UC-NRLF *B S2T sas Bookseller, i ^°"enhamCt.RH ' % I THE DIFFERENTIAL AND INTEGRAL CALCULUS. In the Press, AN ANALYTICAL TREATISE ON PLANE AND SPHERICAL TRIGONOMETRY. BY THE REV. DIONYSIUS LARDNER, A. M. LONDON : PRINTED BY THOMAS DAVISON, WHITEFRIAfiS. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS. BY THK REV. DidNYSIUS LARDNER, A. M. \* LONDON: PRINTED FOR JOHN TAYLOR, WATERLOO-PLACE, PALL-MALL. 1825. U^6 i. lii PREFACE. Analytical science, after having been long neg- lected in these countries as an elementary depart- ment of education, has, within a few years, been cultivated by the young aspirants for mathematical celebrity with an ardour, and prosecuted with a ra- pidity and success, which its warmest admirers could scarcely have hoped for. This change would pro- bably have taken place at an earlier period, but for the obstacle opposed to it by the want of treatises on the subject, in our language, of a suf- ficiently elementary nature. The restless activity of the human mind in the pursuit of knowledge was not long to be checked by so trifling an impediment, and our students soon found in foreign works that which our own professors had failed to supply ; and through the medium of these treatises, analytical science began, and has continued, to be cultivated at the universities with singular success. In the mean time, several original works have appeared, which 463024 Vi ' PREFACE. are gradually superseding the works of foreign pro- fessors. For these, the public are indebted to some of the distinguished members of the University of Cambridge ; Woodhouse, Whewell, Herschell, Bab- bage, Peacock, &c., &c. Desirous of contributing to the great work of improvement which was thus in progress, I some time since published the first part of a treatise on Analytic Geometry ; a subject which had not then, nor has been yet treated of by any other English author. The favourable manner in which that work has been received has encouraged me in the pro- secution of my labours, the result of which I now venture to offer to the public. The present Treatise is divided into four parts, the subjects of which are severally, 1. the Differential Calculus; 2. The Integral Calculus; 3. The Cal- culus of Variations ; 4. The Calculus of Differences. The arrangement which I have adopted throughout the work has been to present to the student theory and ilhistration in alternate sections. I have found by repeated experience, that as on the one hand, the total omission of examples, so common in foreign treatises, renders the theory obscure and even unin- telligible ; so, on the other hand, their too frequent PREFACE. Vll recurrence in the progress of the development of the abstract principles of a science is apt to break the continuity and oneness of the reasoning, and to render it difficult for the student to take a general view of the subject as a whole. By the method adopted in this work, I have attempted to remove both these defects. The stu- dent will find in general, that the complete theory of each department of the subject is fully explained before the current of his ideas is stopped by an example. At the same time the subdivisions of the subject are so numerous, and the sections of illustration so frequent, that none of the confusion which is apt to arise from a long exposition of ab* stract principles without examples of their appli- cation can ensue. Another advantage of this method is, that it is suited to students of different capacities. The sec- tions of illustration will receive only that degree of attention which is necessary to fix clearly in the mind the general principles which have been esta- blished in the preceding sections. There is one part of the work, the calculus of dif- ferences, which I am sensible of having written under considerable disadvantage. The treatise on this sub- yiii PREFACE. ject by Mr. Herschell, which forms the appendix to the translation of the Calculus of Lacroix, together with the collection of examples by the same author, which accompanies Mr. Peacock's examples on the differential and integral calculus, form a treatise on the calculus of differences so excellent, that it would be useless as well as presumptuous in me to attempt to improve it. Under these circumstances, I have confined myself in the fourth part to a few of the most elementary and generally useful principles of dif- ferences, particularly those connected with the method of interpolation and the summation of series. I have attempted in this treatise to include a more extensive range of analytical science, more fully developed, accompanied by a greater quantity of practical illustration, under a considerably less bulk than will be found in most of the foreign trea- tises on the same subject, particularly those which have hitherto formed the class books at the univer- sities. Whether I have succeeded in this design, I leave the public to decide. CONTENTS. PART I. THE DIFFERENTIAL CALCULUS. {*) SECTION I. Preliminary principles. (1.) Quantities, constant and variable. (2.) Examples of constant and variable quantities. (3.) J f miction, what? (4.) Manner of denoting functions in general. Object of the differential calculus. (5.) The fluxional calculus of Newton. (6.) D'Alembert's method of limits. (7.) Improvements of Lagrange. Derived functions. (8.) The same end attained by these three methods. Notation and nomenclature. (10.) Functions, explicit and implicit. (11.) Functions, algebraical and transcendental. (*) SECTION II. The differentiation of functions of one variable. (12.) If M = F {x) and a: be changed into ^ + /z, m = f (a? + /^) may be developed in a series of positive and integral powers of A, provided that x be an indeterminate quantity. (13.) The coefficient of the first power of h in this series is the differential coefficient. (14.) Abbreviated manner of expressing the series. (15.) The differential is the first term of the expanded differ- ence. (16.) If M ^ F (y) and 7/ =f(xj to determine the differen- tial coefficient of u, considered as an implicit function of x. (I7.) To differentiate a quantity composed of several functions of the same variable united by addition or subtraction. (18.) Constant quantities, combined with a function by ad- dition or subtraction, disappear in jts differential ; but constants, combined as factors, are similarly combined with the differential X CONTENTS. (19.) Exponential functio?is, what.'' (20.) To differentiate an exponential function. (21.) To differentiate a logarithm, (22.) To differentiate a product of several functions of the same variable. (23.) To differentiate a fraction, whose numerator and de- nominator are each products of several functions of the same variable. (24.^ To differentiate a power. (25.) Lemma. The limit of the ratios of the chord of a cir- cular arc, and the arc itself to the tangent, is a ratio of equality. (26.) The limit of the ratio of the sine of an arc to the arc it- self, is a ratio of equality. (27.) To differentiate the sine of an arc. (28.) To differentiate the cosine of an arc. (29.) 'I'o differentiate the tangent and cotangent. (30.) 'i'o differentiate the secant and cosecant. (31.) To differentiate an arc as a function of its sine or cosine. (32.) To differentiate an arc as a function of its tangent or co- tangent. (33.) To differentiate an arc as a function of its secant or co- secant. (*) SECTION III. Praxis on the differ ejitiation of functions of one variable. (*) SECTION IV. On successive differentiation. (34.) Second differential coefficient. Second derived function. Notation. (35.) For any proposed value of x, the differentials of u and x du are indeterminate, though their ratio -t' is determinate. (36.) The independcfit variable, what.^ (37.) Third, fourth, &c. Differential coefficients, or derived functions. Notation. (38.) If u = F (?/), y =.f {x)y given the second differential coefficient of w as a function of y, and of 3/ as a function of Xy to find the second differential coefficient of m as a function of x. (39.) To determine the successive differential coefficients of a power. (40.) To determine the successive differential coefficients of the product of two functions. (41.) To determine the successive differential coefficients of an exponential function. / CONTENTS. XI (42.) To determine the successive differential coefficients of a logarithm. 43.) To determine the successive differential coefficients of the sine and cosine, as functions of the arc. (44.) To determine the successive differential coefficients of the tangent and cotangent, as functions of the arc. (45.) The successive differentials of the secant and cosecant as functions of the arc. (46.) The successive differential coefficients of the arc as a function of its sine or cosine. (47) The successive differential coefficients of the arc as a function of its tangent or cotangent. (48.) The successive differential coefficients of the arc as a function of its secant or cosecant. SECTION V. Of Development. Theorems of Taylor, Maclauri?ij Lagrange, and Laplace. * (50.) If the variable of a function be supposed to consist of two parts, the differential coefficient will be the same to which- ever part the variation be ascribed. * (51, 52.) Taylor's theorem. * (53.) The difference expressed in a series of its corresponding successive differentials. * (54 — 56.) The cases called exceptions to Taylor's theorem. * (57) Maclaurin's theorem. Exceptions. (58.) Lagrange's theorem. (59.) Taylor's theorem, a particular case of Lagrange's, (60.) Original form in which Lagrange proposed his theorem. (61.) Laplace's theorem. (*) SECTION VI. Praxis in the development of functions . (62.) The binomial theorem. (63.) Expansion of a^ in a series of powers of x. (64.) The value of the hyperbolic base. Fundamental pro- perties of logarithms, (^^.) Expansion of e^. {^^^ Series for the exponential formulae e'^'^~^ + e~~^^~^ and e^^A/^ - ^— ^a/^. (67.) Expansion of a"*^. (68 — 72.) Various series for logarithms. (73.) Series for the sine and cosine in a series of powers of the arc. CONTENTS. (74.) Exponential formnla for the sine and cosine. 2 COS. X = e^'^- ' -f- e—V-i. 2 v/-l sin ,1- — e?AV— I _ aV— 1 (75.) Moivre's theorems. (7f).) Exponential formula for the tangent. (77') Expansion of an arc in terms of the sines of its successive multiples. (78.) Expansion of an arc in a series of powers of its sine. (79.) Value of the circumference of a circle, deduced from this. (80.) Expansion of the tangent in a series of powers of the arc. (81.) Expansion of the arc in a series of powers of its tangent. (82.) Value of the circumference of a circle. (83.) Series for the cotangent in terms of the powers of the arc. (84.) Expansion of mu^ ~ kx = m, in a series of powers of cT. (85.) Value of u in the equation mii^ — a^'^u + w^,^' = o, in a series of descending powers of jc. (86.) Given f (x + A) + f (/k — A) = f (x) f (h) to find the form of the function. (87-) To find the w^'' power of a root of the equation xy'^ -f a — 7/ = o, ^ and a being known. (t) SECTION VII. On the limits of series. (88.) In any series of ascending positive powers of h, a value may be assigned to h, so small, that any proposed term may be made to exceed the sum of all that follow it. (90.) To determine the effect produced upon the function by a given increase of the variable. (91.) To determine the limits of error arising from assuming any portion of the development off (c^' 4- h) by Taylor's series, for the whole value. Examples. (92, 93.) To determine the limit of the ratio of two series, ascending by the powers of the same quantity, (t) SECTION VIII. Differentiation of functions of several variables. (94.) Partial differential coefficients. Notation. (95.) General explanation of this notation. CONTENTS. XIU (96.) Extension of Taylor's theorem to functions of several in- dependent variables.^ (97.) Differential' of the algebraical sum of several functions of several independent variables. (98.) Differential of the product of several independent va- riables. (99.) Examples. (t) SECTION IX. Differentiation of equations of several variables. (100.) Differentiation of equations of two variables that can be solved for either variable. (101.) Differentiation of equations of two variables without solution. (102. ) Differentiation of equations of any number of variables. Total differential partial differential equation. (103.) Successive differentiation of equations of two variables. (*) SECTION X. Effect of particular values of the variable upon a function, and its differential coefficients. (104.) Four remarkable states of the function P = o, 2<* zr %, 3^ imaginary, 4^ infinite. (105.) To determine the values of the successive differential coefficients of a function which correspond to that particular value of the variable which renders the function, or any of its differen- tial coefficients, = o. (106.) Given the function which vanishes when .r = o, to de- termine the highest power of (x — a), which divides the function. (IO7.) To determine the true value of a function which a par- ticular value of X renders e, or infinite. (108.) Examples. (109 — 113.) To determine the conditions under which any dif- ferential coefficient of an implicit function becomes = o =§, or = §. (114.) A more particular examination of this state of the func- tion, with reference to the geometrical application of these prin- ciples. (*) SECTION XI. On maxima and minima. (H5.) Definition of ma 0. 20. when c < 0. (221.) The sixth class — binomial differentials.— Two cases in which they are immediately integrable. (222, — 226.) Four formulae for reducing the integration of given binomial differentials to that of other binomial differentials, with lower exponents. (227.) The seventh class ; how rationalised. (228.) The eighth class ; how rationalised. (229.) The ninth class, reduced to the fifth. (t) SECTION V. Praxis on the integration of differentials, whose coefficients are irrational. (t) SECTION VI. Integration by series. (230.) Integration by series always possible. In what case useful. Sometimes useful even when the finite integral can be assigned. (231.) To develop an arc in a series of powers of its sine. (232.) To develop an arc in a series of powers of its tangent. (233.) To develop an arc in a series of powers of its cosine or cotangent. (234.) To develop the versed-line of an arc in a series of powers of the arc itself. (235.) To develop the logarithm of a given number in a series. (236.) Examples of integration by converging series. (237, — 240.) General method of approximating to the values of integrals by series. (238.) Origin of the integral — indefinite and definite integrals. (t) SECTION VII. Of the integratio7i of differentials whose coefficients are expO' nential or logarithmic functions of the variable. (*) (241.) General principlesof the integrationof transcendental functions. CONTENTS. Xix (*) (242.) Elementary integral of exponential functions, fttxdx = -7-. la (243.) To integrate a differential of the form, za^dx. (244.) To integrate a differential of the form, f^F {x) dx. (246.) (247.) To integrate the formula, udx = a^x^'dx. (*) (248.) Elementary integral of logarithmic functions, •^ X " (249.) To integrate a differential of the form x(lxydx. (t) SECTION VIII. Praxis on the integration of exponential and logarithmic dif- ferentials. (t) SECTION IX. The integration of differentials whose coefficients are circular functions of the variable. (*) (250.) Six elementary integrals of differentials whose co- efficients are circular functions. ' (*) (251.) The arc or angle is disengaged from the coefficient by integration by parts. (*) (253.) All functions of trigonometrical lines may be re- duced to functions of the sine and cosine. (254.) When powers of the sine and cosine occur in the dif- ferential coefficients, they are developed in series. (255.) When sines and cosines are connected by multiplication, they can be disengaged by the elementary trigonometrical formulae. (256.) Functions of the sine or cosine may be converted into exponential functions. (257.) Differentials of the form sin."'^. cos."0 d(p, may be re- duced to binomial differentials. (t) SECTION X. Praxis on the integration of circular functions. (t) SECTION XI. On successive Integration. (259.) By the successive integration of a differential coefficient of any order superior to the first, the final integral is expressed in a series of descending powers of the variable whose coefficients are arbitrary constants. b2 XX CONTENTS. (*) SECTION XII. Of rectification, quadrature, and cuhature. I. Rectification, (260.) To find the length of the arc of a given curve, related to rectangular coordinates. (261.) The length of the arc, related to polar coordinates. (262.) The length of the arc of a curve of double curvature. II. Quadrature. (263.) To determine the area of a plane curve, related to rec- tangular or polar coordinates. (264.) To determine the area of a curved surface. (265.) To determine the area of a surface of revolution. III. Cubature. (266.) To find the volume of a surface in general. (267.) To find the volume of a surfece of revolution. (*) SECTION XIII. Examples of rectification, quadrature, and cubature. (268.) To determine the arc of the parabolic curve, y = pa:". (269.) Rectification of the common parabola. (270.) To rectify the hyperbolic curve, y =:px~^. (271.) To determine the arc of an ellipse. (272.) To determine the area of the parabolic or hyperbolic curve, y = /^x— ". ( 273. ) To find the area included by two radii vectores from the centre of an equilateral hyperbola. (274.) To determine the surface and volume of a cylinder. (275.) To determine the surface and volume of a cone. (276.) Of the surface of a sphere. (277) Of the volume of a sphere. (278.) To determine the volume of an ellipsoid. (t) SECTION XIV. Of the integration of differentials of functions of several variables. (279.) Integration of a partial differential of two variables of the first order, and of superior orders taken with respect to the same variable. One partial diflferential insufficient to determine the primitive function. • (280.) Integration of a partial differential of a superior order taken with respect to diflferent variables. (281.) The integral of a total diflferential is the sum of the in- tegrals of the partial diflferentials. A total differential, when in- tegrable. \ CONTENTS. XXI (282;, 283.) Given two functions of two variables, to deter- mine whether they are partial differential coefficients of the same functions, and if so, to find the primitive function. /') =0. (35(3.) 4. Integration of the form ^{y'y'a') = 0. (357.) Three methods of integrating the equation F{y.vc) = 0. (358.) 5. Integration of the form T(y"y'y) = 0. (359, 360.) Two methods of integrating the equation F(yyc) (361.) Three cases of differential equations of the second order, including both variables, which may be integrated. CONTENTS. XXV (362.) 1. Integration of the equation, (363.) 2. Integration of the equation, (364.) 3. Integration of equations homogeneous with respect to the variables and their diiferentials. II. Jritegration of differential equations which do not contain either variable. (365.) Two integrable forms of such equations. (366.) 1. Integration of the form, p(^, ^^) = 0. (367.) Integration of the form, rfy-^, ^^) = 0. III. Integratio?i of equations which include one variable only. (368.) Two integrable forms of such equations : 1° integration of the form, F(^,-, - ^„) = 0. 2 '. If the equation include the dependent variable only. (369.) A differential of the nth order including no variable, may be reduced to one of the {n — l)th order, including one variable, or to one of the (?2— 2)th order, including two variables. IV. Iidegri'tion of komogtneous equations of the first degree with respect to the dependent variable, and its differentials. (370.) Equations of this class may be reduced to equations in- cluding no variable. (371 to 373.) Integration of the equation, r/»w d^-—'u dy d.v^'^ dcP"-^ d^ ^ in the case where a, b are all constant. (374.) D'Alembert's method. V. Linear equations of the first degree with respect to y and its differentials, (375.) Two methods of reducing the integration of such equa- tions to the resolution of algebraic equations. (376, 377- ) Euler's method. (378 ) Lagrange's method. XXVI CONTENTS. (t) SECTION XXIV. Praxis on the integration of equations of the second and superior orders. SECTION XXV. On the integration of simultaneous differential equations of the Jirst degree. (387-) General principle of simultaneous integration. (388.) To integrate simultaneously the equations dy dx My 4- N^ 4- P-f^ 4- Q— - = T. ^ dt dt , , ,dy ,dx •^ dt dt (389, 390.) The same effected otherwise. (391.) The same principles applied to two differential equations between three variables. SECTION XXVI. The integration (>f equations by approximation, (392.) General theory. (393.) Examples. (394.) Method of approximating to the integrals of equations by a continued fraction. (395.) Hence a method of converting functions into continued ' fractions. SECTION XXVII. Integration of differential equations of two variables by the geometry of plane curves. (396.) These methods used only in the infancy of the calculus. Origin of the calculus — its original names, and objects. (397, 398.) Manner of representing the integral geometri- cally. (S99, 400.) Manner of representing the integrals of equations of the second and higher orders. (401.) When the variables are separable, the manner of repre- senting it is different. (t) SECTION XXVIII. The problem of trajectories, and other geometrical applications of the integral calculus. (402.) The problem of trajectories— its origin and enunciation. (403 — 405.) General principles of its solution. CONTENTS. XXVll (406.) A system of parabolas having a common vertex and axis, and hyperbolas having a common centre and asymptotes, are given, to find the trajectory intersecting them at a given angle. (407-) To find the trajectory of a system of circles touching a given right line at a given point. (408.) The class of problems from which the integral calculus derived the name of " inverse method of tangents." (409.) To find the curve whose normal is a given function of the intercept of the axis of x between it and the origin. (411.) To find the curve in which the radius of curvature is a given function of the normal. (413.) Given a system of parabolas having a common vertex and axis, or hyperbolas having common asymptotes, to find the curve which intersects them all, so that the areas included by the coordinates of the point of intersection and the arc of the para- bola or hyperbola, between that point and the axis of y, shall be constant. SECTION XXIX. Of the integration of total differential equations of the first degree of several variables, which satisfy the conditions of integra- hility. (414.) Integration of a total differential equation of the first order between three variables, if it satisfy the criterion of inte- grability, or if one of its variables be separable from the other two. (415.) If it be not an exact differential, it may be rendered so by a factor. (416.) Cases in which the integration of the proposed equa- tion of three variables may be made to depend on the integration of an equation of two variables. Examples. (417) Integration of equations of superior degrees between several variables. SECTION XXX. Integration of total differential equations which do not satisfy the criterion of integrahility . (418.) Such equations not absurd or impossible relations. Their integrals expressed by several equations which must subsist to- gether. (419.) Geometrical representation of the integral of an equa- tion of those variables. (420.) General principles on which such equations are inte- grated. (421.) The proposed equation has an infinite number of systems of integrals. Geometrical representation. y XXVlll CONTENTS. SECTION XXXI. Of the integration of partial differential equations of the first order. (422 — 424.) Integration of partial differential equations in- volving but one partial differential coefficient. (425 — 432.) Integration of partial differential equations in- volving two partial differential coefficients. (433.) Another process for integrating partial differential equa- tion of the first order. (434.) Integration of partial differential equations of the first order, by the introduction of an indeterminate quantity. (435.) Integration of the equation vp -{- (iqzz v homogeneous with respect to the three variables. SECTION XXXII. Of the integration of partial differential equations of the higher orders. (436.) Subject of this section. (437-) Equations between three variables of the form ^ ^ di/"" dx df dx'^dy''^ may be reduced to the mXh. order. (438.) Equations of the nth. order, which include partial dif- ferential coefficients with respect to one variable only. (439.) Examples. (440, 441.) Integration of partial differential equations of the second order and first degree. (442.) Examples. (443.) Case in which one of the conditions which the integral must satisfy is a numerical equation. (444.) P^xamples. (445.) Manner of integrating partial equations of the second order by the introduction of an indeterminate function. SECTION XXXIII. On the integration of partial differential equations by series. (446.) General principle of the process. (447.) Application of this to partial equations of the first order. (449.) Example. CONTENTS. XXIX (450.) Integration of partial equations in series by the method of indeterminate coefficients. SECTION XXXIV. Of arbitrary functions. (451. ) Nature and signification of arbitrary functions. (452.) Examples on the manner of determining arbitrary functions involved in the integrals of partial differential equa- tions. (453.) General rule for determining the arbitrary function. (454.) Example in which there are two arbitrary functions to be determined. (455.) Signification of the arbitrary functions, when they can- not be determined by the data of the question. PART III. THE CALCULUS OP VARIATIONS. SECTION I. Preliminary observations and deflniiions. (456.) Origin of the calculus of variations. Examples of the class of problems for which it is necessary. Isoperimetrical pro- blems. (457-) Notation for the calculus of variations. (458.) Explanation of the symbols, M"yj d"Ey, §/u, /^u, SECTION II. Of the variation of a function. (459.) In any formula to which d and ^ are prefixed, the transposition of these characters does not affect the value of the quantity. (460.) In any formula to which y" and o are prefixed, the trans- position of these characters does not aflfect the value of the quantity. (461.) To determine the variation of a function of several variables, and their successive dififerentials. (462.) To determine the variation of a function of two vari- ables only. XXX CONTENTS. (463.) To obtain the variations of the successive differential coefficients in terms of the variations of the variables. (464.) To determine the variation of the integral of a function of several variables, and their differentials. (465.) Conditions on which the variation of Ju will be free from the integral sign. Criterion of integrability. (466.) To determine the variation of the integral of a given differential vdx, when v is a function of several variables, and the differential coefficients considered as functions of one common variable x. SECTION III. On the maxima and minima of indeterminate integrals. (468.) The powers of the differential calculus are inadequate to the investigation of the maxima and minima of indeterminate integrals. (469.) Conditions necessary in order that the indeterminate function may be a maximum or minimum. (470.) To determine what relation between the variables will render an indeterminate integral, taken between assigned limits, a maximum or minimum. (471 — 475.) Effect of the several conditions which may affect the limits of the integral. (476.) The higher the order of the formula u, whose maximum or minimum is sought, the greater number of conditions may be imposed upon the constants. (477-) Manner of treating the variable co-ordinates of the limits in taking the variation of u, and in integrating with re- spect to the variables x, y, z . . . . (478.) When the variations 5x, Sy, $z . . . are restricted by conditions independent of the limits of the integral. SECTION IV. Examples on the calculus of variations. (479.) To find the shortest line between two points. (480.) To find the shortest line between two given surfaces. (481.) The shortest line between a fixed point and a surface. (482.) The shortest line between two plane curves. (483.) To find the shortest line, joining two points of a given curved surface, drawn on the surface. (484.) To find that curve of a given length, drawn between two given points, which will include with its extreme ordinates and the intercept of the axis of x between them, the greatest pos- sible area. (485.) Of all solids of revolution having equal surfaces in- CONTENTS. XXXI eluded between given limits, to determine that which has the greatest volume. (486 ) Of all plane curves of a given length joining two given points, to determine that which produces by its revolution the solid of greatest volume. (487-) Of all plane curves of a given length drawn between two given points, to determine that which produces by its re- volution the solid of the greatest surface. (488.) Given two points at different perpendicular distances from the horizon to find the line of swiftest descent from one to the other, or the hrachystochronoiis curve joining them. (489.) To find the line of swiftest descent from a fixed point to a given curve ; or from one curve to another. (490.) A solid of revolution moves in a fluid in the direction of its axis ; to determine its figure to that, cceteris paribus, it will suffer least resistance. (491.) To determine the curve of a given length joining two points of which the centre of gravity is the lowest. PART IV. THE CALCULUS OF DIFFERENCES. SECTION I. Definitions and notation. (492.) General principle — notation. (493.) Explanation of terms — Increasing or decreasing series generated — general term : index. (494.) Difference of the function — Notation. (495.) Difference of the diflference of a function — successive differences. Notation. (406.) The calculus of differences — divided into the direct and inverse calculus. SECTION II. Of the direct method of differences. (497.) To determine the difference of the algebraical sum of several functions of the same variable. (498.) The constant quantities connected with the variable of a function by addition or subtraction disappear in its difference, and XXXll CONTENTS. those united by multiplication or divisioi^ are united in the same way with its difference. (499.) To determine the values of «., and AUt in a series of w„ and its successive differences. (500.) To determine A"Uj. in a series of u,„ m„_i, u^—2- (501.) Given the function to find its successive differences. — The successive differences of u,, = (3/ + nh)'"^. (502.) To determine the successive differences of a rational and integral function of x. (503.) Every rational and integral function of ^ has a con- stant difference, whose order is expressed by the exponent of the highest powers of x, which enters the function. (504.) Every function which admits of being expanded in a finite series of ascending integral and positive powers of jc has a constant difference of the nth order^ n being the exponent of x in the last term. (505.) No other species of function can have a constant dif- ference of any order. (50(3.) Example. (507.) Examples of the application of the calculus of dif- ferences to approximate to the value of transcendental functions. SECTION III. Of interpolation. (508.) The method of interpolation — what. (509.) General principle of the method of interpolation when the particular values of the variable are in arithmetical pro- gression. (510.) Application of this process to algebraic functions. (511.) Application to functions not algebraic. (512.) IMethod of conducting the process when the particular values of the variable are not in arithmetical progression. (513.) This more general case includes that in which the values of x are in arithmetical progression. (514.) The general formula for u may be expressed in another way. (515.) Application of the method of interpolation, to qua- dratures, or to approximate to the value of the integral yxc^x. SECTION IV. The inverse calculus of differences . (516.) Its object. General form of differences which are ex- pressed as immediate functions of the independent variable, A" Wj = y{x). CONTENTS. XXXin (517.) Three theorems derived from the inversion of the rules in the direct calculus of differences. P. 2(Am^ ) = w^ + c 20. 1:(ax) = A'Ex (518.) When the difference is a rational and integral function of the independent variable, its exact integral may always be determined. (519.) Formula by which every function which can be reduced to a product of equidifferent factors may be integrated. (520.) Formula for integrating all fractions whose numerators are constant, and whose denominators can be reduced to a pro- duct of equidifferent factors. (521.) Integration of functions of the form, AX^ + BX^' -f CX'^ (522.) Given one of the integrals, Dzco, "Ex, Xx% Ex^ ^x'^-^ Sx"* the succeeding ones can be determined. Table of their values. (523.) General series for 2a;'». Numbers of Bernoulli. (524.) To integrate exponential functions. (525.) To integrate circular functions and their powers. (526.) Integration by parts Sx'x" = x'Sx" — 2:[Ax'.2(x" + Ax")] (527.) The integral of a function considered as a difference may generally be expressed by a series. SECTION V. Summation of series. (528.) Notation for expressing the sum of any number of con- secutive terms of a given series. (529.) The summation of the series su, depends on the inte- gration of Uj+i and w^ considered as equal differences. (530.) The sum of the series from the wth to the xili term in- clusive. (531.) Examples of the summation of series. The marks prefixed to the articles in the table of contents divide the work into three different courses suited to three dif- ferent classes of readers. Those who wish merely to acquire a knowledge of the calculus sufficient to enable them to under- stand the elementary parts of physical science may confine their attention to the articles and sections thus (*) marked. Those who are desirous of penetrating somewhat further, may add to the former the articles and sections to which the mark (t) is prefixed. Those whose views are still higher, will, perhaps, find it expedient to read the entire work. ERRATA. P. 40, line 12 from bottom, for d¥, read t>¥. P. 46, line 21, for -h, read — d^u J du P. 79, line 16, for -Y-,read y . P. 89, line 12, for z, read ^. line 17, for a — a', read A' - a. line 19, for w, readw — 1. p. 151, line 4, for r, read r — 1. PART I. THE DIFFERENTIAL CALCULUS. THE ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS. • PART I. THE DIFFERENTIAL CALCULUS. SECTION I. Preliminary Principles. (1 ) Quantities engaged in this science are considered as constant or variable. A quantity, which is supposed to retain the same value tlirouohout the whole of any investigation, is said to be constant. On the contrary, a quantity to which in any in- vestigation different values may successively be ascribed, is said to be variable. Constant quantities are usually expressed by the first letters of the alphabet, and variable quantities by the last. Constant and variable quantities are not, however, analogous to Jcnown and unlaiozai quantities in common algebra, since a constant quantity may be unknown. (2.) The following may 'serve as examples of constant and variable quantities. A point being given within a circle given in magnitude and position, a line drawn from the given point to the circumference of the circle is in general a variable quantity, as its length will change with the point B 2 THE DIFFERENTIAL CALCULUS. SECT. I. in the circumference to which it is drawn. But if the given point within the circle be the centre, the same line becomes a constant quantity, being the same length to whatever point in the circle it is supposed to be drawn. Again, if the base and vertical angle of a triangle be given, the radius of the inscribed circle, and the distance of its centre from the vertex, are variable quantities ; but the radius of the circum- scribed circle, and the distance of its centre from the vertex, are constant quantities. (3.) When two variable quantities enter the same in- vestigation, they are frequently so related that the variation of either may be determined by that of the other. In other words, a relation may subsist between them, such, that any particular value being assigned to either, the corresponding value of the other will be determined. In this case, either of the variables is said to be a function of the other. Thus, for example, in the equation u = 4^ sin. ^, any variation in r produces a corresponding variation in u, and vice versa. Also, any particular value, as 30°, being assigned to x^ the corresponding value (2.) of u is determined. In this case, therefore, u is said to be a function of a:, or a: a function of u indifferently. The same may be observed of the equations u = 10j7% u = log. a:, u = «% &c. (4.) As it is necessary to express functions without regard to any particular form, a peculiar notation has been invented for this purpose. The character f or^' signifies a function, and f{x) ory(jr) signifies a function of .r, jr being considered the variable. Thus, u = F(«r) signifies that m is a function of.r*. If the variable ,r be supposed successively to assume all values from zero to infinity, the function f(^) or u assumes * The characters (p{a:) and ^{x), and others, are also used to express functions. SECT. I. THE DIFFERENTIAL CALCULUS. 3 a succession of corresponding values. The rate of the varia- tion of u compared with that of x in general will change with the value of x. There is but one case in which their rates of variation will have an invariable ratio, which is when u = ax, a being a constant quantity. In this case it is obvious that u varies as x. The immediate object of the Differential Calculus is to determine the rate of variation of a function relatively to that of its variable. (5.) It was nearly under this point of view that Newton presented the first principles of the Fluxional Calculus. He considers quantities to be generated by motion, as lines are produced by the motion of a point, surfaces by that of a line, &c. The quantity \\\vx^ Jloiving or varying he called a Jluent, and the rate or velocity of its increase or decrease he called Its Jluxion. The Fluxional Calculus was therefore a method of determining the velocity with which a function varies at any point of time compared with the velocity with with its variable changes. (6.) The conceptions of motion and time, which are in- volved in this method, were considered inconsistent with the rigour of mathematical reasoning, and wholly foreign to that science. As an improvement upon the principle, D'Alembert proposed the method of limits. Considering w as a function of JT, let the variable x be supposed to receive any finite in- crement h, so that it becomes or + A, and let the corre- sponding value of u be u, so that we shall have the equations u = r(x), u = f(x -\- h). Let the value of -^ — be found. This will be in general a quantity whose value will depend on those of a; and h, and it will express the ratio of the finite increment {u' — y) of the function, to {Ji) that of the variable. If in this quantity b2 4 THE DIFFERENTIAL CALCULUS. SECT. I. h be supposed to be = 0, it will express the limit of the ratio * of the corresponding variations of the function and variable, these variations being reduced to infinite minute- ness. It is not difficult to perceive that this method attains the same end as the former ; but in rejecting the mechanical ideas of time and motion introduces those quantities or in- crements infinitely minute. (7.) The last improvement in the principles on which the calculus is founded is that of Lagrange. He equally rejects the limits of the ratios of D'Alembert and the motions and velocities of Newton, and has proposed fundamental prin- ciples for the calculus at once rigorously demonstrable and purely analytical. Let u = f(.i) and u' = F(a; + h). By developing u' — u in a series of ascending integral and positive powers of h (which may be proved when ^ is variable to be always possible), let the series u' ^u = A'/i + aW' f A"7i' .... be obtained. In this series the coefficients a', a", a'", &c. are functions of X. The function a' is called by Lagrange the Jirsf derived function. This may be shown to be the same quantity which D'Alembert calls the limit of the ratio of the corresponding increments of the function and variable. Let both members of the preceding equation be divided by h, ni' ni = a' + a"/* + A'7^« h If 7t = this becomes a', which is therefore the hmit of the ratio. (8.) Thus these three methods of presenting the first * A limit is a state to which a quantity continually approaches, and nearer to which it comes than any assignable difference, but to which it cannot actually attain. SECT. I. THE DIFFERENTIAL CALCULUS. 5 principles of the calculus to the student arrive at the same end, though by different means. Newton proposed to determine the ratio of the velocities With which the function and variable increase or decrease, and called these velocities iheir Jluxions, The notation by which he expressed the fluxions was, u, x, the function u and its variable x being called fluents. The quantity u . . , — is the fluxional coefficient. It may be observed here X that the fluxions are not quantities absolutely determinate, but may have any values, provided that their ratio is that of the velocities with which the function and variable change. The fluxional coefficient, however, is given for any par- ticular value of jr, and, in general, only varies with x, D'Alembert proposed to determine the value of the frac- tion having for its numerator and denominator the simul- taneous increments of the function and variable, when both these increments are = 0. The value thus determined is called the differential coefficient, and two indeterminate quantities, du and dx being assumed, so that the fraction — shall dx have this value, are called the differential (du) of the func- tion, and the differential {dx) of the variable. Tlie no- tation du, dx, is not meant to express d x u, d x x, but simply "the differential of w'^ and '< the differential of 07." It is evident that the " differentials" of the function and variable, according to this system, are the same quan- tities as the " fluxions'' in the Newtonian method, differing only in notation and name. Lagrange attempted to set aside both the notation and nomenclature of the differential and fluxional calculus. He showed that the true principles of this science consisted in the methods of developing functions in series, and v^exe alto- gether independent of the ideas of velocities or of infinitesimal 6 THE DIFFERENTIAL CALCULUS. SECT. I. or evanescent quantities, or even of the limits of ratios. He proved that if in any function w of a variable jr, the variable be supposed to be changed to x -{- h, the function ¥(x + h) or u' could be always expanded in a series of ascending in- tegral and positive powers of //, provided that the variable j; is not supposed to have any particular value. If this de- velopment be W' = A -h a'A + A'7i« -h aW -h &c. . . . he called the coefficient a' of the second term, the first de- rived function of the function u. From what has been already observed, it appears that this is the same as the " fluxional coefficient"" of Newton, and the " limit of the ratio" or differential coefficient of D'Alembert. It is also evident, that the second term of the series is the differential of the function w, h being assumed as dx *. (9.) We shall in the following treatise adopt the notation of the differential calculus in preference to that of the fluxional, as well because it is generally received by the scientific world at present, as because of its superior sim- plicity and power. We shall, however, use the principles of all the three methods as they may seem best suited to the subject of investigation ■(•. (10.) Functions are explicit or implicit. * In this enumeration of the methods of the different founders of the calculus, I have omitted Leibnitz's infinitesimal method, because, although I believe it was the first promulged and pub- lished, yet it is inferior in rigour to the others. Its validity con- sists in a kind of compensation of errors. t Wherever it can be used without too great complexity for so elementary a treatise as the present, I have preferred the method of Lagrange, as being most rigorous, and free from metaphysical objections. t SECT. I. THE DIFFERENTIAL CALCULUS. 7 An explicit function is one whose form is known. Thus, 0'"', log. X, sin. jr, a^, are exphcit functions of x. An imphcit function is one whose form is unknown, or at least not expressed. Thus, if u^ -f u^ax -{- ux -\- 1 = 0, u is an implicit function of x, being a root of this equation. Also, if u = sin. 7/, and t^x^ + bx^ -\- ex -^ d = 0, u is an implicit function of jr; h, r, and fZ being supposed to be constant quantities. The roots of an equation are implicit functions of its coefficients. Functions also are of one or several variables. If wssj"*, w is a function of one variable, m being supposed constant. If M = x^, u is a function of two variables x and i/. Again, if m = a: - , w is a function of three variables, and so on. In these cases the variables are supposed to be in- dependent^ that is, the variation of either or any one of them is independent of the others, which, at the same time, may or may not be varied. If, however, any two of the variables be connected by any equation or condition, they cease to be independent variables, as any change in either produces a corresponding change in the other. Thus, if m = x^^ and at the same time x = 2y, x and y are not independent variables, and the function in this case, though apparently a function of two variables, is implicitly a function of one variable, and becomes an explicit function of one variable by eliminating 2/, whereby u~x . (11.) Functions are also divided with respect to their form into algebraic and transcendental. Those in which the variable is united with the constants by common algebraical operations, are called algebraic functions. Such are ax, — » X x'\ l/x, &c. But those in which the variable is connected otherwise with the constants, are called transcendentaljunc^ tions. Such are x% a log. .r, a sin. x, &c. 8 THE DIFFERENTIAL CALCULUS. SECT. 11. The process by which the differential of a function is found, is called '* differentiation," and the function is said to be " differentiated." We shall commence by explaining the methods of dif- ferentiating functions, whether explicil or implicit of a single variable. SECTION II. The differentiation of functions of one variable. PROP. I. (12.) Ifu = r(jr) and x he changed into x + h, so that u' = f(x -\- h) id may be developed in a series (f positive and integral powers of /z, provided that x be an indeterminate quantity. Let u! = A7i« + v,h^ + oh' .... the quantities a^b^c . , . must be positive and integral, for 1^. If any of these exponents were negative, the supposition 7i = would render u (which then becomes equal to u) in- finite. Hence x must have that determinate value which renders —r— = 0, which is contrary to hypothesis. Also, since when h — 0, li! ~ w, it follows that one of the terms of the series must be independent of A, and that the value of that term must be f(^). Hence the series must be of the form %i — F(.r) 4- aA" -f B/i'' -f cA'". .... SECT. II. THE DIFFERENTIAL CALCULUS. 9 2^ If any of the exponents were fractional, there would be as many values of the term which involved that power as there were units in the denominator of the fractional ex- ponent. Now it is plain, that the radicals affecting h can only arise from radicals included in the primitive function F(a7), and that the substitution o^ x -\- h for x can neither increase nor diminish the number of these radicals, nor change their nature, so long as x and h remain indeterminate. On the other hand, it appears from the theory of equations, that every radical has as many different values as there are units in its exponent, and that every irrational function has consequently as many different values as there are different combinations of the values of the radicals which it includes. Therefore, if the development of v{x -f- //) could contain a term of the form gh~'^ , the function f(^) must necessarily be irrational, and must have consequently a certain number of different values, and therefore r(.r -l h) must have the same number ; but the development of this last in a series being f(j7 + Ji) =z y(x) + Ah" + Bh .... ^7i «... . each value of ¥{x) is successively combined with the n values of gjy/t", so that the function f(x -f- h) has a greater number of values when developed, than it has when not developed, which is absurd *. Hence no power of h can occur in the development, except such as have positive integers as ex- ponents. The series must therefore have the form, T(X -f h) = t{x) + a'Ji f A"r- + aV. . . . (13.) Cor, Hence u' -u== A'h + a!^¥ -f A"7i3 By dividing both sides by A, and supposing h — 0, it ap- pears that the coefficient a' of h is the limit of the ratio of * Theorie des Fonctions Analytique. I^agrange^ p. 7. ■ 10 THE DIFFERENTIAL CALCULUS. SECT. II. the increment u' ^ u o^ the function to the corresponding increment h of the variable. This is therefore the dif- du ferential coefficient, and -7- — a'. dx (14.) Cor. 2. As 7t* is a common factor of the terms of the series after the first, the series may be expressed thus, yj -^ u — A'h -\- s/i®, where s =r a" + a"7i + A^'h^ (15.) Cor. S. The first term A'h of the expanded dif- ference u' — u of the function may alv/ays be considered as its differential. PROP. II. (16) If u — f(^) and y =y(x) to defe?'7mnc the dif- ferential coefficient of u considered as an iinjdicit function ofx. Lety=/(.r + A), and if —y — A!h -f- s/i^. If y _ ?/ =: A', and '.• y^ = y -\- Jc,v! ^ T(y + k), u' — u =: b7v + S7x2, substituting for k in this its value given by the former, the result arranged by the dimensions of h will be of the form id - u — A'n'h -1- s''h\ By th. cse three series we find -^ - A' dx -'"' du , ^^ = A'., dx From which it follows, that du du dy dx " dy d I r SECT. II. THE DIFFERENTIAL CALCULUS. 11 Therefore, if there be three quantities u, y, x, each a function of the other, the differential coefficient of any one u considered as a function of another x is equal to the pro- duct of the differential coefficients of that one u considered as a function of j/, and cf 1/ considered as a function of the remaining one x. It is obvious that by continuing the process, the same principle may be shown to be applicable to any number of differential coefficients. PROP. III. (17.) To differentiate a quantity which is composed of sexwral functions of the same variable united by addition or subtraction, the differentials of the component functions being given. Let u = V ■\- y — z, v, y, and z being functions of the same variable x\ Let i/ - r = aA + sA^, y - ^ = A'A + s%s «' - z = a"A H- s'^h\ Adding the first two and subtracting the third, observing the conditions, u =v -^ry - Zy yj ■= V^ -\-y^ — jSf', the result will be w' — w = aA + A!h — a!% + (s + s' - s")/^«. Hence du ■= dv -\- dy — dz. Hence it is clear that the result is in general the dif- ferentials of the several functions united together in the same manner, and with the same signs as the functions themselves. 12 THE DIFFERENTIAL CALCULUS. SECT, II. PROP. IV. (18.) Constant quantities combined with a function by addition or subtraction disappear in its differential, and all constaiit quantities which are comtnned with it as factors are similarly combined with its differential. P. Let u -- F[a:) ± a, '.' it ~ f{.v -j- h) ± a, •.• u' -71 _F{x^-h) — T{a:) In which a dc^es not appear, and therefore it does not appear in the differential of w, which is deduced from this. Hence it follows that u has the same differential, whatever the constant a may be. 2". Let u — '^'F(.r), •.• u' — aF(x -\- h\ '.' u' — u = a[F(x + h) - F(.r)], u'~ti F{x-\-h) — F{x) ■•■-T-='' h ' from which the differential coefficient being derived, it is evident that a is a factor of it. The same observation obviously applies to constant di- visors, since a may be -r-. (19.) Def. Functions of the form u — a'^ are called ex- ponential functions. PROP. V. (20.) To differentiate an exponential function. Let u = a\ '.' u' = a"+^'^ = a' - a\ Let a = 1 -'r b, \- fl* = (1 -f- by. This being expanded by the binomial theorem, gives h , h.h--l^^ h'h^l .7^-2 „ SECT. II. THE DIFFERENTIAL CALCULUS. 13 which arranged by the dimensions of h, is of the form a* = 1 -f M -h s7i«, where _^ h'' b^ h^ ^- I'^T+li" ~T multiplying both by a% and substituting u for a% and m' for a'"«^, the result after transposing u is u' ~u — kuh -f sw/i®. Hence da = A;wc?.r — ha'dx, and •.* -r— :=: Tcu ~ ka", dx The value of tlie series h will be determined in finite terms hereafter (64.). PROP. VI. (21.) T'o differentiate a logarithm. Assuming the logarithms of the equation u = a"" relatively to the base a, we find lu — a-, '.' d • lu = dx. Ehminating dx by this, and the equation du = kudx found in the last proposition, we fiwd 1 du d ' lu = ~r • — . /c u It is obvious that the value of k depends upon that of the base a. The base which renders 7c = I is called the Neperian base, and sometimes the hyperbolic base ; and the corresponding logarithms are called Neperian or hyperbolic logarithms*. The value of the Neperian base will be de- termined hereafter (6i.). The quantity -v is called the modulus of the system, whose base is a. Hence the modulus of hyperbolic logarithms is unity. * For the origin of the term hyperbolic logarithm, see my Algebraic Geometry, Art. (385.). 14 THE DIFFERENTIAL CALCULUS. SECT. II. Logarithms of the hyperbohc system are sometimes de- noted thus, /' ; and those related to any other base a, thus, /or L *. Hence d - I'u = —, and, in general, if m be the modulus u a ' lu — m ' — . u PROP. VII. (22.) To differentiate a function which is the product of several functions of the same variable. Let u = i/y^y^^ .... y^'"'\ n being the number of factors, and the factors being all functions of x. Assuming the hyperbolic logarithms, Tu = /y + ly 4- v • • • • ^!y^"^- Differentiating this (21.), dM_d£ d^f d^^ d^ w - y + y" "^ /' • • • • «/(") * Multiplying this by the original equation, the result is du = i/'y 2/(") . d7/' + y'f y^"^dy + t/'fi/'" .... j/<"-'> di/^^'K Therefore " the differential of the product of several func- tions is equal to the sum of the products formed by mul- tiplying the differential of each function by the product of the remaining functions." Thus, if u = i/y, du -ydf +fdi/^\ that is, " the differential of the product of two functions is equal to the sum of the products of the functions into their alternate differentials." ifr.=j/yy', du = yiydy'" -f-j/yjy 4- yfdf, * We shall generally use the Neperian logarithms without any distinguishing mark. Whenever any other logarithm is used it will be expressly mentioned. SECT. II. THE DIFFERENTIAL CALCULUS. 15 PROP. VIII. (23.) To differentiate a fraction whose numerator and denominator are each products of several Junctions of the saine variable, i, and since u — z"^^ *.