'x the least positive arc having w for the tangent, we find 3 5 n n-+1ld. pies oo ‘ee tan. | @ == & 3 1s $F, [cag The series 1 — x? + «*— «a®... ceases to be converging for 3 5 2 ==1; but the series x — = -{- " --- is still converging for this value of x: hence ae te tan. oy ae gtk - INTEGRATION BY SERIES. 389 da ee in ho Ex. 3. ip Vi = = sin We have 1 135 Vi-a 1+ 5245 ig aa So From this, by multiplying by dz, and integrating, we find Agr earns on cea (tae aes —1 PE, Ne re tae ope Sek aa ace iy se a : ; on Lee : ; , a converging series when 22, ,) since Series 1 is converging between these limits. The series 35 ne 2 a ea ae ice eR we Lt 50 +5 gets ae? is not converging when «=1; but since, for s=1=sin. _ the series 1 x3 E39. 3° Ls OTmipy acs Samo y ka eo ee is converging (Art. 231),* we have 2 AK es 131.1351 salts stoastaag@t™ A still more converging series is found by making whence * Space does not allow the proof of convergence or divergence when these con- ditions are asserted relative to the series involved in the last three examples. (See Art. 68.) 390 INTEGRAL CALCULUS. 2338. By integrating /(x)dx by parts, we have [Ff (2)du = af (x) — fxf’ (x)da, fap (ade =" p(x) — f Spr(a)d, fe of cde =f" (@) = fda The combination of these results gives ue a* JF (ada = af(x) — tat () =i ta37(*) “fa ee ae ee ae fO-Y (a) el i ma 5 Sent (wd This is the series of John Bernouilli, and may be advanta- geously used in many cases: for example, if /(x) be a rational algebraic function of (n —1)™ degree, f™(x) is 0, and the series will terminate; or there may be cases when can be more readily found than f f(a)dx, or when only an approximate value of if “f(«)dx may be required, and the in- 0 tegral fe sr@de may be small enough to be neglected without sensible error. 234, Assuming f /(«)de = g(x), and making «=2#-h, we have, by Taylor’s Formula, (+ h) — 9(%) = he’ (x) + m9" (2) “Era But, because f /(x)da = g(a), we have J(%) = 9'(%), f'(&) = 9" (x), f" (@) = g” (2). INTEGRATION BY SERIES. 391 These values, substituted in (1), give h? h? g(a +h) — 9(2) = Bf a) + 5 (2) tog (a) + In this series, making «=a, h=6—a, and denoting by A,, Ay, A;..., what f(x), f(x), f”(x)..., become under this supposition, then g(x +h) — g(a) becomes 7 p(0)—g(a)= ff (x)de, and we have b mr. A, 2 A, preg 3 J f@de= Ab SUD aa ee (B greet ae) SPs This series enables us to find the approximate value of the definite integral f fede when 6 — ais sufficiently small to make the series converging. When this is not the case, or when the series does not converge rapidly enough for our purposes, put b— a= na, and take the integral successively between the limits a and a+ a,a-+aanda- 2a, and so on, denoting the results by B. B Brat eG a? + as aire i 9 C; 3 Cia+ 19% +753% ret D, 2 D, 3 e POE ee ENE I ah then (Art. 224) we have [ Sla)da= (Bit G4 Dy + +-+)0+ (Bat Cot Dat +-)o? + (B3-0;+D,+ +: )a°-+---, a series that may be made to converge as rapidly as we please by making @ sufficiently small. SGT O Nee GEOMETRICAL APPLICATIONS. QUADRATURE OF PLANE CURVES REFERRED TO RECTILINEAR CO- ORDINATES.— QUADRATURE OF PLANE CURVES REFERRED TO POLAR CO-ORDINATES. 235. The quadrature of a curve is the operation of finding the area bounded in whole or in part by the curve. If w denote the indefinite area limited by the curve, the axis of x, and any two ordinates, it was found (Art. 164) that du = ydx =f (x)da ; y =f (x) being the equation of the curve referred to the rec- tangular axis Ox, Oy. If it is desired to have the area limited on one side by the fixed ordinate C'A, correspond- ing to the abscissa « = OA =a, > the integral must begin at x=a; and we have U =" f(2)de. Finally, if.the area is to be limited on the other side by the ordinate BD, corresponding to x = OB =), we have u = area ACDB Bale J (x)dax. When the co-ordinate axes are oblique, making with each other the angle o, then u— area ACDB = sin. ie I (au)dz. 392 QUADRATURE OF CURVES. 393 236. The definite integral is the limit of the sum, taken between assigned limits, of an infinite number of infinitely small areas (Art. 192). Observing that /(x) dx is equivalent to f(a) Aa, if we suppose Aw = da to be positive, the element J (x) Ax will have the sign of f(x). Consequently the integral will represent the difference between the sum of the segments situated above the axis of x and the sum of the segments situated below. If, for example, the ordinate changes, as in the figure, from positive to negative, and then from negative to positive, the area between the ordinates AC, BD, will be f f(@)dx = ACL — LMN+ NBD; and if OL =h, ON=k, the sum of these segments will be expressed by ff sede —f sode+ f so)dn, 237. If y=/f(x) is the equation of the curve CM, and y, = w(x) that of the curve C/I, and the area bounded by these curves and the ordinates AC, BM, corresponding to «=a, x2 = b, is required, we have Area COMM =f f(a)de —f w(a)de =f} 7@)—v@) | an, 50 394 . INTEGRAL CALCULUS. ° EXAMPLES. Examp.eE 1. The family of parabolas is represented by the equation y”= px”, m and n being positive. We have 1 om du = ydx = p” x" dz, and xz lom é fiptdhan= pg which may be written Le ae tie n MS en be ee eee But xy measures the area of the rec- tangle OPMN, contained by the co-ordi- nates of the point J Hence, from the above formula, we have OPM: OPMN::n: m+n, OPM: OMN::n that is, the arc of the parabola divides the rectangle con- structed on the co-ordinates of its extreme point into parts having the ratio of n: m. Reciprocally, the property just enunciated belo to the parabolas alone ; for the proportion OPM: OMN:: 0 may be written UsrXY—UIENSM. Hence (m+ n) u = ney, and, by differentiation, we have (m+ n)du = nady +- nydzx ; or, since du = ydza, mydx = nady : dx dy whence m—_—=n x x QUADRATURE OF CURVES. 395 Integrating nly = mlx 4+- C, or ly® = la™ + C, putting lp for C, we have ge (pe or. a po™ for the general equation of the curves which possess the property in question. For the ordinary parabola in which n = 2,m=1, we have Ex. 2. The hyperbolas referred to their asymptotes are rep- resented by the equation x” y” = p, m and n being entire and positive numbers. Assume the asymptotes to be rec- tangular, and let WCW be the branch of the curve situated in the angle xOy. Suppose n> m, and let w= area AOMP, OA =a, OP=2; then x zr 1 _m w= f yd = | nn” n da ni pie Fide n LOY! Adee! aoe or a Bee eee hs, 3 | ak As x increases, so also does wu, or the area ACMP; and x and u become infinite at the same time. If, however, we suppose PM to be fixed, and a to decrease, the surface, while continu- ally increasing, will remain finite ; and at the limit, when a= 0, 1 n—m it reduces to ee pn a ". Hence the surface PUNLZ n—-m approaches a fixed limit as the point NV approaches the asymptote Oy. 896 INTEGRAL CALCULUS. This limit, which may be written xy, bears to the m— mM rectangle PMBO the constant ratio of n to n—m: since, denoting this limit by u, we have nN A—MIni: cy, Uu= n—m LY. The converse of this is also true; that is, no curves, ex- cept those represented by the equation «” y” = p, possess this property: for, from the preceding proportion, we have u(n —m) = nxy, which, differentiated, gives (n — m) du = nady + nydz ; from which, by substitution and reduction, we have Integrating . ed40 nly = — mlx ; making C= lp, then ly” = I= : hence 2p? ae When m=n, the general equation takes the form zy = p, which is that of the equilateral hyperbola of the second degree ; and we have da Yer a ydx = p Bie and therefore u=ple+ C= pl= by making C =a When p = 1 anda=1,we have u=—Iz; and the area is then equal to the Napierian logarithm of the abscissa. Ex. 3. The equation of the circle, referred to its centre and rectangular axes, is oi 9? eee, of — A/G ee QUADRATURE OF CURVES. 397 and ydxc = */a*— «dx is the differential of the area of a segment limited by the axis of y and an arbitrary ordinate PM. Denoting this area by wu, we have af / a — x de. Hence (Ex. 2, p. 326) 1 Se le x Us 2 2 YE a eed u=zaVa — 2 + 9 sin.~! 7 From this we deduce the area of the sector OBM; for the area of the triangle OMP is measured by 5m ¥/ a? — x, which, subtracted from the expression for w, gives 2 sector OBM = ee sin,—! za prey} = sin.—! = — ie 2 a 2 Gen that is, the area of a circular sector is measured by its arc arc MB: multiplied by one-half of the radius. Ex. 4. If a and 6 denote the semi-axes of an ellipse, the equation of the curve referred to its centre and axes is Gey 0m ater. fs Let w denote the area of a seg- ment bounded by the axis of y and any ordinate, as PI; then b ar w= | / a? — xv? de. ad 0 Describe a circle on 2a as a diame- ter, and denote by w/ the area of the segment BMPO; then My 5) wf VJ/ a? — xv? da. 398 INTEGRAL CALCULUS. ? b ub ows Me Ue sad OTe ee a we That is, the segment of the ellipse is, to the segment of the circle which corresponds to the same abscissa, in the constant ratio of 6 to a; and therefore, denoting the entire area of the ellipse by A, and that of the circle by 4’, we have Ais ee Dee and, since 4’ = za’, it follows that b A=-—2a* = zab. a Hence the area of the ellipse is a mean proportional be- tween the areas of two circles, having for diameters, the one the transverse, and the other the conjugate, axis of the ellipse. The ordinates PM, PN, are to each other as a to b, and hence the triangles OPM, OPN, are in the same proportion ; that is, OPN tan Neem 2 em OPM PM =a)" ut wa u—OPN_b | OON_d. w — OPM a’ ° OBM™ a’ and thus the area of the elliptical sector may be found in g terms of the area of the corresponding circular sector. An ellipse may be divided into any number of equal sectors when we know how to effect this division in a circle. It would only be necessary to describe a circle on the major axis of the ellipse as a diameter, then divide the circle into the required number of equal sectors, and through the points in which the circumference is divided draw ordinates to the major axis of the ellipse. The sectors formed by joining the centre with the points in which these ordinates cut the ellipse will be equal. QUADRATURE OF CURVES. 