:~~~~~~. ELEMEN TTS OF TIHE DIFFERENTIAL AND INTE GRAL CALCULUS. ARRANGED BY ALBERT E. CHURCH, LL.D., PROFESSOR OF MATIEMATICS IN TIE U. S. MILITARY ACADEMY. IMPROVED EDITION, CONTAINING THE ELEMENTS OF THE CALCULUS OF VARIATIONS. NEW YORK: PUBLISHED BY A. S. BARNES & CO. 51 JOHN STR E-ET. 1855. Entered according to the act of Congrecs, in the year 1S50 by ALBERT E. CtiUi'CIT, An tne Clerk's Office of the District Court of the Soutlhern District of New-York. P RE F A. C E. AN experience of more than fifteen years, in teaching large Classes in the U. S. Military Academy, has afforded the Author of the following pages unusual opportunities to become familiar with the difficulties encountered by most pupils, in the study of the Differential and Integral Calculus. The results of previous endeavours to remove these difficulties were given to the Public in a former edition. The favour with which that edition has been received, induces him to offer a new one, containing, not only such modifications as have been suggested by a thorough trial in the recitation room, but, in addition, an elementary treatise on the Calculus of Variations. That he has, in some degree, realized the hope of advancing a more general and thorough study of one of the most important auxiliaries to scientific research, is an ample reward for the labour which he has bestowed upon the work. The Author has in preparation, and expects soon to publish, an Elementary Treatise on Analytical Geometry. U. S. Military Academy, West Point, N. Y., August 1, 1850. CONTENTS. PART I. DIFFERENTIAL CALCULUS. Page Definition and classification of functions............................ 1 Definition of the differential, and differential coefficient.............. 6 Rules for obtaining them....................................... 8 Expression for the new or second state of a function..................10 Manner of making first term of a series greater than the sum of all the others.......................................................... 10 Differential coefficient of an increasing function is always positive, &c. I1 Equal functions of the same variable have equal differentials..........12 Differenatation of the product of a constant by a variable....... 13 Differential coefficient of one variable the reciprocal of that of the other.14 Differential coefficient of an implicit function...............15 Differentiation of the sum or difference of several functions...........17 " of the product...................1... 8.........1 of the power of a function.................... 20 of radicals......................................... 22 of a fraction....................................... 23 " of miscellaneous examples...........................24 Successive differentiation..................................... Maclaurin's Theorem..................................... 29 Taylor's Theorem...........................33 Failing case of Taylor's Theorem.................................38 Development of the second state of a function of one variable..........39 Differentiation of the transcendental function a............. 40 " of logarithmic functions............... o.....42 vi CONTENTS. Page Differentiation of the circular functions............................ 47 " of the arc in terms of its sine, cosine, &c............ 51 Development of the sine and cosine in terms of the arc.............. 53 " of the arc in terms of its sine, and tangent............. 55 " of the second state of a function of two variables........ 57 Differentiation of functions of two or more variables................. 62 Development of any function of two variables....................... 66 Differential equations..................................... 67 Immediate differential equations.........6................... 69 Partial differential equations...................................... 72 Vanishing fractions.......................................... 74 Maxima and minima in functions of a single variable............. 84 Solution of problems in maxima and minima...................... 95 Maxima and minima in functions of two or more variables........... 99 Application of the differential calculus to curves................. 103 General equation of the tangent line................... 107 " " of the normal.............................. 109 Expressions for the subtangent, subnormal, tangent, &c.............. 109 Mode of determining whether a curve is concave or convex......... 111 Asymptotes..................................................... 12 Differentials of an arc, area, &c........................ 116 General remark................................ 124 Definition of singular points..................................... 127 Points of inflexion............................................ 130 C usps..........................................................132 Multiple points.................,.............. 134 Conjugate points.................................................135 Osculatory curves............................................... 137 Order of contact determined.....................................141 General expression for radius of curvature......................147 Definition of curvature........................................ 149 Value of radius of curvature of the conic sections.............151 Evolutes.........................................................153 Rule for finding the equation of the evolute..........................156 Logarithmic curve..........................................159 Cycloid..........................................................161 Spiral of Archimedes....................................... 169 Parabolic spiral................................................. 171 Hyperbolic spiral............................................... 171 Logarithmic spiral................................. 173 CONT TENTS. vii PART II. INTEGRAL CALCULUS. Page Object and first principles........................................ 175 Integration of monomial differentials............................ 176 " of particular binomial differentials................. 178 " of fractions, in which the numerator is a constant into the differential of the denominator..............................180 Discussion of the arbitrary constant and integration between limits....183 Integration of differentials of circular arcs......................186 of rational fractions..............................189 by parts............2..01 " of certain irrational differentials........................203 of those containing V/a+.bxc...