• (24.) -7- = mz"'-^, dx ' ' ^ ' dz diL \- (16.) --,— = mb{a + bx)"^^. Ex. 4. If 2^ = (a + i^)* (a' + Vxf, Lety == « + ^>.r, and y = ft' + ^^'-^^j SECT. III. THE DIFFERENTIAL CALCULUS. 21 But di/ = bdx, dj/' = b'da?, •/ du_ / U W \ dx'~\a^bx'^ {a!-\-Vx)} or — = 9.b{a + bx) (a! + b'xY + 36'(« + bxY {a! + VxY, du or --=(« + 6^) (a' + b'xf [2b{a! + 6'a;) + U\a + M}- Ex. 5. If M = (a + bxY (a' + VxY {a!' + ¥xY\ Let J/ = (a + ^>^)'", y = (a' + ^>'^r)'™', y = (a'' + 5"^:)'"". Hence rfz, = ^^ + ^ + ^^J^j/^/, Jz^ —. mb[a + bxY~^dx^ dj/ = m!bXa^ + VxY'-'dx, df = m"b"{a" + 6"a;)'«"-^f?ar, •.• — = (« + 6a:)'" (a' + 6'^)'"' (a' + b"xY' X i mb idU wl^W \ I a^bx '^ a!+Ux ^ a" + Z>"^ i* -\-bx a' + b'x Ex. 6. If w = (ajr*"-" -f- ^)''. Let ^ = ax""-^ + ^>, du . Also u — z'Jy \' -J = q^fl-^, du Ex. 7. If M = — — , ••• M = ax-^\ *.• -7— = — max- x"" dx by (24.). - "H du m ^-t Ex. 8. If M = a^'V^^i *.* 2^ = cf^", *.• -7^= — ao;'* dx n by (24.). Ex. 9. If w = Va"*— a:^ Let z = a"^ — >r, dz m— 1 •.• -5- = — mx ax gg THE DIFFERENTIAL CALCULUS. SECT. III. Also u = z', ••• -^ = - ^" "• H^^^^ ^y (^^•)' du m rn-\ , ™ m\r~* Ex. 10. If w = — =. Let ^ = « - a:S ax ^L du 1 -L-i - ^ . Ah0U=: Z n, ',' j^=^-—'Z n , •/ (16.), du__2_ dx n x(a — x^y Ex. 11. Ifw = y{a- A 4-Vic^ -^~)} . Let — - = «/ ; l/c^ — x^ z= z. a/X Hence we find M = (a - ?/ + 2)4-, ... du = iia--i/ + z)^-\-- dy + dz), -'6dy+Sdz or cZw But aliO, 4t/«— 3/ + ^ 6 ^tZ^ 1 _| _ O 2'/ — 1 sin. a.-}. SECT. III. THE DIFFERENTIAL CALCULUS 25 = C0S.2 X — sm.^ X — COS. So;. HInce in this case du z^ */ — i - udx. It appears from this and Ex. 16, that the differentials of the function e 5 and the above are the same. It will appear by the integral calculus, that these functions are actually equal. Ex. 21. Ifw = sin..r cos.jr, *.- du = Q,o?>.x,d^\x\, x -j- ^\xi,x.d cos. J7, which, by substituting for d sin. x and J cos. x^ their values (27.), (28.), gives du dx Ex. 22. If w = sin. a; cos. a + sin.a cos. 07, •.• du = cos.a.dl sin. a: + sin.fl.^ cos. jr, or y- = cos.a cos.jr T sin.a sm.jr. dx The differential and integral calculus is of very extensive use in the deduction of the formulae one from another in Trigonometry. There are many parts, such as the ex- pansions of series, &c. in which its application is indispensa- bly necessary ; but many even of those parts which are usually proved independently of its principles, may be much more concisely and elegantly deduced by their aid. We shall give here a few obvious examples. In the last, u — sin. {x + fl), '.' du = cos. {x ± a)dx, • .• du -r ~ cos. (x -f a). dx \ — J Hence cos. {x ±: a) = cos.^ cos.a + sin. a: sin.«. Ex. 23. If M = sin.2:r, •.• du — 2 cos.2^ • dx, •.• -^- = 2 cos. 2^. dx By this and Ex. 21, it follows, that if sin. 2^ =:2 sin-jrcos-a:*, cos.2.r = cos.2 X — sin.2 J7. Ex. 24. Let u = cos..r -\- cos.2a? -f cos.3^ . . . cos.w^. Since by Ex. 17, d cos.?/.r = — sm.nx.d{nx) = — n sm.nxdx. gb THE DIFFERENTIAL CALCULUS. SECT. III. i.ave # .= — j sin.{dPz-\-d':i;) {dz-dx) ■\-(z-'x) {d!^Z'\-d^x) = 2 [ zd^z -}- A^dzd^z + QdHd^z-xd^x -Mxd^^-M^xd^x]. PROP. XX. (41.) To determine the successive differential coefficients of an exponential function. Let u = a'. By (20.) -^ — ku\ -r being the modulus to the base a. Hence d^u -— - = Icdu. dx And by substituting for du its value kudx^ and dividing by dx^ d^u ^^ S^ = ^ ^- In like manner d^u and in general, d'^u If a be the hyperbolic base /t = 1 ; and in this case the differential coefficients are all equal to the primitive func- tion w. PROP. XXI. (42.) To determine the successive differential coefficients of a logarithm. X ,, du \ , TT JLet u = I'x. \' -T- = — = x-\ Hence dx X d^u _ _l_ __ dx'^ -" x'"''^ ' 34f THE DIFFERENTIAL CALCULUS. SECT. IV. d'u _ ^ _3 The differential coefficients are therefore alternately po- sitive and negative, and that of the nth order is, 1.23.... n - 1 . x-% which is + if 71 be odd, and — if w be even. In this case the logarithm is assumed to be hyperbolic. If it be not, the successive differential coefficients should be affected by the modulus as a factor. PROP. XXII. (43.) To determine the successive differential coefficients of the sine and cosine asfuncticms of the arc. Let vb — sin.^ > '•* du Tx = ^"^-^^ • d^u - sin.^> d^u dx^ ~ — cos./r. And in general, if n be an odd number, d^u d^^ ±cos.ar; + being taken when —^ is even, and -, when — ^ is odd. And when n is even, b SECT. IV. THE DIFFERENTIAL CALCULUS. 35 + being taken when — is even, and — , when -^ is odd. It It . Since cos. x = sin. (-^ — x), and —dx — d(-^ — x) it fol- lows that by changing the sine into the cosine, and + into — and vice versa, the preceding observations may be applied to the successive differential coefficients of the cosine. Hence if u = cos. X, ^ = ± s.n.^; when n is odd, + being used when —r — is odd, and — when —pr— IS even. And d"u _ -—- = + cos.a:. n When n is even, + being used when -^ is even, and •— , when -^ is odd. PROP. XXIII. (44.) To determine the successive differential coefficients of the tangent and cotangent as functions of the arc. Let u = tan. J, *.• :r"= T" = sec.^a?; hence dx cos.*a7 d^u -T^ = 2 sec.xd secx = 2 sec.^j: tan.xdx, d^u •.• — = 2 tan.j: (1 + tan.^ar), D 2 36 THE BIFFEREKTIAL CALCULUS. SECT. IV. Hence the third differential is But dtt = (1 + u-)dx, :• ^ = ^(1 + «») (1 + 3«n- By continuing this process, the succeeding coefficients may in hke manner be found. The differential coefficients of the cot. x may be deduced rff from those of the tangent by changing iv into (— — x)j and changing the sign of dx\ PROP. XXIV. (45.) Tojind the successive differentials of the secant and cosecant as functions of the arc. Let w = secar, '.' du :=. tan.a^ sec.jrc^jr, •.• du —■ ^ = » v«« - 1. d^u , ti^du *.• ^j— = a/w« — 1 • 0, a^' -^ when h = 0, and since the exponents a, b, c, . . , . are ascending if the first be > 0, they must all be > 0, *.* ^ = renders every term of the series = •.• f^x) = 0. Such a value of X being excluded, a cannot in this case be > 0. If a < 0, Ah"" would be infinite when h = 0, '.' f(x) would be infinite. But such a value being excluded from this case, a cannot be < 0. Since therefore a cannot be > nor < 0, '.' a ~ 0. All the succeeding exponents being > 0, ?i = renders all the succeeding terms =: *.• a = f(^). Thus the first coefficient and exponent is determined. The series [2] becomes therefore F^^ + h) = bBh^^ + cch'-^ + do^-^ .... If ^ = 0, f\x + h) becomes f\x), and since by sup- position no value is assigned to x which renders f\x) = 0, b — 1 cannot be > ; and since no value is assigned to x 42 THE DIFFERENTIAL CALCULUS. SECT. V. which renders F^a?) infinite, 6—1 cannot be < 0. These follow in the same manner as for the first exponent a. Hence 6 — 1 = 0*.-^ = 1. Since the exponents ascend c — 1, d — 1, . ... are > 0, •.* A = gives B = t\x). Thus the second coefficient and exponent are determined. The series [3] therefore becomes f''(^ 4- A) = c . c - 1 . ch'-^ + nor < 0, •,• c -- 2 = *.• c = 2. Since d •— ^, e — 2 . . , . axe > 0, ••• ^ = Ogives 2c = f^cc) •.• c = f*(^) . -^. Thus the third coefficient and exponent are determined. The series [4] becomes F%x-{-h)=d.d-l . d-2 . D7i'^-» + 6 . e-1 . ^-2 . Eh'-^ If A = 0, F^(a7 4- It) becomes f"(^), and it follows in a similar way that d ^ S can neither be > nor < 0, *•• cZ = 3. And also d = f\x) . fg-^- Thus the fourth ex- ponent and coefficient are determined. In the same way the others may be found, and the several values being substituted in [1] for the coefficients and ex- ponents, the series becomes F(x-\-h) = F(^) + fH^). Y + F'(^). j^ + ^X^)-JJ^ + ^'^^h-iki t^J- Or if z^ = f(.2?) and u! = ¥{x + h) the series may be ex- pressed ^_ du 7i ^ h^ d'u h^ d^u h^ '' = ^'^^* T^d^^'\79>'^d^'\j:s^d?' YMA"" [8]. (53.) If A?f=:w'~M, andi^= Ax^ and the arbitrary quan- SECT. V. THE DIFFERENTIAL CALCULUS. 43 tity dx be supposed to equal A a?, the quantities du, d^u, d^u, &c. consequently, having such values as will render du d^U d^U ^ , ^ . v/r. • 1 — , ;:, — r, &c. equal to the successive amerential coefficients, we have du d^u dhi d^u ^ ^" = T+T:2+i:2j+i:2r4 + ^'=---- which expresses the difference of the function in a series of its corresponding successive differentials. The character A before a variable signifies its finite difference. The series which is the result of this investigation was first published by Dr. Brook Taylor in his Methodus In- crementorum in 1715. Taylor was a profound mathe- matician of the old school ; he does not, however, seem at all aware of the immense importance of his own discovery. Lagrange has made it the basis of his theory of analytic functions. On it depend almost the entire application of the calculus to geometry, the principles of contact, osculation, singular points, &c. &c. Some very elegant applications of it have been made by another able modern mathematician in finding fluxions per saltum, in approximating to the roots of equations, &c. * (54.) II. If the value of <2? be a root of the equation F( 0, and we shall presently explain the method of determining it. If h = do not render ¥{a' -{■ h) — Ah~^ either = or in- finite, then 6 = 0, and the value of F(,r + A) — Ah~^ when h — is the value of b. The other exponents, when ne- gative, and coefficients may be determined in a similar way. If 0? be a root ofy^(^) = 0, but not ofj'(^) = 0, then the series [1] and [2] become f(.«' ■{- h) — ¥{.v) + Bh^ -f ch^ + !)¥.... F^{a^ + ^) = ^B^''-' + cc^*^-* + dok^-' .... Since f\ 0, *.• b must be a proper fraction and positive. To determine its value, let f(<2?) be brought over in the first, ••• y(^ -}- h) - f{x) = Bh^ + ch'' 4- D/i" . . . . let that power of h be found, by which f(x + h) ~ f(t) being divided, the quote will neither vanish nor become in- finite when h = 0. The exponent of that power will be = b. For let it be k, F(x-\-h)--F(2) ,, ^ , ^ -^^ ^'"-^ = ^^ + ^^ -f- . . . . If ^ < 6, ^ = renders this = 0, which is contrary to hypothesis, and '\£ k > b, 7t, = renders it infinite, which is also contrary to hypothesis, *.• k — b. If a? be a root off^(x) = 0, but not of/(:r) = 0, nor f^{x) - 0, then the series [1], [2], [3], become 46 THE DIFFERENTIAL CALCULUS. SECT. V. f(^ + h) = ¥(x) -h F^(^) Y + ch' + d¥ . . . . f\x + h) = f\x) + cch'-^ + dD^-^ .... f2(^ + 7i) =: C . C - 1 . C^^-=^ + d . C^ - 1 . D'^-* Since f\x + ^) becomes infinite when ^ = 0, •.* c— 2<0, ••• c <2. But since the exponents ascend, c > 1. Hence the value of c is between 1 and 2. It may be thus determined in the same manner as b. The left side of this equation is known. Let that frac- tional power of h be found, by which this being divided, gives a quote which neither vanishes nor becomes infinite when ^ = 0. The exponent of this power is c — 1. Hence c becomes known, and also c. It follows therefore in general, that if a value be assigned to X which is a root ofy " {x) — 0, or which renders the wth differential coefficient infinite, but none of the preceding ones, the series [7] gives the true development as far as the Tith term inclusive ; but the exponent of ^ in the (w 4- l)th term is a fraction, whose value is between the integers w and w -f- 1, and which may be determined by the method already explained*. • The method of determining the exponents o£^ given above is taken from the Theorie des Fonctions Analytiques of La- grange. This method applied to negative exponents may be somewhat improved by the application of the Integral Calculus. Let a be negative, and the series is v{x + ^) = hh-^ + B^^ + cA*^ -f- .... Multiply both by dh, and integrate fwix -\-h)dh^- .Kh'-"-\- --—.bA' -''+-; cA'+«^ SECT. V, THE DIFFERENTIAL CALCULUS. 47 (56.) There are some peculiar circumstances attendant upon the state of the function when x receives any value which is to be found among the roots of the equations fix) = 0,/'(a:) = 0,f-ix) = . . . . which merit examination. If the denominator of any fractional exponents which occur in the development of F(ar + h) be an even integer, the numerator (the fraction being supposed in its least terms) must be odd. The power of h therefore being the even root of an odd power is imaginary if 7t be negative, and has two real values with different signs if h be affirmative. Hence the particular state of the function r(.r 4- h) is one at which it passes from a real to an impossible value or vice versa by the variation of .r. In this transition it is plain that two values become equal and then impossible, which must happen by a radical disappearing in the value of the function corresponding to the particular value of x, which renders the differential coefficient infinite. This cir- cumstance is similar to that which will be shown to happen (114), when some differential coefficient assumes the form -TT, But there is a very important distinction to be ob- served between the cases. In the one case the radical Let fF{x + h)dh — f^(x + h). Multiply again by dh, and integrating Let this process be continued until an integral be found, which will neither vanish or be finite when h = 0. If one be found which vanishes when ^ = 0, a is a fraction whose value is be- tween the number of integrations and the integer next below it. If it be finite, then a is equal to the number of integrations. This is evident. 48 THE DIFFERENTIAL CALCULUS. SECT. V. passes through zero without becoming impossible on either side of it, therefore it must vanish in the primitive function, not by its suffix vanishing, for that would infer a change of sign in the suffix, and therefore a transition from a real to imaginary value, but by a coefficient of the radical vanishing which produces a change of sign in the term in which the radical was engaged without rendering the radical imaginary. In the present case, however, the function passes from a real to an imaginary state, and therefore the particular value of X must make the suffix of the radical vanish, and not a co- efficient of it, and the suffix changing its sign in passing through zero, there is a transition of the function from a real to an imaginary state, or vice versa. If the denominator of the lowest fractional power which occurs in the development be an odd number, the nume- rator may be either even or odd. First suppose it even. The sign of the fractional power of li in this case is the same whether h be positive or negative, and therefore this term of the development of F(>r -f h) and ^{x — h) is the same ; and in each case there is but one real value for the power of h. If the numerator be an odd number, the sign of the power of k changes with the sign of h, and therefore for ¥(x + h), and f{x — h), the fractional power of h has different values, but in each case has but one real value. PROP. XXX. maclaurin's theorem. (57.) To expand a function hi a series of ascending in- tegral and positive powers of the variable. Let u = ¥{x) and id = F(.r + h). If jr = in the equation [8], it becomes SECT, V. THE DIFFERENTIAL CALCULUS. 49 h h^ h^ h^ u — Aq -f A/ . ~ + ^2. — + A3 j-^ + A4 ^23A V ■ " where Aq, A/, Aj, A3, &c. are what w, j-, y-^, -7—3, &c. be- come when jr = 0. When a' =.- 0, w = F(/i), and therefore the differential coefficients of this function must be the same functions of h as those of i^{pc) are of x. Hence it follows that the latter when jr — become identical with the former when h = 0. The quantities Aq, Ay, a,, &c. are what the function f(//) and its differential coefficients become when ^ = 0. Hence the above series, considering u = f(A), solves the problem. In general, therefore, FCr) = A,, + Ay . y- 4- A, • j;^ H- ^3 . j^g-g + where Aq, Ay, y^* A3, ... . are what the function F(.r) and its differential coefficients become when x = 0. This theorem, like that of Taylor, is liable to exceptions; but the exceptions arise here from the form of the function, and not, as in the former case, from the particular value assigned to the Variable, ^laylor's series, if x be inde- terminate, holds good zcithout exception ; but that of Mac- laurin, even though x be considered indeterminate, is liable to exceptions, because the coefficients are not functions of ^, but are what certain functions of x become when ^r == 0, in which case they may happen to be infinite or impossible. Thus, if the function to be expanded be — , the first term X Aq being — is infinite, and the function cannot be expanded in the required form. This, however, ought not to be called a y»?//^ ox failure in the theorem, because in these cases the function does not admit of an expansion in positive integral powers of the variable. The cases which form exceptions to Maclaurin's series E 50 THE DIFFERENTIAL CALCULUS. SECT. V. may sometimes be solved by a transformation. The sub- stitution of x'^z for w, k being arbitrary, frequently effects this. Such a value should be assigned to k that none of the quantities u -j-, -7-^ .... should be innnite when x = i). An example of this is given in (83). Maclaurin's theorem may likewise be applied to develop a function by descending powers of the variable. Let u = f(x) be the function, and let zx 3= 1, or x = — . Sub- stitute this for .r, and •.•m=f( — ), or =J'(z), Let this be developed by Maclaurin's theorem according to the ascending powers of », and then substitute — for z, the result will be a series of descending powers of x. For an example of this see (85). prop. xxxi. Lagrange's theorem. (58.) Given u = F(y) and y = z + xf(y) to expand u in a series of ascending integral and positive porters of il^z not being a function qf^. Considering w as a function of x by Maclaurin's theorem, X x'^ x"^ a:* M = Aq + A^ . -r- + ^I'J-l. + A3 . T-^-^ + A4 ' • 1 ' ''1.2 ' '• 1.2.3 ^ "' L2.3.4 • • * • du dhi d^u Where Aq, a^, Aj, A3, Sec. are what u, — , j-^, ji, Sec become when X = 0, The problem will therefore be solved if the values of these be determined. If in the equation 1/ = z -{- ^(«/)> S/ ^^ considered as a function of x, we have SECT. V. THE DIFFERENTIAL CALCULUS. Si dx -'^^^^ ^ ^ dx dy ' dx'^ '^dx dy "^ dx"-' dy "^ dx^' df~ And by the equation u = ^(y), du _ dF{y) dy dx~~ dy ' dx^ dH^^c^y dF(y) ^ d^f(y) dx" dofi' dy dx'^ ' dy^ ' If .r = 0, the function y and its differential coefficients become ^' •^^^^' dz ' dz^ '•■■■ And by these substitutions, we find A„ = F(2), And in general, A, = A« = A„ = 4/w-'-?} rZ^ -U'^m dz' .►-■{/W".*^'] ^2^ dF(z) Therefore if —-— = q and /"(z) = p, we obtain the series, e2 62 THE DIFFERENTIAL CALCULUS. SECT. V. or which is the solution of the proposed problem. (59.) Cor. 1. If/Cj/) = 1, and '//{z) = 1, and x = h, this series becomes . = F(.4-70 = F(.) + -^.-^+-^.j-^.... which is Taylor's series. Taylor's theorem is therefore a particular case of Lagrange's, which, therefore, also includes Maclaurin's. (60.) Cor. 2. If a: = 1, d(p^q) 1 , d%p^q) ' 1 , It was in this form that Lagrange delivered the series. prop. xxxii. Laplace's theorem. (61.) Given u = F(y) and y = f'[z + xf(y)], to expand the Junction m in a series of ascending positive and integral powers ofx. Let F { f'[z + xf{y)] ] = f"[z +xf(t/)]. Hence u = f"{i/') and y = ;s + xf(y). Hence by Taylor's theorem, „ = p»(^) + __. __ + __ .___ + .... Also/( 2/) =/! f'[z + ^(t/)] } =f\z + .r/(3/)]. Hence by Taylor's series, SlSCT. V. THE DIFFERENTIAL CALCULUS. 53 f(y) -/'W + ^n^) ^f{y) . ^!/''(^) ^'fW dz 1 "^ dz 1.2 Let q, q,, q,, represent f"(^), —^, —^ and let v —f{ij)^ and let jt?, ^^, 'p^^ representy'(jz) and its successive differential coefficients. Hence the preceding series become XV x'^v'^ x^v^ XV 2,,2 X'-V Also , . ^(jo«) XV d\f-) x^v"- P ^ dp ' \^ dp^ ' \.^^ .3 „ ^3 . difl XV d\f) x^ Hence + \c^W' + ^va?^) x^ d(v^) This series must be equivalent to that of Maclaurin, which gives M = Ao + Aj-r- + A2 r-^ + A3.- x-" 1 ' "M.2 • "^'1.2.3 ' and therefore the corresponding coefficients must become equal on the condition .r = 0. In this case v, t;®, ^5 • • • • 54 THE DIFFEllENTIAL CALCULUS. SECT. V. become p^ p^y p^ Hence we obtain the following equations : , ^ d(p^) dz ^ ^ ^ dz dz^ Making these substitutions, the result is ■^ dz"- ' 1.2.3^ dz^ ' 1.2.3.4* The Tith term of this series being d"-Y\^y~^' dF"(z) dz X n—\ dz^^ ' 1.2.5.... (w-1) Lagrange'*s theorem is evidently a particular case of this. For in this theorem f"[;3; + ^(^)] is considered as a func- tion of another function of z, z scil. f{f'[;s: -f ^(j/)]}> and Lagrange's is the particular case where SECT. VI. THE DIFFERENTIAL CALCULUS. 55* Laplace has extended this theorem to functions of several variables. This generalization, however, is not suited to so elementary a treatise as the present. The preceding proofs of the series of Lagrange and Laplace are taken from the notes of the Cambridge trans- lation of Lacroix, in which the student will find many useful applications of them. SECTION VI. Praxis in the development of Junctions, PROP. XXXIII. (62,) To expand (x + h)"^ in a series according to the powers qfh. Let u = X"' and u' = (x + h)'". By differentiation we obtain (39.), du , d^u ^ „. 2 Hence by Taylor's theorem, m.m — 1 , m.m — l.m — 2 ,, u^ = x-+mx"^-Kh±-j^x"^-\h^ + j^ 0^-3^3 the n\h term being jn.m—l,m-9, m-(w — 2) . a;'"-<"-i) . ^"~^ 1.2.3.... (/J-1) As the value of m is not restricted, this example contains the binomial theorem in its most general state. ^ #56 THE DIFFERENTIAL CALCULUS. SECT. VI. PROP. XXXIV. (63.) To expand a'' in a series of powers qf\. Letw = a^-.- (41.), du dhi _ and, in general, -^ = ^"m. When x = 0,u = 1. Hence the successive coefficients of Maclaurin's series are 1, k^ k'^, .... A:", from which it follows that «^ : = 1 + kx 1.2 + 1.2.3"^ k^'x^ 1.2.3.4 "^ (64.) C(yr, 1 . if^ 1 ^- A;' ♦.- lex = : 1, hence this being a converging series, we can approximate inde- finitely to its value. Its value continued to seven places of decimals, is 2*7182818. By (21.) if appears that this is the hyperbolic base. Let it be e',' a^ =e, '.'a = e^. Assuming the logarithms la = kle, •.• k = -1—, hence, since a and e are known, k is known. If a be the base of a system of logarithms le = ~j-. Hence At it appears that the modulus of a given system is the loga- rithm of the hyperbolic base in the given system. Also k = I'a because Ue — I. Hence the modulus of any system is the reciprocal of the hyperbolic logarithm of the base of the system. The logarithms of the same number (y) in different systeqMLare as their moduli. For c^'^ = a^^ a being the base 'W SECT. VI. THE DIFFERENTIAL CALCULUS. 57- of the system. Taking the logarithms relatively to the base a, le . Vy = ly\ since the number^ is given, ty is constant, therefore ly x le\ that is, the logarithm of a given number is proportional to the modulus of the system. Hence being given the logarithms of any one system, we can find the corresponding logarithms in any other system whose modulus is given. {Q5.) Cor. % If a = ^ *.• ^ = 1, and the series becomes X x^ x^ x^ ^' = 1 + -^+1-7; + 1 ' 1.2 ' 1.2.3 ' 1.2.3.4 • • • • {jaQ.) Cor. 3. If in this series x become successively -\- X y/ — \ and — X V — I, and the results be added and subtracted, we find xV—l —xx/—l r ^ X- X e -{■ e =2? 1-TT.+ r , X' 2 ' 1.2.3.4 1.2.3.4.5.6 5 xV—\ — *V^I f X x' = ^^-^- It- 1:2:3 x^ x'^ i "*" 1.2.3.4.5 ~ 1.2.3.4.5.6.7 S (67.) Cor. 4. If in the series for a% x become rnx, Tcmx Ji'^m^x^ k^m^x^ ^ ^ 1 ^ 1.2 ^ 1.2.3 Hence it follows that {i + y+X2+r2r3----l =^+-r+-i:3- + k^rri^x^ PROP. XXXV. (68.) To expand the Junction l(x -p h) in a series of powers qfh. Let u = Ix and u' = l(x + h). Substituting for the co- efficients in Taylor's series their values determined in (42.), the result is # 58 THE DIFFERENTIAL CALCULUS. SECT. VI. , r h h"- h^ h* 7 (69.) Cor. 1. Hence.wefind When ^ is small compared with x, this series converges rapidly, and therefore serves, when the logarithm of one number is known, to compute the logarithms of a series which varies by a very small difference. (70.) Cor. 2. If in this series x = 1 , it becomes Z(l+/0 = /.{p-^ + ^-^....} which, when 7t is negative, becomes Hence by subtraction, (71.) Co.. 3. If i±-J=l+-^.v^=^. Hence the last series becomes this gives the logarithm of n + z when that of n is known. Let 71 = 1, and 2 = 1, and le = 1, hence f 1 1 1 1 7 this rapidly converges, and therefore gives the hyperbolic logarithm of 2 to any required degree of accuracy. For higher numbers it is still more rapidly convergent. The modulus may be obtained by calculating the loga- rithm of the same number in the Neperian or hyperbolic system, and in the system which we wish to adopt. The function Ix cannot be expanded in a series of positive SECT. VI. THE DIFFERENTIAL CALCULUS. 59 powers of the variable x. For the first term of Maclaurin's series, being what Ix becomes when a? = 0, is infinite. See (57.). (72.) Cor. 4. If in the series h be changed into h~^, it becomes l}-^=l{\+h)-lh = leV^- Subtracting the latter from the former, the result is Ir^ Ir^ Ir^ ~2"^~S 4" ^i~/i-i lh=:lei 7. + 7i T- ■\ PROP. XXXVI. (73.) To express the sine and cosine of an arc in a series of powers of the arc itself. Let u = sin .a; and u' = sin.(^ + h). By substituting in Taylor''s series the values of the differential coefficients given in (43.), we find h . ^« h^ sin.(jr + h) — sin.a: + cos..r-Tj sm-^-y-^ — cos.jc-j-^ + ^* sin.;r + COS.^-; 7^5 1.2.3.4 ' ^^""^ 1.1^.3.4.5 Arranging this by the factors sin. a-, cos.d7, we obtain h^ A* h^ sm, {X + 70 = sm.^| 1 - - +j^^^^_^-^^g^j^... I + ( 7* ^^ 7j' j But by trigonometry, sin.(ar 4- A) = sin.o; cos./i \- sin.^ cos.^* Since the value of h is independent of x\ the equations must hold for all values of x ; hence 60 THE DIFFERENTIAL CALCULUS. SECT. VI. . , h ?l' h' Sin./l = -z -, r, n + 1 1.23 ' l.'2JdAM C0S.7i = 1 - r-- + 1.2 ' 1.2.3.4 1.2.a45.6 These series might be also deduced by Maclaurin's theorem, and thence might be obtained by the preceding in- vestigation the trigonometrical formula sin. (07 ± h) = sm.x cos.^ =b sin./fc cos.^. (74.) Cor. 1. Since by {QQ.)y we have e -\re =2>1 ='{ 1.2 • 1.2.3.4 5 .T\Airi -^a/-i — r r a; ^^ ^^ ^ -e =2./-l^y-y-^ + j^^^^. And by the series found in this proposition, cos.^= 1 - ijj + i";^;^;^ sin..r = X 1.2.3 x^ "*" 1.2.3.4.5 It follows thaf 2 cos.a? = e "' + e-^~. 2^^- -l.sin.a7 = eV- — 1 e-^-\ and hence CQ%,x + V — 1 sm.T = e (75.) Cor. 2. Hence also, ' . ±mx,J — I cos.7?i.r + v^ — 1 sm.wor = e , •.• cos.r/io: ±\/ — 1 sin. wzo; = (cos.^+ V — 1 sin.a:)"*. Also, if ^ = ;2;, it follows that when 2 cos.a? = z H , '.* 2 cos.m.r = ;^"» + -— and 2 a/ - 1 . sin.TTio; — s'" -. SECT. VI. THE DIFFERENTIAL CALCULUS. 61 (76.) Cor. 2. By division of the results of Cor. 1, we find , — = ^ e — e e - 1 //— 1 tan..r = — = =_= = . e^^~' + ^■-^^-' e^''^-' + 1 (77.) Cor. 4. If in the series found in (72.) / '~ be sub- stituted for hy and the equation divided by 2 V —1, {le being supposed = 1), the result is X e — e e — e 5^ 2^/-l ' S^/-l 3xJ—i —3x^'^ e — e Making here the substitutions suggested by Cor. 1, we find X sm.x sin. 2^ sin.3^ sin.4x + 2 1 2 ' 3 4 PROP. XXXVII. (78.) To express a circular arc in a series of powers of' its sine. Let u = sin.~^ x. Substituting for the coefficients in Maclaurin's series what the differential coefficients found in (46.) become when x = 0, the result is sin.w P.sin.% l-.3\sin.^?^ . V.S^.B^.sm.'' u " "" T" "^ 1.2.3 "^ 'L^IaX ^ 1.2.^.4.5.6,7' ^^' (79.) Cor. 1. If w = 20% ♦.• sin.w - -i, •.• _ C ^ _^ , 3^- 7 '^ - ^r-^ + ^ • 1.2.3 "^ "^'^ • 1 2.3.4.5 "^ 3 See Geometry (375.). 62 THE DIFFERENTIAL CALCULUS. SECT. VI. PROP. XXXVIII. (80.) To expand the tangent of an arc in a series of powers of the arcitself. Let u = tan.jr. Substituting in Maclaurin's series for the coefficients the values of the differential coefficients of this function found in (44.), the result is tan.a7 = -— + =-— r + 1 ' l.M ' 1.2.3.4.5 PROP. XXXIX, (81.) To expand a circular arc in a series of powers of its tangent. Let It = tan.~^ x. Substituting in Maclaurin's series the values which the differential coefficients found in (47.) as- sume when a: = 0, the result is tan.w tan. ^2^ \.an.^ii tan.'^w (82.) Cor. 1. l^u= ^ ••• XxmM = 1, •.• ^ ^ = 4[l-i- + i-^ 1 This series is not sufficiently convergent for the purpose of computing the circumference. One may, however, be deduced, which will be sufficiently convergent. See Geo- metry, vol. i. Art. (375.). PROP. XL. (83.) To express the cotangent of a circular arc in a series of powers of the arc itself. Let u = cot.^. In this case the first term of Maclaurin's series becomes infinite. Let u — c^i ~ cota?, •.* ^=a:"~*cot.r. SECT. VI. THE DIFFERENTIAL CALCULUS. 6S If Ti be assumed > 0, a; = renders z infinite. Therefore sin.^ Substituting for cos.,r and sin.jr their developments ob- tained in (73.), 1- 1 :f^ 1 x^^ l.<2.3.4.5 XT ^^ Hence z, -7-, dx dH • become 1, 0, — T9 • • X=zO. Therefore z = \ . - 3'^.5 • • whence we find z cot.jr = x-^ - X x" 2^ x-i X 3\5 3^5.7 32 ■5«.7 This process fails in giving the law of the series. when PKOP. XLI. (84.) To express the value ofu in mu.^ — ux = m in a series of powers qfx. By differentiating we obtain the values of -=-, -^-^ .... 1 2 which, when x = 0, become 5—, 0, — and it is 3m' ' 27^3 evident that when x = 0, u = 1, \'hy Maclaurin's series X X'' ^* "" ^ "^ 3w 3*^3 "^ S^m* 64 THE DIFFERENTIAL CALCULUS. SECT. VI. PROP. XLII. (85.) To express the value of u in the equation m\x^ — x"u — mx^ =^ in a series of descending powers ofx. Let zx^ = 1, '.' ar^ = — , '.• mu^z ~ u — m = 0. The ' z ' successive differential coefficients of u with respect to z being found, and their values when z = substituted for the co- efficients in Maclaurin's series, give u=^m — m*z — SmJz^- - ISttz^V + 55m^^z^ .... PROP. XLIII. (86.) Given f(x + h) + f(x — h) = F(x)F(h), to find the form of the function. Let u — F(jr), and id = y{x -t h), and u, = f(x - h). By Taylor's theorem, ^ du h d^u _A^ (Pu J^ d^u M du h d^u h^ dhi h^ d^u A* ^'"^^^""5^' I'^di^' \:^'^d^''lJ^'^db^''\J^^'''' adding and dividing by 2iy "V --^[^ "^ dx'^'l.^' u'^dx^'lXSA' u S But u' -{■ Ui= F{x)F{h) = uF(h), hence X ^f. d^u h"- \ d'u ¥ 1 > But f(^) being independent of x, it follows that d'^u 1 d*w J^ c^z^ 1 ^ * 17' ^ * 2i' 'd^' li' are constant quantities. Let SECT. VI. THE DIFFERENTIAL CALCULUS, 6^1 d^ J_ . . ^ _ , U dx'^ u ' dx By successive differentiation, d^' ~^"dx^~^'''''"d?' M ~ ' ^^ -h — ~h^ • • ^^'^* i- - 73 5^^ "~ * 5^* ~ ^^' ' dx^' 'u ^ Hencc we find F(7i) =2^ X ^ ^ . ^^^ , 2g^5^ or substituting for b the constant — a^, F(/0 _: 2 ^ 1 - _ + 2.3.4"" 2.3.4.5.6 Hence by (73.), this gives F(/i) = 2 COS.tt/t. It is upon this theorem that Poisson founds his proof of tlic composition of force. (Mecanique, torn. I. p. 15). PROP. XLIV. (87.) To determine the mth power of a root of the equation xy" + a — y = 0, X and a being considered as known. Let u = y", and by the given equation i/ = a -\- xip. Comparing this with Lagrange's theorem, u = f(3/) = ^^ 2 = a, f{y) = z/». dF(z) Hence f(;s) =: a"',/(2) = a", -^ = ma'^-^ &c. which is the development required. G6 THE DIFFERENTIAL CALCULUS. SECT. VH. SECTION VII. Of the limits of series. PROP. XLV. (88.) In any series composed of ascending and positive powers of\ a value may he assigned to h, so small, that any proposed term may he made to exceed the sum of all that follow it. Let m' = F(a; + A), and this being expanded, let ii=iiiJ^ aA« + B/i^ + c/fc^' • • • • a^h^c ' ' ' • being in an increasing order. Let s = M + n/a"-^ + oh"-"" .... s7i"' = M/i"' + n7*" + oh' ' • ' ' Therefore «/ = 11 + aA« + bA^ . . . . Lh^ + s7r, w' = w + A^« + b7i* .... ^^(l 4- sA'^-O. , Since m > I, h may be obviously assumed so small that ^Ji^m-i ^^y i)g indefinitely diminished, and may therefore be rendered less than l. In this case then L7i' > s7r, that is, the term L7i^ is greater than the sum of those which susr ceed it. (89.) Cor, Hence by assuming h sufficiently small, li — u will take the sign of a^", and if a have an even numerator, 7^ - u will take the sign of a, and will be consequently the same whether ^ be + or — . SECT. Vn. THE DIFFERENTIAL CALCULUS. 67 PEOP. XLVI. (90.) To determine the effect which the increase of the variable x = a^ox = a4-h produces upon the Junction. Let v! = f{x -|- h)j dii h , u — u = -J- ' T + s/i®. dx 1 du h may be assumed so small, that -^ > s^*, consequently du u! — u will have the sign of -j-. Hence, if the first dif- ferential coefficient be positive, the function increases, and if it be negative, the function diminishes. Thus the state of the function for all values of the variable may be determined du by finding the roots of the equation -7- = 0. If therefore a and a + ^ be between two roots of this equation, -7- does not change its sign between those values of x, and the function increases or decreases according as -j- is positive or negative. PROP. XLVII. (91.) To determine the limits of the error arising from assuming the first term, the first two terms, or any number of successive terms of the development of {{yi-^-h), by Taylor'' s theorem, for the "whole value qff(x + h). du Let ~ — = f'(^)- In this function let x be supposed gra- dually to increase from x to j: -{- h, h being taken of any f2 68 THE DIFFERENTIAL CALCULUS. SECT. VII. finite value. While x varies between these limits, the function J'\x) suffers a corresponding variation ; let x' and a:" be the values of .r, which, between the proposed limits, render y'(ir) greatest and least. The quantities f(x + h) -/V), /'(^') -/V + h) are both positive. These are the differential coefficients of f(^ + h) -f{x) -f{al')-h, fix) +f{x').h -/(T + h), h being taken as variable. Hence it follows by (90.), that these quantities must increase from x to x -^ h. Now, since they are both = when A = 0, it follows that they must be both positive between the proposed limits, and therefore /(r + h) >J\x) +/'(^") • h, and u+^'{ +f%r^')^. Hence if w + -i q- be taken as the value of f{x + h)j ax 1 the error is comprised between the limits y"(J?'') ^^ ^^^l In general, therefore, if n terms of the series be taken for the whole value of 7/ orjf(x + ^), the error is comprised between the limits of the greatest and least values of '' d"'U h"" dx"" * 1.2.. -w' d"u X being supposed to vary in the function -7- from .r to ^ + h. It is, however, to be understood, that there are no values of X comprised between x and x •\' h, which render the function u or any of its differential coefficients infinite; in other words, it is necessary that there should be no values of X between these limits which furnish exceptions to Taylor's series. Ex. 1. Let u — a^, •.* -t-„ = Va% and if w' = a'"'*"*, dx^ d"u' The greatest and least value of ^-;7 between x and xi h arc 70 THE DIFFERENTIAL CALCULUS. SECT. VII. the values corresponding tox + h and x themselves. There- fore /^"Ka^O = k"a''+^ and /^")(^') = ^"a^. Hence the limits of error are included between the quantities " 1.2.3.. ..7l' "" 1.2.3 . . • ♦ 71 Ex.2. Let u = /(.r + h). The limits of error in this case are i» h'' 1 1 7t (x-^-hy PROP. XLVIII. (92.) Two series ascending hy the powers of the same quantity (h), being given, to determine the limit of their ratio, h being indefinitely diminished. Let the two series be S = A^« + bA" -^ ch' S' = a'¥' + B7i*' + c'A'^. . . . When h is diminished without limit, the limit of s . Ah" s' '^ A7^«'• For s k'iA-hBM-^+ch'-" ) S' - h"'{A' + B"''-'' '+c7a^'-«' )• When h = 0, the factors within the parenthesis become a and a', and therefore in the limit S AC" s' """l^^* /^ If a = a', the limit is —r- If a > aK the limit is ; and a' ' ' if a < a', the limit is infinity. SECT. VIII. THE DIFFERENTIAL CALCULUS. 71 (93.) Car. Hence i£ a = a\ a value may be assigned to h so small as to render s > or < s' according as A is > or < a', and s will continue > or < s' for all values between that assigned value of h and 0. If a > a! J a value may be assigned to h so small as to render s < s', and s will continue less than s' for all lesser values of k. If « < «', a value may be assigned to h so small as to render s > s', and s will continue greater for all lesser values of^. If any number of successive terms of the two series com- mencing from the first be respectively equal, that is, if A = a', a = a', B = b', b = V, &c. then the relative values of s and s', h being supposed to be indefinitely diminished, may be determined by the first pair of corresponding terms of the series which are not equal, in the same manner as above. Thus, if the terms of the series be respectively equal as far as hh} and ilN', inclusive, let the common values of the series thus far be s. Then s— s _ uhr'-\-^h'' • ' ' s'—s ~ m'^^"' + n'A"' * If m = wz', •.* s — 5 is > or < s' — s, or s is > or < s', according as m is > or < m', h being assumed sufficiently small. And all that has been observed before applies here mutatis mutandis. SECTION VIII. Of the differentiation of functions of several variables. (94.) The functions which have been subjected to in- vestigation in the preceding sections have been supposed to 72 THE DIFFEEENTIAL CALCULUS. SECT. VIII. be composed of one variable quantity, connected by some given or supposed relation with constant quantities, or if more than one variable has been introduced, they have been always understood to be connected by some condition, which being expressed by an equation, an elimination might be effected, by which we might finally arrive at a function of a single variable. We shall, however, now proceed to con- sider a more extensive class of functions, namely, those "whose variation depends on the variation of several quan- tities, which are independent of one another. As the variation of functions of several variables is pro- duced by the several variations of each of the variables, there are as many differential coefficients of the first order as there are variables independent of each other. In ge- neral, when w is a function of several variables x, .r', .r" • • • the differential coefficient determined by considering w as a function of the variable x alone, all the other variables being supposed to be constant, is called the partial differential coefficient of the first order of u differentiated with respect dij/ to ;r, and this coefficient is expressed -j- , as if the function u were a function of jt alone. This differentiation being continued, a series of successive partial differential co- efficients will be found which are expressed, -7-^, -y-^ .... ■YT9 ^s ^^ ^ ^^^^ ^ function of x alone. In the use of these symbols, therefore, their true meaning should be carefully attended to, and the student should be cautious not to use them as if they referred to the entire variation of the func- tion u, but only to that part of it which depends upon the ^ ,., du d-u d"u variation or x. In like manner -7--: -r—- • • • —1—,- are tlie dx^ dx'^ dx'" partial differential coefficients depending on the variation of SECT. Vlir. THE DIFFERENTIAL CALCULUS. 78 x^ alone ; and in a similar way the partial differential co- efficients depending on the variation of any other variable may be expressed. There are therefore as many partial differential coefficients of this kind of any proposed order as there are variables. Besides the species of differential coefficients above men- tioned, there are also others obtained in a different manner. The partial differential coefficients of the first order are themselves functions of the original variables. The dif- (iu ferentiation of the function -7- has been continued as a func- dx tion of X. But as it is a function of the other variables x\ x" ' ' ' - as well as of .v, it is susceptible of differentiation (111 with respect to any one them. If y be considered as a function of x', all the other variables being considered con- stant, it may be differentiated. The differential coefficient resulting from this process would, according to the system of notation used in functions of a single variable, be — , , • To avoid the complexity of these symbols, it is, however, expressed thus, , , , which signifies, therefore, the dif- ferential coefficient obtained by differentiating the function zi with respect to x, and again differentiating the result of that operation with respect to x'. If the function u had been first differentiated with respect to x', and next with respect to x, the differential coefficient would be expressed thus, , . , . ^ dx'dx The differential coefficient obtained by two successive d^u differentiations with respect to .r being j--; if this be consi- 74 THE DIFFERENTIAL CALCULUS. SECT. VIII. dered as a function of ^r', and as such be differentiated, the coefficient resulting from this operation is expressed thus, -z — -,, If, on the other hand, the function u be first dif- ferentiated with respect to x', and the result -rj be twice differentiated with respect to x^ the differential coefficient d^u resulting from this operation is expressed thus, , , , ^ . In general, if the function u be differentiated successively m times with respect to the variable x, and the result 7i limes with respect to the variable x', the differential co- efficient which results from these operations is expressed thus, dx"'d:c"' ' But if the function be first differentiated n times with respect to x^, and then m times with respect to ;r, the result- ing coefficient is expressed thus, da^'^dx'" ' (95.) To explain this notation generally, let m be a func- tion of the variables of, x", a/" • • • • x^^\ and let it be dif- ferentiated m' times successively with respect to x', and the result of that operation 7/i" times successively with respect to x" and so on. The differential coefficient which results from this system of operations is expressed thus. dx-^'-dx^'""" • . . .^<")'" If in the function ?/, the several variables x', x'', • • • • a:^") be supposed to receive the increments h', h", • • • • ?i^'^\ and the function to become u', let it be supposed to be expanded SECT. VIII. THE DIFFERENTIAL CALCULUS. 75 in a series according to the positive and integral powers of the increments ^', A", • • • • ^(">. The first term, which is independent of the increments, must be the original function ?/; for it is what the series which is equal to u' becomes when the several increments = 0. Let the series be u^ = u-\- A%' 4- M" 4- • • • • A^").^^"), + dh'h" + c"h"h'<' + • • • • c(^W"-'W^\ + nW + jy^^m + . . . . D<")^<">^ • ; •. • f^i- The sum of the terms of this series which involve the first powers of the increments is called the total differential of the function m, and each of these terms are called pp,rtial dif- ferentials. Thus, du = A'h' + A"h" a(")^(">, or, as it is more usually expressed, du = A'dz' 4- A"dx" A<«)d^("). If in the series [1] h", k'" • • • • /i'"^ be supposed severally = 0, the series becomes u' = ?^ + A'h' + B'h"- + d'^'3 . • • • This is equivalent to supposing a^ alone variable, and x'', ^", uu • • • • o:^"^ constant. Hence it follows that -j-j = a'. In like manner -7-r, = a", and so on. Hence dx" - du ^ , du ^ ,, du ^ , ^ du = -T-7^^' + ~T7idx • ' ' ' ■j-T-dx^^'K dr dz" dx"^ In the use of this notation, it should be observed that -j-jda^, does not signify a division and subsequent mul- du tiplication of du by dxK In this case y-7 is not the repre- sentative of a quote, but represents a function of the several 76 THE DIFFERENTIAL CALCULUS. SECT. VIII. variables Jt', .r", • • • • cr*"^ derived from the primitive func- du tion u in the manner already explained ; and -j-^-rfj7' signifies this function multiplied by dx\ This product, as has been already observed, is called a partial differential of u. PROP. XLIX. (96.) It is required to express a function u' = f(x' -|- h', x" + h" • • • • x*"> + h<"^) in a series of positive and integral powers of the quantities h', h", • • • • h<''>, the quantities x', x", .... x<") being independent variables. We shall first consider the case in which there are but two independent variables a/, ^''. In this case w' = f(^' + h\ ^" + /z"). As the exponents of the quantities h\ li\ in the required series, are positive integers, the series arranged by the powers of h^ must have the form M' = A + a'A" + a"F2 + a"72"3 + B . /?/ ^- C ^ + B W3 ( 4- c'W3 i:- d'"F3 4-d'^" I J If w' be considered a function of x^ only, a?" being treated as a constant quantity, it may be expanded in a series of powers of /i' by Taylor's theorem, and if m" = r(^, a;" + /*"), this expansion will be _f . . . . dx^ This series must be indentical with the former, inde- pendently of/?', and therefore ,ECT. VITI. THE DIFFERENTIAL CALCULUS. 77 w" = A + a!W + a'7/'2 + a'W^ + • . [2], 1 rf/z" Y . ^ = B + B'/i" + bW2 + b'"A"3 + [3], 1 //«//" If ic = F{x'a/'), and this be considered as a function of cc", a^ being considered constant, w" may be expanded by Taylor's theorem, the result is du W d^u m d'u h""^ If u and its partial differential coefficients -7-^, -fij^i &c. be considered as functions of iv\ this series, differentiated successively, gives du" _ du^ dru h" d'u h'"" d}u _^ dx^'-~ dx'"-^ dx'doc'"- ' T'^dx"'da;^ ' 1.2'^ dx"'dx'^ ' 1^3 [8], d'u!' d'u d'u h" d'u M' dhi ¥^ J [9]. dx^~da^^^da^^dx^^>' 1 ^ dx'^'^dsd^' 1.9.'^ dx'Hx'^ 1.2.3 These series [6], [7], [8], [9], &c. must be identical re- spectively with [2], [3], [4], [5], &c. independently of /t". Hence '^~''' ^ ~ 1^'' "^ "= 5^«* 2 ' "^ "^ d^' ' f2.3 " " 78 THE DIFFERENTIAL CALCULUS. SECT. VIII. B = dll ,' — dru , d-u dhi d*u dh 1 d'u ,b'"= da^^-dx^ 1.2'" ~dx''dx' 1.2.3 d'u 1 d' c"= i-7ijC J^"2£?^'2 1.2'" -^/.xy'^c/^^^ 1.2.3 __^ ,__dSi_ ^,_ (Z^^ _1_ ,„__d^u_ I ^ ""5^' ^ "dw'da;''' ^ " dx"^da;'^'T^'^' ~ dj^'W' 1.2.3' Hence the expansion of w', arranged according to the powers of 7i' and ^", is u'=u du h' ^ ^ d^ m "^^Tf ■^^' L2 dx'' tS + c?3 7i'3 (Za7"(Z2^ 1 '^d^'^ ' 1.2 c?^'3 1.2.3 !^ "^c/^"W 1.2 d^u WW^ + ....[10] J-^ +■ A' A'2 J^"3 1.2.3 As the variables in the function u = f(A") are not distinguished by any particular condition, this series will still represent w', if x' and 7i' be changed into x" and A", and vice versa. This change gives duh"^ d\t A"^ 1 dHi_ _hP_ ^ +....[11] dx''^' 1.2.3 d'u hH" du h" >v dhi. h'"' ^ ^-^+rf^Tl+5^'lT2 + du^M^t d^u WU I d^u dx^^ 1.2 dxHx}^ 1.2 ^ da^dx'^'^ 1.2 (Z. By the rules already laid down, if the partial differentials be found, and added, tjie equation du ^ , du ^ „ du ^ , ^ du = -T-rd^ + -Tli'doo" 3-^/^^"^ dr' dx^^ dx^""^ becomes in this case, du = x"x'" • • • x^»>dx'-{-x'x"' • . x^'^^dx" • • • x'x'' • • • a?<"Wic<">, which is the same as the result of (22.), where a^, ^', • • • • are not independent variables, but all functions of a common variable. In a similar way the result of (23.) may be extended to fractions, of which the numerator and denominator are pro- ducts of independent variables. (99.) The following examples will illustrate the principles on which functions of two variables are differentiated. Ex. 1. Let u = x'^f, '.' du dx — mx^^^-y^y dy ^ . Hence du = mx'^-^y " -dx \ nx"\y^~^ • dy = x'^~^ 'y^~\mydx -f- nxdy). 8S THE DIFFERENTIAL CALCULUS. SECT. VIII. X DC Ex. 2. Let w = /'tan. — . \i z — — and «' = tan.^;, then y y u = l':^, and by (16) du __ du d^ dz "" 6?2' dz ' ^ du I ^dz' 1 But -T7 = --r, and -r- == -. Hence dz' z dz cos.®z du QO\,.Z 1 dz cos.^z sm.2cos.;2 But since %——.'.' y ydx — xdy ^ ~ If * Hence , yds-ivdy 2/2 sm. — cos, — ^ y y X X Ex. 3. Let u = tan.~^ sin.""^ — . As before, let ^ = — y ' y and z' = sin.-^s;, •/ u — tan.~V. Hence (16.), du du d^ dz ~ dz' dz' _ du 1 . dz' 1 But -77 = 7-7— To and 3— = . Hence dz' l-f-2'* rf2 ^l-;22 du^__ 1 dz ~(l+;^2)^/Trii* But since 2; = — , •.• y ydx—xdz dz = ^ . y Hence - _ ydx — xdy ydx— xdy SCET. IX. THE DIFFERENTIAL CALCULUS. 88 SECTION IX. The differentiation of equations of several variables. (100.) When an equation ¥{xy) — 0, involving two variables, is given, either variable^ may be considered as a function of the other x. The resolution of the equation for y would change it to the form y =./(^). In this state the function // might be dift'erentiated by the rules already given for the differentiation of functions of one variable, and thence the value of the successive differential _ . dy d~u d"y ^ . coefficients --,—...._. found. This method, however, would in general be of no prac- tical use, as it would require the general resolution of equa- tions. It will be therefore necessary to find a method of determining the successive differential coefficients of y with respect to T, without resolving the proposed equation for 3/. (101.) For this purpose let x and j/ be first supposed to be independent variables, and let u ~ ^[xy)^ the values of y and x not being necessarily limited by the condition u — 0. If j/ and X become y -\- Ic and x -^ h, the function becomes ?/.' = F[(.r -{- h)f {y -{■ ^)], and by the equation [11] (96.) ri o 84 THE DIFFERENTIAL CALCULUS. SECT. IX. If in the equation y =f(x)^ x become oc ■]- h and y be- come y, we have by Taylor's series dy h d'y A* d^y h^ ^~^ ^~ dv 1 ^ dx^ 1.9,^ dx^ 1.2.3 Making this substitution for A: in the series already found, and arranging the result by the dimensions of A, it assumes the form, ,/ = « + v.A + v. ^ + A". ^^ + ....[.]. Where du du dy fdu du dx dy . d v~~\^x dy ^Jdv'' d^u 1 d^u dy d^u dy^ 1 57« ' ¥ "^ dx'dy "dx^ dy'^'d^"^' fdhi ^ d^u ^ ^ dHi ^ \ 1 H ^dx^' d^^-^' -^ ''•d^y^'^''^^'''d^^^^^^' + dy'^ J ' l.^.S.dx^' And in general the series of coefficients of the powers of dx and dy in each term is evident from their analogy to the co- efficients of an expanded binomial. Let the variables x and y be now restricted so as always to satisfy the equation w = 0, so that whatever be the value of A, the condition u' = must be fulfilled. In this case the series [1] must = independently of h; hence its several coefficients must separately = 0, which gives the equations u = 0, -j-dx 4- -T-du = 0, dx dy '^ d^u d^u , , d^u , , SECT. IX. THE DIFFERENTIAL CALCULUS. 85 d^u (Pu d^u d^u The first of these equations is only a repetition of F(.rj/) = 0. The second, however, determines the value of the differential coefficient -7~» if the functions -r- and 3- be ax ax ay known. Let these be a and b, •.• kdy -f- Bdx — 0, ' dx A * Hence it follows, that " to find the first differential co- efficient of an implicit function y, given by an equation of two variables x and j/, the equation must be differentiated as a function of two independent variables, and the total dif- ferential being equated with zero, will determine the sought differential coefficient." (102.) In a similar way it may be shown that an equation of any number of variables may be treated as a function of the variables, and differentiated. Let u = F(d7', x^-'-x^'"'^) =0, by differentiation we obtain , du , , da ^ ,, du ^ ^ . dx' dx" ' (ir<") This is called the total differential of the proposed equation. The partial differential equations may be obtained by considering the given equation successively as a function of each combination of two variables. This process will give as many partial differential equations as there are different combinations of two variables in the primitive equation, and each of these equations will determine a partial differential coefficient of one of the variables as a function of another. As however the differentials of the variables severally enter 86 THE DIFFERENTIAL CALCULUS. SECT. IX. as multipliers of all the terms of these equations, any one of them may be deduced from the others. (103.) To return to the equations of two variables, the differential coefficient -,— being expressed by ^, the dif- ferential equation becomes Ap -f B = [1]. In this case a and b being functions of the variables x and y, this may be treated as an equation of three variables, ^, 7/, and jp, and being expressed by u\ its differential is dii^ , du! , dii , d'^y . . . Since dp = -~, it is evident that this equation, combined with the first differential equation, will determine the dif- dhj ferential coefficient ,- - as a function of the variables x, y. Hence " to obtain the second diff'erential coefficient, the first differential equation must be differentiated, considering dy ;r, ^, and -y- as variables.'*'' Again the equation [2] being differentiated, considering dy d^y X, y, -J- and -^ as variable, will give a third equation, which, combined with the other two, will determine the d^ij third differential coefficient -^. ax^ Thus, in general, " the equations which determine the successive differential coefficients -r^, -7^ • • • •- r^, of an dx' dx"- dx''' implicit function given by an equation of two variables, are deduced by successive differentiations, each differential co- efficient being considered as an additional variable." SECT. X. THE DIFFEEENTIAL CALCULUS. '8'7 SECTION X. Of the effect of 'particular values of the variable upon a function^ and its differential coefficients. (104.) A function is in general rendered either positive or negative by the real values which may be assigned to the variable. There are, however, four states of the function which are attended with peculiar circumstances, and which require some examination. Certain particular values of x may render the function, or its differential coefficients, 10 = 0, 2° — ~, 3" imaginary, 4° infinite. We shall con- sider these four cases first in explicit, and next in implicit functions. PROP. LII. (105.) To determine the values of the successive differential coefficients of a function (u) which correspond to any par- ticular value (a) of the variable (x), which renders the func- tion or any of its differential coefficients ~ 0. P. Let X ~ a render the function itself — 0. By the principles of Algebra, it follows that x — a, or some positive power of it must be a factor of u ; so that ii must be of the form u — v{x — a)'", m being a positive integer or fraction, and p being a function of x not divisible by {x — a), or any power of it. From the process of differentiation it appears that {x — «)"*-% (x — a)*"-^, &c. are factors of the successive differential coefficients of w. Let these coefficients be u', /<", • • • • u^"\ they must be of the forms 88 THE DIFFERENTIAL CALCULUS. SECT. X. where p', p" • • • • are quantities not divisible by any power of {x — a). If 7n be an integer, these successive differential coefficients will = when x = a a.?, far as the {m — l)th inclusive ; but the wth differential coefficient will be of the form which not being divisible by any power of {x — a), will not vanish when x — a. The same may be observed of the dif- ferential coefficients which succeed the mih. It is plain that if m — 1,- the function vanishes, but none of its differential coefficients do. If w be a fraction, let n be the next integer below it, and '.' n -\-\ the next above it. In this case the differential co- efficients as far as the wth inclusive vanish with the function, and those that succeed it all become infinite. This is evi- dent from considering that m—n is positive, and m-'{n^\) negative. If m be a proper fraction, then n — 0'^ and in this case all the differential coefficients are infinite. 2". Let X — a render any proposed differential coefficient — 0. If the first differential coefficient which it renders = be of the nth order, it follows that ^(n) ^ p(n) ^^ _ ^)m^ p^"^ not being divisible by a power of ^ — a. In this case it may be proved by the process already used, that when m is an integer, the differential coefficients from the wth to the {n \ m — l)th inclusive vanish when.r = «, and those which succeed them do not. If m be a fraction be- tween / and Z 4- 1, then the differential coefficients from the SECT. X. THE DIFFERENTIAL CALCULUS. 