399 Ex. 5. The equation of hyperbola is a? y? — b? x? = — a’b’, or y=" Va? a2; and the area of J M the segment 4//P is expressed by Oh ieee w=— f Va — a de. 0} A P x Hence (Ex. 6, p. 328) AMP = 90 Vm a —Fi(2t VEO), 2 a Ex. 6. The differential equation of the cycloid (Art. 146) is dy Tete aides ee, 4 Nos; : / 2ry — y? a Oe Pea ay . U = fyte= J Fo This integral may be found by Ex. 1, p. 370: the following, however, is a more simple process. Put NM = 2r —y =z; then, denoting the area OLNM by wu’, we have ul = fada = [(2r —y) da = f/2ry — dy ; observing that the limits between which these integrals are taken must correspond to z= 2r and z=>2r—y. But [V2 —¥ dy is evidently the expression for the area of the segment of a circle of which ris the radius; the segment taking its origin at the ex- tremity of a diameter, and having y for its base. This segment is represented by ADB. The area OL NM takes its origin from OZ, and the cir- cular segment from the point A, and both areas are zero when 400 INTEGRAL CALCULUS. y =0: hence the constant of integration is 0,and we constantly have area OLNM= segment ADB. When y = 2r, the segment becomes the semicircle ADC, ig ee 1 which is measured by a But rectangle OLCA = OA X AC =ar X 2r = 277"; that is, the rectangle is twice the area of the generating cir- cle: hence g 2 and therefore the area bounded by a single branch of the area semi-cycloid OMCA = 2 cycloid and its base is three times the area of the generating circle; or, in other words, this area is three-fourths of the rec- tangle having for its base the circumference, and its altitude the diameter, of the generating circle. Although the area of the cycloid may be said to be thus represented by a part of a rectangle, it is not a quadrable curve; for the base of the rectangle cannot be accurately determined by geometrical processes. 238. Quadrature of curves referred to polar co-ordinates. The differential of the area PRM (Art. 165) is du = ; rdo: hence S U == f rray, the limits of this integral being g the values of 6 corresponding to the points & and I. Example 1. Applying this formula to the logarithmic spiral, of which the equation is 7 = ae”?, we have QUADRATURE OF CURVES. AQ] one a* ; 2m@g dé — a? 2meg neat ad wat fe et o=_ te Put PI =,', and in the formula make r= r’; then R yl? y!? A —————_———. 1 ike eee ao ae ") ay, The figure supposes PA to be the initial position of the radius vector; that is, the position at which 6 = 0 and r = PA =a, and also that 6 is positive when the motion of the radius vector is in the direction of the motion of the hands of a watch. Hence, when the generating point moves in the direc- tion from A towards B, 0 is negative. Let the motion take place in this direction from the fixed radius vector PR = 7; then, after an infinite number of revolutions, 7’ becomes 0, yr? and the expression for wu reduces to u = i m 2. When the length of the radius vector of a spiral is pro- portional to the angle through which it has moved from its initial position, its extremity describes the spiral of Archi- medes. The equation of this ‘spiral is r= a6; and hence ry = a when 6 = 57°.29578 of the circumference of a circle to the radius 1. For this spiral, we have ee 2 pil 292 peel 293 : w= 5) rido = 5 | a20%do= a9 + 0; and, if the area begins when 6=0, C= 0, and u = = ag? When 6 = 2z, u — area PAB a a’ is the area de- scribed by the radius vector during the first revolution. In 51 402 INTEGRAL CALCULUS. the second revolution, the radius vector again describes this area, and also the area PBA'B' included between the first and second spires. Hence the area PBA’B' is measured by ey giag 3 __ 28 2 93 ri (477) (2x)?! = a mt. It is evident that during any, as the m, revolution, the radius vector describes the whole area out to the m™ spire, and that, to find this area, the integral eee: 242 ra: 293 waz farords == a0 must be taken between the limits (m — 1)2a and 2ma, which will give for this area denoted by w” alae a*(m2n)* — ; a*(m — 1)? (2)? ~ 702(2n)?| m® — (m —1)3I. In like manner, we have for the entire area denoted by w’, out to the (m— 1)" spire, w' == a? (2n)? (m — 1)? —(m — 2)*t: “ui —4' = 7 a3(2n)?| m* — 2(m— 1)? + (m — 2)%, which is the expression for the area included between the (m — 1)™ and the m"™ spires. If we suppose a = , this formula becomes wu" — 14! = % Oat | m?— 2 (m— 1)8-+ (m —2)! 2 ae AS. 3 am 3 maddie aaa cad Tu bik 4) at ee and in this, making m= 2, we find 2m for the area included between the 1* and 2° spires. Hence the area included be- QUADRATURE OF CURVES. 403 tween the (m—1)™ and m™ spires is m— 1 times that included between the 1*t and 2° spires. 239. The quadrature of curvilinear areas is sometimes facilitated by transforming rectilinear into polar co-ordinates. Take, for example, the foliwm of Descartes, which, referred to rectangular axes, is represent- ed by the equation v* + y* — axy = 0. This curve is composed of two branches, infinite in extent, which intersect at the origin of co-ordi- nates, and which have for a com- mon asymptote the straight line of which the equation is a To determine the area of any portion of this curve in terms of the primitive co-ordinates, we must find what the integral of ydx becomes when in it the value of y derived from the equa- tion of the curve is substituted. This requires the solution of an equation of the third degree ; but if rectilinear be changed into polar co-ordinates, the pole being at O, there will be but one value of the radius vector in any assumed direction; for, the origin being a double point, two values of 7, each equal to zero, must satisfy the polar equation of the curve, and the first member of this equation must be divisible by 7”. Ox being the polar axis, the transformed equation is r*(cos.? 6 + sin.? 0) — ay* sin. 0 cos. 6 = 0: whence __ asin. 6 cos. 6 ~ gin. é + cos. 6 6 1 , For the area of the segment OMN, we have u = a J ro, 404 INTEGRAL CALCULUS. which, by substituting for 7 its value above, becomes a ke ? sin. cos.? 4 Dak pe sin.” 6d0 2 » (cos.20-+-sin.29)? ~~ 2/ , cos.40(1 + tan36)? 6 i] 2 a? ee cos. 6 "21 (tan 0 To effect this integration, put 1-+ tant O0=2: .°. dz=3 tan2¢6 — and hence Sane ee cos."6 1 pdz fal (isttanZ¢}e7 8 Pees 1! i; Bevo RE TAT) a? 1 . a ee o*. uf 6 Taotanee Y a? The area beginning when 9 = 0, we have C==, and con- sequently fr a? tan.* 0 7a) 6 det tans The entire area OMDZ is found by making 6 = 5 in the 2 “valuc of u, which then becomes alg for then the fraction 6 TaD. 0) hae “Aertan 0° SECTION VI. RECTIFICATION OF PLANE CURVES. 240. Tue rectification of a curve is the operation of find- ing its length, and the curve is said to be rectufiable when this length can be represented by a straight line. Denoting by s the arc of a curve comprised between a fixed point and an arbitrary point (x, y), we have Se eee Se 2 ds = / daz? + dy? = de) + we (Art. 161); and, by integration, a i cael bE ca ATE : J x | eels da? By means of the equation of the curve, ds may be expressed in terms of either x or y; and, the integral being then taken between the assigned limits, we have the length of the curve. ExampPLe 1. The Common Parabola. From the equa- tion y? = 2px of the curve, we find ydy = pdx, dx = ee. This value of dz, substituted in the differential formula, gives Tne Sie Ch ee da [UU + dy = Vy Ep on a ae whence, making the are begin at the vertex of the parabola, 1 py 1 Ee s= = da ata? uk yvy p = VP te t+ gly t Vy FP) + C, by Ex. 5, page 327. 405 406 INTEGRAL CALCULUS. Since the integral is to be zero, for y = 0 we have jes Bey Recon IE O= Ge Te: o= 5 Ps By the substitution of this value of C, the formula becomes ep pane Mawes yt vie Ex. 2. The Ellipse. From the equation of the curve ay? + b2 a? — 7b we get BY Wenn ae dx ary hence ph jgel | pi kbtet Pipe ds = de | pee =e fi ae a?(a2b? — b?)’ PY ane PPC ER) 1 PMS ES 1 — 10 oe a? —e*a or ds =dz.| ( ) tc ae a? (a? — &*) ea in which e= vie is the eccentricity of the ellipse. Suppose the arc CN to be estimated from the vertex C of the minor axis; then, to get the length of the arc CNA, the integral of the expression for ds must be taken between the limits a = 0 and ea; but all the values of & between 0 and a will be given by 2 —asin.g, the angle g varying be- tween 0 and 5" The substitution in the value of ds of these values of x and its differential gives ds=av/1 — e’sin’ gd; RECTIFICATION OF CURVES. (AOT and therefore pF afl — e*sin’ g dg. 0 This integral belongs to a class of functions for which we have no expression except under the sign of integration ; and, to find its approximate value, we must have recourse to a series. The Binomial Formula gives nie $ ae ae ee oe (1 — e?sin2g)° = 1 — ge sing a5 qe sin.’ Lg Ss Sow a hence, for the arc C/N, we have € aeons e®sin.®g — ne 6 let 24 CO] Nr sag — : ae? f sin? gd — 5 sy qe f sins gd moa Jf sin. gdp... The integrals in the second member of this value of s may be found by applying Formula 1 of Art. 221. We should thus get, by taking all the Sa Nee between the limits 0 and 5 a1), in 24/1 — e?sin. *gdg = 5 — es A 246 hence, for the arc CNA, we have rh LNT a EN ~ 1 (1-3 5oN 2 kak Se ea Pere ey Of Yes age ae (*) AG ) ere) SEGRE). aaa ‘) This is a converging series, and the more rapidly so as e becomes less, or as @ and 6 approach equality. When the 408 INTEGRAL CALCULUS. eccentricity is very small, it would be sufficient to compute but a few of the terms of the series. This value of the arc CNA may be found without using Formula 1 of Art. 221; for, assuming the first equation in that nm 6)? 2 article, and taking the integral between the limits 0 and we have Tv 7: ie a (Bs sin.” gdg = ce le sin.” —* pd. ; m 0 In like manner, a er bie a fi sinetgdy =P T = ne 0 he 2 ain m—4 — 2 ain m—6 iE sit. a dg = ; ai) sin. aS, Multiplymg these equations member by member, there results — a __ (m—1)(m — 3)(m —5)...8 mt ip spiel fe las m(m — 2)(m—4)...4.2 2 The values given by this, by making m equal to 2, 4, 6,..., successively substituted in the value of s, lead to the result before found. The angle g is found by the following construction: On the major axis as a diameter describe a circle; produce the ordi- nate PN to meet the circumference at J, and draw OM; then e= OP OM cos. POM = asmib eae hence mg = angle BOM. Kx. 3. The Hyperbola. Assuming the equation ary? 2? — oq) RECTIFICATION OF CURVES. 