-.................206 of binomial differentials...............................212 Formulas A, B, C, D and E..................................... 217 Jntegration by series......................................222 Series of Be'-nouilli......................................... 22G for integrating between limits............................... 22 7 Integration of transcendental differentials........................ 228 " of dilferentials of the higher orders................23 of partial differentials.................................. 39 " of total differentials of the first order containing two variables 242'" of the same when homogeneous......................245 Integration of total differentials of the first order containing three or more variables.................................................247 Mode of differentiating an indicated integral...................... 248 Separation of the variables in differential equations........ 250 Integration of linear equation dy+P'ydx= Qdx..................... 255'" of certain equations which may be made homogeneous.... 256 Equations not directly integrable in consequence of the disappearance of a common factor....................................... 257 Differential equations containizg the higher powers of d..... 259 Singular solutions..............................264 Integration of differential equations of the second order.............. 266 4" of differential equations of the higher orders...........271 " of linear equations.............................. 273 Viii CONTENTS. Page Integration of partial differential equations.................. 277 Rectification of curves.............................281 " of spirals................,......................... 286 Quadrature of curves............................................287 of spirals.......................................... 295 " of surfaces of revolution...........................299 Cubature of solids of revolution................................... 302 Ap plication of the Calculus to suraces................305 AMTaximum inclination or slope of surface.......................... 309 Equation of tangent plane to surfaces..............................310 Distance from any point of the normal to the point of contact.........312 Osculatory surfaces................................312 Circles of least and greatest curvature..............................315 Cubature of solids, in general.....................................317 Area of the projection of a plane area..............................321 Quadrature of surfaces, in general............................... 322 PART ITI. CALCULUS OF VARIATIONS. First principles.........325 Variation of the differential equal to the differential of the variation... 327 Variation of the integral equal to the integ'ral of the variation........ 328 General expression for the variation of a function............328 General expression for the integral of the variation of a function......331 General expression for the variation of fv..........................334 Maxima and minima of indeterminate integrals.....................334 Conditions of maxima and minima................................. 336 Problems in maxima and minima...............................337 Method of reducing the number of independent variations........... 342 PART 1L DIFFERENTIAL CALCULUS. FIRST PRINCIPLES. 1. In the branch of Mathematics here treated, as il Analytical Geometry, two kinds of quantities are considered, viz, variables and constants; the former admitting of an infinite numlber of values in the same algebraic expression, while the latter admit of but one. The variables are generally designated by the last, and the constants by the first letters of the alphabet. 2. One variable quantity is a function of another, when it is so connected with it, that any change of value in the latter necessarily produces a corresponding change in the former. Thus in tie expressions U = bx au =2 C x3 u. is a function of x, and x is also a function of u. One of these variables is usually called the function, and the other the indepcndent variable, or simply the variable; since to one, any arbitrary values may be assigned, and from the connection between the two, the corresponding values of the other dedcced. 1 2 DIFFEIRENTIAL CALCULUS. This relation is expressed generally thus, u =f (x) u = ( ) or f (UZ, ) = 0, f and p being mere symbols, indicating that u is a function ofx, The first two expressions are read, z a function of x, or i equal to a function of x; and the third, a function of u and x ecjual to zero. 3. FIunctions are Jizcreasing and Decreasizng: Increasing, when they are increased if the variable be increased, or decreased if the variable be decreased: Decreasing when they are decreased if the variable be increased, or increased if the variable be decreased. In the expressions u- = a3 u = ( + a)3, u is an increasing function of x. In the expressions y y = y (a- a x) X y is a decreasing function of x. In the expression - (a- y) z is a decreasing function for all values of y less than a, but increasing for all values greater than a. 4. Functions are also Exkplicit and Implicit: Explicit, when the value of the function is directly expressed in terms of the variable: Implicit, when this value is not directly expressed. In the examples a t = ( x)of x. In the exam u and. y are esxplicit functions of x. In lthe examples DIFFERENTIAL CALCULUS. 3 au2 + bx - Cx2 y2= a2, 2 or au2 + bx cx2 == 0 yO + 2 - = 0, they are implicit functions of x. The relation between an implicit function and its variable may be expressed, either by a single equation, as above, or by two or more equations, as u == ay y2 = bx, in which u is an implicit function of x. The first relation is indicated generally by f (u X) = 0, and the other thus, u =f () y = q (x) 5. Functions are also Algebraic and Transcendental: Algebraic, when the relation between the function and variable can be expressed by the ordinary operations of Algebra, that is, by addition, subtraction, multiplication, division, the formation of powers denoted by constant exponents, and the extraction of roots indicated by constant indices: Transcendental, when this relation cannot be so expressed. In the examples tu = log x u = sin (a - x) u = a"'u is a transcendental function of x. If the variable enter any -of the exponents, the function is called Exponential. If the logarithm of a variable enter, the function is Logarithmic. In the expressions ut = sin x u- = cos x u= - tang x 4 DIFFERENTIAL CALCULUS. u is said to be a Circular function. 