89 nth to the (n -f /)th vanish, and the succeeding coefficients become infinite. If w be a proper fraction, then all the coefficients after the nth become infinite. PROP. LIII. (106.) Given a function which vanishes when x = a, to determine the highest power of {x — a), which divides the function. Let u = F(ir), which vanishes when x =^ a. It is mea- sured by [x — aY to determine z. Let the function be differentiated until a differential coefficient w^"^ be found which does not vanish when z = a. This coefficient will be either finite or infinite. If it be finite, the value oi z is an integer, and = w. If it be infinite, the value of z is a fraction, whose value is between the integers n and n — \. To determine it, let «^<'*-^) be divided by such a fractional 2^(w— 1) power A; of « — ,r, that the quote r^ may be finite \(l — X) when X — a. Then the exponent of the sought power will be w + k. This is manifest from the last proposition. PROP. LIV. (107.) To determine the true value of a function which a particular value qfx renders ^, or irifinite. That the first may take place, it is necessary that the nu- merator and denominator be both functions of x^ which vanish when j: = «, and which therefore have factors of the form {x — ay. Let the function then be 90 the differential calculus. sect. x. f{x) and let the highest power of (2^ — a) which divides one be z, and the othier z'. The function may therefore be ex- pressed thus, p(ir— aV The values of z and z^ are to be determined as in the last .proposition. if z > z', u — 0. IF ;s < 2', II is infinite. 1£ z — ^, P u = —. Hence it appears that the method of proceeding to de- termine the value of the function is, to diiferentiate both the numerator and denominator until a dijBPerential coefficient of each be found, which does not vanish when x = a. Let this coefficient be of the n\h order for the numerator, and of the Twth for the denominator. Then 1^. If 71 > m, the function is — 0. 9P. If 7i < m, the function is infinite, as well as all its differential coefficients. 3^. l^n — m, the 71th differential coefficient in each term of the fraction maybe either finite or infinite. This pre- sents four cases, First, If it be infinite in the numerator, and finite in the denominator ; in this case 2; is a fraction less than n and z' = n; hence the function is infinite. Secondly/. If it be infinite in the denominator, and finite in the numerator^ then z — n and 2;' is a fraction less than w, therefore the value of the function is 0. Thirdly. If both be finite ; in this case the value of the function is a fraction whose numerator and denominator are the differential coefficients themselves. For let x -^ k be SECT. X. THE DIFFERENTIAL CALCULUS. 91 substituted for oc in both numerator and denominator, and the results developed, we find h .. A« f(^)+a'.-~+a".j-^- + u = - where a', a", &c. b', b", &c. are the successive differential coefficients. Substituting a for w, the functions and their successive coefficients vanish as far as the ?fcth differential coefficient, which is by hypothesis finite in both numerator and de- nominator. Hence the function becomes ^n ln+ 1 A<«^.,-7^ +A<"+^). _ l,^ ""n 1.2.. ..w+1 b(«). -^ ,_j_b(«+i). l.%"'n ' 1.2.. ..71+1 Dividing both terms by h^, and supposing A = 0, we find which is a fraction whose numerator and denominator are the first differential coefficients which remain finite when £c = a. Fourthly. If the first differential coefficients which do not vanish be of the same {n\h ) order, and both become infinite when X = a. In that case z and 2' are both fractions be- tween the integers n — \ and n. The values of the frac- tions may be determined as in (I06.) ; and if they be equal, both terms of the fraction being divided by the common power of a: — fl, the result will be its true value. If z > z\ u = 0, and if 2 < je/, u is infinite. Or the value may be determined thus. In both numerator and denominator let oc H- h be substituted for ^, and the results expanded ac- cording to increasing powers of h by the ordinary rules of 92 THE DIFFERENTIAL CALCULUS. SECT. X. Algebra, or by the method explained in (55.) ; for in this case the series of Taylor does not apply (55.) ; and let the result be _ aA«+a'^«' + a"A«"...- ^ ~ pA^ + b'A^Tb^-...* The exponents a, al, a" • • - - b, b', b" - - - - being arranged in an increasing order. I£ a > b, the fraction becomes It is evident that all the exponents in this case are positive, and therefore by substituting a for cc, and making ?i = 0, we find u — - 0. If, however, a < b, the fraction becomes _ A + a'A«'-« + A'7i«"-«- . . • Making oo = a, and A = 0, this becomes infinite. If a = ^, by dividing both numerator and denominator by A% _ a4-a'A«'-« + a"A""-°-.-. ^ "" b+b'A^'-«+bW'-«.--.' Substituting « for ^, and supposing h = 0, A u = — . B If in the product of two functions of x one factor become infinite when a; = a, and the other 0, it can be reduced to the form |, and therefore its value may be found by the preceding rules. Let u = r(d7) X f'(^), and let F'ix) be infinite and f(x) = when a: = a. If we supposey(^) = "77~~\' ^^^^" ^^ x = a, fijx) = 0. ^"t '^ — ~Fnro which becomes ° when ^r = «. '.• u SECT. X. THE DIFFERENTIAL CALCULUS. }d6 If M = — — -, and X — a. render both numerator and de- nominator infinite, the value of u may be found by the same rules : for let fix) — — — :, f{oc) — -rrTi ^ ^ f(^)''^ ^ ^ f'(^) f'(x) '.' u =*^7r— tj which becomes ^ when x = a. Also, if M = ¥{x) — f'(^), and these functions be- come infinite when x = a, let f(x) = -pr— :, and f'(x)= -t;:^— f(xy ' f{oc) fix)-f(x) f{x)f'(x) ' which becomes -° when x = a. Hence all combinations of functions of x coming under the preceding forms are regu- lated by the rules already dehvered. (108.) We shall now proceed to give some examples of the application of these rules. ^n 1 Ex. 1. Let u — q-, to find the value of this when x—1 X = 1. By differentiating once, we find the first diff^erential coefficients of the numerator and denominator, Hence u = n. _ ^ ^ ¥{x) ax^-{-ac'^^2acx ^ , , , Ex. 2. Let u = -7-^ = -^—z; — -z-r r— -, to find the value f'{x) bx^~-2bcx-\-bc^ of u when x = c. By diff^erentiating dF{x) dx d'F\x) = ^a{x — c), = ^b(x - c). dx These both = when x = c. Differentiating therefore again, 94 THE DIFFERENTIAL CALCULUS. SECT. .\ d-'F'ia;) which not being affected by the value of x, give a Ex. S. u = , to find u when x -- 0. dpix) -^ = aH^a - bH'b, dp'jx) _ dx ~ Hence u — Va — Vh = I (i> 1 — sin.<«' + cos.a7 , ^ , , * liiX. 4. u — ; -, to find u wnen.r = —r- sin.jr+cos..r— 1 2 d¥(x) — — cos.^ - sin.a:, dx dF\x ] dx = cos..r — sm.a:. 7t Hence when x = 77-, w = 1. 3 (x^ ~ a^\^ Ex. 5. Let 11 = ^^ , to find u when x = a. (x-a)^ The value may easily be found in this case by raising both to the power y, ••• u = — ^ = X -\- a, x — a Hence w^ - 2(7, ••• w = (2a)^. Ex. 6. Let M = — , to determine u SECT. X. THE DIFFERENTIAL CALCULUS. 95 when X — a. In this case, let (« -j- K) be substituted for or, and the result is w = . Developing (a + h)^ by the binomial theorem, and sub- stituting for it its development u' = Dividing both terms of this fraction by Vh, we find Making h = 0, we find _ J_ Ex. 7. Let II =: (1 — .r) tan.i(irj:), to find u when r = l. In this case u assumes the form x x . But 1 1 — X , . 1 1 tan.i-te) = r, •.• u = — -, which becomes ~ ^^ ^ cot.i(7rzi;)' cot.i(7r:r) when X = I. Applying to this the common rule ' dF{x) __ dv'(x) _ iff dx- ~~ sin.®i(ffar)* 2 When X = lf sin.*4(ffjr) — 1, *.• m = — . X tan. Iff — • Ex. 8. Let u = -; — 77—^ ?r~T, to determine the value x^a~^{x^—a^)~^ X of u when x = a. In this case the fraction becomes — . 00 96 THE DIFFERENTIAL CALCULUS. SECT. X. X 1 But since tan.^^Tr • — = , and a a: cot.ir — ^ a u = y- , COt.i^TT- - "" a which becomes ° when x — a. Differentiating, we find dY[x\ —~^ = 2ax-' - 2a(x'- - a'')x-^, dF'(x) _ _ -i-y dx . x' a sm.«iir- — Hence when x =: a, u ~ . Ex, 9. Let u ^ X tan..r — iir seer, to find u when IT In this case u ~ oo — cc , But since '.* u 1 1 — , sec.jT = — it-a? cos X IT xsm.x —~ + B = 0, Ajt/ + B = 0. Since a and b are rational functions of the variables, they cannot have more values than one for a given system of values of the variables. Therefore, the values of a and b in these two equations must be necessarily the same. Sub- tracting, we find A(/7-.y) = o, ••• A = 0, •.• B = 0, (114.) As the state of the function corresponding to such a system of values of the variables as that we have just been considering is attended with circumstances of some import- ance in Geometry, we shall examine it somewhat more particularly. Let y be what y becomes when x becomes x -\- h. By Taylor's series, we find h ¥ h' y = «/ + A^ • -J- + A,- j-g 4- 'Vj-^ • • • • where a^, A2 • • • • are the differential coefficients. Now, if for the same values of?/ and ^, a^ have two unequal values, there will necessarily be two corresponding unequal values of y, whether h be affirmative or negative ; but when h = 0, the value of y becomes single, and equal to ^* Hence it appears that this circumstance must arise from the particular values of the variables which render a^ = o, 10% THE DIFFERENTIAL CALCULUS. SECT. X. making a radical vanish in the value of ?/ derivable from the original equation, and yet not making the same radical vanish in Ay. This follows from the principle, that the roots of an equation can only become equal by a radical disap- pearing. The possibility of a radical disappearing in a function, and yet reappearing in its differential coefficients, may easily be shown. Let u = (x — a) s/¥ + a:« + c. When 07 = «, M = c, and the radical disappears. But du x{x — (i) When X = a, du + vb" + x^. in which the radical appears. See Geometry Art. (368.), note. It appears, therefore, that if the first differential co- efficient becomes ~, and its values be determined by an equation of the second degree, whose roots are real and Unequal, Taylor's series divides itself into two at the second term; thus, h h^ i Ay. Y + A2-j-^ + • (b,y + b,.^^ + .... In like manner, if the value of the differential coefficient were determined by an equation of the third degree, it would divide itself into three, and so on. If the differential coefficient of the n\\\ order assume the form §, the series divides itself at the {n + l)th term; thus, y = 2/ + Ay • Y + vj-gH- • • • • + \ -^n SECT. XI. THE DIFFERENTIAL CALCULUS. 103 And in general the series divides itself into as many dif- ferent series as there are unequal finite and real values of the differential coefficient. SECTION XL Of maxima and minima. (115.) Let u be a function of the variable ^', and let three values of a corresponding to x = a ^ h, a: =i: a, and .r = a -f //, be 71 = F(a — h), W" = F(«), n'" = F{a + ?i). Def. If a be such a value of x, that for any finite value of /i however small, the quantities u"—u' and u" — u"' have the same sign, and continue to have that sign for all values of h between that finite value and 0, then the value ii" is called a maximum or minimum value of the function according as the common sign of the quantities ?/" — w' and w" — zi'" is + or — . (116.) From this, which is a rigorous definition of maxima and minima, it will be perceived that these terms do not necessarily signify the greatest or least value of the func- tion. It is true, that if the function is incapable of un- limited increase or decrease, and therefore has a greatest or least value, this value must be a maximum or minimum, and this case vvill be found to come within the preceding definition. But on the other hand, the function may have maxima and minima values which are not its greatest or least values, and may even have several maxima and several minima of different values. This will be easily conceived, if, while the variable x is supposed continually to increase 104 fllE DIFFERENTIAL CALCULUS. SECT. XL from to infinity, the function be supposed to vary, and in its variation, alternately to increase and decrease, the value of the function, which stands exactly between its increase and decrease, or at which it changes from its increasing state to its decreasing state, is a maooimum ; and that value at which it changes from a decreasing state to an increasing state is a minimum. Upon examining the definition al- ready given, it will be found that these principles are in- volved in it. An example will probably put the matter in a clearer point of view. Let ti = b — (x — ay. If .?> = 0, u = b — a"^; if Z> be supposed > a^, this value of u is positive. As ^ increases from X = to X = a, the quantity (a; — a)- diminishes from a'^ to 0, and therefore tc increases from u =: b — a^ to u — b. When x becomes > a, the quantity {x — g)'^ again increases, and therefore u diminishes, and therefore the value u = b stands between the increase and decrease of the function, and is therefore a maximum. Again, let w — 6 + (^ ~ ay. In this case, when ^ = 0, w = 6 -f ft", the quantity (x — aY decreases from <^' = to X = fl, for which u = b. When a: becomes greater than a, ti begins to increase j hence in this case z« = 6 is a minimum value of the function. (117.) From the definition of maxima and minima, it follov/s that the essential characteristic of a maximum is, that it exceeds those values of the function which imme- diately precede and follow it, while a minimum is less than both these values. (118.) The general method of determining maxima and minima of functions of a single variable is derived from Taylor's series, except when the values of x come under its exceptions. We shall first consider the cases which do not fall within the exceptions. Let ?/'- F(.r), and ^^'=F(.r-/^), ti'" = y{x + //) ; hence SECT. XI. THE DIFFERENTIAL CALCULUS. 105 " ^ - -P 1 P 12 P 1.2.3- ••• y,/y', /?'". . . . expressing the successive difFcrential coefficients of tlic function. Let tliese series be expressed thus, iSucb a value may be assigned to h as will render the sum of all the terms of these series which succeed the first less than the first, and therefore the signs of the entire series will be those of their first terms. If the quantity /?' be not = 0, the value of i^" cannot be either a maximum or mirtimum ; for by assigning a sufficiently small value to 7/, the sign of u" — u' will be that of + />', and u" — u'" will be that of — ^y, these signs being different, the value u'' does not come under the definition. In order that u" should be a maximimi or minimum, it is therefore necessary that /?' = 0, and as p^ is a function of a', it follows that no value of a; but such as are roots of the equation p' = 0, can render the function either a maximum or minimum. Let it therefore be supposed that a value of x, which is a root of this equation, be substituted for z' in the functions ?fc", p\ p", &LC. We shall first suppose that this value of a; is not a root of the equation p" — 0. The series by this substitution become In this case, as })cfbrc, such a value nuiv be asbigned to h 106 THE DIFFERENTIAL CALCULUS. SECT. XI. as will render the first term greater than the remainder of each series. Hence the quantities n" — u^ and w" — w"' will both have the sign of — p'', and will have the same sign for every value of h between that and 0. The corresponding value of the function will therefore be a maximum if />" < 0, and a minimum if;?" > 0. If, however, the root of the equation p' = 0, which is substituted for x, be also a root of the equation /?" = 0, biit not a root of p'" = 0, the series become In this case, as in the first, a value may be assigned to h such, that w'' — m' shall have the sign of + p'^', and w" — ?/'" the sign of — p'", and therefore the corresponding value of the function is not either a maximum or minimum. If, however, the value of x be also a root of y ' = 0, and not of y" = 0, the function is a maximum or minimum, ac- cording as p"" < 0, or > 0, and so on. Hence we conclude, that in order to determine the maxima and minima values of a function, it is necessary first to find the first differential coefficient {^), This being, in general, a function of 07, determines those values of x which render it = 0, or the roots of the equation /?' = 0. No values of the variable x, which are not exceptions to Taylor's series, can render the function u a maximum or minimum, but such as are found amongst the real roots of this equation. Substitute these roots successively for x in the second differential co- efficient //'. Such of them as render p" < 0, being substituted for x in the function w, give maximum values ; such as render y > 0, give minimum values. If, however, any of them render p" = 0, they must be substituted in y ; and if they render it > or < 0, they will not render the function w cither SECT. XI. THE DIFFERENTIAL CALCULUS. 107 maximum or minimum ; but if they also render p'" = 0, they must be substituted in^'", and so on; and if the first differential coefficient, which they render > or < 0, be of an odd order, they do not give either maxima or minima values of the function ; but if it be of an even order, they determine maxima or minima according as they render that differential coefficient negative or positive. (119.) We shall now consider the maxima and minima values of the function u, which are found among those values which form exceptions to Taylor'*s series. Let the values of jr which are roots of the equation — = be determined. In this case the developments become u" - u^ — a(- hy + b(- hf -f c( - hy .... m" - Zi'" = A^« + B/i^ -{- Qh' If any of the exponents have an even denominator, the consideration of maxima and minima becomes inapplicable ; for, the fraction being in its least terms, the numerator must be odd, therefore one series will be real, and the other ima- ginary, since the corresponding term is the even root of an odd power ; and therefore the value of the function does not come under the definition (115.). If all the denominators be odd, the numerator of the ex- ponent a may either be odd or even. If it be odd, (+ h)' and (— hy will have different signs, and *.* w" will be neither a maximum nor minimum. But if the numerator of a be even, (-f hy and (— ItY will be both positive, being the odd root of an even power ; and *.' in this case w" will be a maximum when a is positive, and a minimum when a is negative. If a value of x which renders /?' = 0, render ^" infinite, the developments assume the same forms as in the last case, 108 THE DIFFERENTIAL CALCULUS. SECT. XI. and the same observations exactly will apply ; and will in general apply if a value of x, which renders the differential coefficients from the first to the nth inclusive = 0, render the {n + l)th infinite. (120.) Before the more general investigation of the maxima and minima of functions of several variables, it may be useful to give some examples of the determination of those of functions of a single variable. Ex. 1. Let u = ax^ — bx" + a; -f 9. Hence 3- = '2>ax^- - 9hx + 1. ax The values of Xy which render this = 0, are bA-Vb^- -Sa 3a 9 b- -^b^- -iSa 3a If b"^ < Say these values are both impossible, and therefore the function in this case is not capable of a maximum or minimum. But if 6- be not < Sa, let the function be dif- ferentiated again, and the result is Substituting in this the values of x already found, dhi ". renders < 0. Hence in this case, if If b" > Sa, one value of x renders -j-^ > 0, and the other b- Vb^-Sa 2a SECT. Xr. THE DIFFERENTIAL CALCULUS. 109 be substituted for x in the function u, tlie corresponding value will be a maximum ; and if 6+ yo'^-^a be substituted, the corresponding value is a minimum. If, however, b"- = 3a, and •.* b'^ - 3a = 0; in this case the value of .^' determined by -^ = is ^ = —, which being substituted for x in renders j-^ — 0. It will therefore be necessary to differentiate again, which gives This not depending on x, and not being = 0, the function admits of no maximum or minimum in this case. Ex. % To divide a number a into two farts suchj that the product of the mth power of one, and the nth power of the other, shall be a maximum or minimum. If jT be one of the parts, and -.' a — x the other, the pro- duct is u = ^'"(« — xy\ du '.' -^ = (a ^ ^')"~^ • ^'""^ . [ma — {m -\- n)^], -=^= (a - xy-'^.x'"-^. \ (nia^ [m -f ?i)xy'-m{a — xf - nx" ] . ax The values of x, which render t- = 0, are determined by the equations 110 THE DlFFEllENTIAL CALCULUS. SECT. XI. which give a — 0, 0. ma — {m ', +n ■,y x = a, 0, and that therefore the function is a minimum. Similar observations apply to the value ^ = 0, by con- sidering X — as a factor of the differential coefficients. The value x = beine substituted for x in -r^.ren- ders it negative, and therefore renders the function a maximum. Ex. 3. Let u = :j—; — r. In this case let mw' = 1 ; when m' is a maximum, m is a minimum, and vice versa. But 1 X dx '^^ X'' dhi^ 2 d~^ " x'' SECT. XL THE DIFFERENTIAL CALCULUS. Ill Ihe condition — = 0, gives x = ± I, *-*y:^ = + %. Hence if x = 1, ••• u' = 2, a minimum, and w = ±, a maximum. If ^ = — 1, *.* w = — 2, a maximum, and u = — 4» ^ minimum. The principle used here frequently abridges the process, scil. by investigating the maximum or minimum of the re- ciprocal of the function in place of the function itself. (121.) The maxima and minima values of functions of several variables are determined upon principles similar to those which have been already applied to functions of a single variable. If u = f{x', x^\ x"' , . . .) and i^—Y{a^±,h\ x" ± ?i", jc'" ± 7i"' ....); let such a system of values be supposed to be assigned to the variables x', a:", x'" ... . as renders the sign of w — u' independent of the signs of the quantities hi, h", W .... these quantities having any system of finite values, however small, and such, that the quantity u — v! will preserve the same sign for all systems of values of 7i', /i", ^"' .... between the assumed system and h! = 0, h" = 0, h'" = , . . . the value of w, which corresponds to the system of values of the variables thus assumed, is a maximum if the sign o£ u — u' be positive, and a minimum if its sign be negative. (122.) We shall first consider the case where w is a func- tion of two variables. In this case c _ du h! _ du 7i" ^ c d^u h'^ '^ - ^' = i + ■^' • T + 5^ • T 1 - 1 ^ • T2 ± dhl h%" dhl l^l , rt-, + dx'dx'' 1 ^ dx^"- '1.23 In order that the sign of w — ti' may be independent of h' and /i", it is necessary that such values be assigned to the variables as will render l\2 THE DIFFERENTIAL CALCULUS. SECT. XI. These equations will in general give determinate values of y and y. In order, however, to find whether this system of values of the variables gives a maximum or minimum value of the function, it will be necessary to substitute them in the differential coefficients of the second order, and to de- termine whether the sign of the quantity d^ ^ d'u Ml" dhi hJ^ is independent of the signs of h' and h". For this purpose, let — =z Icj and the preceding formula becomes K'^ c dHv d^u dhi ^^ 2 > djt^ ~'^d?d7''^'^"d^^'^ y If the sign of this be independent of k^ the values of ky which render it — 0, must be imaginary. This gives the condition d~U d-U / d% Y r. FA-I That this condition may be fulfilled, it is necessary that -r-g- and -f-jii should have the same sign. d^u //,'* If/t'' = 0, the quantity [3] becomes , ,2 t q ^ ^"^ ^^^^ sign of this quantity must therefore be the sign of [8], since it always retains the same sign. Hence it follows, that P. If any system of values of ^' and x", determined by . dn d^U , ^ n ' ^ [2], give -j^y -j-rj^ different signs, the function has no cor- responding maximum or minimum. 2^. If any such system of values of ^ and ^ give -t-jz, ',-;77the same sign, and yet render SECT. XI. THE DIFFERENTIAL CALCULUS. 113 l^'d^" \d?d?') < " or = 0, the function has no corresponding maximum or minimum. 3°. If such a system of values of y, a^', give -j-j^ and 7 1,2 ? a negative sign, and also fulfil the condition [4], the corresponding value of the function is a maximum. 4P. If such a system render -y-^, j-^ , both positive, and also fulfil the condition [4], the corresponding value of the function is a minimum. (123.) It may happen that the system of values of .r', a-", determined by [2], also fulfil the conditions 'd^ ~ "' "3^ ~ ^' ^W' ~ In this case it will be necessary to substitute them in the partial differential coefficients of the third order. If they do not render these = 0, the function admits of no corresponding maximum or minimum ; but if they do, it ^ is necessary to examine the effect of the same substitution on the differential coefficients of the fourth order. The terms of the development involving N, h", in four dimensions, being treated as those involving two, and the conditions of ima- ginary roots determined, similar conclusions follow, and so the investigation may be continued as in functions of a single variable. (124.) Similar reasoning may easily be applied to functions of any number of variables. The conditions which determine the system of values of the variables which may give a maximum or minimum, are du du ^ du ^ rir-i But to determine if any and what system of values of the I ' 114) THE DIFFERENTIAL CALCULUS. SECT. XI. variables derived from these equations must give a maximum or minimum, it will be necessary to examine their effects upon the successive partial differential coefficients. It frequently happens that some one or more of the equations [5] can be inferred from the others. In this case the number of independent equations being less than the number of quantities to be determined, it follows that there are an infinite number of systems of values which may all determine maxima or minima. In this case, if the question be geometrical, the solution is a locus, (125.) We shall now give some examples of the in- vestigation of maxima and minima of functions of several variables. Ex. 1. To divide a quantity a into three parts, x, y, and a-- X --yj such that the product u = ^"».y".(a — x ^ yY is a maximum or minimum. The differential coefficients of the first order are du di = 00^-' . z/" . (a - ^ - y)P-i . i^rrna - mx - my -^ p^X^ du The factors within the latter parenthesis of each being put = 0, and solved, give In order to discover whether these correspond to a maxi- mum or a minimum, we must substitute them in the general expressions for dHL d'^u dHi ^" ^' d^y And if W2 + w + ^ be called q, we find SECT. XI. THE DIFFERENTIAL CALCULUS. 115 P— 1 Ihe quantities -5-^, — , are both negative, and fulfil the condition [4], and therefore the corresponding value of the function is a maximum. We shall not pursue here the investigation of the con- sequences of the other factors of the above equations being = 0, as the student can readily do it himself. Ex. % To find the greatest triangle which can he included within a given perimeter. Let the perimeter be 2flf, the sides x, 1/, and ^a — a? — 2/, and the area u. By a well known principle, u = ^/a(a -- iv)(a — «/)(^ -\- ^ — a). Assuming the logarithms, 2lu = la + l{a — x) -{- l(a — 3/) + ^(a; + j/ — a). Differentiating for x and ^, we find du dx dx u a — x~x + «/ — a du 9^a — 2x—v • = Xu — dx * (a—x){x-\-y—aY du ^ 9^—^y — x The conditions under which these = are ■^ 9a -.2a:-2/ = 0, ence 2a' -, % - or = 0. x^\a, 2a - ■ X — I2 116 THE DIFFERENTIAL CALCULUS. SECT. XI. Hence the triangle is equilateral. It is evident from the nature of the question, that this result determines a maxi- mum. However, this may be proved by examining the partial differential coefficients of the second order by the criterion which has been already established. Ex. 3. Let w* = (l + Y^ — Kyf + (m + xz - zxf + (n + zy-YzY. This being differentiated for x, y, and z, and the partial differential coefficients being made = 0, the results, after reduction, and putting r* for x* 4- y'' + z^, are X(xa7 + Yj/ + ZZ) + MZ — LY — R"^ = 0, Y(xa7 + Yy + Zz) + LX — NZ — R'j/ = 0, z(yLX + Yy + zz) + NY — MX— R^2 = 0. If these equations were independent, they would give a determinate system of values of ocyz. But they are not in- dependent ; for if the first be multiplied by x, the second by Y, and the third by z, an equation will result independent of ocyz, whose terms will destroy one another. If the quantity within the parenthesis be eliminated, the equations will become z(lz-|-my + nx) Yd? - XJ/ + L = -^ , x(LZ+MY-fNx) Zy - yz + N = — -, y(lz-|-my+nx) XZ — ZJ + M x= — ^^ • . This question comes under the observation in (124.), and it follows, that there are an infinite number of systems of values which determine the maximum or minimum value of the function. If in this case xyz be the co-ordinates of a point in space, the locus of that point is a straight line re- presented by the above equations, and the value of ic for all values of xyz is SECT. Xir. THE DIFFERENTIAL CALCULUS. LZ + MY+NX 117 (X^ + Y^ + Z^)^ This question is connected with the theory of statical moments, and the right line thus determined is the locus of the points of minimum principal moment. (See Poisson, Traite de Mecanigue, livre i. chap. 3). SECTION XII. Application of the Differential Calculus to the Geometry of Plane Curves. Arcs and Areas. Principles of Contact. OF ARCS AND AREAS. PROP. LVI. (126.) To determine the differential of the arc of a curve considered as a Junction of the co-ordinates of its extremities. By the equation of the curve, ^ is a function of ^. Let AM = x, PM = y, mm'= h, p'm' = 2/' = F(a? + h). By Tay- lor's series. A p'm' _dy h d^y %"• M? X The co-ordinates being rectangular, pp' = VA* -|- (p'/?)*? *.' the value of pp' must have the form bA* 118 THE DIFFERENTIAL CALCULUS. SECT. XII. The limit of the ratio of the arc pp' = s, and its chord being a ratio of equality, it is evident that when ^ = 0, pp'__ ds h ~ dx' But pp[ _ / h "V When /i = 0, this becomes dy'' dx -4/1 4-^ PROP. LVII. (127.) To express the differential of tJie area included by a curve, and the ordinates of any two 'points upon it, as a Junction of the co-ordinates. Since the arc and chord pp' co- incide when ^ or mm' is indefinitely diminished, the limit of the ratio of the area included by the or- dinates and the arc pp', to that included by the ordinates and the chord pp', is a ratio of equahty. It is evident that M M- Let da be the differential of the area when It, = 0, pp'm'm da lut h '^ dx' pp'm'm = mm' X K^M + i''m ') = 4 ^' dy h d^y 1 '^dx^' 1.2'' My + y) =y^t \,%^dx^ h^ 1.22' \h{y + y% SECT. XII. THE DIFFEHENTIAL CALCULUS. 119 pp'm'm_ dy h d^y When /i = therefore we obtain da dx = y, ••• da =z ydx. IL. ' 1.22 1.2^* * OF CONTACT. (128.) Let the equations of three plane curves passing through the same point p be f(^j/) = 0, F'(;r7/) = 0, F"(a:j/)=0, and let these equations be related to the same axes of co-ordinates AY, AX. Let the co-ordinates of the common point p be yx, and let y, y, y, be what j/ becomes in each of the equations when x becomes x + h. Let mm' = /i, and p'm' =y, p"m' = y\ p'"m' = y. These values being severally expressed by Taylor's series, are y 3^ + A, y = 2/ + c, + Aa 1.2 + A, 1.2.3 y h h^ Where a,, Ag, • • • • b,, B2, + C2-J-g + C3 • Cy, C2J • • 1.2.3 • • express the suc- cessive differential coefficients. Since the first terms of the three series are the same, mm' may be assumed so small, that the order of the magnitudes of the three ordinates ?/', ?/", 3^'", will be that of the three co- efficients Aj, Bp c^, if these three be supposed unequal (92.). 120 THE DIFFERENTIAL CALCULUS. SECT. XII. Thus the figure is represented as if A; were the least, and c^ the greatest. If the same negative value mm" be assigned to h, then the order of the magnitudes of y, z/", y, must be the opposite of that of Ay, By, Cj, therefore, of the three points p', p", p'", where the curves meet the parallel to the axes of ^ through m", p^" is the lowest, and p' the highest. This being op- posite to their order on the other side of the point p, it is plain that the curves cross each other at the point p. (129.) It therefore follows, that the position of the curves in the immediate vicinity of their common point P is to be determined by the relation between the magnitudes of the first differential coefficients. If two (Ay and By) of the three coefficients be rendered equal by the co-ordinates of the point p, then the relative magnitudes of the ordinates y and y are to be determined by A2 and B2, and by assuming li or mm' sufficiently small, y will be greater or less than y, according as A^ is > or < B2. In this case, also, 3/'" is at the same time greater or less than both 1/ and y, according as Cy is greater or less than the common value of Ay and By. These conclusions are evident from (92.). Hence, it follows that if Cy have not the common value of Ay and By, the curve pp'" cannot pass between the curves pp' and pp", but must pass either above both or below both, ac- cording as Cy is > or < the common value of Ay and By. The curves pp' and pp" in the vicinity of the point p therefore approach each other more closely than the curve pp'" can to either of them. These curves are said in this case to have contact of the first degree. (130.) Let us now suppose that the point p is such that its co-ordinates render the three coefficients Ay, By, and Cy, equal. Then by diminishing mm' or h sufficiently, the order of the magnitudes of y\ «/", y, will be determined by that SECT. XII. THE DIFFERENTIAL CALCULUS. 121 of the coefficients A25 B2, C2, and the three curves will have contact of the first degree. In this case the change in the sign of 7i not affecting its square, produces no effect upon the order of the magnitudes of ?/', y, 3/'"; therefore the points p', p'', p'", are in the same order on both sides of the point p, and therefore the curves do not cross each other at that point. If the co-ordinates of p render A2 = Bj, the order of the magnitudes of e/, y, must be determined by that of A3, B3. In this case y must be greater or less than both y and y, according as C2 is greater or less than the common value of A2 and B2. Hence as before, it follows that no curve for which C2 is not equal to the common value of Aj and B2 can pass between the curves pp' and pp" in the immediate vicinity of the point p. The two curves pp' and pp" are in this case said to have contact of the second degree. If the co-ordinates of the point p render the three quan- tities A2, B2, C2, equal, then the three curves have contact of the second degree. In this case, as the sign of the third term of the series changes with that of 7t, since it involves h^^ the order of the magnitudes of «/', y, «/'", for + h and — h are opposite, and therefore the points p'p"p'" on different sides of the point are in opposite orders. Hence the three curves cross each other at the point p. Thus contact of the second degree is both contact and intersection, (131.) By pursuing this reasoning, we may conclude in general, that if the co-ordinates of the point p render the suc- cessive differential coefficients from the first to the n\h in- clusive, equal, each to each, no curve which agrees with these in a less number of differential coefficients can pass between them. The two curves are said in this case to have contact of the wth degree. If the contact be of an even degree, the first terms of the two series, which do not agree, involve an odd power of /*, the sign of which changes with that of ^; THE DIFFEIIENTIAL CALCULUS. SECT. XII. and, therefore, contact of an even degree is both contact and intersection ; but if the contact be of an odd degree, the first unequal terms involve an even power of h, of which the sign does not change with that of h, and, therefore, contact of an odd degree is contact without intersection. (132.) If the equations '^■{pcy) = and v^\xy) = be those of right lines, being equations of the first degree with respect to the variables, all the differential coefficients after the first are = ; therefore the series end at the second terms. It follows from what has been already proved, that if Ay = By, and c^ be not equal to Ay, that the second right line cannot pass between the curve and the first, and if Cy becomes equal to the common value of Ay and By, the two right lines become identical, since the two series end at these terms. Substituting or' — x for 7i, it appears that the right line represented by the equation (Geom. (26.)), meets the curve at p' in such a manner, that no other ris:ht line passing through the point p can pass between it and the curve. This right line is therefore a tangent to the curve at the point. If the co-ordinates be rectangular, -~ is the tangent of the angle under the tangent Hne and the axis of x. Geom. (15.) If -^ = ; the tangent will be parallel to the axis of x ; and if -~ be infinite, the tangent is parallel to the axis of y. dx For the values of the subtangent, and subnormal, and the equaUott of the normal, see Geom, (323.)) Gt seq. SECT. XII. THE DIFFERENTIAL CALCULUS. 123 (133.) For a curve, whose equation is of the form, 1/ = a -\- bx + cx\ The series fory'' terminates at the third term, and By = ^ + 2ca7, If in such a curve the coefficients By and Bg be equal to those of the curve f(^z/) = 0, it will touch this curve with contact of the second degree, while no curve of the same kind, that is, whose equation is of the same form, can touch the curve f(x7/) with so intimate a contact. The curve is in this case said to osculate. The nature and principles of osculation are so fully explained in my Geometry (353.), that it would be needless repetition to enter upon the sub- ject here. (134.) The curve represented by the above equation is a parabola (Geom. Sect. VII.). By analogy to this, a class of curves represented by equations of the form 7/ = a + bsc + cx'^ .... g^"*, are called parabolic curves, and the series for y for each of them terminates at the {m + l)th term, all the differential coefficients after the mth being = 0. When such cur^jes have a common point p with any proposed curve, and all the terms of the expansions of j/" agree with the correspond- ing terms in the expansion of y for the proposed curve, they are called osculating parabolas. In this sense the osculating parabola of the first order is the rectilinear tan- gent. The osculating parabola of the second order is the common parabola. The osculating parabola of the third order is the cubical parabola, and so on. It follows, also, from what has been said (131.), that osculating parabolas of even orders both touch and intersect, while those of odd orders touch without intersecting. IM THE DIFFERENTIAL CALCULUS. SECT. XII. (135.) The osculating parabolas furnish means of repre- senting geometrically the successive terms of Taylor's series, or the differential coefficients. Let 2/;, 1/.^, j/a, &;c. be the ordinates of the several oscu- lating parabolas corresponding to ^ + h, so that dy h 3^3 =3/ + dx yz dy 1 '^ dx"- Al 1.2' h?- dhj \.% "^ dx' 1.2.3' ••«// dy y^-di d^y d^y h f Al 1.2' ' 1.2.3* d^y K . . n Let mm' = Jij and p// being the tangent, let pp", pp'", pp"", &c. be the successive osculating para- bolas, then m'?w, vijil^ p^p^\ p^'jJ", p^iip'iii^ &c. are the successive terms of Taylor's series ; and if d:c be assumed = hj then u'm = y, mp' = dy, 1.2{p'f) = d% 1.2.S(py') = d% SECT. XII. THE DIFFERENTIAL CALCULUS. 1^5 (136.) The order of osculation of a curve of any proposed degree depends on the number of constants which enter its equation (Geometry, 353). The curve of the second degree, which osculates any proposed curve, touches it therefore with contact of the fourth order, and the coefficients of the equation of this osculating curve are functions of the con- stants in the equation of the proposed curve and the co- ordinates of the point of contact. Let the equation of the osculating curve be Ay + lixy ■\- cx^ + ny + E^ + F = ; the species of this curve is to be determined by the quantity b2 — 4ac, which being a function of the co-ordinates of the point of contact, varies from point to point of the proposed curve. Suppose ?/ eliminated by means of the equation of the curve, then b^ -- 4ac becomes a function of x alone. Let the roots of the equation b2 — 4ac = be x', .r", x^" • • • • At the points of the curve, which cor- respond to the real roots of this equation, the osculating curve is a parabola. And since b^ — 4ac changes its sign in passing through 0, it follows that the osculating curve on one side of such a point is an ellipse, and on the other side an hyperbola ; the species changing as often as there are real values of 3^ corresponding to the real root of the above equation. If the roots of the equation be all imaginary, the quantity b2 _ 4^^-, always retains the same sign, and therefore the osculating curve always remains of the same species. If the condition b^ — 4AC = be fulfilled independently of xi/ by the constants of the given equation, then the oscu- lating curve for all points is a parabola. Similar observations may be applied to osculating curves of any proposed degree. Although the degree of contact of an osculating curve of 126 THE DIFFERENTIAL CALCULUS. SECT. XIII. any species depends on the number of constants which enter its equation, yet it may happen at particular points of the given curve, that the contact is of a higher degree than that which marks in general the order of its osculation. This circumstance arises from an additional differential coefficient of the given curve being rendered equal to the corresponding differential coefficient of the curve which osculates it, by the peculiar values of the co-ordinates of the point of contact. We shall soon meet an example of this. SECTION XIII. Of osculating circles and evolutes. (137.) The most remarkable osculating curve is the circle. The equation of the circle, involving three constant quan- tities, the order of its osculation is the second. Let {y — i/Y + (^ — *^y = R* be the equation of a circle, whose radius is R, and the co- ordinates of whose centre are ocy\ The first and second differential coefficients are "• " (.y-1/r yx being the co-ordinates of the point of contact, it is neces- sary that the quantities x\ t/, and r, should receive such values (130.), that Ai = B« A, = Bo. SECT. XIII. THE DIFFERENTIAL CALCULUS. 127 To determine the values of y, ^, and R, which fulfil these conditions, let the values of b^ and Bj, already found, be sub- stituted for them, and the equations ^{y - y) + ('^ - ^0 = 0, A2(5^-3/r + r2 = 0, give y=^ + l±V . [1], ^ = a;-ii-^A, [2], "=-(%)■ » or substituting for A/ and A2 their values y = «/ + df^-dx'^ dy d^y dx "" d^y.dx (138.) The equahty A^ = b^, which gives shows that the centre of a circle having a common rectilinear tangent with the curve, must be upon the normal (Geom. 325). The radius of the osculating circle is generally called the radius of curvature. (Geom. 335). (139.) Since «/, a^, and Aj, are functions of x by the equation of the given curve and its differentials, it is evident that y and x^ are implicit functions of x. If, there- fore, X varies by assuming the values corresponding to the different points of the curve, the quantities y^ suffer con- sequent variations, and the centre of the osculating circle 1^8 THE DIFFERENTIAL CALCULUS.. SECT. XIII. assumes different positions accordingly. The locus of this centre is called the evolute of the curve, and its equation may be found by eliminating x by the equations which give s^y^ as functions of a:. (Geom. 337, 338, and notes). (140.) Since ds"- = d7f 4- dx"- (126), ds being the dif- ferential of the arc of the curve, *.• _ ds^ ~~ d~ydx' The expressions already found for the radius of curvature are determined on the supposition that x is the independent variable. To obtain its value, independently of this hypo- thesis, it is only necessary to substitute for y^ (38), dxd~y — dyd^x dx"^ ' which will give (dy^' + dx'-y dxd-y — dyd^x df_ ~ dxd'^y — dyd^x If s be considered as the independent variable d{ds'^) = 0, •.• dyd^y 4- dxd^x = 0. Squaring this, and adding it to the denominator of the above value squared, we find ^_ ds^ "" '~[{d^yy + (d^^y]{dy'' + dx^y or since ds'^ = dy" + dx'^^ '.' ds' ^= --. ., . n.. R = V(d^y)^-{-(d^x)^ This expression is used by Laplace. See Mecanique Celeste^ liv. i. chap. 2. Another expression, frequently used by physical authors for the radius of curvature, is SECT. XIII. THE DIFFERENTIAL CALCULUS. 129 _ ds ^~ V d(p being the angle under the normals through the ex- tremities of ds. As ds may be considered coincident with the arc of the osculating circle, it is evident that since d(p is the angle under the two radii through the extremities of dsy we have rcZ^ = ds. PROP. LVIII. (141.) To determine the 'position of the tangent to the eoolute at any point y'x' corresponding to xy upon the given curve. Let the values y^, already found, be differentiated as functions of x, and the results are di/ ■= dy -^ d ^ , Aj 1 + A* ds^ =^ dx — {^. -^ kf)dx - A^d ^. Aa Multiplying the first by a^, and adding K^y^ + dx' = A.fy — Kfdx ; but dy = A^d;r, •.* A^£?y Jt da} —% jy \ ^ dx ' dod ~ Ay ~ dy Hence the equation of the tangent at the point ijod is (y "" y)% -V {x — a^^dx = 0. Hence the normal to the curve is tangent to the evolute. (Geom. 341). 130 THE DIFFEttENTIAL CALCULUS. SECT. XIII. PROP. LIX. (142.) To determi7ie the change in the arc of the evolute corresponding' to any proposed charige in the radius of curvature. In the equation ^' = (2/- yj + (^' - ^')% ^, y, and y, being functions of x already determined, R may be differentiated as a function of a?, *.* Rc^R = ( j/ - y)(c?z/ —^y) + (^ - x^)(dx — rf^), Substituting for ^ , , its value — ~, the result is cZr= -tZy + ;^.da:'; dy dx? but -J- =— -f-j' Making this substitution, and squaring, we find (di/^+dx'^f V^iy-1/Yr'''- dy^ ' or dn^ = Jy'* + dx^"; '.- Jr = ((^y* + da^^)^, observing that '^ (y-y'T' '^ dx^ -^^ dy^^' Hence it follows, that the increment of the arc of the evolute is equal to the simultaneous increment of the radius of curvature, and the property from whence the evolute has derived its name may be thence deduced. (Geometry, 342). SECT. XIII. THE DIFFERENTIAL CALCULUS. 131 PROP. LX. (143.) To determine the point upon any curve at which the radius of curvature is a maximum or minimum. By (137.), ds' R = — d^ydx' Differentiating and equating the result with zero, we find (y - y)d'y - Sdyd^y = 0. Substituting in this equation fory, z/, and the differentials, their values as functions of ic, the roots will determine the sought points. PROP. LXI. (1 44.) To determine the points of a curve at which the contact of the osculating circle is of the third degree. The third differential coefficient derived from the equation of the osculating circle is _ ^dyd\y ^' ■" ~ {y~y')dx^' d^u * Equating this with -^ derived from the curve, the re- sult is This equation being identical with that found in the last proposition, it follows that the contact is of the third order at the points of greatest and least curvature. K% 132 THE DIFFERENTIAL CALCULUS. SECT. XIV. SECTION XIV. Of asymptotes, (145.) Let the equations of two plane curves which have infinite branches be F(a72/) = 0, F'(^y) = o, y and y being the values of j/ in the two curves correspond- ing to the same value of x. The distance between the curves measured in a direction parallel to the axis of y is y — y\ If, as x increases without limit, either positively or negatively, the distance y — «/' diminishes without limit, but vanishes only when x becomes infinite, the infinite branch of the one curve is said to be an asymptote to the other. (Geom. 345). In order that this should occur, it is necessary that the quantity y — «/', developed according to the powers of x, should contain only negative powers of x. For if it con- tained a positive power, y — y' would be rendered infinite by X becoming infinite, and if it contained a term inde- pendent of X, it would be finite when x is infinite. Hence the development of j/ — ?/' must have the form y — y' z= Ax-° + JiX~^ -f. . . . . the exponents being supposed to descend. It follows, therefore, that if the development o£ y by the descending powers of x contain any positive powers or a term independent of x, all these must also occur in the de- velopment of y, in order that they may disappear by sub- traction. Hence, if the development of j/ be y = a'x"'' -f B^x^' . . . . M + Ao;--" + bx"^ • • • • the development of y must be «ECT. XIV. THE DIFFERENTIAL CALCULUS. 133 y = a!x^' + b'^^' M + • • • • the terms which succeed m, or those which involve negative powers of .r, being unrestricted. (146.) Since the terms of the development which succeed M are arbitrary, it follows that there may be an infinite number of asymptotes to the same curve, and that each of these will be asymptotes to each other. The most simple asymptote which the curve admits, at least that whose de- velopment is simplest, is the curve represented by the equation 3/' = a!x^' + ^^x^' + • • • • M. The curve represented by 3/'' = a'o:"' 4- n'x^' + ••••>! + A;r-« is also an asymptote, and approaches closer to the curve than the former, since, by increasing x^ it is manifest that y approaches nearer to equality with y than y does. In like manner, the curve represented by y" = a'x«' + B':r^' + . . . . M + hx-" + bo;-'' has asymptotism of a still higher order with the given curve. (147.) Thus it appears that there are orders of asym- ptotism in some degree analogous to the orders of contact. Curves which admit asymptotes are sometimes divided into hyperbolic and parabolic. Hyperbolic are those which admit a rectilinear asymptote ; parabolic those which do not. All hyperbolic curves must therefore be involved in the class !/ = a'x -\- b' 4- Aar-« + Bor^ • . . . The equation of the rectilinear asymptote being y = a'x 4- b'. If « = 1 and B, &c. = 0, this curve is the common hyper- bola. If a' = 0, the asymptote is parallel to the axis of x, and if a' = 0, b' = 0, the asymptote is the axis of w itself. 184 THE DIFFERENTIAL CALCULUS. SECT. XIV. (148.) We shall now give some examples illustrative of the preceding theory, h I Ex. 1. Let J/ = ± —{x'^ — a^y. Expanding by the bi- CL nomial theorem 2/ = ± ~x + ^hax-^ Hence the curve has two rectilinear asymptotes repre- sented by the equations y' = ±: —07. •^ a See Geometry (232.). Ex.2. luQiyx = c«, •.• J/ = c^x"'^^ Hence the asymptotes are the axes of co-ordinates them- selves. Ex. 3. y\x'^ — aP-) = h^. By developing, we find 3r = ± yx-"^ + j: = ± a ± k—y"" + Hence the axis of x is an asymptote, and there are two other rectilinear asymptotes parallel to the axis of ^ repre- sented by the equations 07 = + a. There are also two hyperbolae, yx = hr^ and yx =■ — i% which are asymptotes. Ex. 4. Let y^ — Qaxy + x^ = 0, •.• ?/ = — ^ — « — a2j?-i — . . . . There is a rectilinear asymptote represented by y = — tf — ^. Ex. 5. 3/*^; — px* — ^3 _ Q^ Hence y^ :=! px + fl^a:"^. Therefore the asymptote to this curve is a common parabola. SECT. XIV. THE DIFFERENTIAL CALCULUS. 135 (149.) There is, however, another method of determining whether a curve admits a rectiUnear asymptote, which is frequently more easily applied than the general method already given. Let the equation of the tangent through any point ijx^ on the curve be (3'-y)--^(^-^)=o. Let X be the intercept of the axis of x between the origin and the point where the tangent meets the axis of x ; and let Y be the corresponding intercept on the axis of y. It is evident that these are obtained by supposing y and x suc- cessively = in the equation of the tangent. Hence we find '^ ~ dy ' In these quantities let a^ be supposed to be increased without limit. If the limits of x and Y be finite, they will determine a rectilinear asymptote. If X have a limit, but y none, then the asymptote is parallel to the axis oiy at the distance x. If Y have a limit, but x none, then the asymptote is parallel to the axis of x at the distance Y. If neither have a limit, or if their values arc rendered impossible by increasing a:, then the curve has no rectilinear asymptote. If in the limit x = and y = the asymptote passes through the origin, and its direction is found by determining the value of ~ when x is indefinitely increased. (See Geom. 346). These conclusions are founded upon the principle, that 136 THE DIFFERENTIAL CALCULUS. SECT. XIV. the tangent becomes an asymptote when the point of contact is indefinitely removed. (150.) Ex. 1. Let yx ^- hy ■\- ex ^ 0. By difieren- tiating dx^ / ^ ^ ^ ' ^a? x-\-h Hence X — X -T r— = — , y+c y+c .: _ y^ _ y^ ~ y + c' "~ x^-^b' By solving the equation for y, we find ex e y = h+x LjL.\ X Hence, when x is infinite, y =. — c. Also, Y = ■+f which, when x becomes infinite, gives Y = — c. In a similar way, we find X = - 5. Hence there are two asymptotes parallel to the axes of co- ordinates. Geom. (123.). SECT* XV. THE DIFFERENTIAL CALCULUS. 137 SECTION XV. Of the direction of curvature — Of the singular points at which a differential coefficient assumes the form •§. (151.) The development of the value of ^ corresponding to X -\- hhy Taylor's series, conducts us to a method of determining the direction of the curvature of a curve. Let ^" be the value of y in the equation of the tangent cor- responding io X -\-h. Hence dy h dhj h^ d^y h^ -^ "^^ dx l^ dx^ \,^^ dx' 1.2.3 dy h *.^ -^ dx^ \3^ dx' 1.2.3 d^y Hence y — y has the same sign with t^. d^y Therefore, if y andy be > 0,y >y' when -~ >0, and d^ii d^y < y" when j^ < 0, •.' if -v^ > 0, the curve is convex towards the axis of x, and if -j^ < 0, it is concave towards dx^ the axis of x. In like manner, if y and y be negative, it is convex or concave towards the axis of x, according as ^,<0,or>0. d^y In general, therefore, if y and -~ have the same sign, the curve is convex toward the axis of x ; and if they have dif- ferent signs, it is concave in that direction. 138 THE DIFFERENTIAL CALCULUS. SECT. XV. (152.) We shall now consider th6 effect produced upon the curve by the differential coefficients becoming = 0, or = °. If the first differential coefficient be = 0, it has been already shown that the tangent is parallel to the axis of x. Such points are therefore thus determined. Let the dif- ferential of the proposed equation be Ap + B = 0, p being the first differential coefficient. Let the values of x and y which satisfy the equations B = 0, ¥{xy) = 0, be determined, and let such systems of values be selected as do not also satisfy a = 0. Such systems of values, if real, determine the points of the curve where the tangent is parallel to the axis of ;r. (153.) If the second differential coefficient = 0. Since in the equation of the tangent the second differential co- efficient also = 0, the tangent must have contact of the second degree with the curve. Now, since contact of the second degree is accompanied by intersection (131.), it fol- lows that, at the point thus determined, the curve passes from one side of its tangent to the other, as in the annexed figure. Such a point of a curve is called a point of inflexion^ and sometimes a point of contrary flexure. At such a point it is evident that the radius of curvature be- comes infinite, since the second differential coefficient is a factor of its denominator (137.). (154.) If the third differential coefficient be = 0, the curve at the corresponding point has contact of the third order with the osculating parabola (134.) of the second order. SECT. XV. THE DIFFERENTIAL CALCULUS. 