409 of this curve, and proceeding as in the case of the ellipse, we get i San (et b?)x:? ie at a? (x? es a”) To simplify, put a? + 6? = ae; then Atenas v2 ds =da J iba saaed . x —a Now, for one branch of the hyperbola, all admissible values of « are comprised between + a and +o, and for the other branch such values are comprised between — a and — 0; and it is evident that all of these values will be given by the : a . nm tio ch a ad @) kine vary between 0 and — f equation x ee y making 9 y n 5 or one branch, and between : and z for the other. ad Substituting this value of x, we have Di asin. pd cos2g ’ a*e* — a*cos.? ae Cos.” 2p cos.” ~ whence (fig. Ex. 5, p. 899) cer [Ge 1 ij aaa 0 COS.” @ e Developing the radical in this integral, we get sae e sts Cosy qe rai COS. y COS.” p Dees 24 e' 1.1.3.5...(2% — 3) ore ha YP; 2.4,.6...27 en or $8 = ae tan mag 4g Sr ihoaeey @ pel 1 cos’g , 11 3 cos. ahs 2 e? 5 2 G e* 13 oe 52 410 INTEGRAL CALCULUS. The integration now depends on that of expressions of the form cos.” gdg, and may be effected by the application of For- yh mula 1, Art. 221, after changing in it x into 5 Ex. 4. The Cycloid. The differential equation of this curve (Art. 146) is Pare yp: ¢ Vfl); dan = dy || J yor dy = de 2r—y y In the formula ds = da? + dy?, replacing dx by its value, we get ds = V 2r(2r — y) dy: s= — 2W 2r(24r—y)+-C. If s be estimated from the origin O to the right, we must have 0— —4r+C: .*. C=4r, and s= OP = 4r— 20/ 2r(2r —y). In this, making y = 2r, we have OO’, the semiare of the cycloid, equal to 47, and the whole arc therefore equal to 87, or four times the diameter of the generating circle. Estimating the arc from the vertex O/ to the left, then C= 0, since at this point y = 2r; and we have OPS 2r/2r(Qr — y). But / 2r(Qr — yy) =V GNX GO=PGE: hence arc O’P =2 chord PG; that is, the length of the arc of a cycloid, estimated from the vertex, is twice the corre- sponding chord of the generating circle. L F &£ SECTION VIL. DOUBLE INTEGRATION. — TRIPLE INTEGRATION. 241. Double Integrals are expressions involving two integrals with respect to different variables. Suppose it is required to find the value of w which will satisfy the equation ; ? tas = (a, y), the variables x and y being independent. This equation may be written ad du =*, by making v = a - The function v must be such, that its differential co-efficient with respect to y, x being considered as constant, is equal g(x,y). We therefore have du eR =" hence w must be such a function of « and y that its differen- tial co-efficient with respect to x, y being constant, is equal to fo (x, y)dy; and therefore w=} fol y)dy | de. The value of uw is thus obtained by integrating the original expression with respect to y, and then integrating the result with respect to a. The last equation is generally and more concisely written py | 92, y)dady, or u= ff p(x, y)dydz ; 411 412 INTEGRAL CALCULUS. the first form indicating that the first integration is performed with respect to x, and the second integration with respect to y. The second form indicates that the order of integration is reversed. du _ du didy dydx these partial differential co-efficients were the same, in which- 242. It was shown (Art. 91) that , or that ever order, with respect to # and y, the differentiation is per- formed. We will now prove that the result of the integration in the one order can differ from that obtained in the other only by the sum of two arbitrary functions, the one of a, and the other of y. Let u,, w., be two functions of x and y, either of which satisfies the equation iy g(x,y); then d?u dee = 9(#,Y), Tage 9% y): : au, Ay 0 ie dxdy dady~ d /dv : or =f a ante — . ps @ 0, putting v =u, — uy, Now, dv dy cannot be a function o. x, otherwise its differential co-efficient with respect to # could not be 0; but it may be any function of y. Hence we may put a= = f(y); whence v= ffay + x(x), in which ie denotes an arbitrary function of # Putting fiyyy = w(y), w(y) being as arbitrary as f(y), we have finally Y= U—M=Y(y) +7(2), as it was proposed to prove. DOUBLE INTEGRATION. 413 b 243. A double integral J if acy vdety is the HAW Oe avg all the products of the form g(a, y)AxAy between the limits of integration. Let (a, y) be a function of & and y, which remains finite and continuous for values of x between a and b, and for values of y between « and 6. To abbreviate, put g(x,y) =z. Now, if we suppose x to be constant while y varies between the limits « and £, we have (Art. 192) ff zdy = lim. S2ay. a Multiplying both members of this equation by Aa, and sup- posing # to vary between the limits a and 6 while y remains constant, there results zac f" zdy = Sax lim. Seay : hence lim. zac f zdy =lim.£Axlim.Szay=lim. SSzaxay. ot G oa But lim. Sax J 2zdy = zdacd. JJ, J. edeay by the article above referred to: therefore Pere ie if g(x, y)dady = lim. SS q(x, y)Axay. Writers do not agree as to the notation for double integrals ; some making the first sign f refer to the variable whose dif- ferential comes first in the integral, while others make the first sign fe refer to the other variable. In what follows, the first sign Bk: will relate to the variable whose differential is first written in the indicated integral. A414 INTEGRAL CALCULUS. 244. In the last article, it was supposed that the variables x and y were independent. It is sometimes the case, how- ever, that the limits in the first integration are functions of b 8 the other variable. For example, let a} i) p(x, y)dady be a a the required integral in which « = 7(a), and 6 =w(«); then b b (x) ip ips g(x, y)dady = if f9@ y) dady. Suppose J’(x, y) to be the result obtained by integrating, first with respect to y, regarding # as constant; then, for the integral between the assigned limits for y, we have F \ 2, »(2)| =F fa, ye) t; and finally > py(z) : Nile g(x, y)dady = tp (F {e, w(x) —Ff |e, 1(2)} )de. When the limits of a double integral are constant, it is im- material in what order, with respect to the variables, the integration is effected; that is, a change in the order of in- tegration does not require a change in the values of the limits. But when the limits for one variable are functions of the other variable, and the order of integration is changed, a special investigation is necessary to determine what the new limits must be to preserve the equality of the results. A geometri- cal illustration of this will be given in the next section. 245. Triple Integration. Let it be required to de- termine a function w of the three independent variables a, y, z, C Cie : which will satisfy the equation adv V. We may write a ds dhe. dadyda — da dady ~"? d*u d d*u or acai dz =F de dz—= Vdz: TRIPLE INTEGRATION. — A15 hence by integration with respect to z, regarding w and y as constant, du ‘ T” being an arbitrary function of z and y. Again: we have du d du 1 eae d du or ratty, 2 mer att = dy f Vda+ Th ay, which, by integrating with respect to y, « and z being con- stant, gives ot — fay f Vide + T+ 8"; T’ being an arbitrary function of x and y coming from f 7" dy, and §’ an arbitrary function of x and 2. Finally d ux f 7, du= fdefdyfVde+ T+ S+R; T, Rk, and S being arbitrary functions, — the first of w and y resulting from Je T' dx, the second of x and z resulting from fS'da, and the third of y and z. It is usual to write the differentials together after the last sign of integration: the above equation thus becomes u=f ff Vdadyde+ T+ S+ &. This example suffices to show the manner of passing from a differential co-efficient of any order of a function of several variables back to the function itself. When the variables are independent of each other, as has been here supposed, there 416 INTEGRAL CALCULUS. is no dependence between the arbitrary functions 7, 8, 2; but more commonly at the limits of the integral the variables are not independent of each other. For example, the limits of the integral with respect to z may correspond to z = Ex, Y), z= f(x,y); those with respect to y, to y=/(x), y=f, (a); and, finally, those with respect to x, tox=a,x=—). By a demonstration similar to that given in the case of a double integral (Art. 244), it may be shown that BOM PRE; ue here 1s dx J dy p(x, y, 2)dz= lim. SSZArAYy AZ. SECTION VIII. QUADRATURE OF CURVED SURFACES. — CUBATURE OF SOLIDS. 246. Let F(x, y, z)=0 be the equation of any surface whatever, and take on this surface the point P, (x, y, 2), and the adjacent point Q, («+ au, y+ ay,2+ Az). Project these points in P’, Q’, on the plane x, y, and construct the rec- tangle P’Q’ by drawing par- allels to the axes Ox, Oy. The lateral faces of the right prism of which P’Q’ is the base will intercept the element PQ of the curved surface. Denote by 2. the angle that the tangent plane to the surface at the point P makes with the plane (x,y). This plane is determined by the tangent lines drawn to the curves Pq, Pp, at the point P. The tangent line to the dz dic is the tangent, and the tangent line to the second makes with first curve makes with the axis of 2 an angle of which dz dy are the angles which the traces of the plane of these two lines, the axis of y an angle of which is the tangent. These that is, of the tangent plane to the surface at the point P on the planes (z, x), (z,¥), make with the same axes. Now, from_ 58 417 418 INTEGRAL CALCULUS. propositions 1 and 3, chap. ix., Robinson’s “ Analytical Geome- try,” we readily find, without regard to sign, Eas A H+) +@) The rectangle P’Q’ is measured by Away, and is the pro- COs: A= jection on the plane (x, y) of the corresponding element of the AwA Z, hence, cos.2 tangent plane. This element is measured by for the element of the tangent plane, we have oat dz\? dz\? A ° cos;A +(F) is ae meh! = sec.AAvAay. Let S denote any extent of the surface under considera- tion, and assume that the limit of the sum of the terms sec. AAxvAy, for all values of x and y between assigned limits, is the area of the surface; then B= ff 1+ e) Ha) 5 am If the surface is limited by two planes parallel to the plane (z,y) at the distances « =a,x=b, and by the surfaces of two right cylinders whose bases are represented by the equa- tions y = g(x), y= w(x), we should have sofia ffir (EY «(Ne and, when the cylindrical surfaces reduce to planes parallel to the plane (zx), p() and w(x) become constants c and e, and the formula reduces to cafes (2) (ita VOLUMES OF SOLIDS. 