6. A quantity may be a function of two or more variables, as in the examples u = ax2 + by z =- axy2 ux denoted in general thus, u=f (,) = F (x,, u). If in a function of a single variable, the latter be made equal to zero, the function reduces to a constant, as in the examples u = ay u = c- b i2; if y = 0, we have u= 0; if x 0, u —= c. If in a function of two variables, one be made equal to zero, the function, in general, reduces to a function of the other. So in a function of three variables, if one be made equal to zero, the result will be a function of the other two: If all be zero, the function reduces to a constant; as in the example U = ax + y2 +- Cz3 + d, z = 0 gives u = ax + by2 - d = f (, y); = 0 and y 0 give u = ax - d =f (); z = O, y = 0, and x O, give u = d = a constant. If then in a function of one or more variables, a variable be made equal to zero, the result will be entirely independent of that DIFFERENTIAL CALCULUS. 5 variable. If however in a function of several variables, one be a factor of all the terms containing any of the others; when this variable is 0, the function reduces to a constant, as in the example u = c + axy +bzy^ == f (x, y, z), y = 0 gives 7. To explain what is meant by the differential of a quantity or Jfinction, let us take -the simple expression e = a,2........... (1) in which u is a function of x. Suppose x to be increased by another variable h; the original function then becomes a (x +- h)2; calling this new state of the function c', we have' =- a (x + h)2 = cax2 + 2axh -+ ch2. From this, subtracting equation (1), member from member, we have't - Zt 2axh + — c............(2). The second member of this equation is the difference between the primitive and new state of the function ax2, while h is the difference between the two corresponding states of the independent variable x. As the variable h is entirely arbitrary, an infinite number of values may be assigned to it. Let one of these values, which is to remain the saze throughout the Calculus, be denoted by dx, and called digferential of x, to distinguish it from all other values of h. This particular value being substituted in equation (2), gives for the corresponding difference between the two states of u, or ax2, 6 DIFFERENTIAL CALCULUS. U' - u 2 ax.dx + a (dx)2. Now, the first term of this particular difference is called the diferential of u, and is written du = 2ax.dx. The coegicient (2ax) of the differential of x, in this expression, is called, the diferential coe/ficient of the function u, and is evidently obtained by dividing the differential of the function by the differential of the variable; and is in general written duZ 2ax. dx Resuming the expression - u= 2axhl + ah2, and dividing by h, we have' u = 2ax ~+ ai. In the first member of this equation, the denominator is the variable increment of the variable x, and the numerator the corresponding increment of the function u; the second member is then the value of the ratio of these two increments. As h is diminished, this value diminishes and becomes nearer and nearer equal to 2ax, and finally when h = 0, it becomes equal to 2ax. From this we see, that as these increments decrease, their ratio approaches nearer and nearer to the expression 2az, and that by giving to A very small values, this ratio may be made to differ from 2ax, by as small a quantity as we please. This expression is then properly, the limit of this ratio, and is at once obtained fiom the value of the ratio, by making the increment h = 0. It will also be seen, that this limit is precisely the same expression as DIFFERENTIAL CALCULUS. 7 the onel which we have called the differential coefficient of the function it. Whhat appears in this particular example is general, for let u f (x), u being any function of x, and let x be increased by h, then'= f (x + h). Suppose f (x + h) to be developed, and arranged according to the ascending powers of h, and ut to be subtracted from both members, we then have z'- Ph + Qh2 + R + - &c............(3) P, Q, R1 &c., being functions of., and every term of the second member containing h, because u'' - u must reduce to 0 when h - 0. Substituting for h the particular value dx, and taking the first term for the differential of uz, we have du du =- Pdx, and P. dx Dividing both members of equation (3) by h, we have P + Qh + R h2 &c...........(4). Obtaining. the limit of this ratio by making h = 0, and denoting it by L, we have L —P, the same val found above for d; hence, the dierential coeidx cient of a function is always e'qual to the limit of the ratio of the increent of the variable, to the corresponding increment of the function. 8 Dll!`EW,'DIiFFER.ENTIAL CALCULUS. 8. The differenftial of a function of a single variable may then be thus defined. If the variable be increased by a constant qcuantity, called the dclierential of the variable, and the difference between the new and primitive states of the function be developed according to the ascelndin powers of the increment; that term of thJis &diference which contains the first power of the increment is the dciferential of the function. It will in general be found most convenient to obtain first, the differential coeffiient, for which we have the following rule: Give to tAe variacble a variable increment, find the cog'espond'ing state of the function, from which subtract the primitive state, divide the eemcaindccr by the incremenzt, obtaizin the limit of,this ratio by mcaaing tke increment equcal to zero, the result will be the differential co-eficient: This, multiplied by the differential of the variable, will give the differential of the function. The object of the Differential Calculus is, to explain the mode of obtaining anld applying the differentials of functions. 9. Let the preceding principles be illustrated by the following Examples. 1. Let u =- bx3. For x put x -- h, then,':- b (x + h-t) bX3 3 bxh + 33bxh + b3 2 u' - u 3bSh- + 3bxh2 + bha' __,- 3b x - 3b Oxh + b2; passing to the limit, and denoting it by L, we have L = 3bxI (IX2 DIFFERENTIAL CALCULUS. 9 whence du =- 3bx2dx. 2. Let U ax2 CX. Putting x +- h for x, and subtracting, we have u' u = 2axh + ah2 - ch u',-u _ 2ax + ah - c; h making h 0, we have du L = 2ax - c d dx whence du = 2axdx cdx. 3. Let a U = -, x then = a x+ h a a -ah x- h z 2x +q x x + h x x + and C) 10% ~ ~ DIFFERENTIAL CALCULUS. a du L = _ _ x dx whence adx du- - - 4. If I = 3ax3 - mnx4 du (=9ax2 - 4mzn ) dx. 10. Equation (4) article (7) may be put under the form - P - h (Q - Rh + &c.), and if the expression Q + Rh + &c., (whichl is a functionl *it and h,) be represented by P', this becomes h -- P.p + Ph............(1); whence' = Ph +- P'h2; that is, the new state of the function is equal to its 2p;.itiive stat,. plus the differential coeicient of the function into the firs pow7er of the increment of the variable, plus a function of the varible land its increment, into the second power of the increment. This expresssion for the new state of thle function being an iniortiant one nhould be carefully remembered. 