139 And in like manner, if the nth. differential coefficient = 0, the curve at the corresponding point has contact of the Tith degree with the (n — l)th osculating parabola. (155.) If several successive differential coefficients from the nth to the {71 + p)th inclusive = 0, the curve has contact of the (n + ^)th order with its osculating parabola of the (n — l)th order. The effect, therefore, of any combinations whatever of the differential coefficients becoming = will be easily per- ceived. (156.) Let us next examine the curve at those points where a differential coefficient assumes the form §. If the first differential equation be Ap + B = 0, let systems of values of the variables xi/ be selected, which at the same time fulfil the equations A = 0, B = 0, f(xi/) = 0. Such values render the first differential coefficient = ■^. In this case, in order to determine the true value of/?, it will be necessary to proceed to the second diff*erential equation (111.), which will give an equation of the form a'p^ -[- b'/} + c' = to determine p. If this equation be not fulfilled by its coefficients, its roots must either be real and unequal, real and equal, imaginary or infinite. First If they be real and unequal, there being two un- equal values of the first differential coefficient corresponding to the same values of x and y, there will be consequently two tangents to the curve at the corresponding point ; there- fore two branches must intersect at that point. Such a point is called a double point. Secmidh/. If the roots be real and equal, there is but one 140 THE DIFFERENTIAL CALCULUS. SECT- XT. value of the differential coefficient, and this presents no par- ticular circumstance in the course of the curve. Thirdly, If the roots be imaginary, the development re- presenting y becomes imaginary for both + h and — h, and therefore the point whose co-ordinates produce this effect stands alone, insulated, and not continuously connected with any part of the curve. Such is called a conjugate point. The case where a root is infinite will be investigated in the next section. If, however, the equation of the second degree for p be fulfilled by its coefficients, it will be necessary (111.) to pro- ceed one step further in the differentiation, which will give for the determination of p an equation of the form ^Y + »V^ + c'!p + d" = 0. If the roots of this equation be real and unequal, there will be three tangents at the corresponding point, and there- fore three branches of the curve will intersect at it. Such is called a triple point. If two of the roots be real and equal, there will be but two values of p, which will give a double point. If two be imaginary, or all be equal, there will be but one real value oip ; in which case the course of the curve will be marked by no peculiarity. If, however, this equation also be fulfilled by its co- efficients proceeding to a fourth differentiation, we shall find an equation of the fourth degree to determine p. Its roots, if real and unequal, determine a quadruple point; if all imaginary, a conjugate point; and, in general, as many as are real and unequal, determine so many tangents to branches of the curve which intersect at the corresponding point. It will be necessary, therefore, to continue the differen- tiation until some equation is found, which, not being satis- fied by its coefficients, will give determinate values of/?. If it have n real and unequal roots, it will determine a muU SECT. XV. THE DIFFERENTIAL CALCULUS. 141 tiple point, at which n branches of the curve intersect. If it have but one real root, no peculiarity marks the curve at the corresponding point. If all its roots be imaginary, the point is a conjugate point. (157.) Let us next suppose that the co-ordinates of the point render the second differential coefficient = °. In this case its value or values may be determined like those of the first (156.). First. If it have several unequal real values, there will be as many values for the third term of the development of «/', and therefore as many different values of y, and therefore as many different . branches of the curve passing through the corresponding point. Since, however, the several values of y agree as to the second term of the developments, they will all have a common tangent. Such a point comes under the class of multiple points, and is characterised by the number of branches which, thus meeting, touch with con- tact of the first degree. This particular species of multiple point may be called a point of osculation *. Secondly. If the coefficient is found to have but one real value, the corresponding point has no particular character. Thirdly, If all its values be imaginary, it is a conjugate point. Similar conclusions may be applied to the succeeding differential coefficients, observing that the contact of the branches, which form the point of osculation, is of the (n — l)th order, if it be the n\h differential coefficient which has the several real values. (158.) In general, therefore, we find, that in order to determine whether a curve admits a multiple point at which its branches intersect, it will be necessary, P. To Some French authors call it un embrassement. 142 THE DIFFERENTIAL CALCULUS. SECT. XV. find the values of xy^ which satisfy the equations a = 0, B = 0, ^{jxnj) = 0. 2^. To determine the corresponding values of/?. There will be as many intersecting branches as^ has real values. If j? have no real values, the point is a conjugate point. In order to determine whether there be a point of oscula- tion, it will be necessary to apply a similar investigation to the superior differential coefficients. It is obvious from what has been already proved (131.), that at a point of osculation produced by multiple values of a differential coefficient of an odd order, the branches inter- sect as well as touch ; but at one produced by a differential coefficient of an even order, they touch without inter- section. It may happen that the value of p in any of these cases may be infinite. We shall consider the consequences of this in the next section. (159.) Ex. 1. To determine whether the curve repre- sented by the equation ay^ — o(^y ^ bx^ = has a multiple point. By differentiating (8fl2/2 - x^)p - Sx%y + b) = 0. Hence a = Say^-a^, b = — 3x%y-\-b). The only values of xy which render a = 0, b = 0, and also satisfy the equation of the curve are ^r = 0, 3/ = 0. To determine the value of p, let the differentiation be continued, and we find ayp^ — X'p — x(y -\- b) = 0, ap^ - Sxp - {y -{. b) = 0. The values x = 0, y = 0, fulfil the former by its co- efficients, and render the latter yb 7' SECT. XV. THE DIFFERENTIAL CALCULUS. 143 which giving but one real value of p, the point is not a multiple point. Ex. 2. Let the equation of the curve be y - ^ + ^* + 357^3^* = 0. By differentiating, we find A = %/(%* + 3;r*), B = w(4iW^ - Bcc^ + 6/). The only values of x and y which satisfy the equations A = 0, B = 0, as well as that of the curve, are ^ = 0, To determine p, let the successive differentiations be effected, and it will be found that the second and third dif- ferential equations will be satisfied by their coefficients, and that the fourth becomes jD* + 3p' + 1 = 0, the roots of which being impossible, indicates a conjugate point. Ex. 3. Let the equation of the curve be cT* - 2ai/^ - 3a*«/^ - ^a^x"- + a* = 0. By differentiating, we find A = Sai/(a +y), b = 2a:(a^ — ^*). The only values of on/ which fulfil the conditions A = 0, B = 0, as well as the equation of the curve, are X = + ay d7 = — a } w = 0. To determine the corresponding values of p, we proceed to the second differential equation, which gives Sa{a 4- 2^)p^ + ^a' - 6x^ = 0. 2 For the first and second points, therefore, p = ± — =,and /v/3 for the third p = ± Vr- The three corresponding points are therefore double points. The condition b = 0, and the equation of the curve are also fulfilled hy a; = 0, y = |a, which values do not fulfil A = ; therefore they determine a point at which the tangent 144 THE DIFFERENTIAL CALCULUS. SECT. XVI. is parallel to the axis of x. The same conditions are also fulfilled by jr = + fl', y = •— \a, which also determine tangents parallel to the axis of oc. The condition a = 0, and the equation of the curve are fulfilled by ^ = —a and ^ = + a a/2, which do not fulfil B = 0, •.• they indicate two points at which the tangents are parallel to the axis of y. To construct this curve, let ax and AY be the axes of co-ordinates. Assume af = ^a, ab = — a, AE = + «, and ae' = — a, AC = — -la, CD= -\- a, ce/= —a, AG = -f a a/2, ag' = — a a/2, GH = — a and g'h' =— a. The curve is placed as represented in the diagram. The tangents at e', are determined by p = ± \/|- ^ ^ Y L. H h ^J H B* CD Ae double points b, e. and « = + — -. It is not necessary to multiply examples, as the student may easily supply himself with sufficient to illustrate the general theory. The following curves have triple points : ^* — a.r2/* + tT* =0, y + 07* - 3az/3 + ^Jbx'y = 0. SECTION XVI. Of the singular points at which y or any of its differential coefficients become infinite. (160.) We shall now proceed to investigate the figure of a curve at a point whose co-ordinates render the first or any subsequent differential coefficients infinite. SECT. XVI. THE DIFFERENTIAL CALCULUS. 145 If the value assigned to x render y infinite, the first ex- ponent o£ h in the development of y must be negative (BB.), In this case, as h is continually diminished, y is continually increased ; and when h = 0, y^ becomes infinite. Thus it appears that a parallel to the axis of j/ corresponding to this value of X must be an asymptote. If the origin of co-ordinates be removed to the point in question, then h becomes x, and the result immediately fol- lows from Section XIV. (161.) If the value of .r render any of the differential co- efficients infinite, rules have been already given for deter- mining the successive exponents of h in the development of y {55.). We shall not here, therefore, enter into any re- petition of these methods, but assume the development of the form y = 2^ + A/i« 4- B^* -f c^'' • . • • If none of the exponents a, ^, c, • • • • be a fraction with an even denominator, the value of y is real, whether h be + or — . Hence the curve extends on both sides of the ordinate y. There are then two cases to examine, 1^. Where the numerator of the first exponent is odd, and 9P, Where it is even. P. If the numerator of a be odd, the sign of A/t" changes with that of h, and consequently at difi^erent sides of the point yx, the curve lies at diff^erent sides of a parallel to the axis of a; passing through the point. If inthiscasea>l ^=0,-.-(13^.) the tangent is parallel to the axis of X, Hence there is an inflexion which is represented as in the annexed figures, the first when A > 0, and the second when a < 0. W V- 146 THE DIFFERENTIAL CALCULUS. SECT. XVI. i f \ % 1 V. If a < 1, ••• ~ is infinite, and the tangent is parallel to the axis oi y (132.). Since the curve ex- tends on both sides of y^ and crosses the parallel to the axis of .r, the point must be an inflexion, as represented in the first figure when a > 0, and in the second when a < 0. 2^. Let the numerator of the first exponent be even. In this case the sign of A^" does not change with that of ^, and since the denominator is supposed to be odd, there is but one real value ; and since by diminishing ^, the term aA" predominates over those which follow it (88.), 2/ — y has the same sign for a; + A and x — h. Therefore, if a parallel to the axis of x be drawn through the point .ry, the curve lies either above or below this parallel at both sides of the point according as A is > or < 0. In this case, if a > 1, •/ -r =0, dx '.• the parallel to the axis of x is a tangent, and the curve is as represented in the first or se- cond figure, according as A is > or < 0. V^ T^-^N ^ If a < 0, ..^k ; dx is infinite, the tangent is parallel to the axis of y. Hence the figure of the curve at the point in question is as represented in the first figure if A > 0, and in the second if A < 0. If the first exponent a = 1, and the second exponent have SECT. XVI. THE DIFFERENTIAL CALCULUS. ^ 147 an odd numerator, then the position of the tangent pt is de- termined by the vakie of a; and since the sign of the second term of the development changes with the sign of h, it fol- lows that at different sides of the point p the curve lies at different sides of the tangent, as represented in this figure. Hence, in this case the point p is a point of inflexion. The second differential coefficient in this case = 0, if the exponent b > 2, and is infinite if 6 < 2. Thus at a point of inflexion the second differential coefficient may be either nothing or infinite. 1£ a = If and the numerator of h be even, the succeeding exponents not having any even denominator, the point is marked by no peculiarity. (162 ) If amongst the exponents a, &, c, • • • • is found a fraction with an even denominator, then a change in the sign of k changes y from real to imaginary, or vice versa If -\- h render all the terms of the development which are affected by such exponents real, and — /* imaginary, the curve extends only on the positive side of 3/, and is excluded from the negative side ; and if — h render them real, and + h imaginary, it is excluded from the positive, and only extends upon the negative side. If -f ^ render some terms which are affected by such ex- ponents imaginary, and — h others, then the curve is ex- cluded from both sides, and the point is a conjugate point. If + Tfc or — 7i render all the terms whose exponents have even denominators real, each of such terms will have two real values for every value of h, and therefore the number of branches of the curve emerging from the point in question will be double the number of combinations of such powers. The tangent to these branches will be de- l2 148 ^ THE DIFFERENTIAL CALCULUS. SECT. XVI. terminecHSy the value of the lowest exponent of h. If it be > 1, the tangent is parallel to the axis of x, if < 1, it is parallel to the axis of y, and if = 1, its position is deter- mined by the coefficient (132.). (163.) Some particular cases will make this general prin- ciple more apparent. 1°. Let the lowest exponent of ^ be a fraction with an even denominator and *.* with an odd numerator, and sup- pose this the only even denominator which occurs in the series. Then m qj - y zzz aIi^ -f B^i' J^ qJi' ' ' • ' where n is by hyp. even. In this case + h renders ^" real, and — h imaginary. First If A be real, for every positive value of h, there are m two real values of aA" with different signs; and for every m negative value of 7i, hh"^ is imaginary. Also, if — > 1, ^- = 0, *.* the tangent is parallel to the axis of T, and if — < 1, -^ is infinite, and •.* the tangent is parallel to the axis of 2/. The first figure represents the curve at the point in question '< Y. IfYt when — > 1, and the second when n V m — < 1. n If A be such an imaginary quantity, that the term m aA" is real for - hy it will be imaginary for + h. Hence, SECT. XVI. THE DIFFERENTIAL CALCULUS. . 149 in this case the first figure is the case where — > 1, and n the second where ■ — < 1. n If A be any other species of m imaginary quantity, aA** is ima- ginary for both + h and —A, •.* the point is conjugate. (164.) If the first exponent be an integer or a fraction whose denominator is odd, and the second exponent be a fraction — , whose denominator n is even, and that no other n even denominator occurs in the series but w, then The position of the tangent is to be determined by the value of a as before. If B be real, + ^ renders A" real, and — h imaginary. Let PT be the tangent as deter- mined by the term pJf. Since there are two real values of b^** with different signs, the figure of the curve at the point p is this. If B be such an imaginary m quantity, that bA" is real for - h^ and •/ imaginary for + ^, the' figure is this. Such points where two branches lie at opposite sides of the com- mon tangent are called cus]ps of the first Tcind. If B be an imaginary quantity of any other species, the point is conjugate. 150 THE DIFFERENTIAL CALCULUS. SECT. XVI. y -y (165.) If the only even denominator first occurs in the third term, the series is m are the three differential coefficients of the second order derived from the equation of the sphere. Hence _ 1 x — x^^dz ""^ "^ "" J^' "^ (7^5^ • "d^' SECT. XVII. THE DIFFERENTIAL CALCULUS. ^1 x-x^^ dz dy ~ 1^- ■z> pq Substituting these values in [1], the result is (a2 + Scim + 52^2) (z— 2;") + (1 +P'') + ^Mwz + (1 + g-)m2=0. This equation determines the co-ordinate z" of the centre of the sphere, which being known, the equations a; — or" = — p{z — 2/')> 3/-y' = _^(2 — 2;"), determine .r'J^". The equation of the sphere being r2 =:{x - X^Y + {y - y)^ + (2 - /)*, by substituting for {x — ^"), (3/ — y), their values, we find R = (2 - 2") • Vl + /?* + 9'^. The sphere thus determined has contact of the second order with any curve traced upon the given surface through the given point, provided that the projection of that curve upon the plane a?y has its tangent through the projection of the given point inclined to the axis of x at an angle whose tangent is m. m2 164 . THE DIFFERENTIAL CALCULUS. SECT. XVII. PROP. LXIV. (177.) At a given point upon a curved surface, to de- termine upon the normal the limits between which the centres of all osculating spheres lie. # This problem may be solved by finding the values of m, which render r a maximum and minimum. To simplify the investigation, let the given point be as- sumed as origin, and the axes of x and j/ in the. tangent plane, the normal being axis of z. In this position of the co-ordinate axes, ^ = 0, ^ = 0, 2 = 0, R = — z", J9 = 0, gr = 0. Hence 1 + /?^' which being differentiated, and its differential = 0, gives Ao— • Bo m^ + -^ ^m -1=0. The roots of which determine the values of m, which give the greatest and least values of r. Since the product of these roots = 1, the directions of greatest and least curvature are always at right angles. Geometry, vol. i. (34.). The formulae will be still further simplified by taking right lines in the directions of greatest and least curvature as axes of J/ and «r. In this case, one value of m in the above equation becomes infinite, and the other = 0. Hence Cj = 0, which reduces the formula for the radius of cur- vature corresponding to other values of m to 1 +m2 B. ^^ ^** II Aj + BaTW^* i 8ECT. XVII. THE DIFFERENTIAL CALCULUS. 165 Let r', r^', be the radii of the greatest and least oscu- lating spheres. Their values are found by supposing m and successively = 0, *.• m 1 1 R' = Hence it appears that the radii of the greatest and least osculating spheres are the reciprocals of the partial dif- ferential coefficients of the second order. PROP. LXV. (178.) To express the radius of aiiy osculating sphere as a function of the radii of the greatest and least osculating spheres^ and of the angles under the directions in which they osculate. By the last proposition, 1 fw* R = — AaH-BjTW* Let (p\ <}5", be the angles under the directions in which the sphere whose radius is R, osculates, and the directions of the osculation of those whose radii are r', r". Hence cos.2^' _ 1 ^1 COS.2(p"' r' r" Making these substitutions in the value of r, it becomes, after reduction, R^ R'^ "■ r' cos.^^'+r" COS.29"' Hence, if the radii of the greatest and least osculating spheres and the directions of their osculations be given, the radius of a sphere which osculates in any given direction may be found. 166 THE DIFFERENTIAL CALCULUS. SECT. XVII. PROP. LXVI. (179.) To express the differential of the arc of a curve related to three rectangular axes. By reasoning exactly similar to that used in (126), we find ds = ^ dy'' -^ dx'^ -\- d% PROP. LXVIL (180.) To determine the equations of a tangent to a curve related to three rectangular co-ordinates. It is evident that the projections of the tangent upon the co-ordinate planes are the tangents to the projections of the curve upon these planes. Hence the equations of the tangent to a curve passing through the point jr'yz', are dd d^ By substituting for the functions -y-j, -7-^, their values de- rived from the equations of the curve, the equations of a tangent through any given point may be found. (181.) Cor. 1. Let tx^ ty, tz, be the angles under the tangent and the axes of co-ordinates. It is evident that dx dy dz cos.^^=-^, cos.^^=^, ^°^-^^= d7' where ds = Vdy^ -|- dx^ -f dz^. (182.) Cor. % Hence the equation of the normal plane SECT. XVII. THE DIFFERENTIAL CALCULUS. 167 through j^JyV, or a plane perpendicular to the tangent, is -jf(s> - y) + ^(- - ^') + ^(« - -') = 0, or d^'(^ — y) + dx'{x — .r^) + ^^'(^ — 2') = 0. (183.) If the curve be not a plane curve, the successive tangents will not all lie in the same plane. The plane of three points of the curve, assumed indefinitely close to one another, is called the osculating plane. Def. A curve, which is not all in the same plane, is called n curve of double curvature. PROP. LXVIII. (184.) To determine the equation of the osculating -plane at a given point upon a curve of' double curvature. Let two points of the curve, indefinitely near to each other, be xt/z and x't/'z'. The equation of a plane through these is a(3/ - y) + b(^' - ^) + c{z - z') = 0, the point ooi/z being considered as variable, and x[y'z' given. In order that this may be the osculating plane, it should pass through two points contiguous to x'y'^ ; it is necessary, also, that its first and second differentials should equal those of the curve. Let the equation be twice differentiated without assuming any independent variable, the results will be Adt/ + Bdjc + cdz = 0, Ad\i/ + B£Z2.r + cd^z = 0. Hence eliminating — and — , we find {dzUPx' - dx'd^z!) {y - -if) + {dy'd^z^ - dz'd'^i/) (x — x') + {dz'd'^y -' dy'd'-x') (z — z') = 0, which is the equation of the sought plane. 168 THE DIFFERENTIAL CALCULUS. SECT. XVII. (185.) Cor. Since the condition under which two planes intersect perpendicularly is, that the sum of the products of their coricsponding coefficients = 0, the osculating and normal planes are at right angles; for (182.), dif[dU\v' - dafd^z') + d^^di/'d^z! - dz'd^i/) + dzWdy - di/^d^x') = 0. PROP. LXIX. (186.) To determine the radius of curvature to a given point in a curve related to three rectangular co-ordinates. This problem is most easily solved by considering the osculating circle as one passing through three consecutive points of the curve. Under this point of view, its plane must be the osculating plane ; and as its radius passing through the given point must be normal to the curve, its centre must be in the intersection of the osculating and normal planes. If, therefore, ^'z/V be the co-ordinates of its centre, they must satisfy the equations dyiy - y) + d^{x ~ a^) -\- dz(z — z!) =z 0, y(«/ - y) + x(x -a:') + z{z - z') = 0, where Y = dzd^x — dxd'^z^ X = dydH — dzd^y, z = dxd^y — dyd^x , , All circles passing through the given point, and having their centres upon this right line, touch the curve. In order to determine that of most intimate contact, let the inter- section of two consecutive normal planes be founds and the point where this intersection meets the right hne thus de- termined will be the centre of the osculating circle. To effect this, let the equation of the normal plane be dif- ferentiated. Considering x'y^z^ as constant, which gives SECrr. XYII. THE DIFFERENTIAL CALCULUS. 169 d^yiy — y) + droc[a' - .r') + dH{z — ^') - ds^ = 0, where ds^ = dy^ + d.r'^ + dz^. From this and the former equations, we find , _^ (Ydz—zdi/)ds^ Ob —~ 00 — ' * D , {zdx — xdz)ds'^ y-y' = — z - z' = D (xdi/ -'Ydx)ds'^ D where B=iYdz — zdy)d^x + [zdx — xdz)d'^y + (xdy -^ Ydx)d^z, Substituting these values in R« = (a; - ody -h (.7/ - iJY -f (2 - z'Yy we obtain ^ _ [{xdy—YdxY^{ zdx-xdz Y-\-(\dz—zdyY'\ds^ But by the conditions xdx + Ydy + zdz = 0, . x2 4- Y^ + Z2 = DS this gives ds^ -v/x«H-y2 + z« which is the value of the radius of curvature for a curve of double curvature. If ds be taken as the independent variable, by differen- tiating the equation ds^ = dy'^ + dx' + dz', we find dyd^y + dxd^x + dzd^z = 0. This being squared and added to the value of d, gives D = ds^[{d''yy + (d^xY + (d^'zyi Hence we find 170 THE DIFFERExNTIAI. CALCULUS. SECT. XVII. See Mecanique Celeste, liv. i, chap. 2. From the preceding formulae, those of plane curves may easily be deduced. PART 11. THE INTEGRAL CALCULUS. PART II THE INTEGRAL CALCULUS. SECTION I. Fundamental Principles. (187.) The object of the Integral Calculus is the deter- mination of the primitive function or equation from which a given differential, or differential equation, may have been derived. The primitive function is in this case called the integral of the proposed differential, and the process by which it is determined is called integration. These terms " integrar and " integration" are taken from the infinitesimal calculus, and have their origin in notions of this science not consistent with the rigour and purity of mathematical reasoning. As, in the infancy of the science, differentials were considered as infinitely small quantities; so the original functions from which these differentials were obtained, were taken as the sums of the infinitely minute elements ; and the process by which these primitive quan- tities were found from their differentials, was looked upon as the summation or integration of the small component parts, and the operation was expressed by the character y 174 THE INTEGRAL CALCULUS. SECT. I. prefixed to the differential, thus, Jafdr, as the initial of the word '' sum" or " summation." Modern mathematicians ^ have reduced the science to more rigorous principles, but they have retained its former phraseology and symbols. Lagrange alone had the boldness to attempt a revolution, not only in the principles, but in the language and algorithm, or notation of the science ; but he can scarcely be considered to have succeeded, at least in the latter, since all mathe- maticians, almost without an exception, adhere to the old symbols, though some of them use the principles and rea- soning of Lagrange. (188.) According to the language of Lagrange, the ob- ject of the integral calculus is to determine the primitive from the derived function ; or, if applied to equations, to determine the primitive equation to a given derived equa- tion. According to the more commonly received phraseology, this branch of the science consists in the determination of the function, of which a given function is the differential coefficient, or the equation, which differentiated, would pro- duce a given equation. As this process is exactly the re- verse of that which forms the subject of the differential calculus, so the rules and methods to be used in it must be discovered by retracing our steps in that part of the science. (189.) We shall, in the first instance, confine our atten- tion to those differential coefficients which are functions of a single variable; and, as in the Differential Calculus, we shall successively consider the cases where they are algebraic and transcendental functions, algebraic functions being di- vided into, 1^. rational and integral, ^^, rational and frac- tional, and 3^. irrational; and transcendental into, 1". ex- ponential, 9P. logarithmic, and 3^. circular. Before we enter upon the methods of integrating these SECT. I. THE INTEGRAL CALCULUS. 175 functions, it will be necessary to lay down a few principles immediately derivable from the differential calculus, and which may be considered among the fundamental principles of the integral calculus. (190.) I. As an independent constant connected with any function disappears by differentiation, so it should re- appear by integration. Thus, if f'(x) be the differential coefficient of f{x), it is also the differential coefficient of f{x) + c, c being a quantity independent of a:. It is necessary, therefore, to add to every integral a constant, which is generally called the arbitrary constant, because its value cannot be derived from, and does not depend on, the differential coefficient, but must, if discoverable at all, be determined by other means. (191.) II. If the value of the integral corresponding to any particular value of the variable happen to be known, the value of the arbitrary constant may be found. For, let the integral with the arbitrary constant be f(^) + c, and suppose that it is known that the value of the integral is a when the variable .r is = a, '.' a = f(^) + c. Hence c = A — F(a), •.* the integral is f{x) — F(a) -{- a. If the value (a) of the variable which renders the integral = be known, the integral is f(x) — F(a). (192.) III. As a constant factor of a function is not af- fected by differentiation (18.), so neither is it affected by integration. Thus, if F'(a?) be the differential coefficient of F(d;), af'(j7) will be the differential coefficient of AF(a;), or, according to the symbols of the integral calculus, jA'F\x)dx = AfF'\x)doCj A being a quantity independent of x. (193.) IV. As the differential of a function, which is the algebraical sum of several functions of the same variable, is the sum of the differentials of these functions (17.), so the integral of the sum of several differentials of functions of the same variable is the sum of the integrals of these differentials. 176 THE INTEGRAL CALCULUS. SECT. I. Thus, / [ F'(a,^)dx -f F\x)dx - F"(x)dx \ =fFXx)dx ■\-fF\x)dx - fF'\x)dx. (194.) V. As the differential of the product of two func- tions of the same variable is the sum of the alternate pro- ducts of each function into the differential of the other, so the product of two functions is equal to the sum of the in- tegrals of each function into the differential of the other. From this principle an important method of integration is deduced. Let xx' be two functions of x. Hence xx' =:Jxd}d +/x'dx, \'jkdx' = xx' —Jx'dx, By this equation the determination of one integral fxdx! is made to depend on another, viz. J'x'dx. Numerous in- stances of the efficacy of this method will appear hereafter. It is called integration hy parts, (195.) VI. A similar method may be deduced from the form for the differential of a fraction (23.). X dx xdx' x'""V~ ~^' ' x' "^ x' x'^ ' xdx- __ dx X ' x'* ~ }d^ "~ ~x''* This, as in the former case, makes the integration of one differential depend on that of another ; but it is not so ge- nerally useful a formula. (196.) VII. As the differential coefficient of a power is found by diminishing the exponent by unity, and multiply- ing by the first exponent, so a differential, whose coefficient is a power, is integrated by increasing the exponent by unity, and dividing by the increased exponent. Thus, yAx'"dx = r 4- c, being tlie arbitrary constant. SECT. I. THE INTEGRAL CALCULUS. 177 This rule extends to the integration of all differentials which can be reduced to the form xx'^^dx. Such is Ajr"-^(B + cx'^Y'dvi for since x^-^x = — c?(jr"), n ^ if J?" = Zy '.' x'^-^dv = dz, '.' aoj^'-'Cb + cx'^Y'dx — n ^ ^ - (B + czfdz. n ^ ' Again, let b + cz = «/, *.* Qdz — dy. Hence we find •.•/Aa:«-»(B + QX^'Ydx = ~ • ^— -r + d, D being an arbitrary constant. (197.) VIII. The preceding rule is subject to the ex- dx ce^iion fx^^dx, oxf- — ; the value of this being Vx + c, X C being, as usual, an arbitrary constant (190.). Under this case also come all those differentials which can be reduced dx dx by any transformations to the form — . Such as x-\-a J ''\f—T- = K^ + «) + c. x-\-a "^x-j-a Again, 5^_ ^ 12^__ ^ diSx^ + ^) _ Here it may be remarked in general, that when an in- tegral is a logarithm, the arbitrary constant may always be introduced as a factor of the quantity under the logarithm. For in fF>{a:)dx = l[f{x)] + c, let the constant c = Za, •.* fY\x)dx = lf(x) + /a = Z[a/(^)]. (198.) IX. From the differentials of an arc, considered N 178 THE INTEGRAL CALCULUS. SECT. I. successively as a function of its sine, cosine, tangent, co- tangent, secant, cosecant, versed sine, and coversed sine, we deduce the following results. f = sin.~ij: 4- c, doc — f = cos.~^a7 + c, dx dx dx /—====. = sec.-i^ + c, x^/X"—! dx — f — = cosec.~^a: + c, ^ dx . / — =1 = ver. sin.~i4r + c, dx — / i - = cover. sin.~^a7 -J- c. •^ V'^x--x' ^ (199.) Some of the preceding integrals may be made more general by introducing a constant coefficient, and sup- plying a radius different from unity. The student will easily perceive that these modifications will give results of the following forms : A . ex = — sm.-i — + D, A/B^—c'^a?* c B ' hdx A ex — f- -_ = — COS.~^ 1- D, A/B^-C^.r'' c B Kdx A _ cj: _ kdx A ^ ex •""y-T"; — r~5= — cot.~i h d, ^ B^ + C^^'' BC B SECT. II. THE INTEGRAL CALCULUS. 179 Adx A B.r , / = — sec.-i h D, .r-v/B'^;c2_c2 c c /Adx A ^X — = — cosec."^ (- D. ^^/B«^2_c2 C C In all of which d is the arbitrary constant. SECTION II. Of the integration of differentials, whose coefficients are rational Junctions of the variable. (200.) All rational functions of x^ and all which can be reduced to rational functions, are reducible to one or other of the following forms : U = AXf -f B^^ + CJT'^ • • • . [1], Aof + Bx^ + cr*-" • . • • " "^ a'^'+ b'^^'+ dx^ ' * t^]* All the exponents in these series may be considered as in- tegers; for if any fractional powers were found amongst them, they might be thus reduced to integral powers. Let the common denominator of all the fractional exponents be found, and let it be ^ ; and let «/ = a:? , \' yi = x, and m y^ zizx^'^ making these substitutions for x and its powers, the quantity becomes a rational function of «/, and since dx = qif~^dyi it will continue rational when multiplied by the value of dx. This transformation, however, is not always necessary previously to integrating the formula. (201.) We shall first consider the integration of udx when u has the form [1]. By (193.) and (196.), ^ , Aa7"+^ B^''+* Ca?"+^ N a L 180 THE INTEGRAL CALCULUS. SECT. IT. K being an arbitrary constant. If, however, any of the ex- ponents happen to be — 1, the integral will be of the form Mix (197.). This integration includes all cases of the form [1], and is applicable whatever be the nature of the ex- ponents. They may be positive, negative, integral, or frac- tional, no previous transformation being necessary. (202.) To this class may be referred all differentials, whose coefficients can be reduced to a finite series of the form [1], either by expansion, multiplication, or any other process. If the series [1] were supposed unlimited as to the number of its terms, all differentials, whose coefficients are capable of being developed in a series of powers of the variable, would be included. But, as this would not give the inteo-ral in a finite form, we shall not consider it here. It will become the subject of consideration hereafter. (Sect. VI.). All differentials, whose coefficients have the forms, X A A. AAA. Q where m, m\ m", • • • • are positive integers, and x, x', x", .... functions of the form [1], the exponents a, b, c, - - - - being any numbers w^hatever, may be integrated by the above process. For they may be reduced to the form [1] by development and multiplication. (203.) The integration of differentials, whose coefficients come under the form [2], presents greater difficulties. If any of the exponents be negative, they may be removed by multiplying both terms of the fraction by a power of a: with the same positive exponent, and if any exponent be frac- tional, it may be made to disappear by the transformation explained in (200.). Let the terms of the numerator and denominator be then arranged, so that the exponents shall descend. If the first exponent of the numerator be greater than, or equal to, that SECT. II. THE INTEGRAL CALCULUS. 181 of the denominator, the fraction may, by actual division, be resolved into two parts, one of the form [1], and the other of the form [2], the exponent a being less than a', and the exponents being arranged in descending order. The dif- ferential being thus resolved into two, the first is integrable by the method already explained. The second may be resolved into as many fractions, whose numerators are of the form Ax^dx as there are terms in the numerator, and thus the problem is reduced to the integration of a differen- tial, whose coefficient is of the form AX" AV + B'a-^' + c'^'-"'----' the exponents being integral and positive, and aJ > a. (204.) Such a fraction may always (see note, page 183) be reduced to a series of fractions, each of which must come under some one of the following forms : udx udx (Mx-\-^)dx (Mx-\-j:i)dx Hence the problem will be solved in general when methods of integrating these four forms shall have been explained. (205.) I. To integrate the first form, it is only necessary to observe, that dx — d{x + a) ; and, since m is constant, by (19^) and (197), mdx , . ■^^T^ = ''^'^(*+"^' c being an arbitrary constant. (206.) II. In like manner the second formula is integrated by considering dx = d(x + a) and ^„ = (jr + fl)~". Hende by (196.), Mdx _ M ^(x-^ay ~ "" (/2-l)(a7-ha)«-*' (207.) III. In the third formula the integral may be re- solved into two ; thus, 182 THE INTEGRAL CALCULUS. SECT. II. M^ + N _ Mxdx T K and L being indeterminate quantities, whose values may be determined thus. Let this equation be differentiated and the result cleared of fractions, the factor dx being sup- pressed. Hence we find N = K(x' + a") - 2k(w - 1)^* -I- L(a;* 4- a"). Since these quantities must be equal, independently of x, we have N = (k -f- L)a", 3k + L - 2K^^ = 0. Hence determining k and l, and substituting their values, we find SECT. II. THE INTEGRAL CALCULUS. 183 yidx _ ax (gw-3)N dx 2(71 -IK •n^M^^)""'* By repeating this process with the latter integral, we obtain an expression for it, depending on the integration of And thus the process may be pursued until the exponent of x'^ -{- a* shall be reduced to unity, in which case the in- tegral is reduced to Case III. The preceding principles contain all that is necessary for the integration of differentials, whose coefficients are rational. It will be perceived that their integration, when J^ractio7ial, depends on our power of resolving the denominator into simple or quadratic factors. Note on Art. (204.). (209.) The resolution of a rational fraction of the form u A + Bjr + ca;^* • • • Mx"^^ V a'+b'^^-tc'^^ • • • • m'jt"* into a series of fractions of the forms given in (204.), being necessary for the integration of rational fractional functions, we shall here explain a method of effecting this resolution. First, It is necessary to show that the denominator is always capable of being resolved into real factors of the forms, I. (X + a), II. (x + ay, III. {x' + a% IV. {x^ + ay, , I. If the roots of the equation v = a' + b'x + dx^ ' • • -m'^*" = be all real and unequal, it may be resolved into simple and real factors of the form {x + a). 184 THE INTEGRAL CALCULUS. SECT. II. II. If there be any number n of real and equal roots, there will be a factor of the form {x ■{■ a)". III. If there be a pair of imaginary roots, there will be a factor of the form (2® -{■ pz -\- q), p^ — 4 0, let it be expressed by a^ ; the form becomes a?* -f a^, which is the required form. IV. If there be n pairs of equal imaginary roots, there will be a factor of the form ( it is evident that q is an integral and rational quantity, the highest exponent of x in it being less than the highest ex- ponent in V. Hence let u _ A p V "" x-^a q' A and P being undetermined ; but a being independent of x, and p a rational function of r. Since y = (x -\- a)% •.* u = AQ + v(x 4- a). In this equation let a? = — «, and let the corresponding values of the functions u and q be w and q. Hence u 9. and SECT. II. THE INTEGRAL CALCULUS. 185 U — Ad P = which, since a has been determined, is known by actual division. This method cannot fail, if, as has been supposed, the equation v == admits no other root = — a, for in that case, a: = — a cannot render q = 0, and, therefore, renders A finite and determinate, except when — a happens to be a root of the equation u = 0, in which case a = 0. Since the exponent of the highest power of a; in o, and u is at least one less than in v, it is evident that the exponent of the highest power in P is at least two less than in v. Hence, another factor x -{- a! being assumed, we can find p a' R a "" x+a! s ' provided that x + a' is not one of several equal factors. By proceeding thus, the partial fractions corresponding to all the simple, real, and unequal factors of v may be deter- mined, so that we shall have u _ A a' _i_ ^' "v "■ xTa "^ J+a' +••••+ "^j q! being a rational function of x, in which the highest ex- ponent cannot exceed m ^ n, n being the number of simple, real, and unequal factors, and p' being likewise a rational function of w, in which the highest exponent of x cannot exceed m — n — 1. As all the real and unequal factors of V have been disposed of, o! can only admit factors of the forms II., III., and IV. (211.) We shall therefore now explain a method of find- ing the partial fractions which correspond to real factors of V of the form (x + a)\ Let u^ A ^^ 1 ^^ A„_i P B}' reducing these to the same denominator, we find 186 THE INTEttRAL CALCULUS. SECT. IL P _ i^x + a)" Since p must be an integral function of .r, the numerator of this expression must be divisible by (jt + a)'*, and *.• it becomes = when x = — a. But it is obviously reduced in this case to u — qa. Let u and q be what u and o, be- come when jr = — a ; hence u — Aq = 0, u '.• A = . 9 u Hence the quantity u — qa becomes u a. Now smce this is divisible by jt + a, let the quote be u', so that ' u u — ci — = v'(x + a), ^ "" (x-tay-' By applying"^ a similar process to this fraction, a^ may be determined, and similarly all the other numerators, so that the partial fractions corresponding to the case of equal factors become all known. (212.) Methods nearly the same may be applied to the case where the equation v = has imaginary roots. By the transformation indicated in (209.) III. and IV., the denominator will be divisible by a factor of the form a?* -fa*, or (x' + a^y. If it be divisible by a factor of the first kind, let u Air-f-B p V ~a:2 + a2+"^' •.• u = a(Ax + b) + i*(^* + ^*) Since p must be a rational function of ^, u — a(Ajr + b) must be divisible by x^ + a^, and therefore ought to become = when x = a x/ — 1. When fl >v/ — 1 is substituted for j? in u and q, they must SECT. II. THE INTEGRAL CALCULUS. 187 assume the forms u •}- u' V -- 1, q + qW —1. And there- fore we have w + wV -^-{q + q'V'- l)(Aa ^/ - 1 + b) = 0, ••• u — Bq -{■ Aa^ + V — 1(m' — Aaq — B^f') = 0. And since the real and imaginary parts must severally = 0, u — Bq + Aaq^ = 0, u' — B^' — Aaq — 0, which equations are sufficient to determine a and b. (213.) Finally, we shall examine the case where v has several equal pairs of imaginary roots, and therefore, after transformation, admits a factor of the form (x^ -f a^JK Let U Aa:+B Ay^ + By P p = u -- a[( A.a7 + B ) +(a^^ + By)(a;^ + flg) . . . . (a^_^x + B>^_^(J^^ + «')""'] {x'^ + a'-f Since p is a rational and integral function of x, (a;* + a*)" must divide the numerator, and therefore it becomes = when a? = a a/ — 1. By this substitution, let u become w + -v/ — 1 • w', and Q, q±V — \'^y '.' u + a/— 1 • w' — {q + V- 1 -q^} ' {Aa\/'^nr + B} = 0, which are sufficient to determine a and b as before. Having thus found the values of A and b, upon sub- stituting them in the numerator of p, the term u — ci(aj7+b) becomes divisible by x^ + a^ Let the quotient be u', •.• _ u^—a[AiX-\-Bi-\-{A2X'{-B2)(x^-{-a') • • • • ] The values of Ay, By, may hence be deduced by a process similar to that by which a and b were obtained. 188 THE INTEGRAL CALCULUS. SECT. III. SECTION III. Praxis on the integration of differentials, whose coefficients are rational functions of the variable. A ft y* fi y* (214.) Ex.1. Let w^ = -r--z» '-'fudx = A/-r— i- Let 1 M N /r*— a« x—a x + a* '.' 1 = (m + N)a; + (m — N)a, 1 1 1 ... M + N = 0, M - N = -, -.-M =^, N =- ^, * x^ — a® 2^1^ — a x+ay Ex. 2. Let udx = ^ ^. Since ^^ — 5^7 + 6 (^ - 2)(x - 3), •.• 1 M N + x'^-5x-]-6 x-^ x-S' \' 1 = (m + N)a? — 3m - 2n, •.• M + N = 0, 3m + 2n = - 1, •.. M = — 1, N = 1, * It is to be understood that the arbitrary constant is omitted in the examples. It must of course be supplied in particular cases where it can be determined. SECT. III. THE INTEGRAL CALCULUS. 189 Kdx ,x—S _ Ad^ ,a? Ex. 3. Let W6?<27 = -7^ —r-. Hence -2/.(.T« -07-2). Ex. 4. Let w^ = /*-; — =-. The factors of the denoml- ^ji^ — l nator are x — I and :i^ + Jr + 1. By resolving the frac- tion into two by the method of indeterminate coefficients, we find xdx dx ^ ix — V)dx ^x^-\ "^ ^-^^^ ~ ^-^ x"' + x-\-V .../^ = ^ (. - 1) ^ ^/r-'^'^ x^--\ ^ '' ' ^-^U+D^ + f Let 07 + -L = 2, I, = a\ ',' dx = dz, X — 1 = z (x — l)dx _ zdz ^ ^ dz restoring the values of z and g, we find A^x-'Vjdx , . -- ,2^ + 1 St rfrT = ^^'^' + ^+l - v^3tan.-i— 4-, ^Xdx C 07 — 1 - 2o7 + l 7 ^ X'--\ I v'072-f 07 + 1 ^3 3 190 THE INTEGRAL CALCULUS. SECT. III. Ex. 5. Let udx — -^ ^, zr. The denominator m this case may be resolved into the factors .r -f- 1 and x'^ -|- 1, and thence \'fudx = |Z(a: + 1) ~ ^l{x'' + 1) - 4tan.-^^, y«^^ = / J- — i-tan.-^o?. Ex. 6. Let Mdr = ^'"^^^'iffl'"^ V by (213.), And since — ^xdx 1 ^a:t?.r 1 A^Hl)'^ 2(^2+1)'* -^(^^+1)2 "^ "" ^^' Also, ^(x^-{-\Y 4(.«?2 + l)~ "^ ^'^(a:2+l)2' f~m = tan.-i^. Hence by combining these results, we find xdx Ex. 7. Let wc?^ = 7- . By division, we find _ d^ a dx ud.r = -; b b ai-b^^ 'fudx = -^ - -^Ka + bx). SECT. III. THE INTEGRAL CALCULUS. 191 Ex. 8. Let udx — , — —t-Tv Let z = a -\- bx, {a-f-bxy '.' dz = hdx. Multiplying both numerator and denominator by 5% 1 {z'-a)dz I c 1 a ^ '.'juax - - 2^2 • i^a^bxf x'^dx Ex. 9. Let udx — r-— ; if a + 6.r = x, {^a-\-hxy ' The following integrations may be easily effected : •^ x2 - 6x' xdx a 1 •^l?" = ^x + :r^^^-^' x'dx ( x" 2a"\ 1 2a . Ex, 10. Let udx = a + bx = Xj d^ 1 x3 ^ -3 - 2^x^» a7(?.r ^ / 37 « \ 1 r^dx /2ax Sa'\ 1 1 . 192 THE INTEGRAL CALCULUS. SECT. Ill, dx Ex. 11. Let udx = a -i- bx = X, dx / 1 9b 36V\ 1 3^ j^ -^ x^x^~\ 2ax'''^a'-x'^'a^'^~i^Jx' ~ «* ^^' x Ex. 12. Let udx = x'"{a-\-bxy' a -{- bx — Xy _^_/n 5br b'a^"\ 1 1 X ^ dx / 1 226^^ 106^^ 46V\ 1 46^ x_ ^dx^_( 1_ ^ 55^' 256^jr 106V \ 1 106" , x log. - «6 ^* X dvC Ex. 13. LetwJr = . — -^- — — -, a •\- bx ■\- cx^ = X, 4«c — />- = k, r— - r— J X "-^ X ' ^dx _ ^cx + 6 2c dx ^j^ / 1 tjc lOc^._ ^^ 20^3 cfor -^ ~^-\^''^mx''^^k^x-'^ kH ) ^^''''^^) + A;^x '^ X SECT. in. THE INTEftRAL CALCULUS. 193 When X retains its signification in tliese examples, we have in general ^dx 2 , 2cx-\-b - v(//— 4«c) ^^'9.cx-^b^ ^[b^'-^ac)' The first form is real when 4ac — b^ is positive ; the second is so when 4«c — 6* is negative. Hence there arises I. If 4ac — b^ he positive (4flc — 6* = A*). _ _2_ _^ 2a/cx " Vic ^''^- "VT' 2 2a/cx 2 , -v/A: cosec."^ ;: r = —rr cos.~* VA? ' 2c.r + 6 \^/c Sv/cx 2 . , 26.r+6 = — 7 sm. " -r , Vli ' 2 Vex ' 1 . (2cx+b)Vk 1 Y ^ iN = 7^ ^'"•" 2cx ^7"^ ^"^"A^-7 1 (2c^4-^)* = —-r ver. sni.~^ — ^ . ^x And wheny — vanishes by putting ^ = 0, ^dx 2 ^ xx/k 2 2a-{-bx ^ir^Vic '""^' 2^Wx^Vk '^'- -J^TT 2 . 2-v/ax sec Vk 2a-\-bx — 1 2^/«x 2 . , a:\/A,' cosec. * 77-= — ; sm. Vk ' ^Vk Vk 2vax 2 2a-\-bx — —77 COS.""^ -rr— , Vk 2vayi ' 1 . , {2ax^-bx'-^Vk 1 . ^kx"- - —77 sm.-^ ^ = -— r ver. sm.-^^^ — . Vk 2ax *^k 2ax II, If 4ac - ft* be negative (b'^ — 4ae = ^'). 194' THE INTEGRAL CALCULUS. SECT, Til. ^~ ~ ^/ A:' ^^' 9.CX + ^, + v ^^ ~ x/^' ^^* ^ Vc^ ' and when the integral vanishes by putting x = 0, In both kinds of integrals, ^/k and s/k^ may be taken either positive or negative. Ex. 14. Let udx — a+bx+cx"^ a + bx -\- cx^ = X, ^dx __ dx ^icJjC 1 , b dx ^57*^ a; ^ , /h" a\ „dx Ex. 15. Letwf/x = dx x a 4- 6a; + CA'* = X, 4ac — 6^ = A;, xdx __ 1 b dx x^dx __/ x" a\\ ab dx xHx _/ x^ hx"- ax\\ a^ dx '^lc3~ = \ "c 2c^~cVx^'*"?"'^T3* ' dx Ex. 16. Let udx = x'^'ia-^-bx^cx'-y^ a -{■ bx •\- cx"^ = x, SECT. in. THE INTEGRAL CALCULUS. 19/ dx 1 11 1 x^ h dx b dx b dx b dx dx _ 1 4^ d dx 5c <^^ dlr Ex.17. Let?^f/a7 = x^a ■\-bx -\- cx'^f^ a -{• bx ^ cx"^ =■ x, r — — L- -i- 1 1 1 ^ b dx ^^~8ax*"'"6a2x3^4a3x^"^ga*x"^2«^ ^^' x "9x1'^'^ ^Z^ 1 5b dx ^c dx -^ x'^x^~ axx"^ a'^ xx^ a^ x^' ^_/_ 2_ j36w /w_5c\ _^ mc dx •^^ ~ L "■ Sax^ "*■ Ga'x'- "" \ ^^ "^a V -^ J x* /35^^ 30bc\ dx / 63b^c S3c^\ dx^ \ a^ "" a-' y xx^^y^ l^y x' ' /*i?f._ 1 2^ dx Sc dx ^^5^5 4aa^x* « "^ ir*x^ a"' a:^x' o2 196 THE INTEGRAL CALCULUS. SECT. III. x'"dx Ex.18. LetwJ^ = a + bx* = J dx __ X 3 dx •^ ^~4«x ia'^ "x' xdx _^ x"^ 1 xdx x'dx _ x^ 1 x'^dx ,xHx 1 •^ x« ~ 4ix' x*dx _ X 1 dx x^dx x" 1 xdx x^dx X' 3 x'^dx Ex. 19. IjQi udx = — 5-, fl! + ^>.r* = x, '^l^'"Vl6«^"^16^yx2 "^8^«>^T'' x^/5bx^ ^)x^\ 1 5 ^^^ ^ x^ ~\3S«2"^32^yl^"^32^^ir' x^ a^dx f x^ x^'\\ 1 zrc^ •^ 1?" ~ \i6^ ~ Teziy^ "'" s^-^T"' a^dx/^ _x^ \\_ _3_ ^«^ar SECT. III. THE INTEGRAL CALCULUS. 197 dx Ex. 20. Let udx = af"(a-^bxy* dx _log. X log. x_ 1 ^* __ 1 X '' xyi~'~a 4a^~4a ^^'"""""ia ^^' a?*' a7* = x, f.dx ^\ 1 (^ •^ 07x2 ~4ax a •^ J7X* /•_^_/^_Jl 5&^'^\1 _^b x'^dx ^_/ J 3^>^-\ 1 36 ^ ^A^^^V ^a^2 4aVx Sa""' x' _^^^ / I lbx\l lb dx_ f^ x^x^\ '6ax^ 12«2yx ^a^'Z x ' r——-(- - i-\}.^^^r— ^ x'x:' ~ \ 4a.r* 2aV x a* ^ a7x ' '^^6x*~\ SflJT^ 5fl2^ "*" 4!a' Jx 4!a^^ x ' _dr / _1_ 5b 56^\J_ 5b^ ^^f 198 THE INTEGRAL CALCULUS. SECT. IV. SECTION IV. Of the integration of differ entiaUj of' which the coefficients are irrational, (215.) The integration of differentials, of which the co- efficients are irrational functions of the variable, is, in ge- neral, effected by a transformation, by which the function is rationalised. Such transformations must be suggested by the expertness and address of the analyst rather than by any general rules. Our knowledge in this part of the in- tegral calculus is considerably limited, and there are nu- merous classes of differentials, the integrals of which have never yet been assigned under a finite form. In the pre- sent section we shall attempt to reduce to a few compre- hensive classes the principal irrational differentials which liavc been integrated in finite terms. I. The first class includes the elementary differentials dx dx dx of which the integrals have been assigned in (198.). II. All differentials, whose coefficients are of the form F(ar, .r", x''^ x^^f • • ' ■) the functional sign f denoting a rational function ; but «, b, c, ' ' ' ' being mny fractions. III. All differentials, whose coefficients are of the form f(a', x«, x^ x% ). Where F denotes as before a rational funcCton, and x is a function of x of the form a -f- bx, and the exponents are any fractions. SECT. IV. THE INTEGRAL CALCULTJS. 199 IV. All differentials, whose coefficients come under the preceding form, x denoting a function of x of the form A +82- a' + bV V. All differentials, whose coefficients come under the form F[r, (A 4- b:c + cx'^Y\ VI. Differentials, whose coefficients have the form k being a fraction. These are called binomial differentials, VII. Differentials, whose coefficients are of the form f(^"'", x% x^ x% • • • •) X x"-'. Where x = a + B,r", and «, 6, c, • • • • are any fractions. VIII. Differentials, whose coefficients are of the pre- ceding form, X denoting a function of the form A + B^" Z+bV** IX. Differentials, whose coefficients are of the form a?"* X F[a?», (a + B^" + cx^"Y\ In all these classes the functional sign f denotes a rational function of the quantities within the parenthesis which fol- lows it. We shall now proceed to explain the methods of integration used in these cases successively. (216.) I. The first class needs no further observation, as the form of the integrals are immediately determined by the differential calculus. (See 198.). (217.) II. The differentials of this class are of the form F(.r, x^, x^, x% • • • )dx. They may be rationalised by reducing the fractions a, b, Cy • • • • to a common denominator. Let this be d, and let x^ = z, '.' X = 2^, ••• dx = Bz^-^dz, OQO THE INTEGRAL CALCULUS. SECT. IV. It is evident also, that x% x\ x% • • • • are integral powers of 2. These transformations reduce the differential to the form where d, a!, b', d,^ - - - are integers. This being rational, may be integrated by the rules in Section II. (218.) III. The differentials of this class are of the form f(x, x«, x'', X'', • • • ')da:. This class may be reduced to the preceding, thus, X — A X — A + BX, '.' X = , _ ^^ ~ B Hence the differentia oecomes 1 yx A B 'I ' S^x, . or — F < x, X", x% • • • i ax, which is included in class II. (219.) IV. This class may also be reduced to II. For A + B'^ A — a'x a' -\-^x b — b'x ba'— b'a , dx = 7 j-^dyi. (b — b'x)* By these substitutions, the differential assumes the form f(x, x% X*, . . . .)(/x, which comes under class II. (220.) V. This class of differentials is not rationalised with the same facility as the former. It will be necessary to consider two cases, where c > or < 0. If c := 0, the diflbrential comes under class III. 1". If V > 0, let A + Bu: -f c.r^ = c{x -f- J/)'-, SECT. IV. THE INTEGRAL CALCULUS. 201 A— Cj/« , 2c(A--BV-Hc;y'^) •,• jc = , d.V = ; -; — rr — a?/, - A-I5?/+Cy ••• ^/A + Ba: + cx^ = v/c 2c?/-B ' By which substitutions, the differential becomes rational. 2^ If c < 0, let x\ Ji", be the roots of the equation A + B^ ~ ca?^ = 0. Hence Let A + B^ - CJT^ = — C{X -- X)(X — Ot") ^Q,(x - x)(^'' - ^) = Or - ^')C7, cjry + ^r" , 2(a:'--j;")cy , C7/2+1 (c/ + l)'^ •^' (a:''— »r')c?/ ^ ' c^^ + 1 It is obvious, since c < 0, that the roots x\ x\ are real. Under this class are comprehended differentials of the forms F(a:, v/A + cx'^dx^ y(x^ v/b^ + cx'-^dx. The former is the case where b = 0, and the latter where A = 0. (221.) VI. This class of differentials cannot be always rationalised by any known methods. In some cases, how- ever, this can be effected. It will not render the results less general to consider the exponents m and ?/ integers, and 7J > 0. For if they were fractional, let d be their common denominator. After the transformation, effected by sub- stituting 2;" for jr, the exponents would become integral ; and in like manner, if n were negative, by substituting — for X, the exponent of z under the radical would become > 0. 202 THE INTEGRAL CALCULUS. SECT. IV. If then 711 and n be considered as integers, and « > 0, the formula may be rationalised whenever either — or — + ^ is an integer, whether positive or negative. Since k is a fraction, let it = — , jy and (j being integers. 1^ If — be an integer, let a + bx™ = y\ -.' p x"'-Ulx = :t^— . K^ ) n dy. By these substitutions, the differential becomes ^ . yP + 'i-n ?lJi -^ m . n which is rational, since — is an integer. 7)1 p ^. If f- -^ be an integer, let a + ax" — x\f' \' n q ° ^ a a?" = A + B.r" - ^ (a -I- B^")^ = - P p, 0. ^m-n^K+l (m-tl)A ,. •^ {kn-\-m)B {Jm-\-m)vr in which m — 1, /w — 2, m — 3, • • • • being successively substituted for m, the exponent of x will be continually re- duced. II. inc > 0. , , , .r"'x^' knx ,. . , 7 rx"'-'x^'d.v - Jr 7 /x^'-^x^-UL^y ^ kn\m ^ kn\-mr S01< THE INTEGllAL CALCULUS. SECT. IV. in which k — 1, /t — 2, &c. being successively substituted for k, the exponent of x is continually reduced. III. If 7« < 0. •^ VIA JRA "^ where the negative exponent of 0. By the transformation in (220.), this becomes udx = 2 vc ^ B— 2cy which being integrated, gives fudx = - — -l{^cy - b) = — - I -v/c ^/c ^cy~2 After substituting for y its value, and concinnating, we find fudx = -^ I \ 2c.r > B + 2v^cVa + b^t + c^^ f . a/c ^ 20. Let c < 0. The transformation (220.), gives 9dy fudx = -/ c^fl' •.'fudx = — = cot.""^ ^/c.^^, Vc SECT. V. THE INTEGRAL CALCtJLUS. 209 2 , Va^^-oc \' judx = — =cot.~^ — . v/c \/x-x^ The integrals just found may be presented under dif- ferent forms. The following are among the varieties they may assume : 1°. When c > 0, fudx= ± —^ I < 2c.r + B+2v/c^A + Ba7 + c^ = "7= sin.-^-^^ .... [3], \/c 'v/b-' + 4ac 1 ^^a/Ca/aH-bj^-co?' ~ C0S.~^ : [4], 1 , 2c.r— B _^, = — ^ tan.-' r-—= r . . [5], >V/C 2v/C.v/A + Ba? — CO?* 1 ^S^CVT+B^^^^C^ 1 ^b2+4ac sec." Vc 2a/Cv/ a+b^ — ca;* [^], 1 _^V^M-4ac = — :: COSeC. *— 7; • • • • [81, = — = ver. sm.-^ \ . ^ • • [9]. 2v/C B- + 4AC J The constant should be introduced when these are applied to particular cases. 210 THE INTEGRAL CALCULUS. SECT. V. dx Ex. 5. udx = — . In the preceding example let ^^2-+. ^2 v s i- c = 1, B = 0, and a = ± a-^ •/ fudx = I2(x + -/jr2 + ««) :zz /(^ + Va?~ ± a^) + /2. dx Ex. 6. 2/c/^ = — ^ . In Ex.4, let c = - 1, b = 0, and A = fl^, •.* ^(7a: = sin.—' — , this is one of the elementary integrals in (199.). Ex. 7. udx = -v/a* + x^ -. dx. Let Va^ + a?* =3/ - r, •.* w^a; = ydx — ^J,r, '.' Judx znjydx — ^jf^. Substituting for Jjc its value, and integrating, we find Jydx = i/ + Wiy. '.'fudx — Ix ^/a^ + X- 4 ia^ + l(iH{x 4- -v/a* -fa:''). Ex. 8. udx = . This is one of the elementary integrals (198.), and Judx = cos.~~^r = (p. But it may be also put under the form , ■ dx .^^x^-l '.' by Ex. 4, putting a = - 1, b = 0, and c = 1, \- ± (p^ ^l = l[a; ± ^/x" - 1]. Since X = cos. 9, ^x^ - 1 = ^/ — 1 sin. = 0, the constant = 0. Therefore ± (p ^/ ~.\ ~ /[cos.^ ± ^/ — 1 • sin.9], '•• ^ = cos.