419 247. Area of Surfaces of Revolution. If y=/(«x) be the equation of a curve referred to rectangular axes, the differential co-efficient of the area of the surface generated by the revolution of this curve about the axis of a has been _ found (Art. 167) to be ds _ i aN ae tees ede Nl 23 ius an f J1+ (32) ya 248. Volumes of Solids. Consider the volume bounded by the surface of which F(a, y,z)=0 is the equation; and through the point P, (x, y, z), in this surface, pass planes par- allel to the planes (2, x), (z, y); and also through the point Q, (w+ Ax, y+ Ay, a+ Az), adjacent to the point P, pass planes parallel to the same co-ordinate planes. These four planes are the lateral boundaries of a prismatic col- umn, having P’Q for its base, and terminated above by the element PQ of the curved surface. The volume of this column is measured by zaxay, when Az, Ay, are decreased without limit; and the volume _ bounded by any portion of the curved surface, the plane (x, y) and planes parallel to the planes (2, y), (z, x), will be the limit of the sum of a series of terms of which zAwvay 1s the type. Denoting this volume by V, we have V=Zzavay = f fedudy. 42.0 INTEGRAL CALCULUS. From the equation F(x, y, 2) = 0, which is the equation of the surface, we have z= g(a, y). If we integrate first with respect to y, we get the sum of the columns forming a layer, included between two planes perpendicular to the axis of a; and hence the limits of integration with respect to y become functions of x, and we should have fady =f(x); f(x) being, in fact, the area of the section of the solid made by a plane parallel to the plane (z, y). Thence, finally, V= if J (x) dz. 249. Volumes of Solids of Revolution. The differ- ential co-efficient of the volume generated by the revolution, about the axis of x, of the plane area bounded on the one side by the axis of ~, and on the other by the curve having y =/f(«x) for its equation, has been found (Art. 166) to be or my” = f(x): hence, by integration, Vn fyde Ti aoe Here, as was the case at the end of the last article, f(x) = my? is the area of a section of the solid made by a plane perpen- dicular to the axis of x; and the integral is the expression for the sum of the elementary slices into which we may conceive the solid to be divided by such planes. APPLICATIONS. Wu EXAMPLE 1. Required the measure M’ of the zone generated by the revolu- tion of the arc MM' of a circle about B OPPA X_ thediameter BA. The equation of the circle is 2? + y= R*. Denoting the area of the zone by S, if OP =a, OP’ =}, we shall have (Art. 247) EXAMPLES. 0 dy\2 S= 2n f y Jit (fh) de eae on | Ra =2af iy ON, w= On fs x eel (ba) — Ink x PP’ 421 To get the entire surface of the sphere, the integral must be taken between the limits x= — Rk, x= Rh, which will give S=4a2h’. Ex. 2. Suppose the ellipse of which the arc BMA is a quadrant to revolve B M about its transverse axis: required the eee measure of the surface generated by the portion BM of this arc, begining at the extremity of the conjugate axis. We now have gain fy [1+ | +(3 i From the equation of the ellipse, a’y? + b’x’? = a*b’, we 2 get - = — a whence dx yit(2) = Vay ia _ bat (GB ye , ay ay and finally, by making */a? — b? = ae, we have | dy\? _ b/a? — ea? J td ary if ay therefore i atic Seana - Ibe Sane | Jai — eat de = sie oe a? de. 422 INTEGRAL CALCULUS. But (Ex. 2, p. 326) z le? x er 7 |g" la? fo Qo atte aba Eat 5 Soin Se therefore abe a a? ex S= —(a«.J/— —2’? +5 sin- 7 Se 2 a e a If, in this expression, we make «=a, and take twice the result, we get S = 2ab? +. —— ae sin.—le for the entire surface of the ane ellipsoid of revolution. Suppose, now, that a < b, or that the ellipse is revolved about its conjugate axis, and put «/b?—a?—be; then we shall have Gacrlae br/at Oe ee ae 0 ary But (Ex. ‘ p. His ss Be 1 rs iN: a . seade os Prt ns pos) rat ashe ee 13 Sige te ae = TA peta + apa le+ aie therefore mb" e as P at | @ na ee i ° aaa ea Chas hte pate) +e Since this integral should be zero, for « = 0 we have syed hence CUBATURE OF SOLIDS. 433 If in this we make x = a, and take twice the result, we shall have Bean a2 Bie’ qe cso aria , for the entire surface of the oblate ellipsoid of revolution. If we suppose a = b, and therefore e =0, the second term 0 in the last expression for S takes the form’ 4 ; but, by the rule for the evaluation of indeterminate forms, we readily find ge et.) afk a € lim. whence we have 47a? for the surface of the sphere. Ex. 3. Cubature of the Ellipsoid of Revolution. The equation of the ellipse, referred to its major axis and 9 a the tangent line at its vertex, is y? = — (2ax — x”); and there- a“ fore, for the volume of the ellipsoid, we have (Art. 249) ab? mb* ( . =) se : ae Ax — aaa a” 3 To get the entire volume, we make x = 2a; and then This is the volume of the prolate ellipsoid. To get that of the oblate ellipsoid, a and 6 must be interchanged in the last formula. We thus get, for the measure of the entire volume, 4 ere ie: f gta; from which it is seen that this volume is greater than the first. Making a= J, the ellipsoid becomes a sphere, the 424 INTEGRAL CALCULUS. ei 4 : volume of which is expressed by 3 eau ; and, for the volume of a spherical segment of a single base, the expression is 7 3 x*(3a— x). Ex. 4. Volume generated by the Revolution of a Cycloid about its Base. In the formula V= fny?dx, substitute for dx its value ydy (2ry — y?)* and we have ies ; derived from the equation of the cycloid. JULY EO ? a) but (Ex. 1, page 370) (2ry —y?) 3 2 st 2 fp =-f env) 42 (Q2ry —y*)° 3 3 ¢ (2h lis yidy = — “en y +27 f yy 7? (2ry — yy’) . 2) (2ry ae f- ydy == (ary — 9) + rf dy : 2ry — y”)* (27y — y’) = — (2ry — y?) +r ver.sin.—! Zs ii therefore, by substitution and reduction, 3 eH, MSE. — a(2ry— wie a “ry +> 12) (2ry — y?)" as 3 artver. sin.1? +. : EXAMPLES. 426 Taking this integral between the limits y= 0, y = 2r, and doubling the result, we have, for the entire volume generated by the revolution of a single branch of the cycloid, eae one? Ex. 5. Volume of an Ellipsoid. Take for the co- ordinate axes the principal awes of the ellipsoid. The equation ; et ge Be oe of its surface is then at -- 52 + a 1% The section PMM' of the ellipsoid made by a plane parallel to the plane ZOy, and at the dis- tance OP = x from the origin, has for its equation 2 2 2 The semi-axes of this section will be found by making in succession Reel ed 0; they are Hier eees & ae 3 fe ops ot a” a” bence the area of the section is abe we abe (1 = S Se (a? — x”); and, for the volume of the segment included between the planes ZOy and PMM’, we have be 23 v= (a a — 2?)dx =~ (@ ato 5): To get the volume of half the ellipsoid, make in this formula ; 2 : %—@, which gives V = 5 mabe; and hence the entire volume is measured by 5 mabe 54 426 INTEGRAL CALCULUS. Ex. 6. The areas of surfaces and volumes. of solids have thus far been found by single integration. As an example of double integration, let it be required to find the volume bounded by the surface determined by the equation xy = az, and by the four planes having for their equations C= 8, C= B,, Y=HYi, Y= Y>- The expression for this volume is y ‘i fe °S dady = me —y')(«, —«') =F (1 — ma — 41) (G1 + aa + 1s FOI) if ane = ©) (Yo — 91) (41 ee ee in which 2,,%., 23, 24, are the ordinates of the points in which the lateral edges of the volume considered pierce the surface Be) eee, Ex. 7. To illustrate triple integration geometrically, in the figure suppose planes to be passed perpendicular to the axis of z. Let two of these planes be at the distances z and z+ Az respectively from the origin of co-ordinates, cutting from the elementary column PQ’ a rectangular parellelopipedon ab measured by AgvAyAz. This _parallel- opipedon may be considered as an element of the whole volume V: hence V= ff fdadydz. EXAMPLES. 42,7 Required the portion of the volume of the right cylinder that is intercepted by the planes z = xwtan.0, z= a tan.0’; the equation of the base of the cylinder being x? + y¥? — 2ax = 0- Here the limits of the integral are z= tan.0,z—=a tan. 0’, y= — V 2an — 2, y= tv 2ax— a, © == 0; 7 = 2a; there- fore, denoting the values of y by — y,, +y,, 2a a tan, 9! | ae ae f_.. dadyde 0 a= hi x tan. @ 2a p+y, =| He (tan. 6’ — tan. 0) «dady 0 1 = 2(tan.6’ — tan.0) f en 200 — «dx = (tan. 6’ — tan.) za’. The base of this cylinder is a circle in the plane (a, y) tan- gent to the axis of y at the origin of co-ordinates; and the secant planes pass through the origin, and are perpendicular to the plane (z, x). The re- quired volume is therefore the portion of the cylinder included between the sections OP, OP’. It can be seen from this exam- ple why, as was observed in Art. 244, when there is a rela- tion between the variables at the limits of an integral, the order of integration cannot be changed without at the same time ascertaining if it be not necessary to make a corresponding change in the limiting values of the variables. In this case, after integrating with respect to z, we integrate with respect to y, taking the inte- } 4 gral between the limits y= —(2ax—’)", y=-+ (2ax—2a’)’; 428 INTEGRAL CALCULUS. that is, the integral is considered as bounded by the circum- ference of a circle tangent to the axis of y at the origin; but by what portion of the circumference is not specified until the limiting values of w are assigned. The integral with respect to a is then taken from « = 0 to x = 2a, which thus embraces the whole circumference. But it is obvious, that, if the order of integration with respect to « and y be reversed, then, that the integral may embrace the whole base of the cylinder, the limits with respect to x must be e=a—WVa?