1I. If we resume equation (3) Art. (7), divide by h and tra nspose P; we have DIFFERENTIAL CALCULUS. 1 U' "P - P = Qh + Rh2 +- &c. h Since when h 0, the expression for the ratio -/ - reduces to P, Art. (7); we can plainly assign a value to h so small that G! - U /'/ - It - < 2P or - P < P; h h whence Qh + R2 + &c. < P, and multiplying by h Ph > Qh2 + Rh3 + &c., which condition will be fulfilled by any value of h which will U/ - U make < 2P. That is; in ac series arranged according to the ascending powers of a variable, it is always possible to assign to the variable, a value so small as to make the first term numerically greater than the sum of all the others. 12. If u be an increasing function of x, its new state u' will be greater than u, and it/ - U h P + P'h............Art. (10) h will be positive for all values of h. If ut be a decreasing function, the reverse will be the case, and the ratio be negative for all values of h. But we see by the preceding article, that when h is sufficiently small, the sum of all the terms that follow P, in the above ecluation, will be less than P, and therefore the sign of P will be the same as that of the ratio; that is, positive when u is an increasing 12 DIFFERENTIAL CALCULUS. and negative when u is a decreasing function. But P is the differential coefficient of u, Art. (7). Hence, the differential coefficient of an increasing function is always positive; and of a decreasing fanction, negative. It should be observed, that the signs of the differential and differential coefficient are always the same. 13. Let U -'- V? u and v being functions of the variable x, which are equal to each other for every value of x. If x be increased by h, and u' and v' be the new states of u and v, we have U' =' Ut' -t or placing for u' and v' their values as expressed in Art. (10); Ph -+ P'h 2 Qh + Q'h, or P + P'h 0= Q + Qh1, and since P and Q are entirely independent of h, when h = 0 there results P Q or Pdx - Qdx. But P is the differential coefficient of ul, and Q the differential coefficient of v, Art. (10), therefore du- = dv, that is; if two functions of the same variable are equal, their differentials will also be equal. 14. But if u —=v C, DIFFERENTIAL CALCULUS. 13 u and v being functions of x, and C a constant; and x be increased by h, we have u' = v' V C, or placing for v' its value, u' -— v + Qh + Qlh2 _ C u'=v+Qh+Qh~C u Q + QQ'h and passing to the limit, L- = -; dx whence du Qdx. Q being the differential coefficient of v, Qdx is its differential, therefore du = d (v C) = dv, that is; if two diffrentials care equal, it does not follow that the exspiessions fr om which they were derived, are equal. We see also, that a constant connected by the sign i with a variable, disappears by differentiation. In fact, the d'i'erential of a constant is zero; since, as it admits of no increase, there is no difference between two states, and of course no differential, Art. (8). 15. Let u = Av, then'= Av = A(v + Qh + Qh'2)...... At. (10), h- u= A (Q + Q'h) 14 DIFFERENTIAL CALCULUS. dv L = AQ == d; whence du = AQdx. dxa But Qdx is the differential of v; therefore du = d (Av) = Adv, that is, the differential of the product of a constant by a variable function, is equal to the constant multiplied by the differential of the function. 16. When two variable quantities are so connected that one is a function of the other; either may be regarded as the function, and the other as the independent variable. Thus from the expression u -= ax, we readily obtain x = u; in which x may be a considered a function of the variable u, In general, let u=f ()............(1); then by deducing the value of x, x = f' ()............ (2). In this last expression, let the variable u be increased by any variable increment u' - u =- k, x will receive the corresponding increment x' - x, and the ratio of these increments will be x' --...........(3). - x. (3). If the increment x' - x be denoted by h, and we substitute x + h for x, in equation (1), we shall obtain' - u = Ph +- P' P k, and substituting these values of x' - x and k in expression (3), we have DIFFERENTIAL CALCULUS. 15 x- x -E- h I k Ph + P'h P + P'h Passing to the limit by making k, the increment of u, equal to 0, in which case h =, we have 1 dx P dP Y du Biut P,=?, hence dx I1 du du dx that is, the differential coeficient of x regarded as a function of W, is the recipnocal of the diferential coeficient of uz regarded as a function of x. To illustrate, take the example u = aC2; whence In article (7) we have found d- 2ax, then dx dx I I I I du du 2ax /u 2 /au dx 2 a"Ic 17. Let u be an implicit function of x of the second kind, mAt. (4), as, =f (y)...... (...) y ()..... (2). 16 DIFFERENTIAL CALCULUS. If x be increased by h, y will receive an incremenlt - y, which we denote by k; and these increased values of y and x in the second members of (1) and (2) will give u' u + Qk + Q'- y' y + Ph- + P/h2; whence U' - U Y -Y + p'-u Q + Q'k' P -.P'h, k f and by multiplication, u - u x - - QP + Q'Pk + QP'h + &c. k ft or since y' - y - I l - ~ QP -. Q'Pfk + QP'h + &c. h Passing to the limit by making' h O 0, which gives k = O, we have L QP du dx But Q du and P dyj dy dz whence du du dy dx dy dx that is, the diferential coeicient of u regarded as a function of x, is equal to the diferential coegicient of u regarded as a function of DIFFERENTIAL CALCULUS. 17 y, multiplied by the di'erential coeficient of y regarled acs a function of x. If u -f(.).........(3) and v ()............(4); in which case ut is evidently an implicit function of v; we find from equation (4) =' ( )............( ), and applying the preceding principles to equations (3) and (5), we have clu du dx 6) dv dx dv But dx 1..........Art. (16), dv dv dx which value in (6) gives du du dx. dv dv dx DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 18. Let U = V ~ W ~ ZJ in which v w and z are functions of x. Increase x by h, then 3 18 DIFFEIENTIAL CALCULUS. U' = V' -r W' ~ Z' - U = (V' - ) ~ (w' - w) i (Z'- Z), from whiicl, after substituting for (' (' (' -2 w), &c., their values as in article (10), and dividing by h, we have U - "u (Q + Q7'h) -( (R + rlh) ~ (S + s'h). Passing to the limit L-Q ~R I S= d; dx whence du = Qdx.-h hIdx h Sdx; or since, Qdx = dv, Rdx = dw, Sdx = dz............Art. (8), du -dv ~ dw ~z dz, that is; the cdiffrential of the sum or diff'erence of any number of functions of the same varicable is equal to the sum or difference of their differentials taken separately. Thus, if u = ax2 ~ bx' du = d (ax2) - d (bx3) = 2axdx - bdx......... dArts. (7 & 9). 