y' «/ 5/1 ^ /I. ^4 /I. C)^i ~ O /I. ^yi-_^» 4.1* 4 'Zx' 2.4 ^ And by the successive substitution of the numbers of the series 2, 4, • • • • we find dx \/\—x^ x\/l-x' X dx _ y/l-X^ 2a/1— JC* dx _ ^n -2^ 4 y'l- ^^ 2^ ^yl-x"^ x^ ./l^^ ~ ~ 5a;^ ""^ 3^^ ~ 3.5 ' a; The following examples are added for the exercise of the student in the integration of irrational functions. 214 THE INTEGRAL CALCULUS. SECT. V. Ex. 11. Let udx = a + bx = X, xdx ... 2 x^dx 2 ■ x^ vx x^dx 2 Ex. 12. udx = 1' x'^{a-\-bxY a -\- bx = X, •^ I a^x^ a^ x^x" XX %zxl^~\ ao^'flVv/x 2«^'^^^/x' X X f,dx_/ 1_ 76 .^_?^^_1 •^ 4 l-^V Sax^'^l2a'-x^'' Ma^x 8a*J\/x iT X* 35/r _Jjc_ 16a*^^v/x* SECT. V. THE INTEGRAL CALCULUS Ex. 13. udx x'^dx [a + baiy a + 6a: = X, dx J 5 ■" '""^ 2 Av /v' ai5 X^ X^ /^= (ix^ ~ 3ax^- 3a^x + i^Ogi^T^' x^ Ex. 14. udx — -jj (a + bx)^ a + bx = X, dx^ _ 2 X* Xdx , . , 2N ^ x^dx /f!^=(x' + 3ax'-a'x + ^»)^^. X^ Ex. 15. udx = x'''dxs/{a + 6a;), flj 4- fea? = X, 2x^x fdx*/x = -g^, 2xVx fxdx^/x = (ix - T^)—;^' 2x /x 216 THE INTEGRAL CALCULUS. SECT. V. fx^dx^^ = (^x3-fax" + 4a*x — |a^)— ^. Ex. 16. ud^ = ^VX«±if) a 4- 6a; = X, Axt^y. ^ , ^dx X •^.Tv^x' ^dx^x ^ */ji h dx ^ x" ""IT'^Y'^ x^k' dx^x^ %^x b\/x b^ dx •^ x^ 2ax'-'^ 4^ax ""Sa-^ ^Vx' -c, ,^ , dx{a^bxY Ex. 17. «d^ = i^ ' a + ^iT = X, 3 y— = (ix+a)Wx+«y— . 1. 3. dxx^ ^ x'-s/x 3b dxx^ •^ "^ 'ax' ^Ww^ X ' ^rxj_/ 1 6_\ ^ 3^ fc* (Irx^ / 1 5 Z>" \ ^ ^ firx^ Ex. 18. wdo? (a-\-boL a + 6a;^ = X, X* 6. -•^v'x' A- SECT. V. THE INTEGRAL CALCULUS. ^17 X In general, dx 1 or/ — ; -7—?:= r sin. '^Ta/ — — j The first expression is real when h is positive ; the second when h is negative. Both a and h cannot be negative at the same time. Hence, we have dx 1 Tr ^ c?.r 1 • _i ^ 1 a — hx^ 11./—- -7— 7r= — 7 sm. ^ x»J — =—71 cos.~^V > 1 a -26^7^ 1 ^v'^ = 77-77 cos.*^ = -77 tanff."^ — : r-T, __1_ , v/(^-6 ^«)_ 1 cot. ~ * -, — = —77 sec, ~ ' V- y/b ' ^x/b s/b • ^ a-bx^' 1 « 1 . 2/^^2 = —7 cosec.-^ v'in-5~77 vers.sm.-^ . All these circular arcs vanish when x = 0. Particular cases are dx dx — tang. -^-77 -r= cot.-* = sec.-J-777 ^ ^{a — x^} X v^(I— J78) = cosec.~* — = |vers. sin.~*2«r'^ 5218 THE INTEGRAL CALCULUS. SECT. V. The integral /* — — — T"t-7>. can only vanish on the sup- position that X = 0, when the upper sign is taken, and in this case dx 1 ^ / b a-i-bx^ Ex. 19. ifdx = dx __ dx xdx >v/x ^ijlrib' xHx^xVx a dx xHx_/x- ^\ xHx_fx^ ^ax\ 3a'^ dx J^~V^-\^'Sb'^r'''^Sb^'^ ,/x* xHx _ / x^ 4tt^ Sa^ N •>^~7^- V5^""156' "^1 5/>V ^'''* Ex. 20. wrf^ = x"*V{(i-\- bx^y a + 6a;® = x, ' x^y.~^ x>^x' dx _ \/x •^jr«>/x aa:' r ^ -. ^^, ^ ^ <^^ SECT. V. THE INTEGRAL CALCULUS. 219 In general, __^r 1_ V(g-f5:t^)-^^a -^ x^{a-{-bx'')^2s/a ^^' ^ {a -\-hx^) + ^ a' y, dx 1 / 6\ the first of which is real when a is positive ; the second when a is negative : a and h cannot both be negative at the same time. J ^ dx ^J[_ ^{a-^bx'')-^a 1 , s/{a^lx'^)'-s/ci where x/a bx^ All these integrals vanish, when x = \^-j-', when 07 = they cannot vanish. Particular cases are dx v/(l+^^)-l ^ ^ , V(l-a:^)-l , 1 - Vg -^') •^^7(1^^)==^''^* — :^ — = ^^^* — ^ ' 29.0 THE INTEGRAL CALCULUS. SECT. VI. = cosec.-' —m — i^ = COS. ^ — = fcos. ' — j— , = sin.- = iver. sin. ^ ^ . SECTION VI. Integration by series. (230.) A series representing the integral of any dif- ferential may always be found by developing the differential coefficient in a series of powers of the variable, and inte- grating each term of the series after multiplying by dx. Thus, if X represent any function of x, and X = ±\X'' -f TiX^' -\- CX"" .... ^ , AX""-^' T&X^-^^ CZt^"+» • •/^'''^ = irTr + iTr+7TT- ■ • • Although such a series is always an analytical representation of the integral, yet it is of no use in obtaining a value, or approximate value of it, except when it converges. If the value assigned to the variable be very small, this will be the case if the exponents continually increase, or if the series ascends. But if the value assigned to the variable be very great, it will only be the case when the series descends, and involves negative powers of the variable. Various analytical contrivances have therefore been used for developing functions in series of these kinds. This method of integration is also useful even where the SECT. VI. THE INTEGRAL CALCULUS. 221 integral can be assigned by other methods under a finite form. If the two integrals thus found be equated, a develop- ment of the finite integral will be obtained, and, in general, this process is attended with much greater facility than the direct development of the integral itself. We shall there- fore give examples of this method of development and in- tegration. PROP. LXXX. (231.) To develop an arc (p in a series of powers of its sine. Let sin.(3 — X. \- d(d = . Let (1 ~ jr^) * be ^ ' ^ v^l-^2 developed by the binomial theorem, .1 . -^ 1 1 1-3 . l-^-5 « . L3.5.7 , (1-.^-) =i+_.,.+ _,.4.._^,a + __,s.... Multiplying by dx, and integrating both sides, 1 ^ 2 3 ' 2.4 5 ' 2.4.6 7 which is the development required. No constant is added, since x = renders ^ = 0. PROP. LXXXI. (232.) To develop an arc in a series of powers of its tangent. Let (p = tan.-i.r, •.• dp = j---^. Developing j^^ by common division, we find THE INTEGRAL CALCULUS. SECT. VI. 1 -. o . « J-p-^^ = 1 - ^~ + ^' - -^-^ — Multiplying by dx and integrating^ we find x x^ x^ x'^ "^ ^ T'""3"^"5 7" No constant is added, since when ^ =:: 0, .r = 0. PROP. LXXXII. (233.) To develop an arc in a series of' pozvers of its cosine or cotangent. dx If C5 = cos."^^:, ••• do = . And if 0' = cot.-^j-, dx ',' dp' r=— - . Hence the developments in these cases X ~j~ X differ only in sign from the two former. But since in these cases

and multiplying each term of the development by u du, and integrating, we find ._du__ - li^u^ h3_2^ 1.3.5 u' •^ v'2iH:^^ ^ ^^ 2.3*2 "*■ 2.4.5* 4 "2.4.6.7 8 ' Since ?4 = .r — 1, this series converges very rapidly. Ex. 3. To integrate — dr bv a series, e being very small. By the binomial theorem sZ-^-eV = 1 - 4.V - l-^e*"^- ~^^ the series to be integrated is therefore SECT. VI. THE INTEGRAL CALCULUS. 225 dx c- 1 „ . 1.1 . . 1.1.3 ^ dJC C , 1 , , 1.1 . . 1.1.3 « « 7 VI -^*^ ^ ^•4' 2.^.6 3 Each term of this development comes under the form which has been integrated in Sect. V. Ex. 9. Substituting therefore for dx x^dx x*dx their values thus determined, we find dx^\-e^x" . 1 ^1 1 . •^ — TfT^"^ sm .-^0?+ -e^ [^^a/1 - x^ --sm -^^ 1.1 _ 1 , 1.3 , q -„ 1.3 . ^ , 1.1.3 ,r,l . 1.5 , 1.3.5 1.3.5 . , , + 2A6^ ^(5"^ + O^^ + 2A6^) ^^ - "' - 2A6^^"-"^^J + • Ex. 4. To integrate the formula dx V(^cx-x^){b-x) Developing, we find (6-.)-* = r*(i-|)-^ _a"^5i i_-^ 1:?^ 1.3.5 x^ X ~ 1 ^"*" '2"b'^2A b^ "^ 2.4.6' 6^ " * " i The question will then be resolved by the integration of a series of differentials of the form x'"dx ^^cx — x^ which is done by the second case of (220.). (237.) The following is a very general method of ap- proximating to the values of integrals by series. THE INTEGRAL CALCULUS. SECT. VI. Let z =fudx + c, c being the arbitrary constant; and let 2' be what z becomes when x becomes ' nd(p _ n cos.nip . d • tan.w^ = — r- , d cosec.w

tan.?i(p cos.nod

n ' "^ sin.^7i(p w (251.) When the arc or angle enters the differential co- efficient, it is generally disengaged from it by integration by parts, either immediately or by the continual reduction of its exponent. The following formula will illustrate this: fK SECT. X. THE INTEGRAL CALCULUS. S41 ^ 1 . 1 . 1.^ fy sm.*<}> cos,(pd(p = -^ sm.;/^— y sln.(7^-2)ir + -yg-sin.(7i-4)a; .... ^. Multiplying by dx, and integrating, we obtain 1 C cos.//.r n cos.(w — 2),r W'W— 1 cos.(w--4)^ i 1.2 w-4 In like manner, if n be even, , . „ _ _ 1 C sin.??.r n sin.(7i-2)a? / sm.^ra.r=: +7r-— r^ — -r- — + 7i.?i~l sin.(w — 4).r 7 1.2 ^T~ r By substituting successively for n the integers 1, 2, 3,"< we find f S42 THE INTEGRAL CALCULUS. SECT. X. J^^m,xda: = — cos..r, jsvn^xda; = i • — %cos.Xj Ex. 10. Let «dc = cos.'*;r • da;. Multiplying the series 71 W'W — 1 2^~^ COS."x = COS.nX + -:rCOS.(n — ^)iV-\ r-^— COS.(7l — 4)07 + by da:, and integrating, the result is _ , „ , sin.nx , n sm.(n—^)x 2"-ycos."xd.=-^p+y . -^Y^ + n-n — l sin.(w — 4)a' Hence, by substituting successively 1, 2, 3, • • • • for w, we find J^cos.xdx = sin. a:, y*cos.*.r • c^ = i- sin.Sa: + ^ar, J'cos.^xda; = -^^ sin.So? -|- 1. sin.a:, y*cos.*j? ' dx = ^ sin Ace + ^ sin.2j + |. • ^. Ex. 11. Let udx — mi.x cos.'^xdx. Since rfcos.^? — — su\,xdx, r J r> « 7 cos.'^+^j; Juax = — / cos.".a? d cos..r = r— . Ex. 12. Let z/dj:- = cos.a: sin.'^o: • dr. This, in like man- ner, gives _ , sin."+^zr Juax = 71 + 1 • Ex. 13. Let. udx = sin.«.r cos.^'xdx. Let cos..r = z. SECT. XL THE INTEGRAL CALCULUS. 243 dz = — sm.^A)^ • dx\ 250 THE INTEGRAL CALCULUS. SECT. Xlll. This can be integrated in a finite form only when 2(n — 1) is a submultiple of unity or of n (221). In other cases the integral may be expressed by a converging series. If 71 = I, '.• 27i — 2 = 1. In this case s =/(l + Ip^xYdx, The origin of this integral is a- = — q— . This curve is the semicubical parabola, and is the evolute of the common parabola. (See Geometry, vol. i. (396.) and note.) To determine the general class of parabolas which are rectifiable in finite terms, let m be an integer, and let 1 1 + 2W2 1+2OT m = -^ ',- n — — ^ — . Hence y — pjs ^"» represents the required class in this case. If 2(7i — 1) be a submul- 2 tiple of Uy let m — — — , m being an integer. Hence 2w J^ ^ , n = -, •.• y = px2»»-i. In general, therefore, the number w is a fraction, whose numerator exceeds its de- nominator by unity. If the denominator exceeds the nu- merator by unity, the integration may be effected by changing a: into y, and vice versa. (269.) If the curve be the common parabola n = 2, •.• ds = (I + 4/7*ar*)Vx. Hence by the formula [2] (224.), we have ^l+4fpV but by Sect. V. Ex. 4 [2], SECT. Xin. THE INTEGRAL CALCULUS. S51 i 1 ^p the origin of the integral being a? = 0. PROP. LXXXVI. (270.) To determine the arc of an hyperbolic curve re- presented by y = px~". The equation being differentiated, gives dy = — npx~"~^ dx, '.' Vdj^ + djr« = (1 + ?iy^-2"-*)^rfjr, •.• ds = ^-^^-^(.r2"+2 + n^p'^ydx, '.' s =zfx-''-\n''p'^ + ^"+2)1: . ^^ This does not come under the criterions of integration established in (221.), and can therefore only be obtained by approximation. PROP. LXXXVII. (271.) To determine the arc of an ellipse. Let the equation a^y^ + h^x^ = a^b^ (see Geometry) be differentiated, du — r~c/a7, ^ a'^y a'y^+b^x^j . -df^dx- = -^^—dxK But 0^*2/2 + 6*a« = a^6«(tt^ — e^x% and a^ = a«3«(a2 — x% where ^ represents the eccentricity. Hence s = / — dx. 252 THE INTEGRAL CALCULUS. SECT. XIII. The series which gives the approximate value of this integral is given in {236.), Ex. 3. If j: = a = 1, and e be supposed very small, the series for the quadrant of the ellipse becomes TT/ 1.1 1.1.1.3^ 1.1.1.3.3.5, \ 2\ 2.2 2.2.4.4 2.2.4.4.6.6 ' ' ' ' / which gives the ratio of the circumference of the ellipse to that of a circle quam proooime. PROP. LXXXVIII. (272.) To determine the area of a parabolic or hyperbolic curve represented qy y = px Multiplying by dx, we find da = px dx, ±71+ 1 j; yx If this integral be assumed between the limits yx and """ n+1 • This integration holds good in every case, except when w = ~ 1, in which it becomes «=p^(^) The integral taken between the limits x and x' being ex- pressed thus, rpti+l _^fn+i shows that the area included between the entire curve and the axis of x can only be finite when n + 1 < 0, •.* 7i < — 1. SECT. XIII. THE INTEGRAL CALCULUS. ^3 Thus the common hyperbola is the limit which divides the class of hyperbolas which intercept with their asymptotes finite areas from those which do not, and no parabola can include with its axis a finite area. PROP. LXXXIX. (273.) To find the area included by two radii vectoresjrom the centre of an equilateral hyperbola. The polar equation of this curve, related to the centre, is r^ cos.^w = fl2 Hence a^du) ,r'dm = by Sect. X. Ex. 2, * ""^ gcos.So; a* '.'J\r'^dw --:: — . Ztan (45° i w), the origin of the integral being a; = 0. If it be taken between the limits w and o)', a^ taiL(45M-a;) PROP. XC. (274.) To determine the surface and volume of a cylinder. A cylinder is produced by the revolution of a rectangle round one of its sides. Hence, in the formula a = 9nfJydZy y is constant, •.* a — 2TfySf y being the radius of the base of the cyUnder, and s its altitude. Hence the surface of a 254 THE INTEGRAL CALCULUS. SECT. XIII. cylinder is found by multiplying its altitude into the circum- ference of its base. For the volume . u — itjy^dxy ••• ?^ = vy'^x. The volume is therefore found by multiplying the altitude by the area of the base. PEOP. xci. (275.) To determine the surface and volume of a right cone, A right cone is a surface produced by the revolution of a rectilinear angle round one of its sides. The vertex of the angle being assumed as origin, and the axis of rotation as axis of .r, the equation of the generatrix is 2/ = px, p being the tangent of the semiangle of the cone. Hence, if a be its surface, da = ^Ttfyds'^ but ds = Vl -\- p' ' d^, •.• a = 2*^/1 ^p^fpxdx - Ti a/1 4- f- • px'^, the origin of the integral being x = 0. Or if s represent the side of the cone, a = Ttys, Since ity is the semicircumference of the base, it appears that the surface of a right cone is equal to a triangle, whose altitude is equal to the side of the cone, and whose base equals the circumference of the base of the cone. If the cone be truncated, the integral must be taken between the limits v^ . r, and therefore its side 2 ^3 . r. The conical surface of this cone is, therefore, 6TTr^, or equal to six times a great circle ; and since its base is STTr^, its whole surface is nine times a great circle. Since the cir- cumscribed cylinder, including its bases, is six times a great circle, the three surfaces of the sphere, cylinder, and cone, are in geometrical progression, and in the ratio 2:3. PROP. XCIII. (277.) Of the volume of a sphere. The formula u = Ttfifdx becomes by substituting for j/^ its value r^ - ^', 258 TFIE INTEGRAL CALCULUS. SECT. XIII. the origin of the integral being jr = ; or u = 7r[r^(x - x') — i(a^3 _ j.f3)j^ u-'ir(x — x%r^ ^^{x- + xxf + x"")], the origin being x = a:'. To determine the volume of a spherical segment, let X = r, *.- u = J7i'(r - a/)[2r* — rx^ - x'^]. To extend the integral to the entire sphere, let ^' = — r, ••• which is the volume of a sphere, whose radius is r *. Let a be the surface of the sphere. By the last pro- position a = 47rr®. Hence u = Jar, which is the volume of a cone whose altitude is r, and whose 'base is a. Hence the volume of a sphere is equal to that of a cone, having its base equal to the surface, and its altitude equal to the radius. The volume of the circumscribed cylinder (274.) is ^irr^ ; since 2r is its altitude and irr'^ its base. Also the volume of the circumscribed cone is Sirr^, since its altitude is 3r, and the radius of its base ^/3.r, Hence it appears that the volumes of the sphere, cylinder, and cone, as well as their surfaces, are in geometrical progression, and in the ratio 2:3. This beautiful property was the discovery of Archimedes, who was so charmed with it, that he is said to have ordered it to be engraved upon his tomb. * This formula evidently contains Euclid, lib. xii. prop. 18. SECT. XIII. THE INTEGRAL CALCULUS. 259 % PROP. XCIV. (278.) To determine the volume of an ellipsoid. Let the equation of the elHpsoid be The equation of a section perpendicular to the axis of z, and at a given distance z from the origin, is The semiaxes of this section are a^/c^- -z* c ? B =i b^/c^" -s'-' c The area of the section is therefore {Geometry, 378.), This being multiphed by dz, and the result integrated, giv«s 2 = being the origin of the integral. If the integral be taken between the limits z and z', u '-= ~(z - ;2')[c- ~ h(^' + 22/ + z'^)]. To determine the volume of a segment cut off by a plane at the distance z\ let ;s = c, *.• " == i^^« - «')[2^'' - '" - <'''']■ To extend the integral to the whole ellipsoid, let z' — — c, '.' u = ^Tfabc, s 2 J;i6(> THE iNTEGKAIi CALCULUS. SECT. XIV. Hence the volume of the elhpOTid is equal to that of a sphere, the cube of whose radius is equal to the product of the semiaxes. If the ellipsoid be generated by revolution round the axis a, b = Cf and the volume is 21 = ^irab^. l£ a = b = c, the formula gives the volume of a sphere, -the same as was before obtained (277.). SECTION XIV. Of the integration of differentials offanctiwis of several independent variables. (279.) The differentials of functions of several variables are of two kinds, 'partial and total (94, 95.). The methods of integration are different for these. We shall first con- sider the integration of partial differentials. i\s a partial differential is found by differentiating the primitive function, considering all the variables but one con- stant, so the integration must proceed upon the same hy- pothesis. To render the investigation more simple, we shall first consider functions of two variables only. The prin- ciples, when established, may be easily generalised. Let u be a function of x and ?/, and let the partial differential taken with respect to x be doc In this, N is a function of x and z/; but as it is derived from the function u by considering y constant, so in the in- tegration, N is to be taken as a function of x only. Let the integral of N£^, under this point of view, be u, *.• SECT. XIV. THE INTEGRAL CALCULUS. ^1 U'= \] -{- C, c being an arbitrary constant. This, however, is only con- stant with respect to the variation of jr, and is therefore to be considered as a function of «/, let it be Y, '.• 21 = \j -\- Y. Hence it appears, that one partial differential is insufficient to determine the primitive function, but will determine that part of it which depends on the variable to which the partial differential is related. In a similar way a partial differential of a superior order taken with respect to the same variable may be integrated by a series of successive integrations. (280.) But when the partial differential of a superior order has been taken with respect to different variables, the process is different. Let M be a partial differential co- efficient of the second order taken successively with respect to 9/ and X. Then if 21 be the primitive function cT^u _ dxd?/ du d^u du dx ~ ^ ' dxdy ~ dy* du = udi/. Integrating this, y alone being considered variable, and the arbitrary constant, which is a function of x, being x', we find u =jMdi/ + x', V^=/M^^/+X'. Since the integral Judj/ is known, let it be u', •.* —-dx — u^dx + x'dx. ax Let this be integrated, x only being considered variable, and we find u =ijli'dx -\-Jk'dx -\- Y, y being the arbitrary constant and a function of j/. 262 THE INTEGKAL CALCULUS. SECT. XIV. If/xj'da; = u smdjk'dx = x, •.• Zi =: U + X + Y, As the value of , , is the same, whatever be the order axd^ in which the differentiations may have been performed, so the integral will be the same in whatever order the in- tegrations may be performed. This is expressed analytically thus : fdxJMy --^fdi/jMdx. Such an integral is therefore usually expressed Jfudxdy. In u similar way, if u be a function of three variables, the integral of the differential d^u j-^—jdxdj/dz ■-- mdxdydz may be obtained ; but in this case there will be three arbi- trary functions, and the integral will assume the form J'udxdydz — u+x+Y + z. And similar observations may be applied to differentials of superior orders. (281.) As a total differential of a function of several variables is the sum of its partial differentials, so the in- tegral of a total differential is the sum of the integrals of the partial diiferentials. If, therefore, the partial differentials be all given, the total differential may be found by the rules which have been established. In order, however, that the integration of a given total differential be possible, it will be necessary to ascertain whether the parts which compose it, involving the differentials of the variables respectively, are the several partial differentials of any one function of the variables; for this may not be the case, and if not, the formula is not the total differential of any function, and therefore cannot be integrated. SECT. XIV. THE INTEGRAL CALCULUS. 263 PROP, XCV. (282.) Given two functions of' two variables, to determine whether they he partial differential coefficients of the same function, and fso, to find the primitive function. Let M and n be the two given functions of the variables X and J/, and let the primitive function sought be u, so that we have du du -T- ~ M, -y- = N. dr dy Each of these being integrated, give u=zfiidy + xJlAJ' Y and X being arbitrary functions ofy and x respectively. If the two differential coefficients m and n be derivable from the same primitive function, it is necessary that these two values of u should be identical independently of the variables. Now since y is independent of x, and x inde- pendent of 3/, it follows that y must be identical with that part of the functionyN^i/, which is independent of .r, and x must be identical with that part of the funciionfudx, which is independent of 3/. These substitutions being made for x and Y, if the two values of u become perfectly identical, the two differential coefficients m and n must be derivable from the same primitive function, and that function is the common value of u thus found, an arbitrary constant being annexed. On this condition, therefore, and not otherwise, the dif- ferential Mdx + tidy is capable of integration ; and if this condition be not satis- fied, the proposed differential is not the exact differential (a phrase implying an integrable differential) of any function. 264 THE INTEGRAL CALCULUS. SECT. XIV. (283.) The process for determining the functions x and y may also be explained thus. Let the first of the equations [1] be differentiated for e/, we find du d\ dr dy~ dy dy' where v =Judx. Hence Hence the complete integral will be or otherwise by the second equation u =Jiidy +ffM - -^jdi where v' —J'-sdy, (284.) The condition of integrability, already determined, may be otherwise expressed. It follows from what has been established, that if the two given partial differential co- efficients be derivable from the same function, the formula __ dw_ dy must be a function of y, and independent oi x. Therefore, if it be differentiated for x, its differential coefficient must = 0, •.• rfN d^\ dx dydx "~ ' d ^^ rfx d^\ dx dx ~^ dydx "~ dy " = M, :•• du du " dx " dy' SECT. XIV. THE INTEGRAL CALCULUS.' ^65 a condition which must be fulfilled, in order that thg formula ■M(Lc -|- Tsdi/ should be a complete differential. On the other hand, it is evident from the differential cal- culus, that if this be the complete differential of a function u, the above criterion must be fulfilled, for by (96.), dru __ (T-u dxdy dydx* du du doc dy dy '" dx ^ dM. _ rfw ' ' dy dx' This is usually called the criterion of integr ability, (285.) The theorem expressed by the formula df2v _ dM_ dxdy ~~ dy ' may be expressed also thus, d.— dx du — 5 • dx = -rr-dx. dy dy By integrating, we find dv ^ ^M , dy ^ dy ' d-fudx ^du ^ dy -^ dy ' which indicates a method of obtaining one partial differential coefficient of a function of two variables from the other, the arbitrary function being understood to be annexed. (286.) The rules for the integration of differentials of several variables may be easily found by generalising those already given. Let m, n, l, be three functions, each being a function of x, y, and z; it is required to determine whe- 266 THE INTEGRAL CALCULUS. S£CT. XIV. |riier they be partial difterential coefficients of tlie same function, and to determine that function; in other words, it is required to assign the conditions of integrability and the integral of du = udx + Nfl?y + i.dz, \' u =f(isidx 4- ^dij -\r i.dz). Since m and n must be the partial differential coefficients of u, considered as a function of d^ and «/, the conditions dM _ dN dj/ ~~ dx^ must be fulfilled. In like mannei conditions Jm dh dz" dx' dN Jl dz ~ df/ ' must be also fulfilled. If the given differential coefficients fulfil these conditions, they must be derivable from the same function of x, y, z; for by the first, m and n are derivable from the same function of ^, 3/ ; by the second, m and l are derivable from the same function of x, z; and by the last, N and L are derivable from the same function of 3/, z. Hence the three have the same integral. It also follows, that mdx + Ndf^, Ndfj/ + Ldz, i.dz + udx^ are respectively exact differentials, and the integral of the proposed differential may be obtained by integrating any one of these, annexing an arbitrary function of the re- maining variable. Thus the sought integral would be ob- tained under the forms w = u 4- Zj u = u' -f X, 71 = u" + Y, SECT. XV. THE INTEGRAL CALCULUS. 267 z, x, and y being arbitrary functions of z, x, and y re-, spectively. The function z may be determined at once by substituting for it that part of the functions u' or u'' which is independent of ^ or j/ ; for since the values of u must be identical independently of the variables, those parts of them which are independent of oc and y must be identical. In a similar manner, the arbitrary functions x and y may be found. These functions may also be determined thus. Let the first equation be differentiated for z. The result is dz •■-=■/'(''• -i)-^'- And, in like manner, The process for integrating differentials of any number of variables will now be evident. The number of equations which give the criterion of integrability is, in general, the number of different combinations of two variables, and is therefore — ^ , n expressing the number of variables. SECTION XV. Praxis on the integration qf differentials of several variables, Ex. 1. Let du =-^-4—^. In this case, 5268 THE INTEGRAL CALCULUS. SECT. XV. y ^ '.*J\iidx = tan.~* — + Y, X f^dy — tan.~^— + x. X Hence y = and x = 0, \' u — tan.~* — . y Ex. % Let du — (3a?^ + %ixy)dx + (a^* + '^y'')dy^ *.• M — 8.r* + %ixy^ N = «^^ + 3^\ '.'fudx = a^ -\- ax^y + y, f^dy = aT> + z/3 + X. Hence y — y^ and x = a?^ ; and by these substitutions, the two integrals become identical. The differential is therefore integrable, and its integral is u = x^ + ax^y 4- y^. Ex. 3. Let du = (2Ay + B;r + D)dy + (2c.r + bj/ + E)dx, ',' M = Sco; 4- Bj/ + E, N = 2a^ + B^ -f D, \'Jkdx = CJ7' 4- Bj^j; + Eo; + Y, ykJz/ = Aj^* + Bj/cT + DJ/ + X. Hence y = Ay^ + By and x = cj;* + Ear, by which al- ternate substitution, the formula becoming identical, proves that the differential is integrable, and that its integral is u = Ay"- -{■ Bxy + cx^ + i>y + ex, Ex. 4. Let y X ' x'^ y"^' Hence _ 1 y \ X SECT. XV. THE INTEGRAL CALCULUS. 269 u. y Hence x = and y = 0, by which the equations becoming identical, the proposed differential is integrable, and its in- tegral is y X X y Ex. 5. Let du — xdy 4- ydx r, "^.5^ y^ 1 1 ^ yx^ xy^ -rfudx = yx -^ —^ H- Y, f^dy = //^ + — + X. yjc Hence x = 0, y = 0, and 1 Ex.6.J.^?^t^^. Hence y^x'--y'- _ 2 _ Six 2 ,,» '.'fudx = 9>l[x 4- ^x"- - y'l + y, f^dy = 2Z[.r -{- v^" --2/'] - % 4- X. Hence x = and y = — 2/j/, •.• u = 2I[x + ^x^ - y^] - 2/y = 21 x-\- Vx'^—y'- y Ex. 7. Let Jd? yrfjr ydy {ydx — xdy) 's/ x"- ^y"- dy Hence K/x'-\-y^( x/^*+y' + J/) M = ; , 870 THE INTECRAL CAT.CUr.lIS. SECT. XVI. _ 1 Vx^+y^+y fi[ence y -- \ly and x — llx, and the other parts of these integrals being identical, the proposed differential is in- tegrable, and its integral is Ex. 8. Let „ = M-^(./.r-,-^^-^)]-|--^-^g-^- y'^ — xu , x'^—ocy , s/x^'^-'if' Hence x = and y — 0, '.* x-\-y u ~ Vx'*-^y- SECTION XVI. The general theory of differential equations and arbitrary C07istants. (287.) Having in the preceding sections explained the methods of obtaining the integrals of differentials of one and of several variables, under all the varieties of form in which SECT. XVI. THE INTEGRAL CALCULUS. 271 they present themselves, we now come to the consideration of the methods of integrating differential equations *. A?, however, this part of the science is of considerable import- ance and difficulty, before we enter upon the details of the methods, we shall offer some general observations on the nature of differential equations, on the connexion of dif- ferential equations of different orders with each other, and with the primitive equation or integral from whence they are derived, and on the constant quantities upon which that connexion depends. (288.) If an equation between two variables x and y be differentiated, a differential equation will be obtained in- volving the quantities .r, «/, and -j— , the last occurring only in the first degree. If this again be differentiated, an equation will be found involving a*, ?/, -t~, and -r^* the last, as before, entering it only in the first degree. In like manner the process may be continued and a series of differential equations obtained, each of which come under the form where a and b are, in general, functions of the variables, and the differential coefficients of orders inferior to the 7ith. (289.) The order of a differential equation is determined by the highest differential coefficient which it contains, as the degree of an algebraic equation is determined by the * The differential equations considered in this section are those between but two variables. Differential equations of several variables will be investigated in a subsequent section. 272 THK INTEGRAL CALCULUS. SECT. XVl, highest power of the unknown quantity. Thus, a dif- ferential equation, which contains no differential coefficient higher than the first, is said to be a differential equation of the first order. If it contain the second differential co- efficient and none higher, it is called a differential equation of the second order, and so on. (290.) Differential equations, like common algebraic equa- tions, are also distinguished by degrees. These are marked by the highest power of the differential coefficient that marks their order, which enters them. Thus, a differential equation which involves no differential coefficient but the first, and that only in the first power, is called a differential equation of the first order and first degree. But if the differential coefficient enter in the second or third power, it is called a differential equation of the^r^^ order and second degree or third degree, and so on. It appears from the process of differentiation, that no dif- ferential equation which is directly obtained from the pri- mitive equation by differentiation alone can be of any degree but the first. Whenever, therefore, we meet a differential equation of a superior degree, it may at once be assumed not to be the immediate differential of any primitive equa- tion. The origin of differential equations of superior degrees we shall find presently. (291.) As an equation and its differential are deduced the one from the other, the same values of the variables which satisfy the former must also satisfy the latter. Hence it follows that other equations may be deduced by their combination. This circumstance indicates the existence of several differential equations of the same order depending upon the same primitive equation. Let v = be the pri- mitive equation between the variables w and «/. By dif- ferentiating this, we obtain v' = 0, v' being a function of .r, t^j and Tj— . In general, v' involves the same constant quan- SECT. XVr. THE INTEGRAL CALCULUS. 27o titles as v, except that constant of v which is independent of the variables x and y^ for this disappears by differentiation. If V should not contain such a constant, its form is not general enough, and it is only a particular case of the in- tegral of v' — 0. We shall, however, consider the integral V = in its most general form, and shall therefore consider v' = as containing all the constants of v = 0, except one. Thus, v' = is the immediate differential equation of the first order derived from v = 0. Now if any one of the constants of v' = be eliminated between the two equations, we shall obtain another differential equation of the first order between x. ?/: and ~-. ^ dx In this equation the constant which disappeared by dif- ferentiation will reappear, and another will disappear by elimination. This latter differential equation will be perfectly distinct from the former, since a constant is involved in it which is excluded from the former, and since it excludes one which is involved in the former. The differential equation ob- tained by elimination may also differ in degree from that ob- tained by differentiation alone. If the constant which is eliminated enter the primitive equation in any dimensions higher than the first, this will necessarily be the case, as will presently appear. Hence the origin of differential equations of superior degrees. A similar elimination taay be practised upon each of the constants common to the two equations v = and v' = 0, and as many different differential equations of the first order may be thus obtained as there are independent constants in the primitive equation. (292.) If the differential coefficient be eliminated by any two of the differential equations of the first order, the result will be the primitive equation in which the two constants, T 274 THE IXTE{JRAL CALCULUS. SECT. XVI, one of which is excluded from each of the differential equa- tions, will appear. Also, if either of the variables be eliminated by any two of these equations, the value of the differential coefficient will be obtained in terras of the other. In this case, also, if the variable eliminated exceed the first degree, the resulting differential equation will be of a superior degree also. (293. ) The several differential equations of the first order, all of which but one have been obtained by elimination, may also be obtained by differentiation alone, by slightly modify- ing the primitive equation. It has been shown that each of the differential equations of the first order excludes a constant of the primitive equation. In order to obtain the differen- tial equation immediately by differentiation, let the primitive equation be supposed to be solved for the constant as if it were an unknown quantity, so that if a be the constant, the equation will assume the form F(a7;z/) — a = 0. Under this form, the equation being differentiated, a will disappear, and an equation between the variables, the first differential coefficient, and all the other constants of the primitive equa- tion will be found. The equation thus obtained must be necessarily identical with, or reducible to, that obtained by elimination, since they involve the same variables and constants. Irt the same way all the differential equations of the first order which were before found by elimination, or their equivalents, may be immediately obtained by differentiation alone. If the constant which is thus made to disappear by dif- ferentiation rise to the second or an higher degree in the primitive equation, then when the equation is solved it will have more values than one, and radicals will appear in the primitive equation which did not enter it before. These radicals will, therefore, also appear in the differential equa- tion obtained from it, and therefore the differential coefficient SECT. XVI. THE INTEGRAL CALCULUS. S75 which must occur in the simple dimension will have several values. Now as this equation must be equivalent to that obtained by elimination, which does not include the above- mentioned radicals, it follows that they must be produced by solving it for the first differential coefficient, so as to reduce it to the same form as that obtained from mere dif- ferentiation. Hence it follows, that in this case the dif- ferential equation obtained by elimination must rise to the same degree as that of the constant in the primitive equation by whose elimination it was produced. (294.) By differentiating the first differential equation v' = 0, the second differential equation v" = may be found. This will be the immediate differential equation of the second order of the proposed equation, but it will not be the only one. By what has been already observed of the first differential equation, it follows that the second dif- ferential equation v" = contains all the constants of the first v' = 0, except one, and therefore all the constants of the primitive equation v = 0, except two. The two which disappear by differentiation alone are those which are in- dependent of the variables in the two equations v = and V = 0. A differential equation may, however, be obtained independent of any two constants a and b of the primitive equation, and may be obtained from two, and only two, of the differential equations of the first order. 1°. By differentiating the equation of the first order which excludes the constant a, and by it and its differential eli- minating B, a differential equation, independent of a and b, will result; or the same may be obtained by solving the equation for the constant b, and then differentiating it (293.). 2°. By differentiating the equation of the first order which is independent of the constant b, and then eliminating a, or, as before, first solving the equation for a, and then dif- ferentiating the result, the same equation as before will be T J^ 276 THE INTEGRAL CALCULUS. SECT. XVI, obtained. Thus, this equation may be considered as a common differential of the two Equations of the first order, the one independent of a, and the other of B. (295.) In this way any two constants of v = may be eUminated ; and, therefore, there are as many different equations of the second order derived from the same pri- mitive equation as there are different combinations of two constants in the original equation v = 0. If w be the number of constants, therefore, \ „ will be the number of different differential equations of the second order, each of which may be considered as a common differential of the two equations of the first order, which are independent severally of the two constants which are excluded from it. It is evident that these equations are all perfectly distinct, since they differ in their constants. T VI 1 1 n.n- 1.71-2 -.„ . , In like manner there may be ^ dirrerential equations of the third order derived from the primitive equation v = 0, each of them excluding three constants of the primitive equation. Each of these may indifferently be derived from three of the differential equations of the second order, scil. those three which exclude severally the three pairs of constants which may be combined from the three constants excluded from the differential equation of the third order. These several differential equations of the third order may be obtained, either by obtaining one by differentiation, and the others by eliminating successively the constants between that and the equation of the second order ; or they may be obtained without elimination by solving the dif- ferential equations of the second order for the constants successively, and then differentiating. (296.) By continuing this reasoning, it follows, SKGT. XVI. THE INTEGRAL CALCULUS. ^ 277 1°. That in a differential equation of the mih order there are a number of constants equal to n —' m, n being, as before, the number in the original equation. 2°. That a differential equation of the with order may always be obtained either by elimination united with dif- ferentiation, by which any combination of m constants shall be excluded, or by successively solving the equations for the constants and differentiating. 3^. That therefore there will be n.n—l.n — ^ n—(m—l) dijBPerential equations of the mth order derived from the same primitive equation, perfectly distinct from one another, since they differ in their constants. 4P. That each of these differential equations may be derived indifferently by differentiating m of the differential equations of the (m — l)th order, scil. those which exclude the m combinations of (m — 1) of the constants excluded from the differential equation of the w/th order. 5°. That the differential equations of any order obtained by differentiation alone are always of the Jirst degree with respect to the differential coefficient which marks their order, while tliose which are obtained by elimination are of the same degree as the constant by whose elimination they were obtained. The two equations will become identical by solving the latter for the differential coefficient. 6^. That if by two different differential equations of the wzth order the mih differential coefficient be eliminated, a dif- ferential equation of the (?w — 1 )th order will be obtained, including one constant more than either of those from which it was deduced, and therefore only excluding {m ^ 1) con- stants of the primitive equation, and this equation must therefore be identical with that differential equation deduced by differentiation and elimination, which includes the same 278 THE INTEGRAL CALCULUS. SECT. XVI. constants. It is evident that this elimination may be con- tinued upwards until we arrive at the primitive equation. 7*^. A differential equation of the nth order will include no constant, since, in that case, the number of constants eliminated is 7i. There will also be but one differential equation of this order, since, in this case, the formula ex- pressing the number of differential equations becomes n.n—l.n—^, ?z— (?i — 1) (297.) The conclusions at which we have just arrived resulted from the consideration of the process by which the several orders of differential equations are derived from a primitive equation between two variables. Let us now con- sider what these results suggest in returning upon our steps and ascending through the differential equations of the several orders to their original or primitive equation. (298.) As by differentiating an equation, a constant dis- appears, so it should reappear upon integrating ; and as only one constant can be removed by one differentiation, so one only should be introduced by one integration. The value of the constant introduced in any integration cannot be determined by the differential equation alone, since a dif- ferential equation is the same, whatever be the value of the constant which has been eliminated. Hence, as far as the differential equation is concerned, this constant is arbitrary, and any value whatever may be ascribed to it. In ascend- ing, therefore, from a differential equation of the first order to its primitive or integral, one arbitrary constant, and but one, ought to be introduced, otherwise, the integral which will be obtained will not have all the generality which it ought to have. (299.) If two different differential equations of the first order derived from the same primitive equation be given, the integration may be effected by eliminating the first dif- SECT. XVI. THE INTEGRAL CALCULUS. 279 ferential coefficient between them ; the resulting equation between the two variables, and independent of differentials, will be the sought integral. (300.) As a differential equation of the second order is immediately obtained from one of the first order, and is related to it in the same way as that of the first order is related to the primitive equation, it follows from what has been said, that the first integral of a differential equation of the second order is a differential equation of the first order, and that one, and only one arbitrary constant must be introduced in the integral. The primitive absolute equation, or final integral, is to be obtained by the in- tegration of the differential equation of the first order thus obtained, in which integration a second arbitrary constant must appear. There is, however, another method of ascend- ing to the final integral. Since each differential equation of the second order may be indifferently derived from two of the first order, it follows that a differential equation of this kind has two first integrals. If both of these can be obtained, each including an arbitrary constant, the primitive absolute equation, or final integral, may be obtained by eliminating the first dif- ferential coefficient between them. (301.) This principle of differential equations of the se- cond order admitting two integrals, also furnishes a method of integrating differential equations of the first order. If an equation of the first order be differentiated, and thence one of the second order obtained, this admitting of another in- tegral different from that from which it was derived by dif- ferentiation, this other integral may be found by integrating and introducing an arbitrary constant. Thus two differen- tial equations of the first order will be obtained involving one arbitrary constant ; by these the differential coefficient being eliminated, the final integral, including an arbitrary 280 THE INTEGEAL CALCULUS. SECT. XVI, constant, will be found. This is frequently the easiest method of integrating an equation of the first order and any degree superior to the first. (302.) In like manner the first integral of a differential equation of the third order is a differential equation of the second order, including one arbitrary constant, and each differential equation of the third order has three different integrals of the second order. And, in general, the first integral of a differential equation of the mth order is a dif- ferential equation of the (m — l)th order; and each dif- ferential equation of the mth. order admits m different first integrals, which are all differential equations of the (m — l)th order, and include w different arbitrary constants. If these m first integrals be obtained, the final integral may be found by mere elimination without further integration. For the m differential equations of the (m — l)th order include in general (w — 1) differential coefficients, scil. all the differential coefficients from the first to the (m — l)th order inclusive. These {m — 1) quantities may be eli- minated by the m equations, and the result will be an equation independent of differentials including m arbitrary constants. This is the final integral in its most general state. (30,'^.) The integration of a differential equation of the first order may be effected by deducing from it by suc- cessive differentiation a differential equation of the mth. order. If a first integral of this can be obtained different from the differential equation of the (m — l)th order from which it was derived, and including an arbitrary constant, the final integral can thence be obtained by elimination alone ; for there are the differential equations from the first to the (m — l)th order inclusive obtained by differentiation, and also another of the {m -- l)th order obtained by in- tegration, making in all 7n equations to eliminate (m — 1) SECT. XVI. THE INTEGKAL CALCULUS. 281 differential coefficients. The result being an equation free from differentials, and including one arbitrary constant, will be the integral of the proposed equation. (304.) In the preceding observations we have assumed two propositions, P. That the final integral or primitive absolute equation, of a differential equation of the mth order, should include m arbitrary constants, in order to have all the generality which is due to it; and 2^ That a differential equation of the mth order admits of m different first integrals. Although these propositions seem sufficiently evident by retracing the process of differentiation, yet, as it is desirable to give to the theory established in the present section all the perfection and rigour possible, we shall subjoin direct demonstrations of these two principles. PROP. xcvi. (305.) Every differential equation between two variables has an integral^ and the integral of a differential equation of the mth order must, if in its most general state, include m arbitrary constants, and no more. The differential equation of the mih order determines the m\h differential coefficient a,„ as a function of the variables and the differential coefficients Ai) A2, • • • • A„j_2, of the inferior orders. By successive differentiation the differential equations of the superior orders may be found, and these will therefore be also determined as functions of the variables and of the differential coefficients of orders inferior to the 7»th ; for the differential coefficients of the intermediate orders may be successively eliminated. 282 THE INTEGRAL CALCULUS. SECT. XVI. By Taylor's series we have Let b be any value of x which does not render any of the differential coefficients derivable from the primitive equation infinite {55), and let A — a: — b. Let b be supposed to be substituted for .v in the func- tions ^> Aj, A25 • • • • so that they will become constant quantities, CLq, tti, flaj • • • • , and let the value of y corresponding to j: be 2/. ' Thus the series becomes x-b (x-bY (a:-by y = a^ + a^.-^-ffl^. ^^ +«3- ^^3 + the coefficients of which are all constant. The coefficients of this series from the {m + l)th term forward are given functions of the coefficients of the first m terms, since they are what -^»?5 -A-OT+ij A„,.)-2j • • • • become when x rsb\ but these are determinate functions of y, Aj, A2, A3, • • • • and, therefore, ^»j? ^»j+l> ^»i+2J ^»»f3J ■ • • • are determinate functions of Aq, Qxi fl^2» Ojzi ' ' ' ' The series expressing the value of ^ is therefore the in- tegral of the proposed equation, and contains m arbitrary constants, scil. ^05 ^15 ^2J • • • ^ni— 15 and no more. This series is the development of the value of y in the final integral of the proposed equation, and may therefore represent that integral. SECT. XVI. THE INTEGRAL CALCULUS. 28B PROP. XCVII. (306.) Every differential equation of the mth order has m different first integrals, which are differential equations of the (m — A)th order. The final integral gives y as an implicit function of x. Let it be expressed as an explicit function of oc, so that y = f(x). By Taylor's series, h 7i^ h^ F(^ + A) = y + Ai y + A2 J-^ + A3j-^ If A be supposed to become — — Xj \' x -{■ h =^ 0, and therefore y[x + h) becomes what the value of y is when :c = 0. Let this value be z/% •.* f = y - A,-^- + A,£l - A^j|g + [1]. In the same manner, by successively considering Ai, Ag, A3, • • • • functions of ^, we obtain X x~ Ai = Ai — Aa-J- -h ^3=-^ — [2], .r JT A3 ~ A-i — Agy -f A4j-g — [3], X Tc" Al = A3 — A^Y + A5 J^ - [4]. If a diiferential equation of the first order be given, it will determine the first and all the succeeding differential coefficients as functions of the variables. In this case the equation [1] will represent the primitive equation involving but one arbitrary constant y^. If a differential equation of the second order be given, it will determine the second differential coefficient and all the 284 THE INTEGRAL CALCULUS. SECT. XVII. coefficients of superior orders as functions of the variables and the first differential coefficient. In this case [1] and [2] represent two integrals of this equation, each including an arbitrary constant if and a" ; all the other terms being functions of the variables and the first differential co- efficient. In like manner, if a differential equation of the third order be given, the three equations [1], [2], [3], represent its first integrals, each involving an arbitrary constant y^^ a", Aa, and being differential equations of the second order, and so on. SECTION XVII. Of the integration of differential equations of the first order and first degree^ in which the variables are separable. (307) As the rules for differentiating functions of two variables equally apply to equations of two variables, so also the rules for integrating differentials of two variables apply to the integration of equations of two variables ; and as there are many differentials of two variables which are not exact differentials, so also there are many differential equations which are not the immediate differentials of any primitive equation, and which are not therefore immediately in- tegrable. When a differential equation has been reduced to the form udx -f ^dy = 0, its immediate integrability may be ascertained by the cri- terion (284.), and its integral found by the rules already established for differentials of two variables. SECT. XVIT. THE INTEGRAL CALCULtTS. 285 (308.) But although an equation may not come under the criterion of integrabihty o^ Junctions of two variables, we are not therefore to conclude that it is not iiitegrable. We may, indeed, pronounce it at once not to be the immediate differential of any equation between the variables, because, if it were, it must come under the criterion. But it may be one of those differential equations which are not obtained by mere differentiation, but by eliminating some constant between the primitive equation and its immediate differential ; or it may have happened, that some function of the variables having been a factor of the immediate differential, it was ex- punged after differentiation. Thus, for example, if the differential equation of a given equation between x and y were {y" + x^)F{xy)dy + (^ + s')A^y)dj: = 0; we should immediately expunge the common factor j/^-f- .^2 ^ and although the above equation would come under the criterion of integrability, yet, after division by (y^ + ^-), it might no longer come under it. Thus, though the cri- terion apphes to differentials, yet it does not to differential equations ; at least, it does not apply as a criterion, properly so called. Because, although every equation which comes under the criterion can be immediately integrated, yet we cannot infer the converse, as has been shown. (309.) Various analytical contrivances have been there- fore invented for rendering integrable differential equations which do not fulfil the criterion of integrability. One of the most simple, when it can be effected, is the separation of the variables, or the reduction of tlie equation to the form xdr -f \dy — 0. In which state it is immediately integrable by the rules for integrating differentials of a single variable, the integral being 2S6 THE INTEGRAL CALCULUS. SECT. XVII. fxdx -VfYdy = *. (310.) In differential equations of this kind, the variables are said to be separated^ and therefore all equations in which such a separation can be effected may be considered as integrable by the above method. The most remarkable classes of equations, in which this can be effected, are the following : 1^. All differential equations coming under the form xc?2/ + Ydx = 0. 2^. All differential equations of the form XYc(y -H x.'Y'dx = 0. 8^ All homogeneous equations; that is, all algebraic equations in which the sum of the dimensions of r and 7/ in every term is the same, and which, therefore, come under the form f(^"^, ^'"-'3/, a7'»-y . . ")di/ + F'(.r"S a7'»-^3/, a;'"-y • "•)dx = 0. 4P. Linear equations; that is, equations which involve 2/ and di^ only in the simple dimension, and which, therefore, come under the form dj/ -1- (x7/ + .x.')da: = 0. ♦ 5^. The equation of Riccati (an Italian mathematician), dy + (aj/* + Bx"')dx = 0, in which, in certain cases, the variables may be separated. There are other equations in which the variables may be * This would be, according to the usual custom, expressed J%.dx -\-fs.dy = c, c being an arbitrary constant. This, how- ever, I conceive superfluous, if not positively wrong, since the introduction of the constant is a part of the operation indicated by the signy, I have, therefore, generally neglected the con- stant, except where the integration has been actually effected j then it is proper and necessary to introduce it, because the sym- bol which implies its introduction has disappeared. SECT. XVII. THE INTEGRAL CALCULUS. 287 separated, but these will sufficiently illustrate the principle. It is evident that all equations which, by any transformation, may be reduced to any of the preceding forms, may be in- tegrated in the same manner. (311.) 1". The equation Jidy -f- Ydx = being divided by xy, is reduced to du dx -^ 4- — r= 0, Y X the integral of which is immediately obtained, , dy djr (312.) 2^. The equation y^Ydy + y^Y^dx — being divided by xy', becomes Y X' —rdy H dx = 0, y' "^ X which is immediately integrable, (313.) 3^ Each term of an homogeneous equation being of the form ky^ ai"''~'^ ^ the constant sum of the exponents being m ; if every term of the equation be divided by x"^, the form of each term will become a( — ) . If ' - =^, the \ ^ / X ' equation will assume the form 's{z)dy -}- Y\z)dx = 0. But since y — xz, *.• dy = xdz + zdx. Which being substituted for j/, gives x^{z)dz + \y\z) + zv{z)\dx — 0, viz) , dx f'(z)+2F(z) ^ X ' 288 THE INTEGRAL CALCULUS. SECT. XVII. dz + — ~ 0, F(2) X t(z) which is of the form zdz -f- xdx = 0, in which the variables are separated. Equations are frequently rendered homogeneous by sub- stituting for ^ and z/, a:' -f a and j/' + b, and disposing of the arbitrary quantities a and b, so as to take out the terms which destroy the homogeneity, changing dx and dj/ into dx' and dj/'. The analyst, however, must be determined in the choice of a fit transformation by the nature of the equation in each particular case. (314.) 