—y?, e=a+WVa?—y?; and those with respect to y must be y= —a, y= +a. We now have, denoting the limiting values of x by x,, — &, esti hk Hid alee i dydaxdz iy fi (tan.0’ — tan. 6) adydx 2 2af (tana! — tan.0)V a? — y?dy = (tan.6’ — tan.@)ma* (Ex. 2, page 326); which agrees with the first result. 250. Polar Formula. The polar equation of a plane curve being r = (9), if s denote the length of an are of the curve estimated from a fixed point, the differential co-efficient of this are (Art. ae is s=frs(Gyp fiee(ia on or, by taking r as the independent variable, ea fin ) it. 1a (2). POLAR FORMULAE. - 429 ExampLE 1. Applying Formula 1 to the spiral of Archime- des, the equation of which is r = a4, we have s= f(r? + a?)8 dd =af(1 + 0°)? do Gf] 1 ‘7 =F (L+oyrtotia+(+oy +e. If the are considered begins at the pole where @ = 0, then =). Ex. 2. For the logarithmic spiral, we have r= da’, or r = bee by making a = ae Hopi page! a and a = O whence, by Formula 2, = f/(lte)dr=/f/(l+ec)r+C. If the limits of the integral correspond to the radii vectores 7, 11, the length of the arc is 8 /(1 +’) (7) — 79). : dd , : Since rv — is the expression for the tangent of the angle dr made by the radius vector of the curve at any point and the tangent line at that point, we have, calling this angle @q, . ds tan.a@—c; hence sec.a = 4/(1 + c”), and Fp a BOG. a: there- r fore s=rsec.a-+C,and the definite portion of the arc an- swering to 19, 7;, 18 (7; —7o) Sec. &. 251. To find the length of a curve in terms of the radius vector and the perpendicular demitted from the pole to the : tangent line to the curve at any point, we have cos.¢ = en 8 (cor. Art. 163) : hence, if p denotes the length of the perpen- dicular, p te Pj AUN Tp 0. © — >» cos. = , r 4 ak r : 430 INTEGRAL CALCULUS. ia therefore 3 —~ a/ 7? — p? aed byes oe r? — pe : 252. The ae of a curve may also be expressed in terms of the perpendicular and its inclination to the initial line. Let w and y be the co-ordinates of any point VM of the curve, and denote by s the length of the curve included be- tween the fixed point 4 and the point MM. From the origin Jet fall the per- pendicular OP upon the tangent to the curve at the * point M,and make OP=yp, MP =u, and the angle POx=0; then, from the figure, we readily find p—=x«cos.6 + ysin. 6, u = x sin. 0 — y cos. 6. Also we have dy t. 0 a sec. 0 — = — cot, 6,—-— =) eoseed. dx aa: therefore d d. oie —xsin.@ + ycos.6-+ cos. 0 >t sin. 0 ! oh But, since 6 is the independent variable, 7 cot.@ may dy : do cos. 6 be written dx = See whence dd da cy Sige sin. 0! 5 t 008: py POLAR FORMULZ. * 430 oF in. 0 cos. 0 = — sin. o0=— dé s oe Y U; d’*p du : _ ae 9%. > x cos.@ —ysin. # — sin. 6 16 -+ cos. fa dy But, from ae cot. 6, we get dy , da COS. 0 ap aoa cos.” cosec. 0 at a LL re FUN MUP cok eco. A - — sin. eet oS: fake pee aac Mee dx /sin.?6 +- cos.?6 dc eee) COSC. 7 * do sin. 0 ) do ods ; ds dx The equation i ae cosec.@ gives nee cosec. 0 Th hence, by substitution, oh ae e i ae as am cos.d — y sin. LET ds —=—p + PS therefore dp dp ° ee 4 a 8 sat J pa ; or st+u= fpdo. Taking the integral between the limits 4), 6,, 80) $1) Uo, M15 being the corresponding values of s and wu, we have stm —u = fi pd. The sign of w will be positive or Ne. according as the angle POx = 6 is greater or less than the angle MOx. These 432, INTEGRAL CALCULUS. results may be used for several purposes, the most important of which are, — First, To find the length of any portion of a curve, the equation of the curve being given. In this case, from the arene! equation of the curve and the equation oF aa 6,cand y, dx and therefore p = «cos.@-+ ysin.6, can be determined in terms of 0; and, by integration, s may be found from the dp equation s = aA + [pdo. Second, To find a curve, the length of a portion ‘of which shall represent a proposed integral. Here, if the integral be fpdo, p being a function of 6, the equation of the curve is found by eliminating 6 between the equations dp. : d == p COS, 0 do S10. 0, app sin. 0 + 47 cos. 6, hich we get from the equations p=xcos.6+ ysin.6, oe = — xsin.6 + y cos. 6. The proposed integral will then be represented by s — a0 APPLICATION. Let BUC be a quadrant of an ellipse of which the equation referred to its centre and axes is ak fb b2 a? — a” 52, This equation, by making *= a?(1 — e?), may be put under the form y? = (1 — e’) (a? — a’). POLAR FORMULZ. 433 Make POx — #, Then, from the properties of this curve, we have tse 4 | mae Ne On? = a; tan. PTO = cot.?6= Coed. x — 2 From the last equation, we get ers: eri 62 sin? =~, dhe ue ae a* — e* a Coe Cee? aa? aH : a?(1—e? “te OP’ = 0T" x costo = a2 St = ©), eee} AMD aE A or OP =p=a [eG = ayia e ante Therefore OM+ MP =s+usafV1— eosin? di. It is here supposed that the integral takes its origin at C, the vertex of the transverse axis. Now, if the point A be so taken that the angle BOA = 6, it has been shown (Art. 240) that Are BA=afV1—e? sin?0do: ag OM + MP = BA. Also we have dp ae*sin.@cos.6 , dp /1 —e? sin? and, w being the abscissa of the point I, EE as po eink dé ae? sin. 6 cos. on a cos. 6@ = a(1— e?sin.0)* cos. 6+ eee SS ae ia —e’sin. 29)? (1 — e? sin.? 0) 55 434 INTEGRAL .CALCULUS. Therefore UP = e*xsin.6; and, x’ being the abscissa of A, era | we have x’=asin.6: .*, MP= ae and hence 2 BA — CM = UP =~ ax’, a result known as Fagnani’s Theorem. From the values of # and x’, we get 2 2 a—a’sin2?¢@ a?—ae’ | x ae 1 — e’ sin.2 6 : 2 e2a/ a” which gives 2 2 e* a xe! —a*(a?+4+ a’ )+a*=0, an equation which is symmetrical with respect to # and a’: hence, if we have 24 BA— CU= oa’, a we also have 2 BU CAs “- ae’. 253. Curves of Double Curvature. A curve of double curvature is one, three of the consecutive elements of which do not lie in the same plane. Such a curve must be referred to three co-ordinate axes, and requires for its expression two equations which represent the projections of the curve on two of the co-ordinate planes. Let the equations of the curve be y=f(x) (1), %=9(%) (2); (1) being the equation of the projection on the plane (a, y), and (2) the equation of the projection on the plane: (@,/2)7 alt x, y, %, are the co-ordinates of a point of the curve, and CURVES OF DOUBLE CURVATURE. 435. e+Ax,y+Aay, z+ 2, the co-ordinates of an adjacent point, then, by the principles of solid geometry, the length of the ‘ chord connecting these points is { (amy? + (ay)? + (ae)? Then, if s is the length of an arc of the curve estimated from a fixed point up to the point (a, y, 2), that of the arc from the same fixed point up to the point (7+ aa, y+ Ay, z+ 42) will be expressed by s+As. We shall assume Jit a bee IER el (aa)? -+ (ay)? + (az)? Ag a= lim. eure Ove N 7 Te 1, fe ea a Lean es emp fa (2) + (2) ta The two equations of the curve enable us to express and therefore dy dz dx’ da’ in terms of «; and, by integrating, s will then be known in terms of «. Any one of the three variables may be taken as independ- ent; and the above formula may be changed into ff < (i) +) = nay 436 INTEGRAL CALCULUS. When 2, y, and z are each a known function of an auxiliary variable, ¢, as may be the case, then dy dz dy __ Ke at dz __ at dx dz dx dx dt dt and we may have J}3+ Gz) +(@) }" oe) +) « aff) +) +G) pa 254. To convert the formule of the last article into polar s | formule, take the pole at the origin of co-ordinates, and denote by 6 the angle that the radius vector makes with the axis of z,and by g the angle that its projection on the plane (a, y) makes with the axis of w; then we have the relations =r sin. cos. g,. ¥ =T'sin. 0 Sin. @, 2 =F iCosee These three equations, together with the two equations of the curve, make five between which we may conceive r and to be eliminated, leaving three equations between 2, y, 2, and 6: hence, x, y, and z may be regarded as known functions of 6. Therefore dr : Hae 7g BID 6 cos. p — or — sin. 6 sin. p 7 + 1 cos. 8 Cos. g, = sin. O sin. a + 7 sin.6 Cos. 9 - + rcos. sin. g, Z dr ; —=—cos.9— — 7 s1n. @: 1 ON do POLAR FORMULA. AZ, aCe) dé which, by changing the independent variable, may become s=fin( 3) tite mela) 5 dr, or c= fir (GZ) + +(Z) 49 cnt dp 255. Polar Formula for Plane Areas. In the curve BM of which the polar equation is r= (0), let r, 0, be the co-ordinates of the point WM, and denote by A the area bounded by the curve, the radius vector PB drawn to the fixed point B,and the F radius vector PM. Then (Art. 165) ak : =\9 (a) } eee A=, [{o(a) }'a9 Let w(6) be ae function having (6) for its differential co-efficient; then 4=w(6)-+ (: and if A,, A,, denote the areas corresponding to the values 0,, 0,, of the vectorial angle, we have . da\2) 4 0 N C m™ % +r sino (TF) t dé ; A, = (9) +C, Ay= (2) + C; te A A= ¥(%)— 9H) =a fo fo} 438 INTEGRAL CALCULUS. EXAMPLE 1. For the parabola, when the pole is at the focus, and the variable angle, measured from the axis, begins at the ver- 9 tex, we have x=p—rcos.6, y=rsin.0; R from which, and the equation y? = 2pm of the curve, we get ners. L/L ps a ee ae ae esl c08 a 2s ae ot andes prin” "2 ng p? 6 0 2. gl pean — F f(2 + tan.? 5) sec. 5 Cope a tan. 5 = 79 18D 5 + C; p? 0, 6 2 6 6 . 4,— A, == (tan 7 — tan. yl a 5 (tam ener tm), : 7 we We p? Making 6,=—0, 6, mage have for the area 7 + 172 3 8 Ex. 2. The equation of the logarithmic spiral being r = bee» we find 1 *0 Dee —=— | bec SS Bie A 5 it ec do 7? + C, le. 20g! *he Mae jan} fined e 2 te 256. A polar formula involving double integration may also be con- structed for plane areas. Suppose the area included be- tween the curves BME, bme, and the POLAR FORMULZ.. 