19. Let uv be the product of any two functions of x, then, if x be increased by h, u'v' will be the new state of the product. But u' ~ Ph - P'2, V = + Qh + Q/'2, and by multiplication, DIFFERENTIAL CALCULUS. 9 LVU'v = uV + vPh + uQh + PQh2 + &C., thence hUV = vP -+ uQ + terms containing h. IPassing to the limit we have L = vP + u d(uv). dx whence duv = vPdx + uQdx = vdu + udv, that is; The differential of the product of two functions of the same variable, is equal to the sum) of the products obtained by mlultiplyin each fanction by the differential of the other. 20. Let uvs be the product of three functions. Place uv = r, then uvs = 7'S, and d(uvs) = d(rs)- rds + sdr....-...... (1). But since r = Uv, dr = udv + vdul; hence by substitution in equation (1) d(.vs) =- vds + suldv +- svdu. If we ha\ve the product of four functions vvsw, we may place sw = r, and by a process precisely similar to the above, obtain d.(uvsw) = uvsdw.4- ds + ivd sdv +- vwssdu. (...........(2), and we readily see, that by increasing the number of functions, we may in the same way prove, that the diflerential of the pro 20 DIFFERENTIAL CALCULUS. duct of acny nzumber of functions of the same variable, is equal to the sum of the products obtained by multiplying the differential of each into all the others. 21. If we divide both members of equation (2) of the preceding article by uvsw, we have d(uvsw) _ dw ds dv du — + - + —, 1..VSVW 10 V U and we should have a similar result for any number of functions; whence we may conclude in general, that the differential of the product of any number of functions divided by the product, is equal to the sum of the quotients obtained by dividing the differential of each function by the function itself. 22. Let u. = v" v being any function of x, and m any number, entire or fractional, positive or negative. Increase x by h, then u' =v' = (v + Qh + Q,2)...........Art. (10), or placing in the binomial formula (x + a)' = x" + max'-' + (- x + &c., 1.2 v for x, and (Qh -- Q'h2) for a; we have i = [v + (Q7h + Q1h)],m = v + m(Qht + Q/h)v'l-l + &c.,^ _ =. m(Q + Qth)v'- + &c., h DIFFERENTIAL CALCULUS. 21 each of the following terms containing h as a factor. Then du L = mvm-IQ; du dv"' = lmvm-lQdx = mv-ldv...........(1), since Qdx = dv, Art. (8). That is, to obtain the differential of any power of a function: Diminish the exponent of the power by unity, and then multiply by the primitive exponent, and by the di~fferential ~f the fuection. Examples. 1. If u = ax then Art. (15) du - a.dx -= a.4xmdx 4 axcdx. 2. If u = bx3 2 2 2b0dx du &'bIdx - bx- dx 3 3 33 x 3. If u cx-3 3cdx du -- -- 3c -edx -- 4. If u = (ax - x) du = 5(ax - x,)4 d (ax - x'2) but d(ax - x2) = adx - 2xdx.A........... Art. (18); hence du = 5(ax - x)4 (a - 2x) dx. 22 DIFFERENTIAL CALCULUS. 23. If in equation (1) of the preceding article we make m = we have n L 1 1- l- dv (dv = -.V dv - -v civ ( I, n n or d V/'v=,~n n V, Vv If n 2, we have _ dv dv that is, the differential of a radical of the second degree is equal to, the differential of the quantity ulner the radical sign, divided by twice the radical. If n - 3, we have dv V3 - dv d-Vv = - 7 3Vv and in general the differential of a radical of the nth degree is equal to, the ditferential of the quantity under the radical sign divided by n times the (n - l)th power of the radical. Exzampleso 1. If u= = 3/a d daSx 3ac2dx 3 2'V'ax v'ai); 2 DIFFERENTIAL CALCULUS. 23 2. If u= /b- du = 3. Let u=.-bx2 4. Let u = V 2ax- x2. 24. Let u = -= sv-l v s antd v being functions of the same variable, then Art. (19) du = v-'ds + sdv-l= v-lds - sv-2dv, or ds sdv due - -; v v1 whence by reducing to a common denominator s vds - sdv du -- 2 -- -v............(1) V V2 that is, the differential of a fraction is equal to, the denominator into the differential of the numerator, minus the numerator into the dciferential of the denominator, divided by the square of the denominator. If the denominator be constant, dv d== 0, and equation (1) becomes vds ds du= = v? v If the numerator be constant, ds 0, and equation (1) becomes sdv du = - In this last case it is evident that u is a decreasing' function of v and that its differential should be negative, Art. (12). 24 DIFFERENTIAL CALCULUS. Examples. 1. If u -- a -- du (ax) dx-xd _( -x) _ (a-x) dx + xdx adx (a-x)2 (a- )2 -( a-x)2 ax4 2. If U - b daax4 4ax3dx du - b b 3. If u Cax3 cdax3 3cdx du =-= ~. — (axt)2 ax 25. By a proper application of the preceding principles every algebraic function may be differentiated. Let them be applied to the following Miscellaneous Examples. 1. If u = (a + bxn") dult = p(a + bxn) P-ld(a + b)............Art. (22); PBut d(a + bxZ) = nbxn-dx; hence du = brip(a + bx")"'xl-ldx. The solution of this example and many others may be simplifled by applying the rule of article (17) thus: make DIFFERENTIAL CALCULUS. a + bx" =- then u = z, d= zbx~-_ du p cdx dz whence du dx dz dx- z x pz4 x xbx~1"-' =lbp(a + bx") -xnL and du = bn (a + bx")' x-ldx. 2. I- e) U = 2) d - ( 3(1- 2)2 (1 - x2) = -6 (I - 2)2xdx. 3. Let ax x + -/a + x2 Place ax y = x + V/a + x', then - xdx aydx - axdy dy =dx V + -a + x d'd y — hence a x + v- a + t ) dx - xd + T/___ )\ du =- / + a2) a (' + V/a +.'M) or after reduction a2dx 4 ( + ~ + xY- ~ + LI 26 1IIDIFFERENTIAL CALCULUS. 4 f =(b d X)2 ( (L2 - b2)dx X X 4.~, 3 f td - - -, azt - n7C 1 2 4 5, U - /a - I J dx - x- (a -x T-)'-d Va - X 7. u = dx -.2 x~ - V-I. V I- ~-x (x - 18. Let u (a - - b))3 9. Let u (1+ x) 10. I e. I - (a-._ c Vx~Llx" V xV ix)' 12, 2 ~ 13. u __- X2 + + i X 1 ~vi SUCCESSIVE DIFFERENTIATION. 26. It is readily seen from what precedes, that the differential coefficient of a function of a single variable is, in general, a funetion of the same variable. It may then be differentiated, and its differential coefficient obtained. Thus in the example u= ax3 -f = 8aa.(1), u a ZC s 3d 3axr......(), ~fax' is a function of x, different fiom the primitive function. Tf we differentiate both members of equation (1), we have (d -- 6aaxdx: dX) DIFFERENTIAL CALCULUS. 2 But since dx is a constant, Art. (24), d d(u$ d (du) _ dhzt \dx^ dx dx the symbol d2u, (which is read second differential of u) being used to indicate that the function l has been diffcrentiated twice, or that the differential of the differential of u, has been taken. Hence d2U d' -_ 6axdx, or t- 6ax, dx dx' in which dx2 represents the square of dx, and is the same as if written (dx)2. The expression, 6ax, being the differential coeficient of the first differential coefficient, is called, the second dicferential coefficient. To make the discussion general, let u = f (x) and p be its differential coefficient, then du...........