4". In the linear equation di/ + (xy + X V^ = 0. Let x";2 = z/, '.' dj/ — x"dz -\-zd}i'\ by which substitutions the proposed equation becomes x"dz -h zd-a!' + xx"zdx -f- x'dx — 0, in which x'"' is an arbitrary function of x. Let x" be such as to fullil the condition zdx" + x'dx = 0, •.* dz + xzdjc = 0, •.* — - = ~ xdr, ••• z -- e'~-'''^'^\ Hence we find ^x" _ _ ^/xrf^x'c?^', (315.) ifm = 0, •.• X" :::= - fe-^^'^X^dx, -,- 7/ = - e-^^'^fe^^'^'x!dx. 5". In the equation of Riccati, dy + (a/ + ■BC(r)dx = ; it becomes dj: + f =0, A2/2 + B in which the variables are sepaiated. SECT. XVir. THE INTEGRAL CALCULUS. S89 But if m be not = 0, let ^ a:* "*" A^ ' xdz — 2;2£7ir rf^ By which substitutions the given equation becomes x^dz + Az^dx + B^+^^-c = 0. If in this w = — 2, it is homogeneous ; and if m = — 4, the variables may be immediately separated by dividing the whole equation by x^kz"' + b). If, however, m be not = — 2, nor = — 4, a further transformation must be effected. Let 1 and let t m4-4 , — B w + 3 w+3' ) A_ ^ ~ W2 + 3* We find by these substitutions, that the equation becomes dt + (a'^2^b'?^")J?/ = 0; this being similar to the first equation, can be integrated when w = — 2, or n = — 4. If n be not = — 2, nor = — 4, by continually repeating the same transformation, the equation may successively be reduced to a series of equations of the same form as the given one, and in which the exponent of the variable becomes successively equal to m+4 71 + 4 p4-4 77Z + 4 Sm^S 5m + 12 __ ImA-l^ ^^' ~m+3' "" 2m + 5' "" 3w + 7' 4m +9 The equation can only be integrated by the methods above or ^0 THE INTEGRAL CALCULUS. SECT. XVIII. given, when some one of these is either = 0, = — 2, or = — 4, that is, when m is a number coming under the formula — 4w n being any positive integer, or = 0. 1 If the transformation y = — , a:'^+* = z had been made in the given equation, the same process would show that the integration could be effected when m = -^ =-• ^^^ criterion of the integrability of the given equation by this method is then m = ^ n bemg a positive integer = 0. SECTION XVIII. On the multipliers which render differential equations integrable, (316.) In order that a differential equation of any order should be immediately integrable, it is necessary that it should be of the first degree with respect to the differential coefficient, which marks its order (290.). Otherwise, it has been the result, not of differentiation, but of elimination. But if it be of the first degree, it may always be considered as proceeding from the immediate differentiation of the dif- ferential equation of the next degree inferior to it, solved for the constant which has been eliminated. (317.) Let a differential equation of the with order be then supposed to be reduced to the form 2+"-» PI. U' + ^' = . . . . [3], SECT. XVIII. THE INTEGRAL CALCULUS. 291 where u is in general a function of the variables and the differential coefficients of the inferior orders. Let the differential equation, from which this is conceived to have been derived, be u' = « [2], a being supposed to be the constant which has disappeared by differentiation, and u' being a function of the variables and the differential coefficients of orders inferior to the mth. This being differentiated, gives an equation of the form dry dcd"' u and u^ being likewise functions of the variables and the differential coefficients of orders inferior to the mth. Since this equation must be equivalent to the first, we have u' u = — •.• vu = u\ u Hence the latter equation becomes "•^ + "" = "' which is an immediate differential, and therefore integrable. But this is the given equation [1] multipUed by the func- tion u. (318.) This multiplier is not the only one which will render the equation integrable. Let the equation [2] be multiplied by any function of a. This function being con- stant, the equation [3] becomes or F{a)u-~ -T F{a)uu = ; but by [2] tl^is becomes u 2 292 THE INTEGRAL CALCULUS. SECT. XVIII, which is the exact differential of the equation u'r(u') = aF(u'). But it is the equation [1] multiphed by f(v')u. Since the function f(u') is arbitrary, there are an infinite variety of multipliers which will render the proposed equation in- tegrable, scil. all those of which, u being one factor, the other is any function of u'. (319.) Since by (306.) a differential equation of the mth order has m different first integrals, we may obtain a class of multipliers from each of them, which will render integrable the proposed equation of the mth order. (320.) Having explained the general principle, we shall now apply it to differential equations of the first order and first degree. Let udo! -f N% = be the proposed equation, of which the primitive or integral is u' = a. By comparing this with the general formula M already established (31 T.)? we find u = — . The equation is rendered integrable by multiplying it by wf(u'). First, suppose f'(u) = 1, the equation becomes liudcc + u^di/ — 0. Subjecting this to the criterion of integrability (284.), we find dfuu) _ d(Nu) dy ~ dr * /dM. d^\ du du \dy dxj dx dy' Since m and n are supposed to be given functions of .r and y, this equation, when integrated and solved for w, would determine its value. It being, however, an equation of partial differentials, its solution can very seldom be effected ; and even when it can, it presents generally greater dif- ficulties than the proposed equation. SECT. Xvm. THE INTEGEAL CALCULUS. 293 (321.) Although we cannot therefore in general determine a factor which will render an equation integrable ; yet there are some properties of these factors which merit attention. 1°. If the integral of the differential umcIj: + u^dy were known, the factor u could be found ; for the above formula is identical with du . du ^ therefore we should be able to deduce the value of u by comparing them. 2^. If the factor u were known, an infinite number of other factors which would render the equation integrable could be found, as has been already shown. 3°. The factor u may, in some cases, be a function of one of the variables only. It may be easily discovered whe- ther this be the case, and if it be found so, the factor u may be determined. Let u be supposed to be a function of the variable x. If so, -^ = 0, -.' dm d^ J _ du d\j dx y dx^ ^ da __ dx/du d^\ u ~ ii\di/ dx)' If the second member of this equation be independent oiy, then ?^ is a function of x alone, and not otherwise. Since m and N are given functions, this can always be determined, if it be so, the value of m is determined by the equation '{ ^'^=^K^''^'^""£^)"-^''^'^' (322.) Homogeneous functions have a remarkable pro- perty, which enables us to assign the factor which renders an homogeneous equation integrable. To explain this pro- ^94 THE INTEGEAL CALCULUS. SECT. XVllI. perty, let u represent an homogeneous function of j: and j/. In u, let X be changed into x(\ + Ti)^ and y into yQ. + h), and let the function become w', so that w = Y(xy), u' = F(jr + hx^y-}- hy). Since w is an homogeneous function, u' = (1 -f /i)"*?^, ??i being the number of dimensions of the variables in each term of u, let these two values of u' be developed, the one in powers of ^^ and hy by (96.), the other in powers of h by the binomial theorem. Hence du ^ du , d^u AV d^u ¥xy d^u AV v! = u(l +mh+ -^j- h^ + 1X3""" ^' '^' Hence, by equating the corresponding coefficients, we find du du m.m — 1 d^u x"^ d^u xy d^u y^ 1.2 '""^S^ 'Ul'^'dbcdy T"*"^ L2 It is evident that this property belongs to homogeneous functions of any number of variables. (323.) Let the equation to be integrated be mdx + i^dy — 0, where m and n represent homogeneous functions of the variables. Let u be the sought factor and also an homo- geneous function, and let it be supposed that wudx + N2^^ = is an exact differential. Hence J(mw) d{^u) dy ~' dx ' Let the dimensions of m be jp, and those of u^ 71, '.' diuu) d(Mu) SECT. XIX. THE INTEGRAL CALCULUS. d(Mu) (Z(nm) ••■ ^p + ")^" = -IT'' + -ir^' *.• (p + n)MU = -^ MW, This equation is fulfilled by the conditions p = - (71 + 1), 1 U = ; — -. Hence the equation udx + N J^ = MX + N^ is integrable. SECTION XIX. Praxis on the integration of differential eqiiaiionsof'tlie first order and first degree. I. Differential equations which a/nswer the criterion qf'in- tegrahUity, Ex. 1. (^axy - 'if)dx + {ax" - ^xy'')dy = 0, M = 9 -f- -!— ( = c. I x^y—x — a S Ex. 3. dy + (j/2 __ a'^x ^)(lx = 0. In this case the ex- ponent — o- comes under the character ^^ ^ , since they agree when n =z 1 (315.). Let X = t-'\ y =■ — Sa^z-^, '.' dz + (22 _ 9aH'')dt = 0, —6ax% c y(l + Sax^) + Sa^x~^ -> ^y(l-Sax^)-{-Sa^x ^ ^ Equations rendered Integrahle hy a multiplier. Ex. 1. (1 + «v/l + x')dx -\- 9.hyVl + ^^ = 0. This equation will be found not to come within the criterion (284.), since du dn __ Myx dy "dx ~ -v/1+^2* But since n = 2%v/l + ^% 1 / dm ^N \ _ jr N \ % "" "dx/ ~~ ~ 1+^2' 4/ which being a function of J7 alone (321.), Case S^, the equa- tion will become integrable if multiplied by a function of ^r. To determine this function, let it be u. By (321 .), Case S% 1 >/lH-a7« SECT. XX. THE INTEGBAL CALCULUS. 301 Multiplying by this, we find ( — + a\dx + 9.bydy — 0, which is integrable. Ex. % xHy + (4jc2j/ — (1 _ ^2) '^)dx = 0. In this case dm d^ __ <2 dj/ dx ~ ^ \ c dm dN -^ _ 1 N I dy "^ dx S~ ^' dy This being a function of x alone, a factor may be deter- mined, which will render the equation integrable. By (321.), Case 3°, this factor is dl Ir Hence the equation being multiplied by x^ gives x'^dy + l^x'^ij — x{\ - ^")'~^]dr = 0, which comes within the criterion, since — 5— = 4^7^ — 4y^)p'^ — 4ixyp — a;^ = 0. The condition u = — y changes F(xyu) into which is therefore a singular solution of the differential equation. From the preceding observations, we may therefore infer generally, that if the general solution, cleared of radicals, F(xyc) = 0, of any proposed differential equation, be differentiated, considering the arbitrary constant c alone variable, and that the partial differential cdc = THE INTEGRAL CALCULUS. SECT. XX. be thus determined ; the values of c, which satisfy this con- dition, being substituted for c in the general solution f(j7j/c) = 0, will give equations, amongst which all singular solutions will be found. (330.) It does not, however, necessarily follow that all the equations resulting from such substitutions are singular solutions. Such equations may be cases of the general solution in which the arbitrary constant receives a particular value, in which case they are particular integrals, and not singular solutions. The partial differential coefficient c is in general composed of c, the variables .r, ?/, and the constants of the proposed differential equation. It may, however, happen in particular cases, that c does not contain the variables ^, y. In such cases the values of c derived from c = are functions of constant quantities, and are therefore themselves constant. The substitution of such values for c in the general solution would, therefore, only give particular integrals, and not singular solutions. Again, if c contained the variables, or either of them, and the elimination of one of them between the equation c = and the general solution F( 0. Let (c — c')"' = ^, ••• V - X = v'^ + v"A'^ H . ••• V = X 4- v'A + w%^ + . . . . This may be considered as the development of y in the general solution. Let the proposed differential equation resolved for dy be dy = pdx. This equation ought to be satisfied by the general solution independently of li. Let the value of y found from this solution be x + 1c ^ and p being expressed in powers of h, we have p = p 4- p'A;'" -f- p"^'* + where the exponents are ascending and positive. Forjp is not infinite when A; = 0, since the equation «/ = x (which does not render p = x ) renders the equation dy = pdx identical, and *.• dfx = vdx. When ?/ = X + ^, '.* fZx + d^ = (p + p'A;'^ + P"A;" + )dx, dx = pJx, •.• dJc = (p'A;- + iV -I- . . . ')dx. Substituting for k its value, V - X = v'^ + v"A'^ H we obtain MV' + A^Jv"+....= j +p.;,«(^,^ ^7/^-1 ^ )n^^^ 312 THE INTEGRAL CALCULUS. SECT. XX. This equation must be satisfied independently of h when 2/ = X is a particular integral. If this be not possible, it is a singular solution. Equating the terms with the lowest exponents, we obtain which is only independent of Ji when m — \ = 0, '.' m — 1. In this case v' = e^'*^. If ;» — 1 > 0, the terms cannot be identified ; but 7id\' may disappear by supposing dv' = 0, *.• v' constant. Then, if |x = 7/1, •.• dw" = v^dxj '.' v" =Jp'dxy and, in a similar way, the other terms may be found. Thus it appears, that if w — 1 be not < 0, the two series may be identified, and therefore the proposed solution is a particular integral. But this cannot take place if m — 1 <0, that is, if m be a proper fraction ; since, in that case, the term v'Y''"h"'dx cannot be identified with hdv\ or any of the following terms. In this case, therefore, the proposed so- lution is singular. (337.) This investigation furnishes a new criterion for the detection of singular solutions, and one which is altogether independent of the knowledge of the general solution. It appears from what has been proved, that, if upon changing y into 1/ + Jc in the proposed differential equationjr'(^p) = solved for p, and developing the corresponding value of ^ in powers of k, the first exponent of k be less than unity, the equation between the variables which fulfils this con- dition will be a singular solution, provided it satisfy the proposed equation. By (55.), it follows, that the condition on which the first exponent in the development is less than unity is SECT. XX. THE INTEGRAL CALCULUS. 313 -^ = 00 dy And by similar reasoning applied to the other variable, it follows, that singular solutions may be determined by the condition dx Thus, \^ ~- 0T~ = — , every singular solution must render N = 0, and must therefore be a divisor of it. Also, every factor of N, which is not also a factor of m, and which satisfies the proposed equation, is a singular solution. The solution of the proposed differential equation forp may be avoided by differentiating it for .r, y^ and p, Letfixyp) = u = 0, •/ du , dv dx dy dy + ^pdp = 0, Ju ' dy 'dy dp dp dx " dv dx " dv' dp If the cquationy(^j/p) = have been previously cleared of radicals, the condition under which these coefficients will become infinite But otherwise, is the cond dp ilion may also be satisfied by the equations 1 : 0, ^ = ' dv [«]. dx df, 314 THE INTEGRAL CALCULUS. SECT. XX. The elimination ofp by the proposed differential equation f(pcyjp) = 0, and any one of the three preceding equations, will determine a singular solution, provided that the result satisfy the proposed equation. (338.) The former of the equations [a] should be used when the proposed differential equation does not contain y, and the latter when it does not contain x. The one de- termines singular solutions of the form ^ = c, and the other of the form y — c. (339.) From what has been already observed on the method of deducing singular solutions from general ones (333.), it is obvious that the conditions dp dp will always be satisfied by making the radicals which enter the values of p derived from the proposed equation = 0. In applying these conditions, the equation should be pre- viously solved for p, otherwise it will be necessary to eliminate p between either of these and the proposed equation. If the equation Ju , Ju _ d\5 . be solved for dp, we obtain d\5 ^ du ^ Ib'^' + lTv'^ dp — du dp du da: cfe + dv, dy'^y dp d-y dx~dx^ du . -j-dx dp The conditions already established for the determination of singular solutions, SECT. XX. THE INTEGRAL CALCULUS. 315 du dv . du _ du render both numerator and denominator of the former ex- pressioa = 0. Hence, singular solutions must always render the second differential coefficient -^. If the differential coefficient p enter the proposed equation (having been previously cleared of radicals) only in the first power, it is impossible that the equation can admit a singular solution. For p will not, in that case, enter -T— = 0, which will not, therefore, be sufficient to eliminate p from the proposed equation, on which elimination the determination of a singular solution depends. The same remark extends to equations which are linear with respect to p, but which involve radicals; and it may in general be concluded, that no linear equation, properly so called of any order, allows of singular solutions. (340.) The connexion of the singular solution with the general one has been determined by considering the arbi- trary constant in the general solution as a variable. Taking the general solution as the equation of a curve, the character, magnitude, and position of which will depend upon the values of the constants, and among others, of the arbitrary constant, if a succession of values be ascribed to it, the general solution will represent a succession of curves cor- responding to these values ; and the equation may be con- sidered as applying to the consecutive intersections of these curves. If, then, the condition of continuity be introduced, and the arbitrary constant be considered as variable, the equation will represent a curve which will include or ex- clude all the others, and touch them. The general solution. 316 THE INTEGRAL CALCULUS. SECT. XX. when any particular value is ascribed to the constant, repre- sents one of the former curves, and determines a relation between the variables, which is expressed by the co-ordinates of any point upon it. But in the other case, the constant is replaced by a variable function of the co-ordinates of the point of contact. The tangent at the point of contact is the same for both curves, being determined by the value of the differential coefficient /?, which preserves the same value, whether the arbitrary constant be considered as variable or not in the primitive equation; whence it follows, that by eliminating the constant between the primitive equation and its differential with respect to the constant, the resulting equation between the variables, which is the singular so- lution, represents the line of contact of the curves comprised in the general solution. (341.) In general, then, from the results of this section it follows : 1*^. That two conditions are indispensable, in order that any equation between the variables may be a singular so- lution of a given differential equation ; 1st, That it be a solution, that is, that it satisfy the proposed differential equation, for otherwise, it is not a solution at all ; and 2dly, That it be not contahied in the general solution, for if it be, it is a particular integral^ and not a general solution (832.). 9P, That if the general solution be differentiated with respect to the arbitrary constant and its differential co- efficient equated with o, and by this equation and the ge- neral solution the arbitrary constant be eliminated, singular solutions may be found among the factors of the resulting equation (331.). 3**.. If the general solution be solved for the arbitrary constant, so that this constant may be expressed as a func- tion of the two variables, and that its two partial differential coefficients taken with respect to each vajiable be found. SECT. XX. THE INTEGRAL CALCULUS. 317 singular solutions may be found from the equations which render either or both of these infinite (333.). 4fP. The condition under which the factor, which renders the equation integrable, becomes infinite, may contain sin- gular solutions (334.). 5". If a differential equation be differentiated with respect to the differential coefficient, and this coefficient eliminated by the equation thus obtained, and the differential equation itself, the resulting equation between the variables may contain singular solutions (337.). 6". If a differential equation be differentiated with respect to either of the variables, and by the equation which ren- ders the partial differential coefficient thus found infinite, and the proposed differential equation, the differential co- efficient be eliminated, the resulting equation between the variables may contain singular solutions (337.). 7°. If a differential equation be algebraic, and include ir- rational functions, singular solutions may be found amongst the equations which make these radicals disappear. This may be effected by the suffixes or coefficients of the radicals vanishing (339.). 8^. The conditions which render the second differential coefficient -x- may contain singular solutions (339.). (In the last seven observations we have expressed ourselves in a contmg'ent sense, since the results must severally fulfil the two conditions of 1% in order to be singular solutions, which in some cases they do not.) 9^. It is of as much importance to determine the singular solutions as the general solution^ since, in many cases, the true solution of the proposed problem is to be found amongst them, and not in the general solution (340.). See Section XXII, Ex. 11. to Ex. 15. 318 THE INTEGRAL CALCULUS. SECT. XXI. 10^. Geometrical problems, the object of which is the determination of curves touching any number of curves of the same kind, but differing from each other by the para- meter, or some other constant part, are solved by singular solutions (340.). SECTION XXI. Of the integration of differential equations of the first order ^ and which exceed the first degree. (342.) It appears by the ordinary process of differen- tiation, that no differential equation, obtained directly by differentiating the primitive equation, can exceed the first degree. But when between the primitive equation and its immediate differential a constant is eliminated, which enters these equations in any degree superior to the first, the result will be a differential equation of the same order as before, but of a superior degree. (343.) Every differential equation of the first order, whatever its degree may be, must be comprised in the formula + N = 0. (i)"- = ~. Hence which being integrable, gives + Vx"^ ^y^ = X -t-c, •.• 2/« - 2c.r - c* = (1) * • is the general solution. To determine the singular solutions, let the last equation be differentiated for c (331.). This gives c + .r = 0, which, by eliminating c by the general solution, becomes 1/2 ^ _3.2 _ 0. The value of c being variable, and this last solution not being a case of the general one, it is a singular solution. The same result might be obtained by (332.) solving the equation (1 ) for c, and making the radical = ; thus, c = ~ a Vx^ -\- y^, '.' X^ + 2/2 = 0. If we examine the general solution (1) by the tests established in (333.), we find y i2 = X 3^ THE INTEGRAL CALCULUS. SECT. XXH. dc _ C dx ~ c^x" ^ dc y ay c-\-x which give the singular sohition already determined. The singular solution may be obtained immediately from the proposed equation without the general solution by the method explained in (337.). By differentiating for p and X, and p and y, we find -^ = ^ X doc py-\-x ' dp _1 —p'' dy py-^x these conditions are both fulfilled by py -{. X =^^ 0. Eliminating p by this and the proposed differential equa- tion, we obtain the singular solution. The same result may be still more readily found (337.) by differentiating the proposed equation for p only, which gives py ^- X =0, from which we obtain the singular solution as before. Ex. 2. Let the general solution of a differential equa- tion be jf- -\- x^ = ^cx, it is required to assign the singular solution. Differentiating for c, we find ^ = 0. Since this is in- dependent of c, it only gives the particular integral cor- responding to an infinite value of c (330.). Ex. 3. Let the general solution be SECT. XXII. THE INTEGRAL CALCULUS. 32S Differentiating for c, we find "" - l^b ' •••y(i/' +^^ - b) = 0. This being only the case of the general solution in which c = 0, it is a particular integral. Ex. 4. (1 -{-p^)x = 1. Hence , Hence by the formula (347.), 1/ =pr ~fF(p)dp, '.' 7/ = px — tan ~i p + c. Eliminating p by this and the given equation y — ^\v — x^ — tan.~^ ==— + c. Ex. 5. Given a general solution 3/ = ^ 4- (c - 1)V^5 to determine the singular solution. Let it be diffe^pntiated for c. Hence •.•c = l. Hence j/ = x. But since this is contained in the general solution, it is only a particular integral, and not a singular solution. Ex. 6. To determine the singular solution of of which the general solution is ^2 _ 2c^ _ ^ _ c^ = 0. Differentiating for c, we obtain c = — j/. Hence Since this satisfies the proposed differential equation, and is not included in the general solution, it is a singular so- lution. y2 324 THE INTEGRAL CALCULUS. SECT. XXII. The same might be also immediately obtained from the proposed equation by solving it for p, and making the quantity under the radical = (339-). Ex. 7. Let the proposed equation be ydx — xdy — oc^/dx'^ + dy'^. This equation is homogeneous with respect to x and 3/. Let y = nx, '.' dx X u = p \ V\ -^ p\ du , ^ dp = , •.• /^ = - l(p - u) +/— ^, Ix = - /^ 1 + p- - l{p 4- a/1 + p") + Ic, m * y , *' f f^ / 1 1 *^2\ CJ) vj/_ ,,^-Sp- vi +p^). VI +p' EHminating p, we find x = 0, 2^2 _ ^2j«^ which is an ellipse or hyperbola according as If^ is > or < 0, that is, according as the perpendiculars are at the same or different sides of the axis of x. Geometry (19^-)- PROP. CII. Ex. 1 4. Tojind the curve such, that the locus of the jjoiiit where a perpendicular Jrom a given point meets its tangent is a circle. Let the line through the centre of the supposed circle and the given point be the axis of jr, and let the origin be taken at the centre of the circle, the distance of the given point from the centre being c, and the radius of the circle a. Let the angle under the perpendicular, and the axis of x be ^, and the perpendicular z, -.' a^ ~ z^ + (^ + 9.ZC COS. or < 0, that is as a > c or a < c. Geom. (223.). PROP. cm. Ex. 15. To find a cwve such, that the locus of the inter- section of a perpendicular from a given point with its tan- gent shall be a right line. Let a perpendicular through the given point be drawn to the right line which is the supposed locus, and let these be assumed as axes of co-ordinates, the distance of the point from the sup[>osed locus being a. The equation of the per- pendicular to the tangent through the given point is y + — (^ — a) — 0, p The value of y corresponding to .r — is therefore a — . Hence the intercept of the ] given point and supposed locus is — . Hence the intercept of the perpendicular between the V ^ a^ v/l+;?« p V Hence y \-p{a—x) _ yl+p' — a , v/l+f/ P y and x being the co-ordinates of the point of contact. The singular solution of which is the equation of a parabola. * fJSO THE INTEGRAL CALCULUS. SECT. XXIII. SECTION XXIII. Of the integration of differential equations of the second and higher orders. (351.) One of the circumstances which give facihty to the integration of differential equations of the first order, is, that it is immaterial which of the variables is considered as a function of the other, or which is the independent variable. This is not the case when we ascend to the higher orders of differentials (38.), where a transformation is necessary to change the independent variable. As the orders of dif- ferential equations rise, the difficulty of their integration increases. It has been proved that every differential equa- tion of tv/o variables has an integral ; but the discovery of that integral in finite terms when the order of the differential equation exceeds the first, is, in almost every case, attended with considerable difficulty, and, in by far the greater num- ber of cases, has totally baffled the skill of the greatest analysts. It would be impossible in the present state of the science, tlierefore, to reduce the subject of the present section to a systematical exposition of the integration of differential equations of the higher orders. All that could be done, even in a treatise less elementary than the present, would be, to explain the methods of integrating particular classes of equations which have been discovered by JEuler, La- grange, D'Alembert, and others. To enter at large into the details of these methods would, however, be quite unsuitable to the objects of this work. We shall therefore confine ourselves to the investigation of the methods of integrating a few of the most important forms of equations, and par- SECT. XXIII. THE INTEGRAL CALCULUS. 331 ^icularly those of the second order. The subject of the present section may be divided under the following heads : I. The integration of differential equations of tlie second order in the following cases : 1. Where they contain only the second differential co- efficient and the independent variable. 2. Where they contain only the second differential co- efficient and the dependant variable. 3. Where they contain the two differential coefficients and neither of the variables. 4. Where they contain the two differential coefficients and the independent variable. 5. Where they contain the two differential coefficients and the dependant variable. 6. Some of the more simple cases where they include both variables. II. Some cases of the integration of equations of the nth order, which only contain differential coefficients and con- stants. III. Certain cases of differential equations which include only one of the variables. IV. Homogeneous equations of the first degree with re- spect to the dependant variable and its differentials. V. Equations in general of the first degree with respect to y and its differentials. I. The integration of differential equations of the second order. (352.) Let X be taken as the independent variable, and let the first and second differential coefficients be y, y. The most general form for a differential equation of the second order is 3Sf? THE INTEGRAL CALCULUS. SECT. XXIII. ¥(ooyyy) = 0. We propose first to consider the five following cases of this, 1. F(y'a;) = 0, % F(yhj) = 0, 3. r(yy) = 0, 4. Y(yyx) = 0, 5. F(y'yi/) = 0. dy' (353.) 1. By substituting -p for y", the first equation becomes F(dy', dXi x) = 0, which solved for di/, gives a result of the form dy' = xt/jr, where x is a function of x. This being an equation of the first order, may be integrated by the methods already ex- plained, and its integral will be of the form y = x' + A, or di/ = x'dr + Adx, A being an arbitrary constant. This being again integrated, gives an equation of the form y = x" -f A.r + B, B being another arbitrary constant. Thus, let d^y = ax'dx% •.• dy' = ax"dx, dy Substituting -~- for y, we find , ax'^'^^dx dy = -; j- Adx, ••• (354.) 2. Let the form fC/z/) = be supposed to be solved for y'', and therefore reduced to the form ax' SECT. XXIII. THE INTEGRAL CALCULUS. 333 d'y _. dx'' ~ ^' where y is a function of 2/. Let both be multiplied by ^di/ and integrated, and we find A being an arbitrary constant. The integral yVc^j/ may be determined by the established rules, and therefore the equation, after extracting the square root of both sides, assumes the form ^^ -Y, . ax- ^ \ ' du which, when the integration of -^— has been effected, be- comes the integral of the sought equation. Thus, hta'^d'i/ ^ ydx" = 0, •.• d\y y dJF'^'"aF' dx V^""^"^' which is an equation of the first order. (355.) 3. When the differential equation does not include either of the variables, it may immediately be reduced to an dy^ equation of the first order by substituting -~~ for 3/", by which it becomes f{ -1—, ^/' j = 0. This being solved for dx, assumes the form dx = F{y')dy'; 334 THE INTEGRAL CALCULUS. SECT. XXIII. and since tjdx — dy^ by multiplying both by y\ we find . dy r= y^i{i/)dy\ Hence by integration, ^ =f^iy')dy\ eliminating y by these equations, the result, including two arbitrary constants, will be the integral sought. Thus, let d'^y = doc^dy'' + dx'^, \- •/ dy = ^/l -\- y'^^ ' dx^ v/i+y*' which may be integrated by the established rules. The value of y being thence obtained as a function of ^, the sought integral will be y =/y'^'^- (356.) 4. A differential equation of the second order of the form F(yya) == is reduced to one of the first order by dy^ substituting -p for ^". The equation, therefore, assumes the form Ff-^ — jy?-^) = ^> which is an equation of the first order between y^ and x. This being integrated by the established rules, gives an equation of the form ^{y^xc) — 0, c being an arbitrary constant. Again, substituting --^ for y, this becomes a differential equation of the first order between y and x. This being integrated as before, gives an equation of the form F{xycc) = 0, c, c' being the two arbitrary constants. 1 (dar-^dy^)^ a^ ^_ SECT. XXIII. THE INTEGRAL CALCULUS. S35 y' 2.r* Foi-y substitute -—-, and we obtain di/ _ ^xdx 3 yy2 ' (1+yT ••• y = v.5/=/ (07- + ac)dx The integral of which is obtained by the rules already established. This is the equation of the elastic ciirve. (357.) In general, the equation f(j/'^c) — may be integrated by three different methods, which may be chosen according as they may severally be found best suited \o the circumstances of the proposed equation. 1. If the equation admit of being solved for y, it may be reduced to the form dx "^^ •.• ^/ —fyidx. % If it admit of being solved for x^ it may be reduced to the form X = F(y). Butyc?a7 = dy^ '.' y —Jy^dx — y^x —JxdyK Hence y = y^ -/F(y)%'- The latter integral being determined, y may be eliminated by means of this equation and x — F(y), and the result will give the sought integral. 3. If the equation do not admit of solution for either x or 336 THE INTEGRAL CALCULUS. SECT. XXIII. y, it will be necessary by a transforma;tion to express t and y as functions of a third variable z ; let these functions be z, z', so that jr = z, y = z', which is the sought integral. (358.) 4. If the equation have the form ^{li^ijy) — 0, it may be reduced to one of the first order, thus, > dy^ — y^'dx^ y^dx = dy^ Let the proposed form be expressed thus, y --^fiy'y), •■■ y'dt/ =-f{.y'.y)dj/. This being a differential equation of the first order between y and y, may be integrated by the usual methods, and its integral will have the form Y{i/yc) = 0, c being the arbitrary constant. The integration of this presents two cases : (359.) First, Where the variables may be separated, and therefore the equation may be reduced to either of the forms, y = Y, y = y', Y and y' being functions of 3/ and y respectively. In the first case, the integration is effected by reducing it to the form Y ' In the second case, since dy =■ y^dx^ '.' y^dv = dy', SECT. XXni. THE INTEGRAL CALCULUS. 337 If Eliminating y by this and the equation 3/ = y', the resulting equation will be the integral sought. (360.) Secondly, If the variables y and 7/ cannot be separated, a transformation may be effected by expressing y and y as functions, z', z, of another variable z. Since 7/da: = di/, '.• z'da^ = dz, and the integration may be effected as in the last case of (357.). (361.) 6. There are some remarkable cases in which dif- ferential equations of the second order, where they include both the variables, may be integrated without much dif- ficulty. We shall consider the three following cases : (where x, x', x", are any functions of jr.) [3] • • • • Where the equation is homogeneous with respect to the variables and their differentials. (362.) 1. Let^ = e-^"^"-, '.' -^ = ueJ''"^% and dx^ \ ^ dx) Making these substitutions, we find du — + (w^ + ^ti + x') = 0, since the common factor e^"^' disappears. This, being a differential equation of the first order, is integrated by the methods established in Section XVII. (363.) 2. This equation may be reduced to the preceding by 338 THE INTEGRAL CALCULUS. SECT. XXITI. any transformation which will remove the term x''. For this purpose, let j/ = tz. Hence dy dz dt dx ""da: dx^ d'^y _ d^z dzdt dH d^^ ~ *d^ ^ (hd^ "^ ^^* Making these substitutions, we find d'^z dzdt: dH A dz dt\ dx^ dxdx ^dx' \ dx dxj 4- x^tz + x" = 0. Let the variable z be limited by the condition d^z dz Hence the transformed equation, after dividing by z, be- comes dH dt / ^dz \ ^ x" ^ r^^ The first [1] of these equations may be integrated by the preceding article, and thence an equation found of the form dz f(zx) = 0. By this process -j- will become known as a function of x, and thus the equation [2] will become in- tegrable. The process in general may be conducted thus. Let , dv *.• xax" = . V By this substitution, [2] becomes But since dH dt/dv Mz\ xJ^dx ^ ax ax\ V z J ^ = dlypz'^) = -, V z ^ ' vz" the equation may be reduced to the form SECT, xxirr. the integral calculus. 339 dH dt d^^^^ + -T-divz^) + x"vzdjt; = 0. Integrating it under this form, we find dt j-vz'^ +fx"vzdx = 0, But since ^ = =^, •.• z /dx \ In this, V is a given function of x by the equation v = e^^^^ and ^ is a function of x by [1]. Hence this last equation is the integral sought. It will obviously include two arbi- trary constants. (864.) 3. If the equation be homogeneous with respect to Vi ^i ^j/j d'^yt and dx, let dy , J'?y js ^ = "^' -^=^' rfi^= x' w, y', and z, being considered as new variables. By this substitution, every term of the equation will have the same power of jr as a factor, which being removed by division, the equation assumes the form F{7jzu) = 0, or z =^f{yhi). By differentiating ?/ = ux, we find dy = udx -\- xdu, '.' ydx = udx + xdu^ dx du Also, since X y-^i cPl_z_ dx'" x' z% 341) Hence THE INTEGRAL CALCULUS. SECT. XXIIL ..• .r p = zd., dx ' ••• xdt/ = zdx, _ dx dy^ X z dnf du ^y .fluffy-) -^' This being an equation of the first order between y and u, may be integrated by the rules for integrating such equa- tions. The integration will givez/' = f(?/). Hence dx du \' Ix =/ du f(u)—u Eliminating u by this and y = ux, we obtain the integral of the proposed equation. It is obvious that this integral will include two arbitrary constants introduced by the two integrations effected prior to the elimination. Thus, let xd^y = dydr, •/ z = y'j and hence dy^ du y' ~ y-w' ••• y^ =Audy' + y'du) =i/u+ Jc. dx di/ But, also — = —j-, '.' X = ay', EHminating y by these X y equations, the result is x^ — 2axu = c. Eliminating e^ we obtain the integral x^ — 2aj/ = c. SECT. XXin. THE INTEGRAL CALCULUS. 341 II. Integration of differential equations which do not contain either of the variables. (365.) There are two cases in which differential equations, which do not include either variable, may be reduced to a form which is integrable by the rules for the integration of Junctions of a single variable. These two cases may be expressed thus : /^ dJ-^\ _ '^- A^^'«' dx^-O" ' which denote any differential equations which include only two differential coefficients, the order of the one being in the first case lower by one, and in the second lower by two than the order of the other, and which exclude the va- riables. {SQQ») 1. In the first case, let dr-'y _ d^-^ - ""' ^ ^d^t/ _ du dx"" ~~ dx' By which substitution, the form is changed to <-|''') = 0' which being an equation of the first order may be integrated. This being effected, we shall obtain from the resulting equation u — X, dry _ X being a function of iv. The integration may be completed by Section XI. 342 THE INTEGRAL CALCULUS. SECT. XXIII. (367.) 2. An equation of the form may be integrated by making the substitutions d^u __ d'y **' 'dx'^~"dx''' Hence the proposed equation becomes /^'^^ «)=pO, \dx''' which being integrated by (354.), may be reduced to the form u = X. Whence d"-2j/ _ d^^ ~^' this may be integrated by Section XI. III. Integratiofi of deferential equations which include but one of the variables. (368.) Differential equations, which include only one of the variables, may be divided into two classes, those which include only the independent variable, and those which in- clude only the dependant variable. P. The class of differential equations, which include only the independent variable, comes under the form / dj_ ^ ,,,^\-o V' dx' dx^' e/W The exponent of the order of an equation of this kind may be always reduced by an easy and obvious transformation. Let # SECT. XXIII. THE INTEGRAL CALCULUS. 343 y = and, in general. dy diif d-y ~dx' ' ~dx~'dx^' d'''\y^ d'y . d.v" da;''' Hence the formula becomes "^V'^'dx' dx^' dx-'J ""^ which is an equation of the {n — l)th order, including both variables i/ and x. 9P, If the equation include the dependant variable only by the transformation (38.) for changing the independent variable, it may generally be reduced to the preceding . case. By this rule, an equation of the second order, when it includes only one of the variables, may be reduced to the first, one of the third to the second, and so on. (369.) If an equation have the form 'd'y d''-\y d" it may by a similar process be reduced to an equation of the second order, including but one of the variables. For, let _ d"-^y _du d''-\y d^u _ d'[y ''' dx~dx^^' dx'^~dP'' by which the equation becomes (d^u du \ which, by a similar process may be reduced to an equation of the first order, including two variables. In general, therefore, a differential of the wth order, in- cluding no variable, may be reduced to one of the (w — l)th order, including one variable, or to one of the (n — 2)th order, including two variables. 344! THE INTEGRAL CALCULUS. SECT. XXIII. IV. The integration of' homogeneous equations of the first degree with 7'espect to the dependant variable and its dif- ferentials. (370.) Equations of this class come under the general formula where a, b, are functions of .r, the independent variable. Let J/ = ^% and •.* dy = e'^du, d^y = e''(d^u + du% d^y = e^d'u + Qd^udu + du^). By tliese substitutions, the proposed equation becomes di- visible by e''y and the resulting equation will be independent oi'y, and of the form d"u d'^-^u du ^^ r_- which may be further reduced to an equation of the (ti— l)th order, including both variables by (369.). (371.) As it seldom happens that the equation [1] is in- tcgrable when its coefficients are variable, we shall at pre- sent consider the case only in which a, B, • • • • are all con- stant quantities. Thus, if the equation be d^u d^j du ^ + ^^^+^-^ + ^ = «- By the transformation already suggested, this becomes * The coefficients of this equation are not supposed to retain the same values as in [I], but are general representatives of functions of j. - SECT. XXIII. THE INTEGRAL CALCULUS. 345 d^u + Sdud'^u + du" + A(d'u 4- du:^)da: 1 _ q + Bdudx^ + cdx^ j This equation may be reduced to one of the second order (jijj by substituting t for -7—, by which it becomes dH + (3jf + A)dtdx + {f + A^* + B^ 4- c)dx^ = 0. Since a, b, c, are supposed constant, this equation may be satisfied by a constant value of t. For if t be supposed constant, dt — and dH = 0, by which the equation is re- duced to t^ + aP + bj^ + c = 0. In general there are three values of t, which are functions of A, B, c, and therefore constant, which will fulfil this equation. Let the three roots be t = rrii, t = 7/225 t = Wg, And since • du = tdx, \' u = tx + c, \' y = e*^"^"" ; we obtain thus three values of ^, Whence we have yi = c^er, i/2 = c.,er, yz = Ca^, the three arbitrary constants being />C pC pC *^l9 t'SJ *^3» Since each of the equations between^ and adjust determined include but one arbitrary constant, they are only particular integrals. We shall, however, obtain the complete or ge- neral solution by equating the sum of the three particular values of ^ already found with y, y = ci^r + c^^r + €3^^. There is no difficulty in proving that this equation satisfies the proposed differential equation, for if it be differentiated, and its third differential obtained; the three constants being eliminated, the result will be identical with the pro- posed equation. 346 THE INTEGRAL CALCULUS. SECT. XXIII. A principle much more extensive, however, may be esta- blished. Whatever be the nature of the coefficients a, b, c, it may be proved, that if j/i, ^29 Vzj be the three values of y^ which separately satisfy the proposed equation, their sum 7/1 + y% + yz being equated with ?/, will form an equation, y = Ci3^i + C2?/2 + C3J/3, which will satisfy the proposed, and which, including three arbitrary constants, will be its general solution. If any of the particular values j/i, 7/2, or 7/3, contain an arbitrary con- stant, the corresponding multiplier may be omitted. To prove this, let the last equation be thrice successively differentiated, substituting in the proposed equation the values of 3/, dy^ d^y^ d-y^ and collecting together the mul- tipliers of the same constant. The result will be • Ci{d^yi + hd^yydx -f Bdy^daf^ -j- cdx^) + 03(^3^2 -\- A.d'^y^doo ■\- -Bdy^dx^ + cdx^ -\-c^{d'yQ + Kd^y^dx + ^dy^dx'^ -f- c which separately satisfy [1], its general solution is y = C,7/i + CaT/a + C^T/s • • • • + C„7/„. When the coefficients a, b, c, are constant, the particular values of y are of the form y = e'"^, m being constant. Hence dy = me'^dx, d^y = m^e"'''dx • • . • ^"3/ = rrC'e^dx, The proposed equation thus becomes divisible by e'"% by which it is reduced to :')U0. SECT. XXIII. THE INTEGRAL CALCULUS. 34^7 m" -{- A7w«-i + Bm"-2 4. cm"-=* • . . • Mm + n = • • . [3]. Let the n roots of this equation be m^, 777-2, in^' ' ' ' rrini and we find the particular values oiy corresponding to these severally, mx and therefore the general solution is mx y = Ci^T* + C2^r + • • • • c„e~ [4]. (372.) If any pair of values of m deduced from [3] be imaginary, they must have the forms a + h^/ — 1 and a — hV— 1, *.• two o£ the particular integrals, deduced as above, assume the forms (a— &V^)^ 3/ = Cii^ , and, therefore, their sum becomes y = e^\c,e ^ + c^e ^ / But (^56,), _ hx^—l , . , e = COS.OX -|- y/ — 1 sm.oj:, —bx^/^l . e = cos.c>a; —^ — 1 sm.6ar, ••• CiC + Ca^ = (Ci + C2)C0S.^^ + (Cj - Cj) \/--l sin.^a?. Let the constants Ci, Cg, be so assumed, that c, -f Cg and (Ci — C2)v/ — 1 shall be both real, which we are alWed to do since the differential equation is fulfilled independently of them ; and let Ci + Ca = p sin.g', (Ci - Ci)*/—! =:pcos,q, p and q being arbitrary. Hence the corresponding terma^ of [4] become y =z e"'^ . p sin..(ba; f q) ; 848 THE INTEGRAL CALCULUS. SECT. XXIII. and in the same manner, terms of the same form may be found for every pair of imaginary roots. (373.) When the equation [3] has equal roots, the result is only a particular integral, for the corresponding terms of [4] become c^e"^ + C2^"*^ = (ci + Caje"^ = c^. There will be, therefore, one arbitrary constant less than the number necessary to give the integral its full generality. In this case, therefore, the integral must be otherwise obtained. (374.) The following process for obtaining the integral in this case was first proposed by D^Alembert. Let the two equal roots be nii and m^, and first let them be supposed to be unequal, and to differ by A, so that fjii = nil -\- h. Hence the two corresponding terms of [4] would be Ci^r + c^eT = ^r[ci + c^e^]. But by (65.), ^"^ = 1 + -T- 4- 1 ' 1.2 ' 1.2.3 ' Let C| + C2 = e', and cji = e". Hence l./V where e' and e" are arbitrary constants. As this will satisfy the proposed equation, whatever be the values of the arbitrary constants e', e", and independent of h, we may suppose A = 0, which is equivalent to m^ = m^^ This re- duces the expression to which being substituted in [4], will render it a general solution, since it introduces the complete number of arbi- trary constants. It will be easy to extend the same process to the case- SECT. XJCIII. THE INTEGRAL CALCULUS. 340 where three or more of the roots of [4] arc equal. First, let nil = m^, *.' y = er[E' + e''-^^] + Ca^T' • • • • + c„e" . Let ??i3 = Wi + h, \' CIe' + e"^] + Q.^ef = ^r[E' + E".r + Ca^^. Developing e^'' as before, and substituting its value, we find the quantity within the parenthesis become e' + C3 + (e" + Cjl)x + Cay^ + ^^.2^ ' * * ' c h^ Let e' + Cs = f', e" + c-ih - f'', y^ = f"', and it be- comes f' + f"jc + f"^2 _^ j-rff ..... In this, let ^ = 0, and •.* in^ = tWj, *.* C'(e' + e"j) + Ca^*^^ = e7^[F' 4- F"a: + F'"a;2], which being substituted in [4], renders the solution general as before. The same process may obviously be continued and applied to any number of equal roots. Linear equations of the first degree with respect to y and its differentials, (375.) This class of equations are included under the formula d"y d'^-^y dy Let the several coefficients a, b, • • • • n be constant, and X any function of the independent variable x. The integration of equations of this form is reduced to the resolution of algebraic equations, as in the last case, by 5S0 THE INTEGRAL CALCULUS. SECT. XXITl. either of the following methods. The first is given by Euler in his Integral Calculus, and the other by Lagrange. (376.) Let us first consider an equation of the second order, Let e~'"'d.v be the factor which renders this equation inte- grable, and lety*— xe-^'^da; = x' + c. Hence the quantity must have an integral of the form To determine the arbitrary quantities 7i, a, Z>, let this be differentiated, and the result equated term for term with the former; *.* — ha — B, — hb -\- a = A, b = 1, ',' h^ + Ah + B = 0, a = - -^, b = 1. The first equation gives h, and the last two a and b. The immediate integral of the proposed equation is, therefore, ^-^ + ««/ = e'X^' + c). If in this equation the two values of h determined by the equation h^ -f- aA + b = be successively substituted, and diJ by the two equations thus found, -j^ be eliminated, the re- sult will give the complete integral. (377.) If the proposed equation be of the wth order, we may infer in the same manner, that the value of A is a root of the equation h" + A/i«-» +....= 0. And we shall have as many different immediate or first SECT. XXIII. THE INTEGRAL CALCULUS. 351 integrals of the {n — l)tli order as tliere are roots of this equation given. If there be n roots given, by the n corresponding in- tegrals, the (w — 1) differential coefficients may be eli- minated, and the complete integral thus obtained ; and if any number of roots less than the entire number be known, the order of the equation may be reduced by the elimina- tion of as many differential coefficients. (378.) We shall now explain Lagrange's method, which is founded upon the most general theorem which has yet been delivered upon the integration of differential equations. In (371.) it was proved that the integral of the equation d"7/ d^'-h/ d^'-'^y du d + ^^' + ^fe»^ + • • • •«£+''3'=o [1] was of the form y = c^r + c^eT + Cs^r • ' . . [2], where «/i = e"^, y^ = e^, .... were particular values of^ which satisfied the equation [1], the sum of which, involved with the necessary number of arbitrary constants, constituted the general solution. The equation to be integrated at present is more general than [1], being of the form d"y d'^-^y d'-'^y du X being any function of jr. Let it then be proposed to assign the functions of a , into which the arbitrary constants Ci, Ca, • • • • in [2] should be changed, in order that [2] should become the complete integral of [3]. If this can be effected^ it will follow that the several terms of [2] will be so many particular values of y, which will satisfy the pro- posed equation [3], and, therefore, that if n particular values of j/ be given, the integral of the equation [3] may be im- mediately determined. We shall investigate the values of the functions c^, Ca, S52 THE INTEGRAL CALCULUS. SECT. XXIII. • • • • in an equation of the third order, and the principle may thence be easily generalised. Let the equation y = c,2/i + C2J/2 + c.^ [4] be the sought integral, Cj, c^, C3, being arbitrary functions of d7. By differentiating, we obtain di/ =■■ Cxdy^ -f Q^dij^ + c^dy^ + y^do,^ -f y^dc^ + y^dc.^. Let the arbitrary functions Cj, C2, C3, be limited by the condition «/i^Ci + y4^^ 'V yjic^ = 0, which reduces the differential equation to dy = c^dy^ + c^dy^ + c.,dy^, the form it would have had if Ci, Ca, Cg, were constant. Differentiating this again, we find J«j/= c4'y^ + c^d'^y^ + c4^y^ + dc^dy^ + dc^^dy^ + ^c^rfj/g. Again, limiting the functions c^, Cj, C3, by the condition dc^dy^ + fl?C2 '.• ;rT= ;^. The equation, therefore, becomes doo^ dx (i+y»)^ and since c?y = i/dx, '.' ai/di/ dy = 3 (1 +yo Integrating these, we obtain ay' a a? = A ^, 7/ = B 4- ;•. (1+yo^ (1+yT Ehminating y\ we find (A -Ory^-f (B-^)* = ff2. This example proves that the circle is the only curve of which the radius of curvature is constant. Ex. 2. ady — dydx. By the usual substitution, we find dx = ^, •.• X = ay + c, dy = ady', '.' y = ay' -\- c'. Eliminating y, we find X = c -\- al\ (*^> Ex. 3. To find the curve in which the radius of cur- vature varies as the angle under its tangent and the axis qfji. Taking the arc s of the curye as tl>e independent variable, the radius of curvature is SECT. XXIV. THE INTEGRAL CALCULUS. 359 ^o2 R = But since ds^ — dy^ + dx^, •/ di/d'y •{■ dxd*x = 0. Eliminating d'-y by this, we obtain dsdy ^- d^x Hence by hypothesis, a^cT dx But if y = -^, ••• ds = (1 +y*)*J;r, •.• differentiating • ' i+y^- Hence the equation becomes ady* , , dr = ^— jtan.-y. (1+yT Integrating this, we obtain aw' ^ , aiJ X = c ■ tan."" y - — . , y = e -{- - tan.~y — Eliminating y, we obtain the required curve. Ex. 4. -^ = "-7—, ^ - x\ 3_ Ex. 4. uidx"- 4- %'')^ = x^dxd^y. Hence t\r — fl y = »/=/- {cx — a)dx / 9^acx a" -- — -. r- a tCxC\- c^) ~\-ac ^.y^c^x^-icx^ar -I] — 7=T- + A/^'-(cx-a^)^ + c'. Ex. 5. c?.r2fi?z/ — xds^d^y = ack-^.? ^/d^-x^ -\- d'y^, s being the independent variable, and ds = x/dy^ -\- dx^. Hence wc obtain d\'} = 0, •.• 362 THE INTEGRAL CALCULUS. SECT. XXIT. Hence y di/ dy' or ?/ — .r -~ = tt— -, which comes under Clairaut's formula (350.). - 0-7/" = aij\ Ex. 6. adxdy^ -h x^dxd-y~nxdy V dx'^ + a^d'-y'-^ *.• aifHx + ^^dfy = nxif^dx"^ -{- ardif^. This being homogeneous with respect to a; and y', let a? = y^, •.• dx ~ ^ ~ 'ii^uSi^-X ' and dx du X y" — 7i dx du M^«V— 1 (383.) 5. Equations of the form ^{y^y) - 0. Ex. 1. y'(2/y + a) == y(l + y'^-). since y'dy = y^dy\ this is reduced to dy\yy^ -V a) = dy{\ 4 y^). This being integrated by Clairaut's formula (350.), gives J/ = dy + cv/i 4-y% - /'^^ - X =/-^ = apy) + ci{y^ + v^i +y^). Eliminating y by these equations, the integral may be found. Ex. 2. Let the equation be abf = a/j/* + «y * ; this becomes, after substitution. ahy^dif = dyVy^ + «y^ To integrate this, let ^ - ijz, and the equation becomes SECT. XXIV. THE INTEGRAL CALCULUS. 363 ahzdy — abydz — z^dy ^/z^ + a*. The variables in this equation are separable by making ^/z^ -f d^ = tz, by which the values of z and dz being found, the equation is reduced to dy — htdt which is integrable "by rules already given. Ex. 3. Let the equation be yu + Ay + B^ = 0, A and B being constant. This becomes, by the usual sub- stitution, y^dj/ + Ayc?y + ^ydy — 0, which being homogeneous, may be integrated by the sub- stitution y = xiy. Hence dy _ —udu —ud u y "'m*-!- Aw + B~ (m— a)(w — 6)' a and h being the roots of the equation M^ + AM 4- B = 0. Also, • dy __ du —diL y" ""^ ~ (w -a)(M36)' dy , —du •.•—^ adx — ;, y u — o dy , , —du -^ — bdx = , y u—a •.* Ly — ax ■= I J, ^ u~b n ly — bx = I , \' u — a = , u -— — y y Hence yQi — a) = m^^ — mef', which is the complete integral. 364 THE INTEGRAL CALCULUS. SECT. XXIV. This result may also be obtained by the process in (371.), which, when the roots «, b, are imaginary, gives and when they are equal, gives an integral of the form y = cef^'[x + /c). !E)x> 4'« ' . -zz. 7}y. Hence ' ^dy^dx-Vy^dx^—yd^ydx ^' ••• <^j/(y' + y'"t = ^nfydy -}- ny'^dy - ny-'x/d^. This being homogeneous with respect to y and y\ let dy ?iiidu ^ 7idu clx = . (l+0(^<-Vl+w^) y (fy -f- ydy = i/^dy + ydy — zdz, ,, zdz •••y' + i/ = ^, which gives x^dy =^ 2azdy - aydz, \*y= ~ -^ -^ 2a -2 2^_^/«^*^.y^' which is an equation of the first order. (384.) 6. Equations of the second order, which include both variables. SECT. XXIV. THE INTEGRAL CALCULUS. 3G5 Comparing this with the formula [2] (361.), we find x' ^ - — r, x" = x' x"' ^ \-x^' Hence the equation [1] (365.)? becomes dx^'^'x ' dx'~ X'' " ' which, by putting z = ^/w^, gives (362.), du / u 1 \ ^ rf^ + ("^+T-W = '*- This equation is rendered homogeneous by making u = — ; the variables are then separated by putting x = su'. Hence du' _ 5^ + 5—1 s V * — 1 neglecting the constant. Substituting for u' and 5, their values, we find Also, V = f •^■^''•^ = ^^ = ^. Hence we find Jx"vzdx =Jadx = ax -{■ b, and, therefore, _ X'—l {ax-rh)xdx ^ ~ X '^ (^^-1)^ ' which is integrated by the rules for rational differentials. II. (385.) Integration of equations which do not include either variable. . T7 1 "'^V d^y T d^y d'u dy „ 366 THE INTEGRAL CALCULUS. SECT. XXIV. the proposed equation becomes Multiplying by 2du, and integrating, we find /J.2//7/.2 = u" + L Hence dx = dx' adu ^u^Arb And Hence we find -^ =zfudx = a Vii^ + 6 + h^ y = a'u 4- ab"l[u + v'w* + 6] + /{>"', or ^ = ce"« + c'^ « + c"a; -f c"^ III, IV. (386.) Integration of equations, including y only, Ex. 1. Let the equation be da^ -^dx'^^d^^ -^dx^^-^' This equation being homogeneous with respect to y and its differentials, and of the first degree, comes under [1] (370.) By comparing the coefficients, we find A= —% B=iH-2, c=— 2, N=l. The equation [3] (371.) becomes, therefore, w* — 9m^ -f 2m2 - 2w + 1 = 0, or (1 - my(l + ?7i«) = 0. SECT. XXV. THE INTEGRAL CALCULUS. 