439 radii vectores PB, PH, is required. Divide the area up into curvilinear quadrilaterals by drawing a series of radii vectores, and describing a series of circles with the pole as a centre. Let no be one of these quadrilaterals, and denote the co-ordi- nates of n by 7, 6; and of o by r+Ar,6+ 6. Now, the area no is the difference between two circular sectors; and the : pe cee a. : 1 accurate expression for this difference is rArad + 5 (Ar)? Ad, the ratio of the second term of which to the first is : (Ar)?a0 Ae rArdad 2r This ratio diminishes as Av diminishes, and vanishes when Ar =(: therefore we may take raraé as the expression for the elementary area, since, in comparison with it, the neg- lected term (ar) AO ultimately vanishes. _ 257. In the last article, it was shown that raraé might be taken as the expression for the polar element of a plane area. If we suppose this area to be the section of a solid by the plane (x, y), the column perpendicular to this plane, standing on the element raraé as a base, may be regarded as an ele- ment of the solid. The volume of this column is measured by zrarad; and therefore, for the volume V of the solid. we have V= f fzrdrdo. The value of z asa function of rand 0 will be given by the equation of the surface bounding the solid. EixXaMPLE. Required the measure of the volume bounded by the plane (a, y), and the surfaces having e?t+y?—az=0 (1), v?+y?—2x=0 (2), for their respective equations. 440 INTEGRAL CALCULUS. Denoting the polar co-ordinates of a point in the plane (x,y) by rand 6, 0 being measured from the axis of 2, we have = '7cos:8, y =rsin.6 (3) ; therefore x? + y? = r*, which, combined with (1), gives From (2) and (8) we find r= 26cos.@: hence, for this exam- ple, we have | on = J feraran =z drdo. To embrace the entire volume comprised between the sur- faces indicated, the integral must first be taken, with respect to r, between the limits r = 0, r = 2bcos.6, since 6 is assumed as the independent variable; and then the integral of the result must be taken between the limits =F (= —5: Thus 2bcos. 8 ; “ 4 V= oh drdé = 40 cos.4dd 0 a a 0 Bi ies A 2 4 4 pas cos ‘9a9 =° lies (Art. 221). a 2a 258. Suppose the polar element va7raé of a plane area to revolve through the angle 27, around the fixed line from which the angle 6 is estimated. A solid ring will thus be generated, the measure of which is 2zrsin.6raraAd; since, in this revolution, the point whose polar co-ordinates are 7, 0, POLAR FORMULZ. 44] will describe a circumference having rsin.@ for its radius. Denote by the angle which the plane of the generating element in any position makes with its initial position; then g + Ag will be the angle which the element in its consecutive position makes with the initial plane. That part of the whole solid ring which is included between the generating element in these two positions is measured by (po + Ag) r’sin. 6ArAd — pr* sin. Ora = r? sin. OArADAG. This may be assumed as the expression, in terms of polar co-ordinates, for an element of the solid: hence, for the vol- ume V of the whole solid, we have Kane falblan: Shee - in which the limits of integration must be so determined from imposed conditions, that the integral may embrace the entire solid to be found. Example. Required the volume of a tri-rectangular pyra- mid inasphere. Integrating the above formula, with respect to r, between the limits r=0,r—=a,a being the radius of the sphere, we find Beaty [fr sin. odrdoda =/fr sin. dédq. Now, a’ sin.@addqg is an element of the spherical surface; 3 and 5 sin.dA0Aq is therefore the expression for an elementary spherical pyramid having a” sin.@adaqg for its base. By this first integration, therefore, the element of the volume has changed from an element of the solid ring, generated by the revolution of rara§, to an elementary spherical pyramid. 56 442 INTEGRAL CALCULUS. Integrating next, with respect to 0, between the limits nm P=), 6= 5 we have a? Vie = dq ; since 0 fsin.sd9 = — 008.0: Sf, sin, Gob ee By this second integration, the elementary volume has become a semi-ungula, or a spherical pyramid, having a bi- rectangular triangle for its base; the vertical angle of the triangle being Ag. We finally integrate, with respect to g, from — Oto ee 5 and get for our result ma V=—3° SECTION IX. DIFFERENTIATION AND INTEGRATION UNDER THE SIGN /.— EULE- RIAN INTEGRALS. — DETERMINATION OF DEFINITE INTEGRALS BY DIFFERENTIATION, AND BY INTEGRATION UNDER THE SIGN /. (259. Wuarever function of x, f(x) may be, there exists another function, g(x), of «, such that g’(x)=—/(a);.and therefore | S@)dx = oa) + (7 (Art. 191), C being an arbi- trary constant. Denoting by wu the integral of f(x)dx, taken between the limits a and b, we have w= [fade = 9) — 9a). The definite integral w is independent of x, but is a func- tion of the limits a and 0; and its differential co-efficient with respect to either of these limits may be obtained without effecting the integration. For, since u= g(2) — 9(@), we have du ; du ; Ap gp’ (a); db gp’ (0) ; and, because g/(x) = /(zx), d ee 10): HOF du = f(b)db — f(a) da. 443 444 INTEGRAL CALCULUS. ILLUSTRATION. Let y=/(x) be the equation of the curve WN referred to the rectangular axes Ox, Oy. Ifa and 6 are the abscisse of the points Mand N, wu =e J (x)dx will repre- sent the area AMNB. Give to a and 6 the increments AA' = Aa; BBGeEaoe Au AMM’ A’ au BNN'B AGcn aed Ae ee rae et The definite area AMNB is ‘obviously a decreasing func- tion of the first limit a, and an increasing function of the second limit 0: therefore Au... AMM'A' _ lim. < itn Tee ies — f(a), VA ee ea ine lim. Kia lim. PBR wert 0): Regarding the areas 4U/M/'A’, BNN'B’, as elementary, we see that the total increment of the area AMNB is the differ- ence of the increments that it receives at the limits. 260. Suppose /(x) to contain a quantity, ¢, independent of »b x, and that the differential co-efficient of J J(x)dxe with re- spect tov is reauired. Replacing f(a) by /(a, t), we have U = cc, 0) On. If the limits a and 0 are independent of ¢ we have by giv- DIFFERENTIATION UNDER THE SIGN /f. 445 ing to ¢ the increment Aé¢ (Au being the corresponding incre- ment of w), Au =]p f(a, t+ at)dx— f° f(a, t)dex sels (F(@, #442) — f(x, t)) da: A | AU maf J (a, satel! — f(x, Je e e Now, by Art. ay we may write J(%, t+ At) —f(&; t) ae t) At wie in which y is a quantity that vanishes we At vanishes. De- noting by 7’ the greatest of the values of 7, we have, gener- ally, 7 fide OS ays and, when neither a nor 0 is infinite, (b — a)7’, and therefore 6 {| ydx, will ultimately vanish: a est, (db. 2phdf (a, t) lim. = GH. i da. APPLICATION. Resuming the formula du _ dt _ pr df (x,t) po al eK gy & ) just established, suppose g(z,¢) to be the function of which J (x, t) is the differential co-efficient with respect to x, and w(x,t) to be the function of which ae t) is the differential co-efficient with respect to x; then (1) becomes “ee AO = (b,t)—w(a,t) (2). 446 INTEGRAL CALCULUS. If f(x, ¢) and a are both independent of b, (2) may be written se “ait @ fie w(b, t) (3) ; C denoting the sum of the terms which are independent of 6. Since we may give to 0 in (3) any value we please, replace b by «; then (3) becomes vio) = 9G) Ve (a). Dropping the constant C, which may be restored when neces- sary, and putting for the other terms of (4) their equivalents, [ease t) ets © fe, t) dx. we have da Example. Let /(x,t) = tha : d Ta? then J (, t)daxc — Jf, t) dx =e Fe = tan. hte d _ afl, 1, \_ paY(%,t) and 5, J A(t) de = at tan. tn) = f F da d 1 Qt? Valtere) = —Secpeey® Thus, having the value of (2s we find, by differentia- Te Q? tion, that of the more complex integral A paleait « ‘gir i —_—.— dx. 261. When, in the integral vu = f f(a, t)dx, both a and b are functions of ¢, then zh will consist of three terms; since in this case, to obtain the total differential of w, we must dif- ferentiate it with respect to ¢, and also with respect to both DIFFERENTIATION UNDER THE SIGN /. 447 a and b regarded as functions of ¢, and take the sum of the results. Thus we should have du gl t) du db , du da =! ct hoes df (x a " He, Per, te ce = f(a, t) eo (Art. 259). Under the above suppositions, the second and higher differ- ential co-efficients of w with respect to ¢ may be found. Thus, _ by differentiating each of the terms of the last formula with respect to t, we get Gu te) 2 diay oe ero, re =A Wart se t) (a) 49 Te, t) i 10 — a let _ 9 Vlad ILLUSTRATION. Let y = f(x, #) be the equation of the curve OD referred to the rectangular axes Ox, Oy, and y = f(a,t-+ at) that of the curve % x EF. Put One a, ON—'b, Meee Ag, NN‘ = Ab. Then w =p J (x, t)dx denotes the area MNDOC, and u + au the ° area M'N'FE: au = EE'F'F + DNN'F'— MM'E'C, Au EEE’ F DNN'F MM EC. Abba aad At At 448 INTEGRAL CALCULUS. It is plain that the first term in this value of is the ratio of Aé to the increment of the area due to the change from the curve CD to the curve HF. The limit of this ratio is the att of ip ACAI wa 1%) ge So, ‘alec the inn tenes second term is the limit of /(0, ¢) a and the limit of the third term is the limit of /(a, i) = —: hence du Hae t) da sf IAS dex 2+ 7 ,t) a ee which agrees with first formula established in this article. 262. An indefinite integral may also be differentiated with respect to a variable contained in the function under the sign of integration which is independent of the variable to which the integration refers. Let the integral be u =f /(a, t)da, t being independent of a: then, without impairing the generality of this integral, we. may write u= fo f(a, dde+y); w(t) being an arbitrary function of 7. Differentiating with respect to ¢, ¢ not Lae on a, we have (Art. 260) a a= As 2D de + w(t): but, since w/(¢) is a constant with respect to a, it may be included in the constant of the integral ii a dx; and hence the last equation may be written of ZO aa, INTEGRATION UNDER THE SIGN ff. 449 and we have only to differentiate the function under the sign if with respect to ¢. 