(2). dx Since p is usually a function of x, let it be differentiated and its differential coefficient be denoted by q, then.. q.......(3). dx In the same way let q be differentiated and its differential coefficient be r, then dq.....(4). dx By differentiating equation (2), we have d (i z- p, or d2 = 7p, adx b the sstit dx and by the substitution of this value of dp in (3), 28 DIFFERENTIAL CALCULUS. d2u dx d'u -I- Q -or q.~( 0 ( 5), which is the second difiberential coefficient of the function. By differentiating (5), we have d3u 2-. dq, and by the substitution of this value of dq in (4), d3't ^ ~,. d3~ - - __ =', or r dx dx which is the dizerential coeficient of the second dciferential coeficient, and is called the third differential coeficient. Inl the same way the fourth, fifth, &c. may be found, each from the preceding, precisely as the first is obtained from the primitive function. From the differential coefficients, we may at once obtain the corresponding differentials, by multiplying by that power of dc the exponent of which indicates the order of the required differential, thus d2u = dx2 dx2 == q dx,2 d, m —] sn(ni - I)a""-'"m2 (a -I,- )m = a' + a'xz + (i ) + &c 1.2 2. Let u =n ad(b,w^ )b-x By differentiation &c., we hlave 32 DIFFERENTIAL CALCULUS. d /7 \2 a d'm 2a = ~a (b = )- ~ - =-2 2a ( — x)~ =( b^- ) d a ( - 42. dx (b V (~x2 d3iu 2.3.a = 2.3a(b — 2 )-_ _)~ 2............&c. dxc3 (2.a( - ) Maakidg x 0 in the original function, and in each differential.oefficient, we have A —~ A' - A" - &C These values in the formula (3) give a a a a a -' -- -_ ~~ 2 + z ~ x_ + ~... b~ -- x -b b" -' b" 3. Let u - 4. Let u 1+ x 5. u ~ 1~ 0. z = - (l+ ) WVhenever the function to be developed contains the second or lligher power of the variabl, the work will be much abridoed by sublstituting for this power a single variable, then making the de —;elopinent, and in the result resubstituting the power. Thusl, in example 6, by putting z for x2 we have U = (I x2)3 = (1 + ) lvhich is easily developed according to the ascending powers of z. 29. Functions whilc become infinite, whlen the variable on -which they depend is made equal to 0; or any of the differential DIFFERIENTIAL CALCULUS. 33 coefficients of which become infinite, under the same supposition, cannot be developed by Maclaurin's formul]a as in such cases, either the first or some succeeding term of the series would be infinite, while the function itself would not be so. u log x t = cot x t = axare examples of such functions. In the first two A, and in the third A', would be infinite. TAYLOR S THEOREM. 30. A quantity is a function of the sum of two variables, when in the algebraic expression for it, a single variable may be substituted for the sum, and the original function thus reduced, without a change offormr, to a function of the single variable. Thus, ct(X -- +y)n is such a function, for if in the place of x - y we substitute z, the function becomes u' = ct", a function of z of the same form. log (x - y) is also suchl a function of the two variables x and - y, which, when for x - y we put z, becomes log z. 31. Let u' =f( + y). For x + y place z, then u' =f(Z) and =.....(1) 5 34 D'IDIFFERENTIAL CALCULUS. p the differential coefficient, being entirely indclependent of dz. If now x be regarded as a constant d = d(x - y) = dy, and equation (1) becomes dzu' dy If y in turn be now regarded as constant d = d(x + y) = dx, and equation (1) becomes cldu' du' dcx dy That is, if Cafunction of the sum2) of two var)iables be digferentiated as though one of the variables were constant; and then the same faunction be differentiated as though the other vacricabe were constant; and the diferential coelicients be taken; these two coeficients will be equal. To illustrate, let ( = (x + y)", then cuZ'- n,(x + y)-l d(x + y), which if y be regarded as constant becomes, dx du',- (x + y)~-'dx; whence d-~ = n(x + y)"l-.l and if x be regarded as constant, the same expression becomes du' =n(x + y)"-'dy; whence ^(x + y)"-' dy 32. The object of Taylor's Theorem is, to explain the manner of developing a function of the algebraic sum of two variables, into DIFFERENTIAL CALCULUS. 3 a series arranged acco'4ding to the ascending powers of one of the variables, with/ coegicients which, are functions of thle other ant dependent also upon the constants which enter the given function. Let us write a development of the proposed form, u' f(x + y) = P p+ q - f - Sy + &-.....(), in which P, Q, R &c. independent of y, are functions of x. It is required to determine values for them, lwhich substituted in equation (1) will make it true for all values of x and y. If we regard x as constant, differentiate both members of equation (1) with respect to y and divide by dy, we obtain dy If we regard y as a constant, diierentiate equation (1) with respect to x and divide by dx, we obtain du' dP dQ dx -, = L —T y + -V: Y2 + ch' W d-v =:' d, fl'' 1 d' But by the preceding article we have' = d-; tlerefore dy dx 9 dP dQ dR y Q + 2Ry + 3Sy2 &c. d -- + d + +.; and since the coefficients of the like powers of y in the two members must be equal, dP d( CIR Q -.....(2), 2R = -......(3), 3S = -......(4). dx 2 dx dx If in equation (1) we imake y =- 0; f(x + y) will reduce to a function of x, Art. (6), which we denote by u. Then u - P. Substituting this value of P in equation (2), we have 36 DIFFERENTIAL CALCULUS. dx This value of Q in equation (3), gives d /i\ 211= \dx/ d' u dhi 2 dx =-~ =l; whenceR - 2d' and this value of RI in (4) gives d d )u \ 3 3 3S = - -; lene S = 3.1 dx 1.2.dx; 1.2.3.dx' By -the substitution of these values of P, Q, R &c. in equation (1) we have Taylor's formula; du y dcu y2 d "u y?/ u~' -= fx q- Y) - u +~^ q + -7 - +...... ~... a -f(!: 1-i/) c= u l-j l 1.2 x' 1.2.3...,2 By an examination of the several terms of this formula, we see that the first (u) is what the function to be developed becomes, when the variable, according to the ascending powers of which the series is to be arranged, is made equal to 0. The second j(du Y) is the first differential coefficient of the first term, multiplied by the first power of this variable; and the general term is the th digfferential coeffcient of the first term, multiplied by the nth power of the variable, and divided by the product of the consecu tive numbers from 1 to n inclusive. The development off(x - y) is obtained from the formula by changing + y into - y; thus du d2 yI d3tu y3 f(x - y) _ " -, Y - L 1.2. + c. fr — y) 2 -x d 1y2 +.3-~ DIFFIERENTIAL CALCULUS. 37 Examples. 