367 Hence its complete integral is 2/ = (a 4- ox)e'' + ce + c'^ > or y = (a + Z>^)e'' -f a cos.a: + b sin.a?. SECTION XXV. Of the integration of simultaneous differential equations of the first degree. (387.) If m equations be given, involving {m + 1) varia- bles, all these variables, except one, may be considered as determinate functions of that one. The forms of these functions are determined by eliminating every combination of (wi — 1) variables, which can be obtained from the entire number of variables, except that one, on which the others are supposed to depend. This process will give m equa- tions by which each of the m variables are connected with the independent variable, and by which they will be implicit functions of it. By the solution of these equations, they would become explicit functions of it. If the equations between the several variables be dif- ferential equations, the process of elimination would be at- tended with considerable difficulty « Instead, therefore, of eliminating first, and then integrating the several differential equations, so as to obtain each variable as a function of the independent variable, we shall explain a method of inte- grating simultaneous differential equations without any pre- vious elimination. (388.) Let it be proposed to fntegrate simultaneously the two equations 368 THE INTEGRAL CALCULUS. SECT. XXV. du dx MJ, + N^ + r ^ + Q.^ = T, which are the most general equations of the first degree between the variables x^ j/, and the differential coefficients du dx -f-and-j—. In these equations the several coefficients M, m', n, n', • • • • are supposed to be functions of the independent variable ^. Let these equations be expressed thus, (m^ H- iix)dt + Fdif + (^dx — 'idt, (M'y + ^^x)dt + p' Jz/ -f- Qidx = T'dt. Multiplying the second by an arbitrary function (9) of ty and adding the product to the first, we obtain the equation Hijdt + Kxdt -f- iidy 4- sdx == vdt, where H = M + m'5,. K = N + n'5, 11 = i. 4- p'a, s = a + q'5, U — T 4- t'9. This equation may be expressed under the form K S This will become a linear equation of the first order with TJ X and d{ij -\ x)^ dz = dt/ -\ • dxy respect to 7/ + —x and a(// -j x), it XT where z — // -j x ; for in that case we have X n , u , dz +- —zdt.= — -fif^, R R which is of the form integrated in (314.). SECT. XXV. THE INTEGRAL CALCULUS* 360 The condiiion K S ix rv gives <^)-i (/.r, K , ,K S _ ••• — ax ^ xd — = —ax, " d— = — = — H ' H R ' Substituting for k, h, s, and R, their vakics, and dif. ferentiating and ehminating 0, the resulting equation between the coefficients m, m', • • • • will be the condition under which the integration of the proposed equations can be effected by the formula (314.). (389.) The simultaneous integration of the equations {uy + Njr)J^ + vdy + Q.dx = Tdty (u'y -f- T^'x)dl; + v'dy + oldx = T^dt, may also be effected thus : let dy and dx be alternately eliminated, and the results will be two equations of the forms dy + (p7/ 4" QLx)dt = idtj dx -\- (v'y -f a'*')^^ = T'dt, the coefficients representing the functions of the former co- efficients, which are determined by the process of elimina- tion. Multiplying, as before, the second by 6, and adding, we find dy + Qdx + [(p + v^^)y + {o, + Q!^)x]dt = (t + T'Q)dl. Let y + Qx = z, *.* dy + Qdx = dz -- xdQ, y = z — ^x. By these substitutions, the equation becomes B B 370 THE INTEGRAL CALCULUS. SECT. XXV. Let such a value be assigned to the function 9 as will satisfy the equation J9 + [(p + p'e)a - (a + Q!^)]dt = 0; and the equation will be reduced to the form dz + Tzdt = t'^^, T and t' expressing new functions of t. This form is in- tegrated as before by (314.). (390.) If the coefficients P, p', • • • • instead of being functions of t, as we before supposed, be constant quantities, we have the conditions d^ = 0, (p + p'0)9 - (a + a'0) = 0. The function 9 then becomes a constant quantity, and its values are the roots of the latter equation. Let these be fi', 0". The equation dz + (v + ^^)zdt = (t + T^)dt becomes dz + mzdt = \dty by putting ?7i = p + p'9, V = T -f t'S. The integral of which is, (314.), 2 = e''"'lfe^*vdt. Whence we deduce 2/ + 0'^ = e^''[fe^'*v'dty y + 9"^ = (r^"lfe>'"W^dt, by substituting successively the two values of 6, and the cor- responding values of m and v. (391.) We shall now apply the same principles to two differential equations between three variables. These may by alternate elimination be, as before, reduced to the forms du + {vu + Q'^+'^-' -1 + A^ +B«6:i26-i This will be rendered identically by the following conditions, 2a— 1=0, a + 6— l=a, a + c — 1=5.--. A^a = 1, AB(a + 6) + A = Hence a=j, 6=1, c = ^.... A = V^, B = — -g-, C = -jg-, v,V= v'2 ^ "3 ^^ + 18 "^^ * If the law of the exponents t> !> I> • ' ' * had been known l:::;Uo. + 3 374 THE INTEGHAL calculus. sect. XXVI. in the first instance, the coefficients might have been im- mediately deduced, or the series of Maclaurin might have been immediately applied by substituting z^ for x. Ex. 2. Let the equation be ^y -Vydx — mx^dx\ let Differentiating this, and substituting the values of 3/ and dy in the proposed equation, and omitting the factor dx^ the result arranged by the dimensions of x is aAa:«--»+(a+l)Ba7«+(a+2)c^^+^ + (a+3)D^+H....? ^ —mx"+ Ajj°+ Ba7'*+'+ c.r'»+2^ ^ "~ This is rendered identically by m a — 1, •.• a = w + 1, A = — , B c = a' fl(a+l)' m —m a(a-\- l)(a+2)' a(a+l)(a+2)(fl+3)* Hence we obtain the law of which is evident. This series is, however, not the complete integral, unless the arbitrary constant be in- troduced. This is always the case when the arbitrary con- stant in the development of y in powers of x cannot be separated from x. 1^, We may, however, obtain the complete integral in the following manner. Let ^{xyc) = be the integral sought. To determine the constant c, it would be necessary to find some one system of values of the variables x, «/, which will satisfy the primitive equation. Suppose a, b, be such a system. The condition F(a, b, c) = 0, would give c in terms of a and b. Let the expression for ^, derived from the differential equation, be prepared in such a manner, that when x becomes equal to «, y will necessarily be equal SECT. XXVI. THE INTEGRAL CALCULUS. 375 to b. This may be done by substituting 2 4- a for a?, and u + bfoY y, and then developing w in a series of powers of z, so that u and z should = at the same time ; then sub- stituting ^ — a for 2f, and z/ — i for u. Under these cir- cumstances, the resulting condition would give x = a and ^ = i at the same time, and the quantities a and b would supply the place of the arbitrary constant. The integral would therefore have all the necessary generality. The proposed equation dy + t/dx = mx^'dx becomes, by the transformation just explained, du 4- (6 + u)dz = m(a + zy^dz* Let Hence we obtain OA2«-> + (a + 1)bz« -i- (a + 2)c;s«+» -f The condition a — 1 = 0, gives a = 1, A = TTia - 6, B = j-g , - *== 0:3 '^'=- The investigation may also be conducted by Taylor's series. If b be considered as a function of a, it will be- come db z^ d^ il . ^ _?1 when a is changed into a -^ z. And since y =: b ^ u, when a? = a + ;2r, •.* db z d^b z' ^ = ^ + 5S-T+d^^ da* 1.2 + Since a and b are a system of values of x and y, which 376 THE INTEGRAL CALCULUS. SECT. XXVI. satisfy the proposed equation, the same relation must subsist between a, b, and -r, as between x, y, and ^. Hence the value of -r will be found by substituting for a; and y in the differential equation the values a and b, and thence deriving db J-. The successive differentials of this equation will, there- fore, give the values of ^^, d^i' ' ' ' ' When z is small, the series will converge rapidly. To extend numerical calculations to greater values of a:, it will be necessary to substitute successively «, for a + r, and to change x into «, -H z, and then substitute a^ for a^ + z, and ^a + 2 for jr, and so on. This process becomes inapplicable when any differential coefficient becomes infinite. This can only happen when X =. a renders y infinite, or when the development of y contains fractional powers of x. If the series of exponents be known in this case, we may frequently employ Taylor's scries. If the exponents be such, that they are all mul- tiples of any one fraction — , then let x = z « , and the series of Taylor will be applicable. Ex. 3. Let the proposed equation be d\y -I- cx^ydx' = 0. Let + (a + /t)(« + 7i - 1)b^'^+^-* + (a + 2//)(a + 2A — l)c^«+=^*-^ + ]dx''. Also, d^y SECT. XXVI. THE INTEGRAL CALCULUS. 377 It is obvious,, that it is impossible that the terms a(a — 1)a^^-*, — CA^+", can be identified in any other case than that in which ji = — 2, which would only include a particular case of the proposed equation. Therefore, such a value must be as- signed to a as will remove the first term altogether. This is effected either by a = 0, or a = I. We may then identify the terms (a + h){a + 7i — 1)b^+^-^ — ca^+«, by the conditio^ ^ — 2 = w, ••• /i = 7i + 2. The two series will then agree, and the coefficients will be determined by the equations {a + h)(a + A — 1)b + CA = 0, (a + ^h)(a + 2^ - l)c + CB = 0. Since the number of arbitrary quantities a, b, is greater by one than the number of equations, one (a) will remain arbitrary. If the two values of a already obtained be substituted successively for it, we find the two series a — ■" (7^+l)(r^+2)(2AHF3)(2« + 4)(3^^+5)(37^ + 6) ^^ "" (7i+2)(72+3) "^ {71 r}-2)(w + 3)(2?2 + 4)(2/i + 5) Each of these series are particular integrals, since each con- tains only one arbitrary constant; but by changing the a in the last scries into a', and adding them, the result will be the 378 THE INTEGRAL CALCULUS. SECT. XXVI. complete integral, since the proposed equation is homo- geneous with respect to y and d^y, (394.) Another method of approximating to the integrals of equations by a continued fraction merits attention. Let y = 1 +Ba7^ 1 + c^*^ 1 + TjX"^ 1 + where the coefficients and exponents are indeterminate. Let hyf^ and aAof^'^dx be first substituted for y and dy in the proposed differential equation. If the integral corre- sponding to an indefinitely small value of the independent variable be sought, let all the terms of this equation in- volving the powers of x, whose exponents exceed the lowest exponent, be rejected. By a comparison of the corresponding terms, the values of A and a may be determined. Next let A^ be substituted for y, and its differential for dy, and by a similar process, b and h may be determined, and this process may be continued until a sufficiently near approximation to the integral may be found. Ex. 1. Let the proposed equation be mydx + (1 + x)dy = 0. First substitute Aaf for ?/, and aAaf*-^dx for dj/. Hence we find SECT. XXVI. THE INTEGRAL CALCULUS. 379 (m -f a)A^ + aAx'"-^ = 0, •/ {m + a)Ax 4- GA = 0. Neglecting the term ax, we find OA = 0, ••• a = 0, the quantity A remaining arbitrary. Now let _ A be substituted in the proposed equation, and the result is m(l -i- Bx^)dx - (1 + x)d{JiX^) = 0, •.• m — bBx^-^ + (771 — I))Ba/' = 0. Rejecting the last term, we find m = bBx^'^, which is satisfied by b = 1, 3 3= 772. Now let * A 2/ = -- 1 + mx 1 + ca?' Substituting this fori/, and its differential for di/, as be- fore, we obtain c = 1 and c = ^, and by continuing the process, we find A t/=z^ mx 1 + — l-(m-l)|- 1 + i(m + 1). 1 -\(m- ')4 l+4(w + 2). X 1 + . . . . . . 380 THE INTEGRAL CALCULUS. SECT. XXVI. (395.) When, as is frequently the case, the integral of the proposed equation can also be obtained in finite terms, this method furnishes a mean for converting the function which expresses the integral into a continued fraction. Hence, to convert a function of x into a continued fraction, dif- ferentiate it, and integrate the result by the continued fraction, supplying the arbitrary constant; this fraction will represent the proposed function. Thus, in the ex- ample just given, the integral in finite terms is a(1 -f ^7)"""*. Hence this function is equivalent to the continued fraction already found, and dividing both by the arbitrary constant A, we find (1 + x)- = 1 + mx 1 -k™- 1)^ 1 4- 4(« 1 .+l)-f - k™ - «>f l+i(m + 2). X "2 1 + ( By comparing the developments of e'' with that of 1-4 I , we find that they become identical when m is m/ supposed infinite. Hence, if in the fraction just found, ~ be substituted for w, and, in the result, m be supposed infinite, we obtain SECT. XXVI. THE INTEGRAL CALCULUS. e' = 1 + X •-^■f i+i-l ■ .-i.| t...| 1 . 381 Ex. 2. Let the proposed equation be dx - (1 -^ x'')dy = 0. By a similar process to that used in the former example, we obtain dx X y "J 1 _l_ ^n 1-1-0?" , X ^ 1 + 7i + l 1 1- "'"" ^+(» + lK2»+l) (2»»)^^» ' (3ra + l)(4,« + l) 1+ • In this case, if n = 1, we find 882 THE INTEGRAL CALCULUS. 7/1 , I ^\ _ . ^ SECT. XXVI, '+.1 2a: ■ ■+& Un = 1 + ... S, we find 1 ^ tan. a; _ ^3.5 1 + ... There are other methods of approximation, one depend- ing on the method of substitutions used by Newton, to resolve by approximation algebraic equations, combined with the methods of integrating equations of the first degree; also one derived from Lagrange's theory of the variation of arbitrary constants ; but the discussion of these would lead us into details unsuitable to the ends of this treatise. SECT. XXVII. THE INTEGRAL CALCULUS. 383 SECTION XXVII. Integration of differential equations of two variables by the geometry of 'plane curves, (396.) Before the methods of approximation to the roots of algebraic equations were known, a method of representing them by the co-ordinates of the intersections of plane curves was used, (Geometry, Sec. XX.) This method is, how- ever, now introduced into the elements of mathematical science only on account of its elegance, since it has been altogether superseded, for practical purposes, by the more accurate process of approximation. In the same manner the calculus, when in its infancy, borrowed methods of in- tegration from geometry, which, though they have since been abandoned for the more useful and accurate methods of approximation, yet merit notice for their elegance, as well as because they constitute the particular connexion with geometry, which first led philosophers to the discovery of the calculus. The problem which called this science into existence (Geometry, Introduction, p. xxv.), was " to draw a tangent to a given curve," and hence the differential calculus, imme- diately after its first discovery, was called " the method of tangents" Problems of another kind presented themselves, which proposed the discovery of the curve from some given property of its tangent. As the former depended on what is now called " differentiation," so the latter depended on what is now called " integration." The integral calculus, when in its infancy, was therefore B84 THE INTEGRAL CALCULUS. SECT. XXVIT. called " the inverse method of tangents." As the calculus, however, advanced to a greater state of perfection, and became more extended in its applications, these deno- minations were necessarily abandoned, being in no respect adequate to the extent of the science. They include no application of either calculus but a geometric one, and even in that, contemplate no differential coefficient beyond the first. The " inverse method of tangents" consisted in constructing the curve represented by a given differential equation of the dij first order. If the equation be solved for -^ -, let this be y. The subtangent is therefore —, and the tangent is " -j • Hence by means of an equation between the ordinate y and the differential coefficient z/', the curve may be constructed by points, and this will represent the integral of the pro- posed equation. (397.) Let the proposed differential equation be y^x^t/) = 0, y being the first differential coefficient. Let the curve be assumed to pass through a point, of which the co-ordinates are x = a and y = b, a and b being values which do not render ?/ in the equation ¥{xyy^) = imaginary. The equation Y{ah/) = will give the value of ?/', by which the position of the tangent will be known. A point indefinitely near the assumed point, and also upon the tangent, being assumed, and its co-ordinates, in like manner, substituted in the proposed equation, another value of y may be deduced, which will determine the direction of another tangent. Then a third point being assumed upon this second tangent indefinitely near the second assumed point, a third tangent may be found, and by continuing the process, and not pro- ducing the several tangents beyond the several assumed points, a polygon will be determined. The smaller the i SECT. XXVIT. THE INTEGRAL CALCULUS. 385 distances between the several points are assumed, the more nearly will this polj^gon approach to a curve, and the curve, which is its limits when the several distances are supposed actually to vanish, is the geometric representation of the in- tegral of the proposed equation. (398.) A still more accurate and rapid approximation to the curve may be obtained by the following process. Let the equation Y{xyy^) = be differentiated, and the value of the second differential coefficient obtained, as a function of the two variables and the first differential coefficient. Hence may be found the radius of the circle osculating at any pro- posed point. As before, let a point be assumed, and the tangent at that point found by the proposed equation, and thence the normal. The radius of the osculating circle being determined in the manner already described, let a part equal to it be assumed upon the normal in a direction determined by the sign of the second differential coefficient (151.), and let a small arc of the osculating circle passing through the given point be described. Upon this arc, and near the given point, let another point be assumed, and the circle osculating at that point being found as before, a third point may be assumed upon its arc, and so on. By this process a polygon will be found, the sides of which are circular arcs, and the smaller these arcs are as- sumed, so much the nearer will the polygon approach to the curve which represents the integral of the proposed equation. The limit of this polygon, when its sides actually vanish, is the geometric representation of the integral of the proposed equation. The first point, arhitrarily assumed in these cases, represents the arbitrary constant. (399.) If the proposed differential equation be of the second order, it is necessary not only arbitrarily to assume a point through which the curve is supposed to pass, but also the direction of the tangent at that point. This is equivalent to c c 396 THE INTEGRAL CALCULUS. SECT. XXVII. assigning particular values to x, y, and i/\ in the equation v(xyM) = 0. Hence the value of y is determined, and the direction of the curvature and the radius of the oscu- lating circle are known. Proceeding then as in the last case, a polygon, whose sides are small circular arcs, may be determined, the limit of which represents the integral of the sought equation. (400.) In approximating to the integrals of equations of the higher orders, the osculating parabolas (134.) are used, their several parameters representing the arbitrary constants. The osculating parabola of the second order may also sup- ply the place of the osculating circle in the former cases. (401.) When the variables in the proposed differential equation are separable, its integral may be otherwise re- presented by geometrical construction. Let it be reduced to the form Ydy -f xdx = 0, where Y is a known function of ?/, and x of x. Let two curves be constructed relatively to the same axes of co-ordinates, represented by the equations .3/ = x, a? = Y, '.'f(yidx 4- Ydy) =JXydx) -^/{xdy) = 0. But the area of any part of the first curve intercepted between the axis of 3/, and any proposed value of ?/, repre- sents the first integral ; and the area of the second intercepted between the axis of x, and the value of x corresponding to the same value of y^ represents the other. Their com- bination, therefore, represents the integral of the sought equation. The preceding results also show that every differential equation between two variables has an integral, a theorem which was before estabhshcd in Sect. XVI. SECT. XXVIII. THE INTEGRAL CALCULUS. 087 SECTION XXVIII. The problem of trajectories and other geometrical applka^ tions of the integral calculics. (402.) Amongst the different questions to which the in- vention of the calculus gave rise, and which were proposed very soon after its invention, one of the most interesting is the *' problem of trajectories.'' In the correspondence between Bernoulli and Leibnitz, on subjects arising out of the new calculus, Bernoulli proposed the solution of the problem, " to find the curve which intersects at right angles a system of curves of the same kind described according to some given law." This problem, he considered, would lead to the solution of the physical problem, to determine the path of a ray of light through the atmosphere, Hght being supposed to be propagated according to the Huygenian hypothesis. The problem soon became generalised to that of the determination of the curve which intersects a system of similar curves at the same angle ; such a curve is called a trajectory * of the proposed system of curves, and if it intersect them at right angles, it is called the rectangular trajectory. By '' similar curves," is here meant curves whose equa- tions having the same form, differ only in the value of one of the constants, which we shall call in general the variable parameter. * The term " trajectory," used here, has no relation to the same term used in physics, where it signifies an orbit described by a projectile round a centre of force. c c2 •f 388 THE INTEGRAL CALCULUS. SECT. XXVIII. (403.) Let the equation of the proposed system of curves be F(a?j/c) = 0, the constant c representing the variable parameter, and for every particular value of which the equation Y{xyc) = represents some one of the proposed system of curves. Let the equation of the sought trajectory hej'(xj/) = 0. Let the differential coefficient deduced from the equation F( tan.-i— = c, c being an arbitrary constant. Let z^ =: x^ + y^, and m = tan.-^ — , •.• h -{• w cot.^ = c ; or, if when a? = 0, we suppose that ^ = 0, •.• c = 0, and the equation assumes the form — w . COt.(p — COt.f JLet e = a, *.• which is the logarithmic spiral, of which it is a characteristic property to intersect at the same angle all lines through its pole*. In this case, for the rectangular trajectory cot.

0, the trajectories are a system of similar ellipses. * Geom. (433.). SECT, XXVIII. THE INTEGRAL CALCULUS. 391 having a common centre at the common vertex of the system of parabolas, and an axis coincident with the axis of the system, the ratio of their axes being 1: ^/m. If the parabola be the parabola of the sec6nd degree, tt* = S. This case is remarkable for having been the first to which the problem of trajectories was applied. The general problem having been proposed by Bernoulli, Leibnitz gave a general me- thod of solving it, and effected the solution in this in- stance as an example. Leibnitz*'s method was founded upon the variation of the constant c in passing from one curve of the proposed system to another, from which he deduced his method of differentiation de curvd in curvam. If m < 0, the proposed equation represents a system of hyperbolas having a common centre and asymptotes, and the trajectories are also a system of conical hyperbolas, of which the axes coincide with the common asymptotes of the system. If the given system of hyperbolas be equilateral, the tra- jectories are also equilateral hyperbolas. PROP. cv. (407.) To determine the trajectory of a system qfcircks toitching a given right line at a given point. The right line being assumed as axis of «/, and the given point as origin, the equation of the circles is ?/« + a?^ - 9^rx = 0, ••• jOj/ -f (a? ~ r) = 0, ••• r = py ■\' X, This being substituted in the first, we find Hence the difierential equation of the trajectory is S92 THE INTEGRAL CALCULUS. SECT. XXYIII. dy(y'^ - .r« ~ 2 cot-ip • yx) +[%^+cot.(?(2/* - x^yidx- 0. This equation being homogeneous, may be integrated by (313.). If the rectangular trajectories be sought, cot.

K- + f) dy Obtaining from this the value oi y.. , and dividing by the radical, we find dy ""-ydx , , = +1=0. , ^a'^-\-9^ax—y^ Integrating this, and supplying the constant, we find an equation of the form ry^^x^ — ^rx ^- A = 0, which is that of a circle. This is the general solution. 394 THE INTEGRAL CALCULUS. SECT. XXVIII. The singular solution is obtained by putting the radical = (339.)> ••• it is which is the equation of a parabola. This parabola is the curve to which all the circles included in the general so- lution are tangents. If K'^+^^)=^+^' a being a constant quantity, we have \' 3/2 _ 2ax + c, which shows that the parabola is the only curve whose sub- normal is constant. (410.) Geometrical questions which relate to the oscu- lating circle are solved by the integration of differential equations of the second order. The following proposition furnishes an' example of this. PROP. CVII. (411.) To determine tJie curve in which the radius of the osculating circle is a g-ivenjunction of the normal. This problem, reduced to an equation, is d^i/da: ~^n\ +^V 3' the integration of which will solve the problem. If the radius be equal to the normal, the equation be- comes di^^^-dx'^ _ y d^T/da; "~ div* SECT. XXVIII. THE INTEGRAL CALCULUS. 395 ••• d^ + dx" + yd'^y = 0. The first integral of which is ydy + xdx = cdx^ which being again integrated, gives ^« + j?« - 2c.r + c?' = 0, which is the equation of a circle of which the centre is on the axis of x, (412.) The student will find no difficulty in reducing geometrical questions relating to contact or curvature to equations. These equations are generally of the first or second degree, and integrable by the rules already esta- blished. To extend the examples on this farther would occupy more space here than the difficulty of the inves- tigation requires. We shall therefore conclude this section with the following proposition, as an example of another and different species of problem. PROP. cvin. (4)13.) A system of parabolas Itaving a common ver- tex and axis, or hyperbolas havvng common asymptotes, being given, to find the curve which intersects them all, so tJiat the areas included by the co-ordinates of the point of intersection, and the arc of 'the parabola or hyperbola between tlmt point and the axis qfy^ shall be constant. Let the equation of the proposed system of curves be y = px"^. The area included by the co-ordinates and the arc is fydx = pfx^d^ = -^^, No constant is added, as the area is supposed to commence when 07 = 0. THE INTEGRAL CALCULUS. SECT. XXIX. If m = — 1, the integral is fydx = pZ(j7), and if 771 < — 1, the area is infinite when a: = 0, which is also the case when w = — 1. These cases will then be excepted in the following investigation, which will therefore only apply to parabolas, and to such hyperbolas as have ?7l > - 1. Let the given area be a, '.• A = » —=, ^m + 1 Eliminate p by this and the equation of the proposed system, and the result is yx = A(m + 1), which is the equation of a common hyperbola. SECTION XXIX. Of the integration of total differential equations of the first degree of several variables, which satisfy the conditions of integr ability, (414.) A total differential equation of the first order between three variables must always come under the formula vdx + Q.dy + B.dz = 0. If the first member of this equation satisfy the criterion of integrability, (286.), for functions of three variables, its integral may be immediately obtained by the rules for these functions, and will be of the form Ti^xyz) + c = 0, c being an arbitrary constant. SECT. XXIX. THE INTEGRAL CALCULUS. 397 If any one of the three variables be capable of being separated from the other two, the equation may be integrated by the rules for the integration of functions of two variables. For if z be separable from x and y, the equation may be reduced to the form zdz = vdx + Q(iy, where z represents a function of z. This separation is easily effected whenever the given equation has the form z{vdx + Qjdy) + nildz = 0, by dividing the whole equation by zr ; P, Q, and n, being functions of x and y only. (415.) If the proposed equation be not an exact dif- ferential of an equation of three variables, it may sometimes be rendered so by the introduction of a factor. To de- termine the condition under which it is rendered integrable by a multiplier, let Tvdx + Tazdz + dz =0, '.' z'^ + z = c. Hence the integral sought is xy -{■ zy + zx + c = 0. Ex. 2. Let the proposed equation be zdx + xdy + ydz = 0. In this case P = 2;, q = «r, and R = «/, by which it ap- pears that the equation [2] is not fulfilled, and therefore the proposed equation is not integrable. If this equation were submitted to the preceding process, we should find that z could not be disengaged from x and z/, so that we should find z = Y{xyz\ Ex. 3. Let the proposed equation be such, that p = «/'^ + 3^^ + 2*, Q = or- + X2 + z^ R = J78 _|_ jj.^ _}_ ^2^ In this case the criterion [2] is satisfied. If dz = 0, we have dx dy x'i^xz-\-z'^'^ y^-\-yz-{-z'^ ~ * Since z is constant, the integral of this is -A-Jtan.-i±2? + tan.-^^+!^}=/(4 If an arc be a function of 2, its tangent must be also a function of z ; and hence by taking the tangents of both sides, we find z^—zx—zy—^xy Differentiating this, and identifying the result with the pro- posed equation, we find %x^z + '^txyz -\-y^z + z^x + z'^y + ^"y + y^x)dz -|- (2* — zx -- zy ^ ^xyydz = 0. SECT. XXIX. THE INTEGRAL CALCULUS. 401 Eliminating the latter parenthesis by 2* — z.v — zy — %xy = , and expunging the common factor {x -\-y -{- z), we obtain 2{a:y +yz -{- xz)z^dz + (a? + 2/ + 2)2* 6?z = 0. Also by the integral just obtained, we find Making this substitution, and dividing by the common factor 2z(2® — ii/), we obtain z(z — \)dz -{■ zdz = 0, dz dz dz cz z •.• z = . z — c Hence the integral required is (xy + xz + yz) — c{x ■{■ y -\- z) = 0. (417.) If the proposed differential equation exceed the first degree, these methods are only applicable when it can be decomposed into rational factors of the form vdx + Q,dy + Kdz — 0; this being the only form it can have when it is an immediate differential. If for example the proposed equation be vdx^ + Q.dy'^ + Rdz'' + ^sdxdy + 9.Tdxdz + 9.wdydz = 0. When this is solved for dz, the quantity under the ra- dical is (t* — VB)dx^ + 2(tv — K%)dxdy + (v^ — QjC)dy^. It is necessary that this should be a complete square, which can only take place under the condition (tv — Rs)* — (t2 — pr)(v* — qr) = 0. D D 402 THE INTEGRAL CALCULUS. SECT. XXX. SECTION XXX. Integration of total differential equations which do not satisfy the criterion of integr ability, (418.) Differential equations, which do not satisfy the criterion [2] established in the last section, were long con- sidered as absurd or impossible relations ; and all questions, whose solution was reduced to such equations, were con- sidered as involving some contradiction, as is the case when the solution involves the even roots of negative quantities. MoNGE, however, has shown that this is not the case, and that such equations indicate a real relation between the variables. It happens, however, that the integral of such an equation is not, like those which satisfy the criterion, one equation between three variables, but it is expressed by two equations between three variables which must subsist to- gether, and which involve an arbitrary function of one of the variables. (419.) The integral of an ordinary differential equation of three variables, which satisfies the criterion of integrability, would, if represented geometrically, be a curved surface, since the integral is an equation of three variables. The integral of an equation which does not satisfy the criterion, if represented geometrically, would be a class of curves of double curvature, enjoying some common characteristic pro- perty. For each value of the arbitrary function which enters the system of equations, there is a particular curve of double curvature. The part of the equations which does not depend on this function, being common to all particular SECT. XXX. THE INTEGRAL CALCULUS, 403 values of the function, gives the general geometric cha- racter to the class of curves. (420.) To determine the system of equations which re- presents the integral of any given equation of this kind, let z. be considered as constant, •.* vdx + Qidy — 0. Let T be the factor which renders this integrable, and let u + z = be the integral of ^ ivAx + TQciz/ = 0. Diiferentiating u + z = 0, and identifying it with I'edx + iQidy + TRof^ = 0, we obtain u + z = 0, dz du dz dz dz In this case ^r is not a function of the variable z alone, dz for if it were, the equation would be integrable by the process (416.), and would fulfil the criterion, which is contrary to hypothesis. These equations must then subsist together, z being an arbitrary function of z. Let z = f(z), . dz , . and^=F'(^), •.• U -f ¥{Z) = 0, d\5 Tz + * <") - ™ =-<^' which are two equations between the three variables, and taken together, represent a relation between jn/^, which satisfies the proposed differential equation. (42 L) Since the function f(;s:) is absolutely arbitrary, it follows that there are an infinite number of systems of two equations which satisfy the proposed equation, and that, therefore, it has an infinite number of systems of integrals. If the integral be represented geometrically for each form dd2 404 THE INTEGRAL CALCULUS. SECT. XXXI. assigned to the function f(2), there is a different curve of double curvature. The terms u, t, and r, however, not changing with the form of this function, will give some common character to all these curves. As an example, let the proposed equation be dz xdx -{-ydy ^ In this case, vdx + Z--C x{x-'a)-\-y{y-'h) xdx\ydy 1 R = . z — c T =^(^- a) -^ y(y - b). Let We find Hence U = .37* + «/\ . oo(x — «) + 2/Cy — *) = f'(«) • (^ — ^)- in which f(2) is absolutely arbitrary. SECTION XXXI. Of the integration of partial differential equations of the first order, (422.) The integration of partial differential equations is a part of the calculus which has not yet reached that state of perfection which might enable an elementary author to introduce such an exposition of its principles as is suitable to the class of students for whose use his work is intended. In this, as in some other parts of the calculus, the utmost which SECT. XXXI. THE INTEGRAL CALCULUS. 405 can be attempted in the present work is to explain the methods of integrating some particular classes of equa- tions, which are most suited to our object, referring students, desirous of further information, to such works as the com- plete treatise of Lacroix. (423.) The most simple class of partial differential equa- tions are those which involve but one partial differential co- efficient. The integration of these may be always reduced either to the integration of functions of one variable, or to the integration of equations of two variables. Let jp be a partial differential coefficient of z with respect to or, i. e, dz p = y-y and let m be a function of x and several other variables, and let the given partial differential equation be F(p, u) = 0. First, suppose that u does not include the variable 2. Let the equation in this case be solved for p, and its value sub- stituted, •.* dz , ^ T. =•/(")' *.• dz ^/{u) • dx. dz Since the coefficient -^ was obtained by differentiating z as a function of x only, all the other variables being con- sidered constant, so the integration must be effected upon the same supposition. Let oc therefore be considered to be the only variable in w, all the others being taken as con- stants, and let the integral o^ /{u) • dx be found by the rules for the integration of functions of one variable. Let the integral be 2 = u + c, u expressing the function of all the variables obtained by the integration, and c the arbitrary constant. Since all functions of the variables, not including x, which 406 THE INTEGRAL CALCULUS. SECT. XXXI. entered the original function, necessarily disappeared by the dz differentiation which gave -^5 it therefore follows that an arbitrary function of these variables should be introduced in the integration. We must then consider c, not as an ar- bitrary constant, but an arbitrary function of all the variables, except z and x, (424.) If, however, the function u contain z as well as x, the equation may be integrated as a differential equation between z and x, the other variables being considered as constants ; and in place of an arbitrary constant, introducing in the integration an arbitrary function of the other va- riables. (425.) The most general partial differential equation of the first degree, including two partial differential coefficients, is of the form pp + Q^ = V • • . . [1]. We shall consider p and q as the partial differential co- efficients of z with respect to the variations of x and «/, •.• dz dz If other variables enter the functions p, q, v, besides x, y, and z, they are to be treated as constants ; and in place of arbitrary constants, arbitrary functions of these other va- riables should be introduced in the integral. In what fol- lows, we shall consider the equation [1] to include only the variables x^ y, and z. (426.) By the definitions of partial differentials (94.), wc have dz — pdx H- qdy • • • • [2]. Eliminating p by this equation and [1], the result will be vdz -^ \dx = q{vdy - adr) .... [3], This equation must be satisfied independently of q^ since in SECT. XXXI. THE INTEGRAL CALCULUS. 407 the proposed equation q is indeterminate. The integration proposed may be reduced to two cases : l^'. When vdz — ydx does not contain y, nor vdy — odx, z, or what amounts to the same, where these variables may be disengaged from them. 2^. Where one or both of these quantities contain all three variables a;, «/, z. (4^7.) 1^ If the quantity Fdz — vdx do not contain z/, it either is an exact differential of a func- tion of x, z, or may be rendered so by a factor. Let the factor which renders it exact be f^, and let the function of which it is a differential be m, •.• vdz — vdx = — . In like manner, since pdy — Q.dx does not contain z, we have pay — QCld? = — 7 - Hence the equation [3] becomes dm = ^dM'. QIX, This is only integrable when -—■ is a function of m'. Let f'(m') = ^du^, and let the integral of this be f(m'), •.• M = f(m'), where f(m') is an arbitrary function of m'. Had q been ehminated by [1] and [2] instead of p, the result would have been Q,dB — \dy — plQdx — vdy), and the integration would, in this case, depend on the in- tegration of the formulae 408 THE INTEGRAL CALCULUS. SECT. XXXI. adz — \di/y adx — vdy. It therefore follows in general, that if any two of the three equations vdy — adx — 0^ vdz — vdx = > • • • • [4], Q.dz — \dj/ =z OJ be integrated, and that their integrals be m, m', each of which represent functions of a:, i/, z, the integral of [1] Will be M = f(m'), the form of the function being absolutely arbitrary. (428.) We have supposed that each of these formulae [4] excludes one of the variables. The principles we have just established are, however, applicable, even if any two of the formulae [4] included all the three variables, provided that the third contained only the two variables whose differen- tials are engaged in it. For this being integrated as a function of two variables, and its integral being m = 0, either of the two variables may be eliminated by means of this integral; and either of the other two formulae, in- cluding the three variables, by which a formula may be ob- tained, including only two of the three variables and their differentials, whose integral m' being obtained, the integral of [1] will be M = f(m'), the function as before being arbitrary. (429.) Even if the three equations [4] should all contain the three variables, yet, if any two of them, and therefore the third (since they are not independent), be satisfied by the equations m = and m' = 0, the integral of [1] will be M = f(m') as before. To prove this, it will be necessary to show that the differential of m = f(m') satisfies the con- ditions [4] independently of the form of the function. SECT. XXXI. THE INTEGRAL CALCULUS. 40^ Let the difFerential of the equation m = f(m') be dm = f'(m')Jm'. That this may be satisfied independently of f'(m'), the form of which depends on that of f(m'), it is necessary that the conditions cIm = 0, du' = 0, should be fulfilled. Since m and m' are functions of a^ y, , their differentials must be of the form ' Adoc + Bdy 4- cdz = 0, A!dx + B'dy -{- ddz = 0. If the equation m = f(m') be differentiated with respect to z and ^, we shall have Adx 4- cdz = F'(M')(A'dx + ddz); and if it be diflPerentiated with respect to z and^, Bdl/ + cdz = F'(M'){B'dl/ + ddz). Substituting for ^ and |- their values p and ,, we find [C - dF'(M')]p + A - aV(m') = 0, ^ [c - c'f'(m')]^ + B ~ bV{m') = 0. Deducing hence the values of ^ and ,, and substituting them in [1], we find ^ AP + BQ + cv = F'(M')(A'P + b'q + c'v). Substunung in the values of du, du', the values of dr, dy obtamed from [4], and taking out the common facto; S, AP + EQ + CV = J a'p + b'q + c'v = S ' ■ ' ■ f^l- Hence the above equation is satisfied independently of the form o .V). It follows, therefore, that unless L dif! ferent,a Is of M and m' combined with [4] satisfy the con- J^ns [5], the equation m = p(m-) „,„„,, ^e tL integral of the proposed equation. ^ It is obvious that M = « and m' = b are particular in- tegrals, a and b being arbitrary constants, for p(m') may be 410 THE INTEGRAL CALCULUS. SECT. XXXI. considered constant, and *.• the equation m = f(m') becomes, in this particular case, m = ; or f~^(m) * may be constant ; in which case the equation becomes m' = h, (430.) If V = 0, the equation [1] becomes pp 4- Qg- = 0, and the equations [4] become vdy — Qdr = 0, dz = 0. Hence z = m, and there can be only two variables in the first, the integral of which being m', the complete integral will be ;2 = f(m'). For example, let the proposed equation be^^ = qi^, \' xdy — ydx = 0, •.* y = ajt, and a = f(z), *.• y = ¥{z) • a;, oic z =J'('—\ which is the general equation of conical surfaces. l£py = qx, •.• p = 3^, a = - ^, ••• ydy 4- xdx = 0, •.• 2/2 + ^« = m', '.' z = ¥{x" -f /), which is the general equation of surfaces of revolution round the axis of z. Let q =-• Fp, V not containing z. The integral is z = f(m'), m' =fT(dx 4- py), 1 being the factor which renders the proposed equation in- tegrable. If two of the equations [S] contain but two of the three variables, the integration presents no difficulty. For ex- ample, let the proposed equation be px -{• qy = uz. Hence xdz = uzdx, xdy = ydx, F-H^) means a quantity u, such, that b(w) = m. SECT. XXXI, THE INTEGRAL CALCULUS. 411 z X \x J This result, applicable to homogeneous functions, has been already obtained in (322.). (431.) Ex. 1. Let the proposed equation be V^ + qf = ^% •/ x'^dz •=. z^dx, X'dy = ifdx^ * 2 X " "* y X ^ "* z X \y xj x — z /x — iA or =:f( ^). zx \yx J Ex. 2. Let the proposed equation be g^ = x^; + v, where X and V are functions of x only. Hence y.dz + ydx = 0, xdy + cir = 0, „\dx / r.dx\ ••- = -/— + .(3^+/-). Ex. 3. Let the proposed equation be qxy — px^ = ^/^ ••• x^dz -^y^dx = 0, x^dy -f- ^j/f/x = 0. In this case, one of the equations [4] includes but two variables. This being integrated, gives xy = m'. Sub- stituting in the first — for y, it becomes X x*dz H- u'^dx = 0, which being integrated, gives z = Jm'2^-3 _^ ^^ Substituting xy for m', and F(xy) for m, we find, for the integral sought, Szx = 3/2 -f SxF(xi/). 412 THE INTEGRAL CALCULUS. SECT. XXXI, Let the equation be *.* xdz = n*/x^ +y^ dx, xdy — ydx = The latter being integrated, gives y - y^x, by which y being eliminated, the former becomes dz — n y/\ -^r m'2 dx^ \' z — nx^ 1 + m'^ = M, Hence z = n\/x^ + «/* + F( (432.) ^, In general, when each of the equations [4] contains all the variables, they cannot be integrated sepa- rately, because we cannot suppose two of the variables to change, while the third remains constant. Various analytical artifices have been suggested for obtaining the integral in these cases. By integration by parts, the equation dz = pdx + qdy, may assume any of the three following forms : z=^ px +f(qdj/ - xdp), ^ = 9y -^/{pdx - ydq), z =px + qy -/{xdp +ydq). It frequently happens that we can obtain the sought integral by substituting the value of p or q derived from the proposed equation [1] in any of the preceding. For example, if jp be a function of q, so that p = a, the last of the preceding equations becomes z= ax -{- gy —/(xo! -f y)dq, where a' = -y-. Hence dq xq! + y = F'(q), ',' z = Qx ^ qy - F{q), where the function r is arbitrary. The integral for par- SECT. XXXI. THE INTEGRAL CALCULUS. 413 ticular forms of the function F(q) may be found from these equations by eUminating q. (433.) The integration of partial differential equations of the first order is sometimes effected by the following process. Let the given differential equation be Y\xyzpq) — 0. Let this be solved for either of the partial differential coefficients {p), and let the value of jo, thus determined, be substituted in dz = pdcc + qdj/, by which we obtain an equation of the form dz =:f{xyzq)dx + qdy. Let 9 be such a function of q, as being considered constant, this equation will become an exact differential ; and let its integral be II = Y^ocyzQ) = c, c being an arbitrary constant. This equation being dif- ferentiated, 9 being considered constant, ought to reproduce the differential equation from which it was obtained ; and it should also reproduce it, if 9 being considered variable in the first member, c were such a function of 9 as would fulfil the condition For differenitiating, as if B and c were both constant, we should get du . du . du , and differentiating, considering 9 variable, and c a function of fl, we should obtain du ^ du ^ ^ du , du ,. dc ,^ dx ' dy ^ ' dz dS d9 In order that this and the former may be identical, we must therefore have 414 THE INTEGRAL CALCULUS. SECT. XXXI. du ,- dc ,- Hence, if by taking 9 as constant, the equation becomes an exact differential, we obtain by integration an equation of the form u = F{xyz^) =/(9), the function 9 being restricted by the condition du _ df{S} W~ dS ' These two equations will satisfy the proposed equation, the function y(fi) being arbitrary. If this function be deter- mined, the elimination of by the two equations will give the integral of the proposed equation. As an example of the application of this method, let the proposed equation be 2 =^ pq- Substituting in z = pdx + qdi/, the value of p derived from the proposed equation, we find z dz = — da; + qdy, qdz—zdx This will be an exact differential, if = g^ — a: and be considered as constant, '.* _ {^'{■x)dz — zdx . . z ^ df{Q) ' • (A' + d)« d^ ' These two equations conjointly represent the integral of z=zj)q. (434.) The integration of partial differential equations of the first order is often effected by the introduction of an indeterminate quantity. Let the proposed equation be SECT. XXXU THE INTEGRAL CALCULUS. 415 f{px) = F(qi/). hei/ipx) = w, *.• T{qy) = w. Deducing from these the values ofp and q, we obtain equations of the forms p =^f'(Xaj), q = F'(2/a;), ••• dz =.f\x(i)) ' dx 4- ¥\yu))dy. Let the integral of /' (xuj)dx, integrated with respect to x be p, and that of F'(i/uj)di/, integrated with respect to «/ be Q. Hence, considering that p and q must be also functions of the indeterminate w, we have dp dv f^(Xj (jo)dx — -T-dx = dp — '1~^^9 f'(2/, oj)dy = -^dy = dQ.^ -^dcv. Hence we find dz = dp + da — (— + --j—jduj. This equation can only be an exact differential when the quantity within the parentheses is a function of w, i. e. dp da -5;; + "ST = «'(">' Hence the combination of equations Z -f (p(uj) = P + Q, dp do, • ^ ^ duj d(v' where Since all the differential coefficients which enter this equa- tion relate to the variation of a?, of which v is a function, it may be integrated as an equation between two variables V, ^, the variable 7/ being treated as a constant, and intro- ducing m arbitrary functions of j/ in the integration in place E E 418 THE INTEGRAL CALCULUS. SECT. XXXII. of the m arbitrary constants. The quantity v being ob- tained by this process as a function of x and y, the final in- d'^z tegral will be obtained by integrating -r-j;^ = tJ. In this last integration, x being taken as constant, it will be necessary to introduce n arbitrary functions of x. Thus the complete integral will include m arbitrary functions of ^, and n arbi- trary functions of x. (438.) ^, Equations of the rith order, which include partial differential coefficients with respect to one variable only, may be treated as differential equations between two variables, scil. the function and the variable with respect to which the differentials are taken. In this case, however, in place of introducing arbitrary constants, it will be necessary to introduce arbitrary functions of the remaining variables. Under this case come the two following forms of equations of three variables : fU, dz dH dx^' d>'z\ ' dx'O'^ = 0, fUj dz y^^' dy' d^z dy''' d"z\ ' dy-J- : 0. The equations 4-^ d"z % dx^dy^ d''z % where p and q contain no variables. except X andj/, also d'^z come under this class. For let -j-r = V, '.' the former be- comes ' d'^v dy^ + Pt; = Q. And by a similar substitution, the latter assumes the form d"v dx" + PV = Q. SECT. XXXH. THE INTEGRAL CALCULUS. 419 If w = 1, the former equation will become dv ^ + w = ^, which is of the first order with respect to v and ?/, and may be integrated according to the rules already given, sup- plying an arbitrary function of x in place of the constant. (439.) Before we proceed to consider more general equations, we shall illustrate the preceding cases by some examples. Ex. 1. Let the proposed equation be fe _ dz where p and Q are functions of x, y^ and z. Let -=-=p, •,• dp dx '.' dp =i vpdx + Qidx. If w =JvdXf the integral of this equation is (314.), dz f{y) replacing the arbitrary constant. Integrating this again, we obtain the sought integral, in- troducing another arbitrary function of z/. Ex. % Let the proposed equation be d'^z dz •.• z =fdxf<^dx + xji^y) +/'(«/),. /(j/) andy'(«/) being arbitrary functions of y. Ex. 3. Let the equation be dH %« ^xy. Az Xlf +/W, •.* Z = xy^ ■ 6« -l-.y/W +/'W EE 4^ THE INTEGRAL CALCULUS. SECT. XXXII. Ex. 4. Let the proposed equation be d^ dxdj/ ~ ' M being a function of ^, j/, z. First integrating with re- spect to^, we obtain dz ■^=fMdy+f{x). And integrating with respect to x, we find z =fdx/kdt/ •\-ff(x)dx +f'(i/% f(iv) and/'(3/) being arbitrary functions. Ex, 5. Let the proposed equation be dH 5^ = ^^ + *^' ^ = ^a^" + hyx +/(«/), •.• z = lax'^y + \hy^x + f(^) + F'(;r), where f(^)5 F'(:r), are arbitrary functions. Ex. 6. Let the proposed equation be d'^z _ dz_ dxdy ~ dx M and N being functions of x and y. Let -r- = p> '.*^ ^ = MP + N. Integrating this by (314.), we find p = e''\j(x) -\-Je~''Ndi/]. Integrating this with respect to x, we find z ^fie^'dxfe-'^Ndy) +fe''F{x)dx + F'(y), where F (a?), f'(^), are arbitrary functions. Ex. 7. Let the proposed equation be d^z , dz ^^5^ = ^^^ + ""^^ dz ay . , ^ 1 SECT. XXXII. THE INTEGRAL CALCULUS. 421 (440.) We shall now proceed to consider the method of integrating partial differential equations of the second order and first degree. The most general equation of this kind is dH d"~z dH dx^ ^ dxdy dxf- ' where r, s, t, v, are given functions of ^, 3/, z^ and the partial differential coefficients of the first order. Let dz dz d^ "^P' '^^^' d^z __ d^z _ d^z _ d? "" ^* d^y "^^ W*' *.* dp = rdx ■\- zdy^ dq = sdx + tdy. By the last two equations, and the general equation rR + 5S + ^T = V, any two of the three differential coefficients r, s, t, may be eliminated; the third will, however, still remain indeter- minate. If r and t be eliminated, the result will be ndpdi/ + idqdx — \dxdy = siyidy"^ — sdwdy + Tdx% This is simplified by putting dy = mdx; \* dz = pdx + qmdXf by which substitution, it becomes ^mdp + idq — \mdx = ^(rw^ — sm 4- t) • • • • [!]• Since the quantity s must remain absolutely indeterminate' the integral sought must satisfy the conditions Rm2 — sw + T = • . • • [2], ■Rmdp + idq — vjndx = • • • [3]. If M = o, m' = fit', be two equations which satisfy these conditions [2], [3], M, m', being functions of x, y, s, p, and q, and a, a', being arbitrary constants, then the equation 422 THE INTEGRAL CALCULUS. SECT. XXXII. M = r(M') [4], in which the form of the function is arbitrary, will be the first integral of the proposed equation. To prove this, it will only be necessary to show that the differential of [4] will always be satisfied by the conditions [2] and [3], inde- pendently of the form of the function f(m'). Let the differential of [4] be dm = f^(m')^m', f'(m') being the differential coefficient of f(m') with respect to the variation of m'. Since m, m', are functions of the three variables, and the two first differential coefficients, the total differentials dm, dm', have the forms du = Adx + Bdi/ + cdz 4- vdp + T£.dq, dM.' = A'da; + b'% + ddz -f D'dp -{- E'dq, which, by the substitutions of ?w jjr for Jz/, and pdx + qmdx for dz, become (Zm = (a + Bm -\- cp -{- cqm)dx + Jidp + 'Edq, dm' = (a' + B'm -}■ dp + dqm)dx + n'dp + E'dq. Substituting in these the value of dq, deriyed from [3], they become EV DT — ERWl , AM = (a + Bm 4- €/? + cqm H m)ax i dp, ■, , . , . , , , e'v ^ , . d't -- E'Rm , dm' = (a' + Bhn + dp -\- dqm H m)dx -\ ; dp. Since by hypothesis the functions m, m', are constant, these differentials must each = ; and since dx and dp are in- determinate, these conditions must be satisfied by their coefficients. Hence we obtain the four equations of con- dition, t(a + am + cp + cqm) -f Evm = 0, t(a' + B'm + dp 4- dqm) + E'vm = 0, DT — ERm = 0, d't — e'ewx = 0. SECT. XXXII. THE INTEGRAL CALCULUS. The four quantities a, a', d, d', being eliminated by these conditions, and the equation xdx + B^Zy + cdz-\'Jidp-\--E.dq=:^\u\A!dx-\-'B^dy-{-ddz + the result, after substituting pdx + qdy for dz, will be (b +■ cq)(d^ — mdx) -\ — {nmdp + Tdq — \mdx) = e' f'(m')[(b' + dq){dy — indx) -\ (nrndp + idq — vw2J.r)], ••• ^mdp + xd^f — VTwcZa; = w{dy — wdf^), where B + cflT — f'(m')(b' + c'g') «;= J . ~[E-r'(M')E'] Substituting rdx + sdy for Jp, and ^da? + tdy fcr xys + yH = 0. Hence e = ^*5 s = ^y, t = 3/^, •.• mx ^ 1/ = 0, and the equations di/ -- mdx = 0, and [3], become, in this case, i/djc — xdt/ = 0, xdp 4- 7/dq = 0. The former being integrated, gives 1/ = oje; and eliminating 7/ from the second, we find dj) + adq = 0, '.' p + aq =b. Hence the equation M = r(M') becomes px +qt/= ^F (^f^. This being treated by the methods for partial differential equations of the first order, we have dz = F{a)dx, ',' z =z xF{a) + F'(a). y But since a = — , *.• '•="(f)- (443.) If the coefficients of/*, ^,.if, in the equation nr + S5 + T^ = V THE INTEGRAL CALCULUS. SECT. XXXII. be all constant^ and the quantity v be a function of the inde- pendent variables alone, the equation [2] becomes a nu- merical equation, the roots of which are therefore constant. Let these roots be m\ m" ; which, being substituted in d7/ — md^ = and [3], and the results respectively in- tegrated, give the two systems of equations y — m'x = a 7 B,m'p + Tq — nifydx — h\ y — ifri^x = d \ ^rnJ^p -^ Tq — rnJ^Jxdx = 6' j ' in which v may be considered as a function of x alone, since it may be rendered so by substituting mix or ml^x for j/. Hence we have the two first integrals ETTi'p -\- iq — Tflfvdx = Y{y — wlx), B,m"p 4- T5' — m'l/ydx = r'(j/ — mi'x). By integrating either of these equations, we shall obtain the primitive integral of the proposed equation. If the former be solved for p, we find T fvdx 1 , , ^ ^ ^m^ R R -^ T But since by the equation [2] m'm" = — , this becomes ^ = — TriJ^q +*^ 1 F(«/ - 7n!x). Substituting this value oip in the equation dz = pdx + qdy, we obtain -Sidz — dxfvdx — ct3?F(y — 7n!x) = 'Rq{dy — m^'dx). The equations therefore to be integrated' are di/ — m"dx = 0, Rdz — dxjvdx — dxY(y — m^x) = 0. The former gives y — w"j? = a', and the latter becomes R^ — fdxfsdx ^jdx¥(y — rnJx) = h. SECT. XXXII. THE INTEGRAL CALCULUS. 