263. Integration under the Sign of Integra- b tion. Taking the definite integral J J(x,y)dx as the differ- ential co-efficient of y, and integrating, we have b favs. f(%, y) de for our result; and it is proposed to prove that this result is the same in whichever order with respect to x and y the inte- grations are performed; that is, we shall have b b fay f Seyde= J de f f(a, y)dy. df f “ y)dy For aJ. dic f f(a, y)dy = f de eel Integrating the two members of this equation with respect to y, we get Jere ydy = fay f f(x, y) dx and, if the limiting values of y are c and d, we shall have b d f wf Sandy ad b =f dy f fa, y) da. The figure gives the geometri- cal interpretation of this formula. Kither member represents the vol- ume AC’ included between the plane (x,y), the surface A’B'C'D' having z =/(a, y) for its 57 450 INTEGRAL CALCULUS. equation, and the planes whose equations arex=a, «= 68, y=e, y=". ExaMPLE 1. Find the form of the function g(#) such that the area included between the curve y= q(@), the axis of a, and the ordinates y = 0, y = g(a), shall bear a constant ratio, n, to the rectangle contained by the latter ordinate and the corresponding abscissa. By the conditions, we must have : __ ag(a) , Yio 2) to ag and, since this is to hold for all values of a, we may differen- tiate with respect to a: hence _ g(a) g(a) | De) a eee : PU) oa hey ne g(@) a? and by integration lp(a) =(n—1)la+C. Passing from logarithms to numbers, p(a) = Car: +, g(a) = Car; and the equation of the curve is y= Cau"—!. Ex. 2. Determine such a form for g(a) that the integral « g(x)dx ; : shall be independent of a. 0 A (4—2) 5 Put «= az; then, since the limits « = 0, x =a, correspond | Realee peso Ua] mead cae a g(ajdxe — prJ/ag(az)dz =/ iv@=o- J, waieam INTEGRATION UNDER THE SIGN /. 451 By condition, wis to be independent of a: therefore the dif- ferential co-efficient of w, with respect to a, must be zero. But gige) / Ree 2a tN 8) yee Dy da a/(1 — 2) 2A4/(a — x) 0 and, since this last integral is to be zero for all values of a, we must have cp! 1 p(x) + 2ap’(x)=0: .°. ae Horne Therefore l(a) = — ae + 0, or g(x) = . Let AOB be acycloid, with its vertex downwards; and let it be referred to the axis Ox, and the tangent through its vertex, as co-ordinate axes. Px Then, denoting the angle DOP by 6, we ¥ F aco have for the co-ordinates OF = y, OQ == of the point P, gee — Tp Go Cos 6 yY—OF = AR —AK= AR — AD + ED = an — as + asin. 6. Put 6 =x — q, then these values of z and y become x—=a—acos.g (1), y=ap+asin.g (2). From (1) we find a Pi COS.1 —x“ ., if sing = 7 | 2ax— ats 452 INTEGRAL CALCULUS. and thus (2) becomes a—aX Y COS Se ear eae — which is the equation of the cycloid. By differentiation, we get dy Ree — we ae J dy\? oa a te ee ee ee fe da a da eh @ a’ and, by integration, s=+/8ax. We therefore conclude that g(x), in Ex. 2, is the expression for the arc of a cycloid esti- mated from the vertex. This example is the solution of the problem in mechanics for finding the curve down which bodies, starting from dif ferent points, will fall in equal times. 264. The Eulerian Integral of the First Species is an integral of the form 1 i] aP—l(1 — x)2—" da, 0 in which p and q are positive numbers. This is denoted by B(p,; q). The Eulerian Integral of the Second Species is of the form co ii Mie be 2 5 0 and is denoted by I(n). The first species may be put under the two forms oo p—ld 5 @ i) aC 2 sin.??—! @ cos.47—-1 dp, by making «= i zi ioe the first form, and x = sin.”6 for the second. EULERIAN INTEGRALS. 453 The integral of the first species is a symmetrical function of » and q; for, making «= 1 — y, we have 0 Bip, Q=f (Loy yt dy = Bg, p): B(p, 9) = BG, p)- 265. Integrating by parts, we have for (1—a) de x? (1 — ax)! se 4 ~ q-1({ — =e sr ae) 1(1— x) de eee (1 — x)? PL fiee—3 (1 — ode —£. fer (1—2\t—I¢ = Z ca (1—2) x ad 2! x) L Therefore, taking 1 and 0 for the limits, we have AG a a q) = BP, q) 7B +h q): Bip +1, 9) = ma Bp, q). In like manner, B l= Se (pRI+l)= rae B( p,q): In the integral of the first species, therefore, each of the - exponents p and g may be diminished by unity. 266. In the Eulerian integral of the second species oa) if (See sii Sa 9 hal 0 n must be positive, otherwise the integral would be infinite. For if n be negative, and equal to — p, we should have ie a) lo a} ye —2£ pmn—l pee é ip e-*a ot oa {aa : and it is plain, that, when w = o , the differential co-efficient is zero, and therefore the integral is zero; and, when # = 0, the 454 INTEGRAL CALCULUS. differential co-efficient is infinite, and therefore the integral is infinite. The integral '(n-+1) may be made to depend on I(n). For, integrating by parts, we have fe-*atda = — e~*x" + nfe-*x"—" dx. But e-*a” reduces to zero both when « = 0 and when = (Ex. 3, Art. 103): therefore . ie) ie a) J Gap ra Lig ees nf Co * aaa 0 0 or r(n+1)=nI(n). In like manner, I(n)=(n—1)r(n—1), F(n—1) = (n— 2) T(r — 2); and, if 2 is entire, we shall have, finally, r(2)=r(1), r(1) ={), en? dpeeas 0 F Therefore, when 7 is an entire and positive number, we shall have I'(n) =1.2.3...(n—1); and, if ” is a fraction greater than 1, then the formula I'(n) = (n—1)P(n—1) enables us to reduce the integral I'(v) to that of I'(u), u de- noting a number less than 1. Hence, to compute the value I(n), it is sufficient to know the values of this function for values of n between 0 and 1. 267. By putting e~* = y, the integral I(r) may be made to take another form. Thus, from e~* = y, we get NAS nel hae cy we 1 yi sa flected f(y av= py dy n—Il or I'(n) =i (: 4 dy. 0 EULERIAN INTEGRALS. 455 268. Relations between the two Eulerian Inte- grals. Assume the double integral {s 1h gPe+raq—t Dnt e—I+ne dyda, and integrate with respect to 2: it thus becomes riptaf a® ays (Art. 266). Integrating the same double integral with respect to y, it be- comes } © p—x p+q—-ld oo U(p) f= =r(p)f eat dx =I(p)I(q): therefore oe ; Peta), eT (Pyr (a): RU a an BC Pe a ia pie POD = pa that is, Pee tea rip icay || Chass verter, Putting : for x in the first member of this last equation, we have eee Fe a) SC Penne aiy or if 2? -l(q — 2)I—dzg —= gP+¢—1 Pees 269. The last formula in the preceding article is a particu- lar case of a more general formula by which may be expressed, in terms of I functions, the multiple integral Oe 2. Be (c —x—y—%...)* 'dadydaz... extended to all positive values of x, y, z..., which satisfy the condition « +y-+2...< a. 456 INTEGRAL CALCULUS. Limiting ourselves to three variables, let a a—2x a—-x—-y A= p—\ df, I—ld, r—l(q—g — you Bie ip x of y yf, a"—*(a—a@ —y—a2)s dz Now, by the last article, ees = | = —y)rrs-t I'(r)I'(s) r—l wen easy eke s—1 —_ Mas bee: 4 , e°—"(a—x—y—z)* =(Aa—zZ Tires Multiplying this by y?%~'dy, and integrating with respect to y from y = 0 to y= a—z@, the result is (a — w)etr+s—1 LAG) Dna 8 aya riqtr-+s) r(r+s) — (a ie io) ta eee (QI (r) Ts), (ace eee and finally, multiplying this last by «?—1da, and integrating with respect to « from x= 0 to x =a, we have — qgetatrr+s— LTipr(gnr(r)l(s) 6 Te In this, making a= 1, s=1, we have p—1l,,q—lyr—l MAL I( p)U(q)r(r) Sf fe pits dati de Te dy the limits of integration being any positive values of a, y, z, which satisfy the inequality «+y+2< 1. a & y\P Z\7 Assume " sai e Seay () =a fee Pha SLE eee then om 4=— Sf wer nb CY ductndt, rf subject to the condition that u on n+¢ <1: therefore gare HOGG) aby 1 ae (2). st ey EULERIAN INTEGRALS. 457 270. By means of Formula 2 of the last article, we can find the volume bounded by the co-ordinate planes, and the surface having for its equation () +) +) =" PeeeenE When @—f—y7—2, and p=g=r=—1, the surface is that of an ellipsoid of which 2a, 2b, 2c, are the axes. _ Then, by the formula, the volume V of 4 of this ellipsoid will be But Ae a 1)= 7 =r e) =, ip () (Art. 266), and I" os a/2; for, let u al e—** dx, then also u => [\e7* dy, es and ur— [ eda [ e-” dy = wi (il @atacats dady. fer de fet ay= ff Now, flee-* —” dady is obviously } of the volume the 04-9 equation of whose surface is z=e-**—”*, In terms of polar co-ordinates, the expression for the same part of the volume is J fi ardodr = [ IWpen tie 0 0 : : 458 INTEGRAL CALCULUS. sas it and f d=6; 25) d= 48; » : 1 us fe dt = 5 A/T. 1 » Now, 7G} i) e—*x—*dx by definition, 2 0 ==,9 [et dy = 20 = on by putting «= y?: __ abe | ryt __ abe therefore, Vis= 8 Bye brea rst) 271. Differentiation under the sign f enables us to find new integrals from known definite integrals. Thus, ay the nt EXAMPLE 1. il eee nn eee el Differentiating each member of this equation nm times with respect to a, we get 74 HART tee gy () penta 3.9 2A 1 WA »@payt l= 5 a aati? 2 da 1.3.5...(2n—1) a whence Cs i 3s i ("+ a)rrt 2.4.6...2n anes Bx. 2 freseee 0 a After n—1 differentiations of the two members of this equation, it becomes ioe} f e~ * tle 2-3. ..(n — oe 0 thatis ft eon Cee (Art. 266). DIFFERENTIATION UNDER THE SIGN /. 459 The last formula holds good when a is replaced by the imaginary quantity a+b —1, in which a is positive; for ee Ds dic C ares i __ e-* (cos.ba — / —1sin.b2) oe EL 7 aaa + C (Art. 13): therefore 2 a 1 —(a+bY—lex d. oe omer gs > Ls J, : aa + bf —1’ and, by differentiating this equality nm — 1 times with respect to a, we get 1.2.3...(m — 1) (a+br/—1) ’ 272. The formula just found leads to other integrals by ioe) ii e—(atoV—lz e®-\dx= 0 the separation of real from imaginary quantities. Assume a+b —1=p(cos.0+ / — 1sin.@), in which 5 a ; b B=aV G?)- 52, cos.0 = Vato’ REY rs eae. Then z: : 0 = {Pa (cos.ba — s/ — I sin.ba) a*—! dx 0 and Wor, ; 10 —\ boy 27) Lee (atb/ —1)” p” cos.nd+/ —1sin.nd In) = —— (cos.nd — / —1sin.né): p 460 INTEGRAL CALCULUS. .. if e~* (cos. ba — / — 1 sin. bx) 2" dax 0 = me (cos.nd — ny hess} sin. 70) ; an equation which may be separated into the two, P(n) pe eo i} Cy te SIN OL oe sin. 20, 0 n eS I'(na i e—* o"—! cos. bada = (7) cos.n0. 0 Pp 273. Making n=1 in the last formula of the preceding article, it becomes a —ax — ‘i e iP e€ cos. bada = ee therefore, denoting by c a constant less than a, we have if da Nee cos. bada =f se But Ip da je e—* cos. bada =|. diac ip e—* cos. bada @ Saeed = é ax , = i —_—_— cos. bada. 0 4b Again: a ada: 1, cde J @apr3 co? + b?? wo e—c@ __ p—ar 1 a* +b? ip ee COR OL T 5 ee Making b = 0 in the last equation, it becomes 2p 62) ea a (see iC 0 w DIFFERENTIATION UNDER THE SIGN /. 461 a result that may also be obtained by multiplying both mem- bers of the equation (s oe de = by da, and integrating the result between the limits a and ce. 274. In like manner, from the formula 2 ene Woe 1D f, e—“ sin. badx = at EB? we get f fF e-* sin.brde = fier a Cc — tan. — tam! —° b b But a wo A foo) a Fs |e f e~ sin. bada aah da { sin. bxe—“* da c 0 0 c 2 e-—ce __ p—ar =) ee SIT OLS 0 ax aes if ee gin, pvae — tan, + ——,tan.!