1. Let u' = (x +t y) Making y = 0 we obtain u = x", and thence by differentiation, du _ _d'u — 1. dx __N = mzx"~~ ~ 2= mdm ~')___ ax dx m dtm{m-1 ) (n 2)x- d =m _ l) 4- 1_)( 2 ).... These values being substituted in the formula give tu' == ({ + y)M') 7.. -- +J- mx'^ y m "-'........ u'-Qi ~y~)'= x ~y + ~ 1.2 m(m- 1) j)......(m - n + ),x-y'" _ 1.2.3...... n If it were required to develope the function in termas of the ascending powers of x we should make x = O, and obtain for thle first terml yl', fiomn which thie other terms are derived as before. 2. Let u'. x+y x + y dlakting y 0- O, we obtain, ut = - for the first term; thence X du Ct d2u 2a dx s" d x2 d3u 2,3.a d" 2.3......... dX -- x4' dxt x'^+ These values being substituted in the formula give a a a a a: _ _ -- - -y +- y~' "'"...... -- x+ y ax x' 38 Ili'i.'DIFlFE:LENTIAL CALCULUS. 3. Developo u' -= -—...accolding to tohe powers of — y. 4. u'. " " of x. ( - y)2 33. Since in the formula of Taylor, the coefficients of the different powers of one variable are functions of the other, it is plain that if such a value be assigned to the other, as to reduce any of these coefficients to infinity, the second member will become infinite, and the formula fail to give a development for this particular value; as, in this case, the first member will become a function of the first variable, which function cannot possibly be equal to infinity for a particular value of the second variable, on which it in no way depends. Thus, in the example't -= v CI + + y which, when developed according to the ascencing powers of y, gives 2 Va- x sV (a -')3 X the particular value x == — a reduces the coefficients of the powers of y to infinity, while the original function is reduced to -/y: We should thus have /y = co, which cannot be. For every other value of x, however, these coefficients will be finite and the clevelopment true. The difference between this failing case and that of Maclaurin's formula is marked. In this, the failure is only for a particular value of that variable which enters the coefficients, all other values of both variables giving a true development; while in the former 0 - - ~ 1urVLIJIVI rll 1 J~ VIIJ DIFFERENTIAL CALCULUS. 3D case, if the formula fails to develope a function for one value of the variable, it fails for every other value. 34. If u =f(x), and x be increased by h, we have for the second state u' =f(x + h), and by changing y into h in Taylor's formula, we obtain du due h2' f(e + h) = u + 71 + _d _ + &c.; dx dx' 1.2 whlich is the develop2meznt of the second state of a /.lnction. F'rom this we have d'zt b z(P u A2 d',3^ A3 u' - u = h -- — + _- -+- &c. =dx dxl 1.2 dx' 1..2.3 If we now put for I the particular value dx, we have u'- = dut + - +. L- +&c. 1.2 1.2.3 35. If in the development of f(x + y) by Taylor's formula, we suppose x -- 0, and represent by A, A', A", &c. what u d d,'dx dx&e. become under this supposition, we have f(y) = -A - Ay + 1A /t A-2 Y &c A,', A", &c. being constant, and since y is the only variable we may write x for it, and thus have 4C DIFFERENTIAL CALCULUS. A/,Y2 fA/f/ f(x) A + A'x + +A A"-. + &c. 1.2 1.2.3 which is identical with Maclaurin's formula. DIFFERENTIATION OF TRANSCENiDENTAL FUNCTIOINS. 36. Let us take, first, the exponential function and increase x by t, then, u' = = a-a'...............(1). a' being a function of the single variable A, may be developed into a series, the first term of which [being what a' becomes when = 0, Art. (28)], is ac 1. We may then write a — 1 + kh + k'h' + k"1h- + &c., k, k', k", &c. being constants depending upon a. By substituting this value of ac in (1), we obtain u' = a(1 -( - kI + kl'h + k"ha3 + &c.); whence - _~_ -l= a, k +- ak'ht + &c. h Passing to the limit of this ratio, we have L - a"k d, alndl du =- a'dz............ (2). dx To determine the value of k, let us develope u = a" by Maclaurin's formula. We have ~a' du d.dx DIFFERENTIAL CALCULUS. 41 Cd (du) kda- k2a dx; whence d-~ = -; Ad dx" d (d_28 k)ckda' = kaa'dx; whence d3 = k3; &c. Mlaking' x 0 in these expressions, we find for the coefficients in the formula A = a~ =, A' =-~k = k, A" 2, A"' = 3, &c.; whence k2X2 k3x3 u a' 1 + kx +- + - +&c. 1.2 1.2.3 In this, let x =, then al = 1+ + I - -+ + &c. 1.2 1.2.3 Sumlming this series, we have aOk= 2, 7182818............ This number is the base of the Naperian system of logarithms, which is usually denoted by e; we then have a, = e, and a = e; hence k is the Naperian logarithm of a, denoted by la. Then du = daz = axladx, that is, the differential of a constant raised to a power denoted by a variable exponent, is equal to the power, multiplied by the Vaperiacn logarithm of the root into the difgerential of the exponent. 42 DIFFERENTIAL CALCULUS. 37. Resuming the expressions du u-cc -- a a', dx regarding u as the incependent variable and x as the function, -we have, Art. (16), dx 1 du I d 1. ~ awhence dx.. du caxla u la If a be the base of any system of logarithms, then x = lotg; in that system, and d d log zu = 1 N NOTE.-Throughout the book, the symbol I before a quantity will indicate the Naperian Logarithm of that quantity. Since the logarithms of the same number in different systems are as the moduli, we have log a: la:: M: 1, and when a is the base of a system, since log a = 1 1: la:: M: 1; whence M =' la Also, log e: le: M: 1, and since le = 1 M = log e. The modulus of a system, then, admits of two forms of expression, both of which should be remembered. The one is, unity divided b/y the Naperian logarithm7 ofthe base of the system whose modulus is required; the other, the togSrithmi of the Napcrianl base taken in the systema whose mvodulus is required. DIFFERENTIAL CALCULUS. 43 But is the modulus of the system whose base is a; then la d log u = M d For the Naperian system, M 1=, and this expression becomes dlu du'6 The differential of the logarithm of a quantity, is then equal to the modulus of the system into the differential of the quantity divided by the quantity; and this in the Naperian system, becomes the differential of the quantity divided by the quantity. Examples. e1 If'u = (ax) dax3 3ax2dx dx 3du -- _3ax ax 2. if u = a d(a \ adx du a d-a) (a-x) dx a a a - x a-cx a- x Otherwise thus, = ( a - la - 1(a-x) du = dlat - dl(a - x) a - X 44 DIFFERENTIAL CALCULUS. 3. Let = (1+ x2 + Multiply both terms of the fraction by the numerator, then u =l (v'1 + x, + )2 = 2Z(V1 + xt + x) du = 2d(t/1 + x ) + x)= 2dx 4. If = = 1.... x V-w + + VI -a - -d xi + - Vi- 6. Letu = l ~7 7. Letu (= (a + xt )2 (a ~ )2 8. u = i(. 9 I l(a - ) ) 10. Let u = (Zlx). x 11. Let u _ (lx). Make x = z, then U = 7z; dz dx du - - Z Xix DIFFERENTIAL CALCULUS. 