427 In effecting the integrations indicated in this equation, the following circumstances should be attended to. P. In determining Jvdx, y should be replaced in v by 2. Hence THE INTEGRAL CALCULUS. SECT. XXXIII. 7»'=^, ?7i"= — ^, \'y=hW'\rafy=bx-\-a\BXidLjUxJydx==^0, v z = f(2/ — hx) + f'(3/ + bx). (445.) The integration of partial differential equations of the second order is sometimes effected by a process similar to that used in (433.), scil. by the introduction of an indeterminate function 9. The equation of developable surfaces rt = «*, gives r s 7=7 = ^' •.• s = ^9, r = 59, rdx + sdy = 0(5£^ + tdy\ -.' dp = Sdq. This equation is only integrable when 9 is a function of q, and in that case the first integral is ;? = t(^). -The equation dz — pdx + qdy becomes dz = dx Y{q) + qdy. Integrating this by the method in (433.), we find, con- sidering q constant, z = X¥iq) + qy + F'(q), = xAq) +y +f(q\ where y(3'), f\q)y are the differential coefficients of the functions ^{q)^ Y\q), SECTION XXXIII. Of the integration of partial differential equations by series, (446.) By the theorem of Taylor, we are enabled to in- tegrate partial differential equations in series by a method SECT. XXXIII. THE INTEGRAL CALCULUS. 4^9 similar to that explained in Section VI. Let x and y be the independent variables, and % the dependant variable in a differential equation between two variables. Let dz dH J«z ^' dx' dx^ * dc'* be what dz d^z ^' da:' dx'' ' d^z dx"" become when ^r = 0, and which are therefore functions of y, and the constants which enter z. Hence by Maclaurin's series. dz X . dH 1.2 ' dx' 1.2.3 "-J' If it happen that any coefficient when a; = become in- finite, the series may be obtained by substituting a: -{- a for X in the function, and developing by the powers of x, (447.) If the given partial differential equation be of the first order, let it be solved for either of the partial differen- tial coefficients, so that it will assume the form dz f dz\ This being differentiated successively with respect to x^ and X being supposed = in the several coefficients, all the co- efficients of the series [1] will be determined as functions of the first term z, which is an arbitrary function of ^. Thus the series in this case will include one arbitrary function of 2/. If the proposed differential equation be of the second order, let it be dH _ /d^z d'z dz dz \ d^ ~ \d^' d^f 1^' ~d^' ^' V It is plain that all the quantities which are included in the parenthesis depend on, and can be derived from the 480 THE INTEGRAL CALCULUS. SECT. XXXIII. dz values of z and -y-, by making x = after the operations indicated by the different symbols have been effected, and the subsequent coefficients of the series [1] may be ob- tained by continued differentiation. The quantities z, dz . . -J-, are in this case arbitrary functions of «/, and therefore the complete integral of a partial differential equation of the second order requires the introduction of two arbitrary functions. By continuing this process, we find, in general, that a partial differential equation of the nth order requires in its complete integral as many arbitrary functions as there are units in n the exponent of its order. (448.) As an example of integration by series, let the proposed equation of the second order be d^z __ d^z_ do^ "" ^^dy^' By successive differentiation, we find dz\ <£' d^z d^z \dxJ = c* = c® — ^ — da^ dxdif- dy^ d^z d^- d*z d*z dx^ d^% = c*- dx"^ dxHy'^ dif dy^' d^z d^z dx^ dH ,y dz Or »a. = C 'ij. Let p be SECT. I. THE CALCULUS OF VARIATIONS. 445 Y B i A ' \ 5^111^^ P r X C "^ m D any point whose co-ordi- nates are xy. If, while the form of the function remains unchanged, the value of X is increased by Mm, the value of y is changed from pm to ym ; and if these changes be assumed of indefinitely small magnitude, they are expressed, as has been explained in the differential calculus, by om = a: + dx.^ pm =y -\- dy. Thus the sign d implies that variation of x and y, which is made on the supposition that the equation ^(xy) = re- mains unchanged, otherwise than by the change in the variables; or, to speak geometrically, differentiation ex- pressed by the character d implies a transition from one point to another of the same curve. Suppose now that the form of the equation ¥(xy) — undergoes a change. This change producing a change in the relation between x and j/, it follows that for each value of X there will be a corresponding value of ^ different from that value cf 2/ which corresponded to the same value of a: before the change in the equation. Thus pm being the value of ^ corresponding to om before the change, let p'm be the value corresponding to om = a? after the change. Thus we have a variation of 3/ of a kind different from that expressed by dy. This variation of y depending entirely on the change in the equation 'P(pcy) = 0, is usually expressed by ^^, and a similar variation of x by ^x. Thus d and ^ both signify changes in the variables, the former signifying a change in either produced by a corre- sponding change in the other, the relation between them being constant; the latter expressing a change in either variable produced by a change in the relation between them. 446 THE CALCULUS OF VARIATIONS. SECT. I. the other variable being constant. The one is the differen- tial, and the other is called the variation of the variable. In physics, the points of the surface of any body being expressed by xyz referred to three axes of co-ordinates, the variations of ocyz by the transition from one point of the surface to another, the position of the surface being un- altered, is expressed by the differentials dy, dx, dz ; but a change in any point produced by any motion of the body itself is usually expressed by the variations 8x, $j/, Sz, The differentials dx, di/, dz, depend altogether on the figure of the surface, but the variations ^a?, ^y, Sz, depend on the time, or on some function of it. (458.) The differential dy being a function derived from the primitive function, is susceptible of variation from the same causes, as the primitive function, and the same may be said of d'^y • • • • or of d"y. A similar observation applies to the other variables .r, z, &c. Hence the meaning of the symbols Sdy, Sd^y • - - - $dy, $dx, W^x ' • . . Wx, is manifest. Also the variations hj, $x, &c. being functions of the variables, are susceptible of differentiation. Hence we perceive the meaning of the expressions d^y, d^ty . . . • dny, d^x, d^8x • • • • d"Sx. In the same manner the meaning of the symbols ^u, fSv, iff", ffiv. will be readily apprehended. SECT. II. THE CALCULUS OF VARIATIONS. 447 From what has been observed, it is plain that the de- termination of the variation of a function is differentiating it under another point of view, that is to say, ascribing its variation to another cause. SECTION II. Of the variation of a function. PROP. cix. (459.) In any formula to which dP^ and ^ are prefixed the transposition of these characters does not affect the value of the quantity. That is to say, Wi; = d'^Bi/. This might, perhaps, be assumed as true upon the general principle, that when certain given operations are to be per- formed upon a function, the final result must be the same in whatever order the proposed operations may have been effected. It may, however, be considered satisfactory also to establish it independently of this general principle. Since PM=J/, \'pm=^y\-dy, \'p^m = y \dy\ l{y + dy\ 'rp^m^y + dy-\-^y-\-Uy. But also, p'm=2/+^^, •.• /}'w =:_3/ + ^yi-diy + $y), p'm=y-\-$y-^dy^d$y, •.* ^dy~d^y. 448 THE CALCULUS OF VARIATIONS. SECT. II. In general, let d'^'^y ^ u^ *.• d'^y = du. By what has been just established, Mu = d^u, i. e. ^dy = d"$j/. As this is the fundamental principle of the calculus of varia- tions, it may be proper to establish it independently of the consideration of curves. Let j/ = F(a;), and when the function by a change in its form becomes f'{x)^ we have $1^ = ^\x) — f{x). In consequence of the supposed relation between the variables, the difference of these functions must be some function of j/-. Hence we have Let y -y + dy, ■.- ••• Jy + Uy =/(y), ••• % =/(y) -f(y) = df(y), \' $di/ = dSj/. And hence, in general, $d"i/ = d"$i/. PROP. ex. (460.) In any formula to which f^*^ and $ are prefixed, the transposition of these characters does not affect the value of the quantiti/. That is, n signifying 7i successive integrations. Lety=/"z/,v y = dy\ Taking the nth integral of these, we find SECT, ir. THE CALCULUS OF VARIATIONS. 449 PRO!'. CXI. (461.) To determine the variation of a function of several variables and their successive differentials. Wc shall consider the problem applied to functions of two variables, and the result may then be easily generalised. Let u = F{x, y, X,, x^, JTg, . . . . y^^ y^, Vz, ' ' ' •) where x^^ x^^ jTa, • • • • signify dx^ d^x, d^x, .... and «/i5 3/i, 3/3» • • ' • signify dy, d% d'y. • • • • Let u become u' when x, «/, .ri, 3/1, • • • • become x + ^'^r, y + ^5 x-^ -\- ^^J^';r, + zH + 'ilUz + il'W^z . . . . z^^'^Sd^z, Where x, x', • • • • Y, y', • • • • z, z', • • • • signify the several partial differential coefficients of u considered as a function of x, dx, • • • • t/, d[y, - • - - z, dz. - • • - (462.) Cor, If u contain only x, j/, and the successive differential coefficients of y considered as a function of x, scil. dy , d'^y „ d'y „, d"y dx "-•^' dx'^ y^dx^-^' dx- " ^ ' we shall have, as before, Su = x^^ + Y^y + Y'Jy + Y^'Sy" Y'^"^$y^''\ where y', y", • • • • are the partial differential coefficients of u considered as a function of ?/', y'L .... (463.) The variations $y', Sy^', • • • • may easily be ob- tained in terms of $y, ^x'. ^ dx dx dx dx\ Mij-y'd$x (d^y—y'M SECT. IL THE CALCULUS OF VARIATIONS. 451 These being substituted in the value of ^u, we shall have it expressed in terms of ^x and Bi/. PROP. CXII. (464.) To determine the variation of the integral of a given function of several variables and their differentials. Let the function given be u. By (460.), jTu =/^u. Let us first suppose u to be a function of two variables x and y. Hence by (461.), +/[Y^j/ + y'% + Y"W^e/ ]. This may be modified by the following substitutions sug- gested by integrating by parts united with the principle My = d^ij: fK$X=fK$X, f>i'5dx = x'$x-fdK'Sx, fxJ'U^x = x"d$x -flK'^x = x''d$jc - dx'h i-fd^x''^r, fx"Wa: ^ x"'d^lv - dx!hBx -^-/d^x'Wx = x"'d^$jt; - dx"'d$x +dV$x-fdV$j:. fii^''^Sd^x = x^^'>d^-Hx~-dx'^d^-'$x + dVd''-'$a; .... d''-'x'$x±/d"x^"^$a;. And, in like manner, f^^rj^fYhj, G G S 45^ THE CALCULUS OF VARIATIONS. SECT. II. d''-^Y'>5t/±fd''Y^''^Sl/. Making these substitutions in the value of ^u, it be- comes $fv =/(x - dx' + d'x" - d'x'" .... )Sx, + (x' - dx" + d^x"' — )o"a7, + (x" ~ dx'" + (^x^^ - )d$2-, + (x"' - )d'$a;, ^/(Y - dY' + ^V - f/^Y"' + . . . . )^j/, + (y' - dY" + ^^y'" - )$y, + {y" -fZY^' + cZ-^yi^- )%, + (y'" - )d^^y, in This vahie of Sfu consists, therefore, of two parts, the one depending on the variation of jr, and the other on the variation of y. If there were a greater number of variables involved in the function u, we should have as many more series, and each of them of the same form as the preceding. Thus, if u included the variable Zj.as well as x and ^, we should have, in addition to the above, the following, +/(z — dz' + d'^z" - . . . ,yz, + (z' - dz" + r/*z'" - . e . . )h, + (z"- 2/, + + ^C £/y' 1 A" "? r This is the variation sought. (467.) Cor, Since u = 5i/ -- y^^x^ it appears that the co- efficients of ^y and ^Xj under the sign of integration, have a common factor ; and it. therefore follows that the same con- dition will make them both vanish, and leave the variation independent of any integral. This condition is evidently ^y' 1 ,rfY" Y — 3- + :r ^^ = 0- ax ax ax From what has been already observed, it is plain that this is the condition which determines \dx to be integrable. SECTION III. On the maxima and minima of indeterminate integrals. (468.) We shall now proceed to the investigation of the class of maxima and minima problems already mentioned, and to which the methods explained in the differential calculus do not reach. These problems, when reduced to an analy- tical statement, generally come under the following form : " Given a differential expression u between any variables " and their differentials to assign that relation between 456 THE CALCULUS OF VARIATIONS. SECT. III. " the variables for which the integral of the proposed " expression taken between any assigned limits will " have a maximum or minimum value."" To apply the method explained in the differential cal- culus, it would be necessary to know the form of the in- tegral ; whereas, in the present case, the form is the thing sought, and must be deduced from the very circumstance of the integral being a maximum or minimum. If the problem be geometrical, the integral, whose maxi- mum is sought, usually expresses some quantity depending on a curve or surface. Thus the integrals f^dy'- + dx^, fydx, although really indeterminate, since no relation is given between x and j/, yet express quantities depending on the sought curve, the former signifying its length, the latter its area. In like manner, if the question be physical, the inde- terminate integral may express the time, velocity, force, &c. the maximum or minimum of which is sought. The principles of variations already established, however, will enable us to extend the method for finding the maxima and minima of determinate functions to indeterminate in- tegrals. (469.) Let u be the indeterminate function of which the maximum or minimum is sought, and let u' be what this becomes when jr, z/, dx, dy, .... are changed into x -j- ^x^ y -\- oy, dx + Mx, dy -{■ Idy .... In order that u may be a maximum or minimum, it is necessary that the sign of u' — u may be independent of the signs of the increments Jr, ly Hence the term which involves the first powers of these must =0, •.• ^u = 0. Thus, that the in- determinate function may be a maximum or minimum, it is necessary that its variation should vanish. This condition is necessary y but not svjffiviuit. JBcsides this^ it is rc(j[uired SECT. III. THE CALCULUS OF VAKIATIONS. 457 that the terms involving the increments in two dimensions should collectively, as to sign, be independent of ^.r, (^^, . . . . and hence all the circumstances incident on common maxima and minima of functions of several variables are also to be attended to here. PEOP. cxiv. (470.) To determine the relation between the variables which will render an indeterminate integral taken between assigned limits a maximum or minimum. If the proposed indeterminate integral beyu, it is neces- sary that lf\5 = 0. Assuming the value [1] of this, deter- mined in (464.), it is necessary that this should = 0. This value consists of very distinct parts, some affected by the sign of integration y^ others free from it. Since the varia- tions ^^, ^j/, .... are supposed to be independent, the terms affected by the sign f are integrable, and, therefore, of the whole value of 6/u = 0, those parts which are affected by the sign y must separately = 0; for, otherwise, they would be equal to the remaining part, and would be there- fore integrable. Hence the condition Ifxj = requires that the system of equations [1] and [2] should be both satisfied. The number of equations in the system [2] is, in general, equal to that of the independent variations. In case, how- ever, of but two variables, u assuming the form ydx, these equations may be reduced to one (467.). The conditions [2] reduce [1] to the form /^u = x'^^ + x"^^^ + x"'^2j^, . : . . where x', x", • . • • y', y", .... signify the quantities in- 458 THE CALCULUS OF VARIATIONS. SECT. III. eluded in the parentheses in [1]. Let the values of the va- riables corresponding to the limits of the integral beo^yz' •••• a/yv .... and when these are substituted for the variables, let the values of the integral^ Ju become l' and l". Hence y*^U = L"— l'; and, therefore, l" - l' = is a condition of the proposed maximum or minimum. This equation will then contain no variables, except those which correspond to the limits, which, however, may or may not be variable according to the conditions "which regulate the proposed limits. The system of equa- tions [2] express the sought relation between the varia- bles. If the problem be geometrical, these will be the equation of the curve or surface sought, observing, however, that it is to be modified by the conditions of l" — l' = 0, and the relation between the proposed limits. The process of solution will be more readily perceived by considering successively the different conditions which may affect the limits of the integral, and illustrating these conditions by their geometrical application. (471.) 1^. If the limits of the proposed integral are abso- lutely given and fixed. In this case, xif^ .... ^"yV .... being supposed to be the particular values of the variables cor- responding to the limits, are fixed, and subject to no varia- tion. Hence, in l" and l' we must put lot} — 0, d^od = 0, . . . . ^y = 0, d'^'ij = . . . . and since these quantities enter every term of l" and l' as factors, the condition l" — - l' = will be fulfilled independently of the coefficients. In this case the relation between the variables is found by integrating the system of equations [2], and ascribing such values to the arbitrary constants, that the integral will satisfy the con- ditions of the proposed limits. Thus, in geometry, if the curve sought, and which must SECT. III. THE CALCULUS OF VARIATIONS. 4iS9 have the proposed maximum or minimum property, is also required to pass through two given points, the co-ordinates of these points determine the Rmits of the integral. The equations [2] being integrated, give the general equation of the curve sought ; but it will be necessary to assign such values to the arbitrary constants introduced in the integration, that the curve shall pass through tjie two given points. (472.) 2^. If the limits be absolutely arbitrary and in- dependent, it is necessary that the equation l" — l' = shall be fulfilled by its coefficients; that is, that the co- efficient of each variation in it shall separately = 0. (473.) B°. If the values of the variables corresponding to the limits be subject to any conditions expressed by equa- tions, these equations will give, by differentiation, relations between the variations of the variables corresponding to the limits. As many variations may be eliminated from l" ~ l' = as there are independent equations of condition. The remaining variations being absolutely arbitrary and inr dependent, the resulting equation must be fulfilled by its coefficients ; that is, the coefficient of each remaining varia- tion must separately = 0. (474.) The same may be effected upon another principle. Let w = and i; = be the equations by which the par- ticular values of the variables corresponding to the limits are restricted. Hence the conditions Su' = 0, W = 0, must subsist at the same time with l" — l' = 0. These three equations may be expressed by one, thus, l"- L + a'5«' + A''fw" = 0, the coefficients a', a", being supposed to be arbitrary con- stants entirely independent of l" — l', W, or hu". This supposition evidently renders the one equation equivalent to the three former, for, otherwise, it would express a relation 460 THE CALCULUS OF VARIATIONS. SECT. III. between the quantities a', a", and the quantities l" -- l', W, and W, which is contrary to hypothesis. This principle is of very extensive use in the apphca- tion of the calculus of variations to geometry and phy- sics. In place of eliminating the dependant variations, we treat them as independent quantities in the above equation, and equate each of their coefficients with 0, and from the equations thus resulting, the arbitrary quantities a', a", •••• being eliminated, the result, which will be obtained, will be equivalent to that which would have been found by eli- minating the variations by the equations of condition. The method which we have now explained is, however, in most cases preferable. (475.) Thus, in geometry, if the curve sought be not as before restricted to terminate in two fixed points, but only to terminate in two given curves or surfaces : in this case, the co-ordinates of the limits are only restricted to satisfy the equations of the limiting curves or surfaces. In this case, the variations of the co-ordinates at the limits must be re- lated to each other in the same manner as the differentials of the co-ordinates of the given curves or surfaces. These conditions being introduced into l" — l' = by elimination, as already described, the coefficients of those independent variations which remain must be put separately = 0. Again, the limits may be still further restricted. Let the sought curve be not only required to be terminated in given curves or surfaces, but also to touch them. In this case it will not be enough that the co-ordinates of the limits satisfy the equations of the limiting curves or surfaces, but the dif- ferentials of the co-ordinates must also satisfy them. Hence the variations of the diff*erentials of these co-ordinates must be equivalent to the second differentials of the co-ordinates of the limiting surfaces. By these conditions, the number of variations which may be eliminated are mcrcased, and SECT. III. THE CALCULUS OF VARIATIONS. 461 the independent equations involved in l" — l' = are there- fore diminished. In these cases, as in the first, the constants introduced in the integration of [2] must be so assumed as to satisfy the equations resulting from l" — l' = 0. (476.) From the form of the differential equations [2], it is evident that their order may be marked by any number not exceeding twice that which characterises the formula u, and therefore the integral may involve any number of arbitrary constants not exceeding this. The number of terms in the equation l" — l' = in- creases with the order of the formula u, and, therefore, with the number of arbitrary constants in u. In general, then, the highe4' the order of the formula u, the greater number of conditions we are at liberty to impose upon the limits ; these conditions being always satisfied by the values ascribed to the arbitrary constants in the integrals of [2]. (477.) When the co-ordinates of the limits are variable, as in the cases last mentioned, and enter the formula u, which sometimes happens in taking the variation of u, these co-ordinates are to be considered as independent variables, and their variations must enter the total variation of u. But, in integrating ^u with respect to the variables ^, ^, z, • • • • the co-ordinates of the limits, and their variations, are to be regarded as constants, and brought outside the sign of integration, so that any term of the formyk^j;' may be re- placed by ^^yX. This is evident, since the integration may be conceived to respect the variation of ^i/z - • • • through the sought curve, and not from one of its positions to another. An instance of the necessity of attending to this circumstance occurs in investigating the hrachystochronous curve. (478.) It sometimes happens that the variations ^x, ^y, 5;s, • • • • are restricted by equations of condition altogether independent of the limits of the integral. Thus, for ex- 462 THE CALCULUS OF VARIATIONS. SECT. IT. ample, when the curve sought is required to be drawn upon a given curved surface, as in the case of finding the shortest distance between two points upon a given surface. Since in this case a relation subsists between the varia- tions, it is not necessary, in order that the integral sign should disappear from the value of ^u, that the several terms which it affects should severally = 0. The number of these terms may be diminished by eliminating as many of the variations as there are independent equations of condition given, and then putting the coefficients of the remaining variations = 0. The number of equations in the system [2] will, in this case, not be equal to the number of va- riables, but to the number of independent variations. SECTION IV. Examples on the calculus of variations . PROP. cxv. (479.) To find the shortest line between two points. In this case /u =fVdx'' + df + dz^ =fds, dx ^, dy ^ dz ^^ Comparing this with the formula for ^u in (461.), we find X = 0, Y = 0, z = 0, x' - -^ y' - -^ z' - — ds' ^ - ds* ^ -^ ds' and all the other coefficients = 0. SECT. IV. THE CALCULUS OF VARIATIONS. 463 The system of equations [2] become , . • • - a -{^ xh. In this case the general term is u^ = a •{■ ccb. By successively substituting 0, 1, J2, . . . . for X in u^^ the successive terms may be found. If the series do not commence at a, the preceding terms may be found by substituting successively - 1, - 2, - 3, ... . for X. Thus it appears, that the nature or law of an arithmetic series is expressed by the equation Uj. = a -|- xh. Again, if the scries be a, ar, ar', ar^, • • • • SECT. I. THE CALCULUS OF DIFFERENCES. 481 The general term is u^ = «?-*, in which the successive substitution of 0, 1, 2, 3, . • • • for X, gives a and the terms which succeed it ; and the sub- stitution of _ 1, - 2, - 3, ... . gives the terms which precede it. As in geometry, Hnes are always supposed to be extended indefinitely in both directions, unless the contrary be ex- pressed ; so in the calculus of differences, series are sup- posed to be continued through an infinite number of terms, unless the question imposes express limits upon them, or they assume limits from the nature of their general term or generatrix. (494.) The difference between two values of the function which correspond to two successive values of the variable is called the difference of the function., The notation ex- pressing this difference should also express the value of the variable in one of the two states of the function. If then the two successive values of the variable be 1 and 2, the corresponding values of the function are and the difference U2 — Wi, which is usually expressed thus, Awi- In general, if the two successive values of the variable be X and X -{• \y those of the function are and the difference is The several differences «i ~ Wo, ?/2 — ii„ u, - «2, . . . • U 48^ THE CALCULUS OF DIFFERENCES. SECT. I. are therefore expressed, Amq, Amj, AW2, • • • • (495.) It is obvious that the difference aw^ of a function is itself a function of the variable, and receives a succession of different values by the substitution of the successive integers for the variable. It therefore has a difference in the same sense as the function itself. The difference of the difference of a function A/y^ would be therefore expressed thus, a(am^.), or more simply, This being again a function of the variable, we find by continuing the same reasoning, a series of successive dif- ferences, Aw^, A2«^^, ^hi^^ AHi,, .... and, in general, A"«/^, which are called the Jirst difference^ the second difference^ &c. and, in general, the nXh dif- ference. The analogy of this language and notation to those of the differential calculus is sufficiently obvious. (496.) The calculus of differences may be divided into two parts analogous to those of the differential and integral calculus. The direct calculus of differences, the object of which is the determination of the successive differences when the function is given ; and the inverse calculus of differences, the object of which is^ the determination of the function when the difference is given. SECT. II. THE CALCULUS OF DIFFERENCES. 483 SECTION II. Of the direct method of differences, PROP. CXXV. (497.) To determine the difference of the algebraical sum of several functions of the same variable. Let % = < + u\ - w"'^» ••• w^+, = <+, + u\+i — m'",+,. Subtracting, we find And, in general, if u^ = 2«), Uj, = (am'j). PROP. CXXVI, (498.) The constant quantities connected with the variable of a function by addition or subtraction disappear in its difference; and those united by multiplication or division are united in the same manner with its difference. 1°. Let the function be u^ -r a. Hence the diiference is A(u^ + a) = (m^+i + «) - (w^ + «) = Wx+i - «. = -^^x* •.• A[u^ + «) = A%. S^. Let the function be aUjt I I ^ 484 THE CALCULUS OF DIFFERENCES. SECT. II. But '.• A(au^) = aAuj,. PROP. CXXVII. (499.) To determine the values ofu^ and AUx in a series qfuQ, and its successive differences, 'By what has been already explained, we have «^i = ^0 + AMo, •.• Au, = Amq + A^Wq, •/ w^ + Awj = Wo + 2Awo + A^Wj^. Also, u^ z= Ui + Aui '.' U2=- Uq + ^Auq -f A'Mq, •.• Aw2 = Auq 4- SA'-^Wq -f A^Mo> which, by addition, gives ^2 + AU2 = Wo + ^^^0 + SA^j^o -I- A3j/o- But, also, U3 = U2 -\- AWaj •.• M3 = Wo + SAMq + Sa'^o + ^^^05 •.• AM3 = AUq + 3A%o + 3AX + AX» which, by addition and a similar substitution, gives «4 = 2^0 + 4AWo, + 6A2?,Q -f 4AX + A^Mo, •.• Aw^ = Awo + 4A2?^Q -f 6A\ + 4A%o + AX; and, in general, X ^ , x.x — \ „ x.x—\,x — 9>^^ r , m^=«^oH-yAwo+ nr^" " "^ — r¥3 — ^^° * * * f^^' ir „ jc.j: — 1 „ x.x — l.x--^^ ^ ^ , aw,=amo+-yax+-y^ax+ — r;2¥~^ ^^' " f^^' and in Hke manner we have, in general, SECT. ir. THE CALCULUS OF DIFFERENCES. 485 , ^^ .r.jT-l .r.jc — l.jr-2 „ ^ ^ w,+„=tt.+ yAw„+ -j-^-AX + j^g^g AX • • • [o]; or, w ^ , n.n—1 „ . w.n — l.w— 2 . ^ , «W.=w,+ ~Aw^+-y-^AX + j^g AX- . . [d]. This series may easily be retained in the memory, by observing that it is the development of 2^.(1 H-A)"; and the former shows that Uj^+n may be expressed thus, Un{l + Ay. So that any of the four expressions may be indiscriminately used one for the other. PROP. CXXVIII. (500.) To determine AX in a series of Wj+„, Uj,+n—u W^ + n— 2" ♦ • • • By what has been established, we have AWo = w, — Uo, A^u^ = Aw, — AWo ; but, also, AMi = Wa — W,, ••• A*Mo = U.^ - 2^1 -f Wq- By taking the difference of this, we find A'X = AMa — 2AWi + AWo- Substituting for these differences their values, we have A^Mq = Us — Su.^ + SUi — U^, In like manner, taking the differences of these, A^Uo = AUs — 3Aw2 + 3Awi — AWq. Substituting as before, we find n n.n — 1 7i.n — l.n — 2 486 THE CALCULUS OF DIFFERENCES. JsECT. 11. and, in general, n Tl.W— 1 71.71— 1.71 — S Thus the value of A"m^ is equivalent to. the develop- ment of The exponents of u in the successive terms being removed below the letter thus, u^+n- (501.) When the function is given, its successive dif- ferences are easily obtained. Let w„ = (?/ + nh)"", \' u, = {^ + hy\ ^2 = («/ + ^'hy, «3 = (2/ + 3^)^ Hence , 7?^.7?i— 1 ,, 7?l.7?l — 1.771 — 2 •/ Aw,=7w^— *AH — i;^""-^""''^' ■*■ — TK^ — ^ ' ■^** To obtain the second, third, and succeeding differences, it is necessary to change y into y -\- hm Aw, a^m. Hence we obtain AW, = m(y + hy-^h + ^'^~ (3/ + hy-^'h'' + • • • It is evident that by developing Am,, and arranging the result by the ascending powers of A, and subtracting A?^ from the result, the series will have the form A^W = 771.771 - 1 . 7/'»-27ii _}_ M^y^-^h^ + M4a?''»-*A*4- • • • • where Mg, M4 • ♦ • • signify the functions of ttz, which form the successive coefficients. By a similar substitution in this last series, and observing SECT. II. THE CALCULUS OF DIFFERENCES. 487 the condition A^w =-. c^u^ — A^?/, we shall obtain A^w = m.m - l.wi — g.^/'^-Vi^ + M'4y«-'*M+ • • • • It is obvious then that the first term of the development of A^'m must be m-m — 1-m — 2 • • • • (m — ;* F \)y"'-''h'\ It follows, therefore, that when the exponent w is a positive integer, the number of terms of the development of AX arranged by the powers of?/, diminishes by unity as ri increases by unity, and that when n — m^ we have ^"^n = m-m - 1-m - 2 • • • • 3.2.1.^'^. This difference being constant, all the succeeding differences must = 0. We can obtain the general term of the series A"tt by means of the values of w, Wj, w^, • • • • independently of Aw, A^Uj A^u, . • • . We have «^n = («/ + nhy Hence by (500.) T? *it 'ill • ' • I TU~^^ "^ (""^ ^^^]'"+ • ■ • • If i be the exponent of h in any term of this, when each term shall have been separately developed, the general ex- pression for this term will be m.m — 1.171—2' '•' (m — i+ I) .3 •• i ^ ~* ^ But the development of A"m cannot involve any powers of h of which the exponents are less than w, as appears from the lowest exponent in the development of A% being 488 THE CALCULUS OF DIFFERENCES. SECT. II. 71. Hence it follows, that the function n , . n.n — l, ^ . n' - j(n - 1)' + -j^(n-^y consisting of (w + 1) terms must = when i < n. Also the coefficient m.m — l.m — ^- • • • ( m — i-\-l) L2.3 ~^i must vanish when m = e -f 1. It follows, therefore, that no power of h in the development of A^w can have a higher exponent than m. PROP, cxxix. (502.) To determine the successive differences of a rational and integral /unction ofx. The form of the proposed function is u = Aa?" + BX^ + c^'' + Bx'^ .... Taking the 7ith difference A"w = aA^a*" + BA'^a:^ + cA"a?* .... * by (498.). The Tith differences of each of the powers of x must then be separately found by the methods given in (501.). If a be the highest exponent in the series, we have A«^« = 1.2.3 .... ah", A« Wo + j^2'o+ ^2 ^'^" ^ 1.2.3 n.n — 1 .... (n — ?7i-|-l) 1.2.3 m 494 THE CALCULUS OF DIFFEEENCES. SECT. III. By this series, if Wo, AWo? A«Woj • • • • be known, the value of Un will be determined for any value of w. Let .r = ^0 + w/j, *.* n = — 7-^, and if jr — .ro = h', h' •••^ = x- Hence the series becomes h! h}(h!-h) h{}i!-h){hJ-^1i) , . and if w„ — m^ = A'w, •.• ^^--=T^^-+-W-^^"--+ A.^;.^/. — ^-^-^^ + • • • • This being a rational function of A', or ( 1.2.3.4 which being developed and arranged by the powers of ^•, becomes 12 + 11 &i! - 1 llA'^ + 34//3— 3/1'^ -n = j^ . In these cases the law of the series has been rigorously determined, and the values of any proposed term Un can be determined, not approximately, but exactly. This is always the case when we obtain a constant difference, however high its order may be, because, in that case, the successive values can only result from an algebraic function. (511.) The series expressing a'm is generally used when the differences Am^, A^^^o? A'^?*^, .... continually decrease, because, in that case, it is convergent. In case the general term of the series be not an algebraic function, the terms intermediate between any two may be determined ap- proximately by assuming one of the differences AUq, A"?/^, &c. of a sufficiently high order, and considering it as con- stant for all the intermediate terms, and determining the in- termediate terms and their differences by the method already given for the case of algebraic functions. As an example, let it be required to compute the common logarithm of the number 3,1415926536 by means of a table containing the logarithms of all integers from 1 to 1000 to ten decimal places. We shall take these logarithms as particular values of n, and the numbers them- 496 THE CALCULUS OF DIFFERENCES. SECT. III. selves as the indices of the functions ; thus, let t/o = /(3,14), u, = /(8,15), u, = Z(3,16), ti, = /(3,17), 7/4 = 1(S,IS). Hence by the tables, we have u, = 0,4969296481 7H = 0,4983105538 ?*2 = 0,4996870826 W3 = 0,5010592622 W4 = 0,5024271200. Hence we find Auo = 0,0013809057 Au, = 0,0013765288 Au^ = 0,0013721796 All, = 0,0013678578. AHio = - 0,0000043769 A'u, = - 0,0000043492 A^w, = - 0,0000043218. A^uo = 0,0000000277 A%, == 0,0000000274. AX = - 0,0000000003. By continuing the process, and taking from the tables the logarithms of 3,19, 3,20, &c. we should find the dif- ferences A^Uo, A^Uq, &c. still decreasing, and for several suc- cessive numbers we should find the fourth differences AX> A*wi, A%2, A*W3 • • • • as far as the tenth decimal place, the same as that already found, we assume that in calculating A^* to the tenth place, the series expressing it should rigorously terminate at the fourth term. Since, then, h = 3,15-3,14=0,01, h' = 3,1415926536 - 3,14 = 0,0015926536. SECT. III. THE CALCULUS OF DIFFERENCES. 497 Hence, ^ = 0,15926536 ^^ = - 0,42036732, :- 0,61357821, h'-Sh = - 0,71018366. 4A Substituting these in the formula h'{h'-h)(h'-n)ih'-sh) ^ ■^ h.2hMAh ^'''' and effecting the operations indicated by the signs, the result is a'u = 0,0002202245, •.• log. (3,1415926536) = 0,4971498726. (512.) In the preceding cases we have supposed that the given values *^0) "^IJ *^2J "^SJ • • • • were in arithmetical progression. When this is not the case, let the particular values be successively substituted for X in the series U = A -\- BX -\- ex' -\- DX^ 4- . . . . which gives t^^, = A 4- B^o + CXl -h DXl .... Z^j = A + B^'i + COJJ + BXl . . . • M2 = A + B^2 + C^^ + D^l • . . . ?4 = A + BXa 4- C^J + DXl - - - - The number of given values Uq, w^, Wa • • • • ought to be equal at least to the number of coefficients A, b, c, • • • • which it is required to determine. K K 498 THE CALCULUS OF DIFFERENCES. SECT. III. By subtracting successively each equation from that which follows it, and dividing the successive results by x^ — Xq, a*2 — -^ij • • • • the results will be — -^ = B + C(^, + ^o) + D(JC2. + X^Xo + ^o) Xi Xq J J = B 4- cfe + ^i) + D(.r^, +x^i + x^) = B + C(^3 + ^2) 4- I>(^3 + ^3^2 +^l) il>2 """1^2 Let Ui — Wo W3— Wj o = Uo, ^ = U,, &C. Subtracting u^ from u^, u, from Ua, &c. and dividing the successive results by x^ — x^^ x^ — iCg, &c. and calling the quantities X2,'—'Xi 373— J7j u'o, u'l, &c. we obtain U'o = C + D(^a + ^x + ^0) • • • • U'j = C 4- d(x, -{- X^ + X,) . , , , from whence we find u'x - u'o = b(x, — X,). Substituting u" for u\-u^o •^3-^0' we have u"o = d -f , &c. If we suppose that the first four terms give a sufficient approximation to ti, we shall have D = u"o, c = u'o — \j\{x, 4- ^i + ^o), B = Uo — u'o(^, + Xo)'{-v"o(x,Xi-\-X^Xo-\-X^Xo\ A = Wo — Uo^^o + v'oXiX^ — u"x^X^Xo. SECT. III. THE CALCULUS OF DIFFERENCES. Substituting these values in the general expression for m, we find ?/= Wo + Uo(^ - ^o) + u'o[.r*— (oTi -f-a^o)^ +^xaro] By this reasoning, we should have a formula similar to this, whatever may be the number of given values of x, and it may in general be expressed thus : u =Uo 4- Uo(j:- ^o) + ^'o(^—^o)(^—^i)+v"o(3!'-a:o)(a:—a:,)(ar-a;2) + \j'"o{a;-' X o)(x - :v ^){a; - a; ,)(^-^x 3) -{- .... The meaning of the several coefficients being determined by Ur'-Uo U2 Wj U3 — U0 W4 W3 =Uo, = Ui, =U2, = U3, &C. tl?i "~"ti7o «l?2'~'*^l 'p8"~"^2 *p4"~~^t"a U.-Uo , U.-U, U3-U2 = u'o, =u'i, = u'2, &c. = U"o, = u"„ &C. (513.) The series already found for the case in which the values are in arithmetical progression, may, without difficulty, be deduced from the more general formula which we have just established. In this case we have t* J •"" X (j "~" X2 "~" "'I ~~" '^3 "~~ X <^ — • . » . Hence Xi=i X \ h^ x^ — X -\- 9Jii a?3 = a? 4- 3^. a?a~^," u. -u, ^3 — Xi XT'. -< Therefore KK 53 500 THE CALCULUS OF DIFFERENCES. SECT. III. AWo AW I Aw, o Uo = --, u. = -^, u, =^-^,&c. ^Hi^ , A«Wi , A^M ^0 — , aioi Ui U'2 = T-d^i &c. ft — -^^^° ff _ ^^^^ <^, AX u'"o = :r-7r7TTT, &C. Let X = Xq -\- h\ •.* X — Xo = Ji\ X - x^ = h' — h, X — x^=: h' — 2)^5 X — Xj z= h' — 2h ' • • • By which the formula found in the preceding article be- comes which is the same as the result of (509.)' (514.) The general formula for u may also be expressed in another way. Since the values of Mo, w,, Ma, ... . in terms of *^'oj ^l> "^zj . • • . are all simple equations with respect to the several co- efficients A, B, c, . . . . It follows that the expression for u should be such, that by changing x successively into Xq, Xi, Xz, . . > . u should become Wo? Uu u^y . . . Hence we should have U = XWo + XjMi + X2W2 + . . . . provided that the functions Xq, x^, x^, .... be such, that the successive substitutions of .ro, jTi, a^aj . . . • for x give Xo = 1, Xi = 0, Xa = 0, X3 = 0, • . . • Xo = 0, X, = iTi, Xjj = 0, X3 = 0, • • • • Xo = U, Xj = U, X2 = <2^25 X3 = U^ • • • • SECT. III. THE CALCULUS OF DIFFERENCES. 501 which conditions are satisfied by Xn = Xi = Xo = (tg-a?o)(cg--^i)(jr— J73) ' ♦ The numerator and denominator of these several ex- pressions each contains a number of factors one less than the number of given values of x. The formula for inter- polation therefore becomes (x—aCiYx — X2)(x^Xo) .... u = ; ^ ^— — Wo, (a:—Xo){x—X2){x^Xs) (^~37o)(a?— 3?i)(a?~ar3) This formula is particularly adapted for computation, since each term may be calculated by logarithms. See Geometry (617.). (515.) The method of quadratures, or of approximating to the value of the integral ykt/^, is facilitated by inter- polation. Let the curve, of which the ordinate is w = x, and of which the area is therefore yxc?;r, be supposed to be inter- sected in a certain number of given points by a parabolic curve represented by the equation w = A + B^ + cjr® -f DX^ .... the coefficients being indeterminate. The area of this curve will be X x^ fudx = A Y + B^ + c-g- + D-^ . . . . + const. 502 THE CALCULUS OF DIFFERENCES. SECT. IIL By (499.) we have each of these series being continued through as many terms as there are points common to the two curves. Let the number of common points be three. Taking the first three terms of the preceding formulae, we have A = Wo, B = Auo — ^A^Woj c = ^A^u. These quantities depend only on the three successive values Uo, Ui, Wa, which correspond to h' =: 0, h'=z h, ^' = 2^, or jr = 0, ^ = 1, a; = 2. If the integral be taken between the limits of the first and last, its value will be 2Uo + 2{AUo — iA^Uo) + -JA^Wo = 2(u, + Az^o + iA^Wo). The value of the integral thus found is the area of the segment of a parabola meeting the proposed curve in three points, and comprised between the ordinates through the first and third point. It is evident that this parabolic area has a part in common with the area of the proposed curve ; and that the second ordinate divides both areas into two parts, one of the parts of the parabolic area exceeding the corresponding part of the required area, and the other falling short of it, the dif- ference of these differences being the error in the total result. SECT. IV. THE CALCULUS OF DIFFERENCES. 50^ SECTION IV. The inverse calculus qf differences. (516.) The object of the inverse method of differences is to determine the primitive function from its differences. Thus, as has been already observed, this part has the same relation to the direct calculus of differences as the integral has to the differential calculus. We shall here confine ourselves to the integration of that class of differences only which are expressed as immediate functions of the independent variable. All such come under the form A'-w^ = F(.r), the increment li of x we shall suppose given and constant. (517.) There are three theorems which are obvious from the inversion of the corresponding ones in the direct calculus of differences. 1*^. That as constants united to any function by addition or subtraction, disappear in its difference, so in integrating the difference of a function, an arbitrary constant should be added. Thus, 2(Aw^) = w^ + c. ^^ As constants connected by multiplication or division with a function are similarly connected with its difference, so, in integrating, the constants thus connected with the dif- ference should be preserved in its integral. Thus, since A(Ajr) = AAor, ••• 2(a^) = KZx. It should be observed, that the sign 2 before any quan- 504 THE CALCULUS OF DIFFERENCES. SECT. IV. tity implies the integral of which that quantity is the dif- ference. Thus, ^x is the integral of which x is the dif- ference. So that A2^ = X, or 2A.r = x ; the operations indicated by 2 and A being subversive each of the other. 3^ That as the difference of several quantities united by addition or subtraction is found by uniting similarly their several differences, so also the integral of several differences thus united is found by uniting similarly their several in- tegrals. Thus, since A(a: + 3/ — s) = A.r + Ay — A2:, •.• ^{x + 3/ — 2;) = 2^ + Sj/ — 22:. (518.) When the proposed difference is a rational and integral function of the independent variable, its exact in- tegral may always be determined. It appears from what has been already established, that there is a certain order of differences of such a function which are constant. Let the exponent of this order be m. Since, in general, A'M^ = r(T), •.• A'+'"w^ - A'^fCjt). Since this latter is constant, we have n n.7i—\ Un = u^ jAu + ——-A^u .... 7^.7^~l.. ..(7^- r-m + l) ^ 13.2..,. (r-^m) ""' in which ?/, Au, AHi, .... are those values which correspond to 07 ~ a. If « 4- w/i = X, '.' Un becomes u^. If we suppose i;^. = F(a:), *.• A'u = V, A'-^^u = Av A'-+"'?/ = A^'v, u and its differences, as far as the (r — l)th order inclusive, being arbitrary. As an example of this, let Au^ =z x^ - 5x- + 6x — 1, SECT. IV. THE CALCULUS OF DIFFERENCES. 505 the increment of x being unity. In this case r = 1, w = 3, h — 1. If we suppose a — 0, we have t; = - 1, Av = 2, A^v = — 4, A^u = 6, A*ZJ = 0. Hence 2(:r3 - 5^* f 6jr - 1) = w^ = ^ " S ^ "^ 1.2 ^ 1.2.3 ^a;(^-l)(a:-2)(^-3) 3^* -26^' + 69^' -58a: " + ^ 1.2 .3.4 — 12 +-"^- (519.) The method of integrating an extensive class of differences may be derived from the form of the difference of the formula u = x{x + h){x +2^) [^ + (w - \)h\ The difference of this is Aw = (^ + h){x -f 9.h){x +Sh) (j; + wiA) - x{x f 70(^ + 2//) [^ + (m - l)7i] = (^ + /OC*^ + 27i) . . . . [a? + (w - l)A]wA. Hence by taking the integrals, observing that mh is con- stant, we obtain 2:(J? + h){x + 2A)(ji' + 3A) [^ + (m - 1 )^] _ ^(^ + A)(;r + 27i) [a7 + (w — 1)A] ~ mh By changing :r into x ^ h, and tw into 77t -f- Ij this becomes 2^(^ + JiXx -\- 2h) .... [a: + (772 — l)h] {x — h)x {xih){X'^2h) [x^-{m- \)h ] ^ ~ {m + l)h • • • L J- By means of this formula, every function which can be reduced to a product of equidifferent factors may be integrated. The analogy which the formula just found bears to 506 THE CALCULUS OF DIFFERENCES. SECT. IV. is obvious. In both, the number of factors in the numerator is increased by one by the integration, and the factor m + 1 is introduced into the denominator. (520.) A method of integrating another class of differences may be deduced from the difference of 1 u = mh) L'~a:{a;+h){x-\-^h) [^+(7/2- — mh Taking the integrals, and substituting for u its value, we have 2 zi 1 mhx(a:-\-h)(X'\-2h) [a; + {7n-l)h]' In order that m may express the number of factors in the proposed difference, let it be changed into r/z — 1, and the formula becomes 1 4jr+A)(^+2^) [^ + (m--l)7i] -1 •ra- (521.) Functions of the form AX'' + B/r^ -{- CJ7'' • • • • may without difficulty be integrated by the formula [1], which we have just obtained. For such functions may, in general, be transformed into products of equidifferent fac- tors. As an example, let X' = (x i- 7i){.v + n)(x 4- 3/0 + A{x -f ^)(a; + 2A) + B{a; + //,) 4 c. SECT. IV. THE CALCULUS OF DIFFERENCES. 507 h expressing the increment of x. This being developed and arranged by the powers of .r, we find x^ = x^ -{- Qhx'^ -f Wh'^x -I- 6^3 + Aar2 4- SaJix + 2Ah\ + C. That this equation should be identical, it is necessary that 6A + A = 0, ll/i2 + 3aA + B = 0, 6^3 + 2A/i« + B/i + c = 0, which give A = - 6/i, B = 7/i% c=-A^ •.• ^5 = (^ + A)(a7 + %h){x + 37i) — U{x + /i)(ar + S/z) + ^h\x + 70 - //^ which by (519.)) gives 1.ar^ = —x{x -h ^)(^ + ^2h)(x -f S;^) 7 — ^x(x + /z)(j7 + 2A) + -^hx(x + 7i) - 7A«jr + const. Since 2(- 7^^) =^ - 7i'2l = - li'^x. (52^.) Each of the integrals i:^% 2t, 2^^ Sjt^ 2^:''^-% 2^"", depend on those which precede it, in such a manner, that if the (m — l)th be known, the ?wth may immediately be de- termined. If each term of the equation (501 .) ^ 1.2.3.4 '' ^' +/i ^ be integrated, we obtain 508 THE CALCULUS OF DIFFERENCES. SECT. IV, Hence we find w + 1 By the application of this formula, it is evident that by knowing the integral we may successively obtain Xx, 2^7^, 2^5, .... 2^'", by substituting successively 1, 2, 3, • • • • m for ttz. Hence the results in the following, table may easily be obtained : X ^•"°=T' x^ 07 -I- 2 "^6' -=£- " 2 "*" 4 ' . ^ ^-M- ^* kx^ ' 2 '^ S ~ 30' ^-'-eA- x' 5hx* 2 ^ 12 12' ^^'=5- ^6 ^^ 6 "^42' -=.t- x^ Ikx^ ' 2 "^ 12 Wx^ h'x^ 24 "^12' SECT. IV. THE CALCULUS OF DIFFERENCES. 509 ^ 9^"" 2 "^ 3 ■""15~"^~9~""W' ^^= 10A-T+-4 — rr +-2— ^T' ^^- ^^- In applying these formulae to particular cases, the arbitrary constant should be supplied. (523.) In general, let ^ 2.r"» = A^'^^* + B^"" -f cx'"--^ + D^'"-^ By taking the differences, we obtain x"^ — A— .j — x'^h (W f l)m ,„ (W2+l)w(?7Z — 1) + K ^^ ^-^/t^ + A^ f;g^3 ^.T-Vi»+ .... m , m.{m—\) + B-=-a7''*-Vi+B — =-^ — a?'"-Vi«+ • . . . w— 1 + c-Y—^'"~^A+ • • • • , 4- . . . . This will be rendered identical by the conditions 1 m\-\ K B = — aA-^— = - i, C = - A/^*— ^3 B/Z-, 2.3.4 2.3 "' 2 Hence, we find in general im+] ^^ -(m+l)A ^"^ ' + 4.3 X*""" 6.5.4 1.2.3 "^ ^' J__ m(7^-l)(m-2](m-3)(m-4) ., ■^ 36.7 1.2.3.4.5 510 THE CALCULUS OF DIFFERENCES. SECT. IV. ^ 7?z(7?2~l---(m-6) "" 10.9.8 1.2.3 7 "^ '^ ' "^60.11 1.2.3 9 691 7^(m^l)...,(^~10) 210.13.12 1.2.3 11 + 215X5 1.2.3 13 "" ^ ' _ 3617 r/z(m-l)....(m-14) 30.17.16 1.2.3 15 ' 43867 ^m-l)....(m-16) "^ 42.19.17 1.2.3 17 ' _ 1222277 7/z(m~l)....(m-18) _,^^,g 110.21.20 1.2.3 19 ^ In this series, after the first two terms (m-l-l)/^"'*^'"' the succeeding terms may be found by multiplying the even terms (2nd, 4th, 6th ) of the development of {x + hf successively by the numeral factors ^ 4.3' ~ 6.5.4' "^ 3.7.6' 3 5 "~ loias' "^ gooT' ^''• These numeral coefficients are called the numbers of Bernoulli, because they were first determined by James Bernoulli *. They frequently occur in the theory of series. * For a full development of the properties of these remarkable numbers, see Mr. Herschel's excellent Treatise on Differences, with the examples on it. SECT. IV. THE CALCULUS OF DIFFERENCES. 511 In obtaining the above development in this way, the last term IV"^ of the development of {x + Tiy"^ should be omitted, even when it holds an even place. (524.) To determine the method of integrating expo- nential functions, let Uj. = a". Taking the difference, we find tiu, - a'ip}' — 1), •.• u^ = sa'^Ca'* — 1) = a% fl*— 1 Hence the method of integrating an exponential function. (525.) Let u^ = cos. a?, *.• A cos.a? = cos.(a: -f h) — cos.a? = — 2sin.(.r + 4A)sin.-i7t. Integrating this, we find . , ,, COS.iT 2 sm.(^ + ^h) = — gsin.^A ' or. by substituting 7/ for x + \h. By a similar process we obtain sin.{2/--M) 2COS.J/ = -7r^--T^. ^ 2 sm. yi Powers of the sine and cosine are integrated by de- veloping them in a series of sines or cosines of the multiples of the arc (Trigonometry), and then integrating the several terms of the development. (526.) If the integral of the product of two functions x', x", of .r, be expressed thus, Dx'x" = x'2x" -V X, where x is an unknown function of x, let x be changed into * It may in general be observed that an arbitrary constant should be added in these integrations. 512 THE CALCULUS OF DIFFERENCES. SECT. IV. X -}- km x'2x" -}- X, and let x', x" and x become x' + Ax', x" -f Ax", and x + Ax ••• = Ax'.S(x" 4- Ax") -f Ax, ••• X = - 2[Ax'.2(x" + Ax")], •.• Dx'x" = x'Sx" — 2[Ax'.2:(x" + ^x")] This formula corresponds to that found in the integral calculus for integration by parts. (527.) The integral of a function considered as a difference can seldom be found in finite terms. Its value, however, may generally be expressed by a series. By Taylor's theorem, _dz h dH h^ d^z ¥ ^^~d^' T'^d^^' T^'^d^^'TI^s' Taking the integral of both members, we have ''"T^di'^T:^^d?^lJ73^d^'~^ ' ' ' ' If w = -r- •,' z = fudx '.' fudx = h^u + och^^ -T- + /SA^S ^2 -f- • • • • where a, /3, y, • • • represent the successive numerical co- efficients. Hence we infer, 1 ^ , ^^du ^. d^u Taking the differential coefficients of each member, ob- serving that d^ii dx ~ du ^dx obtain dx~ 1 — ahX d'u -/3A^ d'u d^u 1 du ' dx ahl. d^ti dx' 1 d^ dx^ .v^^'^ ^^^d^^- -»-£ SECT. IV. THE CALCULUS OF DIFFERENCES. 513 Eliminating successively the functions da _ dH „ the final result must have the form 1 ^ , y du \, d^u h^ dx dx^ In a similar way we may obtain the values of the integrals 22m or 2'm, 222w or 2^m, and in general for 2"'m. The for- mula •.•25 =^•"2'"-,— + a^'"+»2'" 5 — r; + l3A'"+»2'»+2 Let ;7-^ =zu\- z ^f^vdaf* ',- 1 /< » , <^w ^,, d^u Assuming the differential coefficients of each member of this equation we find successively, du 1 d^u d^u Eliminating the functions Sm^ vm^ .... the final result will have the form ^""u = jip^f^ydx"' + -^.r'^'udx^"' L L 514 THE CALCULUS OF DIFFERENCES. SECT. V. SECTION V. Of the summation of series. (528.) If the successive terms of a series be expressed by the notation explained in (492.) the sum of all the terms from that whose index is 1 to that whose index is x inclusive may be expressed by sw^, ; thus SM^ = Wi + Wa + %• In like manner SW^+„ = Wi + W2 +2^^ -f ^+1 + + ". + «• Subtracting the former from this we have by which we may express the sum of any number of terms of a series commencing and terminating at any proposed terms. (529.) Ifw = 1 we have But by (494.) A(sw^) = s%+i — sw^, ••• s% = 2w,,+i + c, c being the arbitrary constant. When a: = 0, s% = .' = 2Mi + c. Subtracting this equation from the last we find sw^ = Dm^+i - 2w,. Hence the summation of the series depends on the inte- gration of Wj:+i and Ui considered as differences. (530.) In like manner if the sum of the series from the wth to the ath term, including the latter, be required, we have SECT. V. THE CALCULUS OF DIFFERENCES. 515 But by what has just been proved, •.• SM^ — SM„ = ^u,^i - 2w„+,. (531.) We shall now give some examples of the applica- tion of these principles to the summation of series. Ex. 1. To determine the sum of a series ofjigurate num- bers of the first ^ second^ and sitccessive orders^ beginning with unity in each series. The figurates of the first order are the series of integers 1,2,3,4, .... of which the general term is x. Those of the second order are L2 2^ 3^4 4^ 1.2' 1.2' 1.2' 1.2' ic X -4- 1 of which the general term is ' ^. . Those of the third order are 1.2.3 2.3.4 3.4.5 1.2.3' 1.2.3' 1.2.3' X X -\-\ X -4- 2 of which the general term is -^ — YtTa — • x.Ai.O Those of the fourth order are 1.2.3.4 2.3.4.5 3.4.5.6 1.2.3.4' 1.2.3.4' 1.2.3.4' „ , . , , , . x.x + \,x-\-9>.x-\-S of which the general term is Tq^ * And in general the figurates of the nth order are 1.2.3 ' ' ' n 2.3.4 . . • n+ 1 1.2.3 . . . n' 1.2.3 • ' ' n 3A5j_^-f2 4.5.6 ^^_ • n+S ~TXS . . . w ' 1.2.3 ' ■ ' ' n 516 THE CALCULUS OF DIFFEEENCES. SECT. V. of which the general term is X.X + l.a;^2 x + jn— 1 ) 1.2.3 n • For those of the first order we have M^ = jr. Hence by (529.) su, = 2(a; -I- 1) -I- c. By the table in (522.) we have x^ X Changing x into x -^1 and h into 1, we have {x+\f - X , V 2w, = i- + c. Subtracting this from the former, we have ^(^+1) 2?/^+, — 2Mi = — j-^ = SM^ vl+2 + 3....+. = ^:f±i. For the figurates of the second order, ''^ - "TF *•* ""'^ ' - iTs • By the formula estabhshed in (519.), Hence 2«,+. = — j^— + C. no constant being added because when ^ = 0, su^ ~ 0. In general for the sum of the figurates of the wth order a?+l . a: + 2. x+S' ' ' - -x^n W.+. - ~i— ~S . 3 n ' X'X-\-\ x^^ x-\-n '•'^"'+1 = 1-2 . 3 »+r SECT. V. THE CALCULUS OF DIFFERENCES. 517 ;r-;r + l'a:-f-2 £c-{-n *•* ^"- = 1.2 . 3 n-hV Ex. 2. To determine the sum of a series of the reciprocals ofthejigarate numbers heginning from unity. The general terms of the several series are in this case, X 07(0: +1)' 1.2-3 Hence, by (520.) the sums for 2°, 30^ ^^ 1 .2 2_^ %(^ + l)~ 074-1 "^^^ 1 ♦ 2 • 3 3 ^ ^(.r+l)(o;-}-2) ~ (07 + 1X^4-2) + ^' ^*'* The formula (520.) fails for s f — J. The constants being supplied by the condition = ~ -J- + c, = - :j--g + C, &C: 12 2 2 2o: s 07(:r-fl)~" 1 07+1 '~07f 1' 1.2-3 3 3 %(07 4-lX^ + ^)~l -2" (^ + l)(07 + 2)' ^""^ 2 3 When 07 is infinite, these values become -j-, -^ — r, &c. Ex. 3. To determine the sum of 13+23 + 33 07^. By the table in (522.) we have (0;+l)* (07 + 1)3 (07+1)2 2(07 + 1)^== 4 3 ' 4 518 THE CALCULUS OF DIFFERENCES. SECT. V. ^ A(^-fi) Y Hence it follows that 13 + 23 ^ 33 . . . a;3 = (1 + 2 + 3 . • . ^)2. Ex. 4. To determine the sum of an arithmetical series a +(a + (I) +(a + ^d) + [a + (j: - 1 )rf]. In this case u, = a-\-(x- l)d, '.' u^+i = a + xd, ^(;r— 1) - •.• Dm^+i = xa + ^ ^ rf. Hence sw^ = ^a H — ^— a. Ex. 5. To find the sum of a geometrical series In this case Hence (524.), ff, «r, ar*, «r' u^ = ar''~\ '.' Ur+i — «?**". r — 1 When X = 0, su, = 0, •.• a ',' c = r-V a(r'-\) Ex. 6. To find the sum of the series _ 1 J__ _]__ 1 SECT. V. THE CALCULUS OF DIFFERENCES. 519 In this case (520.), 1 1 Whence c = 1, *.* Ex. 7. To determine the sum of the series 12 4- 22 + 3« + .r2. In this case, u,+i = (^ + 1)% _ (^4-1)3 i^ + iy (^'-\-l) 1 1 1 X^ X^ X Ex. 8. To determine the sum of the series sUj. = cos. (p + cos. 2^ + COS. S(p + cos. x(p. Hence (5^5.), U^+l = COS. {x + 1 0"?? ^"^^^ - 2sin.i^ + ^' sin. 4^ 2 sm. i(^ Ex. 9. To find the sum of su^ = sin. (p 4- sin. 2