— 0 x b b- In this formula, making a =o, c= 0, it reduces to = sin. bax f sin. b de —™ faa Beer 2 n when fe: when See the second member becomes — 9° gin. Es from which it is seen that the integral if dx changes abruptly from 5 to 5 when 8, in passing through zero, changes from positive to negative. 462 INTEGRAL CALCULUS. 275. The integral f e-# de = svn (Ex. Art. 270) leads tof e-"da=ar/n; for ihe e—™ da ape e-* dx +f. e—* da. Now, if we change x into — a, we have 0 0 J eda =f edz = ha/n: — 00 0 i Sth tesla VE rg And generally, if f(x) is a function of the even powers of a, that is, such a function that /(x) = /(— «), then fo f@)de =2 [" f(a)dz; for f s@ae=f a flo)de +f" f(a)de. But fo fede =f" s(—2)de=f fe)de: f fla)ae = af f(a)de. In like manner, it may be shown that if f(a) is a function of the odd powers of a, that is, if f(— «) = —/(a), we should have f° S@)de= 0. 276. In the integral [ e—" dx =/n, putting x/a for x, we have which, by n differentiations with respect to a, becomes ie eae" nn Jn oe Sip aN — 1) a (ake). DEFINITE INTEGRALS. 463 In this, making a = 1, we have ii o-# ihdan = a/m 123:5--.(2n = 1), ¢ on 277. Changing x into x +a in the formula ecto — a/ tt of the preceding article, we get if Ga op afar ioe) e 2 2 that is, G8 f Ct I AY Tee —o2 at 2 d. 2 nee e777 —2axr Tp = e*? 1. {jee x4 o 2 0 7 ra) P But {i om Sat =f Cae Oe +f Camis 6 O07, ee mie 0 0 ) ze 2 and if er? da =}/ et +2ar da aap 0 by changing x into — a: ai in AB OLN wens tr eo limcre ede +f eM de =o 0 : 0 =P g-* (e%4* 1 6-392) do « 0 whence i e-=" (ere oo Crs) Oe “= ew Jt. J 0 In this equation, replace a by a4/— 1; then, since ere | g— tar — e—WaV—1 | e2ae~—1 — 2 cos. Zan (Art. 73), we have cs 2 1 °o iP 6 * cosecccds — ae eae / 1. fd 464 INTEGRAL CALCULUS. This example is another instance in which the value of a definite integral is found by passing from real to imaginary quantities. #78. Another process by which f e>#* cos. 2axdz may be 9 * | found consists in differentiating with respect to @ and subse- quent integration: thus, put io 2) U =f e—* Cos. 20005 5 0 then ae = —f sin. 2axe—* 2ada =| sin. 2am, d,e—*. Integrating by parts, and observing, that, at the limits, ° 2 sin. 2@ce—* is zero, we have du v4 2 iy eee ee —& 2actada = — : la J, cos. 2ax2ada 2au du WO, 99 U But, regarding w as a function of a, we have du : ad, do. thing Wee Integrating with respect to a, we get lu=—a@?+C: 2. wae @ +e = Oe-e by making e*-= C. To determine C, make a= 0; then ms — x? — ae 1 — e wet ae dx = s/t = C3 therefore le e—** cos, 2aada = ; e—% n/m. 0 SECTION X. ELLIPTIC FUNCTIONS. 279. Elliptic Functions or Elliptic Integrals is the name given to the following integrals : — 9 dé First order. mommll cll Koper A Ip /1 —c?sin.Z26 — ie ) 9 ee Second order. if /1—c?sin26 d) = E(c, 6). 0 ‘ Third order i : a9 '¥ 0(1+asin20)/1—c?sin26 The constant c is called the modulus of the function, and is SEIU, AE 6). supposed less than unity; the constant a, which appears in the third function, is called the parameter; and the variable 0 is called the amplitude of the function. The function is said to be complete when the limits of the amplitude are 0 and - The integral of the second order expresses the length of the arc of an ellipse estimated from the vertex of the conjugate axis (Art. 240); the semi-transverse axis being unity, and the eccentricity of the ellipse the modulus of the integral. From this fact, and from the relations which exist between the sev- eral functions, the term elliptic functions has been derived. Our limits permit us to investigate but a few propositions relating to such functions. 280. Putting «x for sin.@, the integral of the first order becomes if. dx 0/1 —eV/1—cx 59 465 466 INTEGRAL CALCULUS. In like manner, for another value of « denoted by ,, we have iP dx, Vil oo V1 — ctx? ot Now assume the relation da On Soe a ae iT athbaeur a . L0) ee 2 V1 — a? / 1 — cx? V1 eV 1 ee Multiply through by the product of the ionotaeean divide =0 (1). by 1 —c?a’* ,» and integrate ; then Ne Walaa 7, | , omy a saa /1—c?a c da ia oata? P= cya = constant. Integrating the first term by parts, we get [PSR SG aoa j t/1— ai V/1—c?x - O — : 5) 1—c*x*ax, 1—c?’x*x, + foe, (+e*)(1tc?x?x! was —2c*x; ams At eee /1—2ir/1 —c? 2? es few eat ae V1 — 22/1 — e822? de. In this result, interchanging w and x,, we have the second term. Adding results, observing that by (1) the terms of the sum which are under the sign df reduce to zero, we find eV1—¢8 Vie +0718 V1—oa Z der. al als aa const. (2). Kq. 1 expresses the condition that the variables a and a, are so related that the sum of the integrals | a VI —eVi— V1 — ai Lene shall be constant. ELLIPTIC FUNCTIONS. 467 “é dx Put if! Rae a i tetas — a, 4 os S(a), V1 —«?= O(a), V1— cx? = R(a); # dc Wey SBVisag a? B= 5) ea = O(8), V1—c?a’ = R(8). Then, by Kq. 1, we have da+tds—0: a + p= constant = ». It is also seen from (1) that the constant 7 is the value of also x, when «=0; and, further, when « = 0, we have e—0, B=7, 2, =S(7)—S(a+ 8): therefore, by making the proper substitutions in (2), it be- comes , S( ae) (8) B(6) Se Ta ) S(a + 8) = ce 1 — o?{ S(a){* | S(B which is the fundamental formula as given i Euler in the theory of elliptic functions. 281. Suppose the variables 6, 6,, to be connected by the equation is d9 4 os dd, Kod oa fil 0/1 — c?sin.26 0 /1 —c?sin. 120) 04/1 — ec? sin2 0 (1), or Ec, 6) + F(¢, 6;) = F(c¢, w), in which » isaconstant. If 6, 6,, be regarded as functions of a third variable ¢, and (1) be differentiated with respect to the latter variable, we have ad) dd, dt dt V/1 — c2sin20 5 /1 —c? sin, =0 (2). 468 INTEGRAL CALCULUS. Since the new variable ¢ is arbitrary, let us assume dp = /(1 —c’sin26) (3); Obie whence, from (2), Oe 1 oss mee dt Squaring (3) and (4), and differentiating, we get i hes (Joa wv chy a Pah ye sin.dcos.9, i ee sin.f,Cos.4,: i Nek a has bad, *. on ae a, = — c*(sin.6 cos.6 + sin.6, cos.4,), “ihe CG? tae : OF a5 (9@+6,)=-—- 5 (sin. 29 + sin.20,) (5). Put 6+ 6,= gq, and 6—6,=w; then 2= p+, B,=g—-Y, | sin. 20 = sin. 9 cos. w + Cos. p sin. w, sin. 24, = sin. g Cos. wy — COS. M SiN. w: therefore, from (5), we have 9 Ci GME IG an cos ee c” sin. w cos tae -@Q ~W, Wiiaae -W -Q. We also have dp dy _ au) ~(q) =2 1 —cos.29, _1—cos. 26 dt ada) i) = ) 2 2 ) a COB: ASE, __ cos. 26;\ ain ae =o (S ; ) = _psin. w: d* op d?w dt® aS = COL. —=CObs Ms dp dw ’ dp dw z dt dt dt dt ELLIPTIC FUNCTIONS. 469 d?w d dg. d - dw. “at? But psn. g = copa, alae ar: dt ad /,d¢ d d /,dw GN ee yea pati a) = Roe aA? i) Shard Whence 1? —Isin w+ C, ls 7 isin. p + Ch; or by putting C= LA, Cr—l4y, i passing from logarithms to numbers, fA BIN. tH, oF — A, sin. 9 (6): | . dy *, Asin. w Fiscme A, sin. Pa Acos.w= A,cos.g+C (7). From Eq. 1 wo see that F'(c,6) = F'(c,~) when 6,=0: therefore we then have 6 == gy =y, and (7) then becomes (4 — A,) cos. 4 = C; and therefore A cos. (6 — 6,) — A, cos. (6 + 6,) = (A — A,) cos. pn; whence, by developing cos.(#—6,), cos.(9@+4,), and re- ducing, : (A — A,) cos. 6 cos. 0, + (A + A)) sin. 6 sin. 4, = 4.— A) COs. 6 (8). Now, dy Oiaps. qed fo oa = ieee F = /(1 — c* sin.’ ¢) — 4/(1 — ec? sin.’ 6,), dw e, = (1 — ce’ sin.’ 6) 4+ (1 — c? * sin. rye Substitute these values in (6), and make 6, = 0; then /(1 — ec? sin.) — 1 = Asin. p, a/(1 — ec? sin.) + 1 = 4,sin. pw. 470 INTEGRAL CALCULUS. From these equations, getting the values of d+ 4,,4— A,, and substituting in (8), we get, finally, cos. 6 cos. 6; — sin. @ sin. 6; 4/(1 — ec’ sin.) = cos. (9). This relation, by an easy transformation, may be made to take the form | cos. 6 = cos 9, cos. w + sin. 6; sin. wa/(1 — ec’ sin’) (10). Eqs. (9) and (10) express the connection which exists be- tween the variables in two elliptic functions of the first order which have a common modulus. 282. Let Fc, 0), '(c, 9,), be two elliptic functions in which c, c,, and 6, 6,, are connected by the equations 9 4c sin. 24, It is proposed to prove that 2 ANG Ey ote oe Differentiate Eq. 2, regarding 6, as the independent variable ; then 1 dd _ 2(1 +-ccos. 24) cos.26 d9, (e+ cos. 26,)? — From (2) we also get (c+ cos. 29,)? 1 + 2ccos.20,-+ 0? dp 2(1-+ccos. 26,) db,) di 2eicosee; cos.2 46 = Also, from the same equation, we get c* sin? Bas 1 + 2c cos. 26, + ¢? __ 1+ 2c cos. 26, + ¢? cos.’ 29, ‘ap 1 + 2ccos. 20, +c? 1— ce’? sin? 6=1— ELLIPTIC FUNCTIONS. 471 f da fas (1) — ¢? sin.’ 6) 2(1 + ccos. 24,) / (1 + 2c cos. 20, + c?) a 1+ 2ccos.26,; +c? 1+ 6ccos.24, a9 a/(1 + 2c cos. 26; + c”) anne dd, i pag 7 phe ote ae pia al pe ea sin.?6, Gls e): d Ac , But the last integral, when ito: =c_, becomes 2 do, 9 l+e /(1 — ¢; sin.?4,) 1+ec (C1 91) | 9 E(c, ea eee ole e i: If we suppose 6, = > 2 ae 4 Tt aera e115) = Fle =) =2F ° 5) 283. Having shown (Art. 281) that there exists, between the variables of two elliptic functions of the first order having a common modulus, the relation cos. 6 cos. 0, — sin. sin. 0, 4/(1 — c? sin. w) =cos.u (1), then, between the corresponding functions of the second order, there exists the relation E(c, 6) + E(c,6,) — E(c, v) = ce? sin. 6 sin. 6; sin. From the equation between the amplitudes 6, 6,, 0,, may be considered as a function of 6; that is, we may assume Li(c, 6) oe Lic, a) = Li(¢, 4) = (4), 472 INTEGRAL CALCULUS. and differentiate, thus getting i] /(1 — ce? sin.’ 6) + 4/(1 — ¢* sin.? 6,) a 7G By Kq. 10, Art. 281, the first member of this equation may be put under the form cos. 9 — cos. 6; cos. u , cos. 0, — cos. 6 cos. u dd, sin. 6; sin. u sin. 6 sin. pu do __ @(sin.’6 + sin.’ 6, + 2 cos. 6 cos. 6, Cos. u 1 a do 2sin.@ sin.6, sin.w But putting Eq. 1 under the form cos. 7 cos. 6; — cos. u = 4/(1 — c? sin.? #) sin. 0 sin. 0,, and squaring, we get cos.”? + cos.79: + cos.2 4 — 2 cos. 6 cos. 4, cos. u = (1 —e?sin.? 2) sin.7 08m oy Adding cos.’ 6, cos.’ to both sides of this equation, transpos- ing, and reducing by the relation cos.? = 1 — sin.2, we find sin.’ 6 + sin.’ 0, + 2 cos. 6 cos. 0; Cos. u = 1 + cos.’ u +c? sin.? 6 sin? 6, sin? n, & (1 + cos.” u + c? sin.’ 6 sin.? 6, sin? 1) 2 sin. 6 sin. 0, sin. u d(sin. @ sin. 6;) a9 : == On ~1 . d(sin. 6 sin. 0) pag f (6) =c? sin. u Wi is 4 and therefore, by integration, —f (0) =e? sin. 6 sin. 0; sin. pw. ok ¥ ve % be iee Peter. f. a4 PAP +