45 12. Let u = a. 38. It has been seen, Art. (29), that log x cannot be developed according to the ascending powers of z. To obtain a logarithmic series, let us take u = log (1 + x) and develope it by Maclaurin's formula. By differentiation &c. Mdx du M du = - M(1 q- x)-~; 1 +- x dz I +- x d2u M d3u 2M = - ~M( + x)-2 3 = + )3 +- = (X XI + x) Making x = 0, we have for the values of A, A', A" &c., in the formula, A = log =0 A' =M A" - M A"' = 2M, &c; whence X'2 3 X n u = log (1 + x) = M(x- 2 +......~ -......), in which the logarithm of a quantity is expressed by a series, arranged according to the ascending powers of a quantity less by unity. 39. By the aid of logarithms we may simplify the differentiation of complicated exponential functions. For example: 1. Let u = zY, z and y being any functions of the same variable. Take the Naperian logarithms of both members, then 46 DIFFERENTIAL CALCULUS. lu = lJ = ylz; and by differentiation du dz d- dylz - y-; ff z whence d u(dylz + y dz = zzdy + yzy-'dz, z which is evidently the sumn of the diferentials, taken by first regarding y as the only variable, and then z. 2. Let u = ab Taking the logarithms of both members lu = b'la, -u la. db - lab abdx, u du = a" blalb dx. 3. Let. u = Zt then du tfdz lu = tslz, -~ = ~+ lz(tltds + st ldt), u z du z tt (~ + lzltds -+- zdt). DIFFERENTIAL CALCULUS. 4 DIFFERENTIATION OF THE CIRCULAR FUNCTIONS. 40. Since any arc of a circle, when less than 90~, is greater than its sine, and less than its tangent; we must have for all values of y less than 90~ sin y sill y sin y - < I and < ~ y tangy y But R sin y sin y cos y aang = ~-7 y whence 1? (1 ) cOS y tang y P Making y 0, cos y becomes R, and we have for the limit of the ratio (1) R L --- 1I ansd since -sn cannot exceed unity, nor be less than s-, we y tang y also have its limit = 1; that is, the limit of the ratio of an arc to its sine is unity. 41. Let u = sinll X increase x by h, then u'' sin (x + h), u' - U = sin (x J) - sin x, or by placing x -- h for y and x for q in the formula, 2 sin p - si n q ~= [sin I(p - q)cos -qp + q)], 48 DIFFERENTIAL CALCULUS. 2 u'- u = - ~ sin >hi cos (x + -hl), Dividing both members by h, and then both terms of the fraction in the second member by 2, u' - u sin 7i cos (x + Ih) h h R' and passing to the limit, since ( =_ 1, h J=0 cos x du. R = c' R, dx whence cos x dx du = d sin x- - R ~[T~f g~u = os X, el d = dcos x = d sin (900 - ) cos (90 d O -; whence sin x d cosx - R dx. If u = ver-sin x du = d ver-sin x = d(R - cos x) = - d cos; sin x whence d ver-sin x = dx. * NOTE. —This nctation indicates that the expression for the quantity wiitlin the parenthesis becomes unity when h = 0. DIFFERENTIAL CALCULUS. 49 If u = tang x RI sin x du =d tang x = d COS X (cos xcd sin x - sin xd cos a) dZ(cos2x + sin2x) cos2 cos 2x COSX COS g whence R2dx d tang x:-= cos.S If u == cot x d (900 - x) du = d cot x = d tang (90~ - x) = R2 (oo X) whence R2dx d cotx == —. sinx If = sec x R2c Rsin x d du a d secc = d --- cOS X COS x whence tang x dx tang' x sec x d sec x = - =- dx. cos x - 2 COS X _ ) If u = cosec x du - d cosec x _ d sec (90 - x) cot.cosec d(90 - t2 50 DIFFERENTIAL CALCULUS. whence d cosec x - -cot x. cosec x dx d cosec x= R2 If R = 1, these formulas become d sin x = cos x dx, d cos x = - sin x dx, dx d ver-sin x = sin x dx, d tang x = d &c. cos x' and should be remembered. Examples. 1. If u -- il b a bx bx b bx du = cos d = - cos dx. a C a C a 2. If u = cos x du = sin I d - sin dx. x x x x 3. If u = tang (a — )2 du d(a- )2 2 (a-x)dx cos2 (a - x)2 cos (a -~ )" 4. If u = cot2x 2 cot x dx du = 2 cot x d cot x = 2 cot x d, Sillx DIFFERENTIAL CALCULUS. 51 s. If u = (cos X)si"' make cos x = z, sin x - y; then u = z, and Art. (39) dt = ylzdy -+ yzY-~d =z dx (cos x)in' cos x cos xz- sin ) ^ ~~ cos x,) 6, Let u sil (1 +x. Let u = tang ( — m V ). 42. In the preceding article we have found the differentials of the sine, cosine, &c. in terms of the arc as an independent variable; let it now be required to find the differential of the arc, in terms of its sine, cosine, &c. If u = sin x, then x = sin-~u, cos x dx 7du cos x dcu, and ~- It~ 1 —B —' dx IR If now x be regarded as the function, and u as the independent variable, we have, Art. (16), dx 1 I du du cos x dx and since cos x /2 -- sin2 -= V R2 - dx _ 4 i_;du _ __ _;. whence dx = du V/ R2 u2 V u/ 2 _ ~ NOTE.-The notation sin — u, tang-l i, &c., is used to designate the arc whose sine is 2; whose tangent is ue, &c. 52 DIFFERENTIAL CALCULUS. If i du sin x U = COS X, X =- COS-, d. ~ dx 11 dx -R R R. du sin x VR2 _ os2x v R2u2 whence Rdu VR2 - z If U = ver-sin x, x = ver-sin-2 u; du sin x dr x_ dx tR du sin x or since sin r = (2R ve-sin) ver-sin x v(2R Zu); dx 11 Rd& ~-r - _;___; whence dx- =. du (2R- u)u )/2 zu -~ W If du R2 u = tang' x, x tang-' u ~ _; dx cos'x dx cos2x R2 R2 du R2 sec2x ^-2 + tang whence R2du dx 2,+ 2. DIFFERENTIAL CALCULUS. 53 If R - 1, these formulas become dx = d u_ d, du du dz = du= d_ dx ~ d _dx 2/ _ —_u2 1 + u2 Examples. 1. If = sin-' 2v/1 — u2, dx = d(2uVx1 - i2) 2du V1 - (2uVI - u2)2 VT/-_2 2. If.= tangi-( c), d - -c dx- Y cdy \ 3. If u -cos-' Y dY - ~-ady a y' (a - y) v-/a- 2ay 4. e If U -- ver-s in- du = d x -x/2x -~ 1 43, We are now able to develope sin x, cos x, &c., in terms of the ascending powers of x, by Maclaurin's formula. 1. If u= sin x, and R =1; 54 DIFFERENTIAL CALCULUS. du d2u d3u L d_ = Cos x ~- sin x -- ~ cos x, &C. dx dx dx Making x = 0, we obtain for the values of A, A', &c. in the formula, A = 0, A' 1 A" = 0, A/" = - 1, &c. thence x X3 X5 u =sinx=- - ~ &+c 1 1.2.3 1.2.3.4.5 2. If u = cos x, du d2u d3u _= - sin x, ~ = cos x, -- sin x, &c. dx dx2 - dx3 in which, making x = 0, we obtain A=1, A' = O, A"/ A -. 0 &,c. and thence x2 x4 u = cosx = - - _ &c. 1.2 1.2.3.4 These series, for small values of x, are very converging, and will give with great accuracy the values of sin x and cos x for small arcs, and may therefore be used in the calculation of a table of natural sines, &c. Thus, RI being unity, we have for the semicircumference or r, the number 3,14159...; this divided by 180, and the quotient by 60, will give the length of the are 1', which value substituted for x in the series, will give the sine and cosine of one minute. DIFFERENTIAL CALCULUS. 55 44. We can also develope the arc in terms of its sine, tangent, &c. If dx 1 -sn- u=- d. ----......Art. (42), dl't2 _ u(l _ (2) &/, du. ~Ud Making u = 0, we obtain A _0 A =- I A" 0 A' = 1, &c and by substitution in Maclaurin's formula i3 3us. Sill`U = + t 3 + &C. 1.2.3 1.2.4.5 If u = = sin 30~ this series becomes x =sin 1 30 i 1 + 3 2 2 1.2.3.23 1.2.4.5.2" by the summation of which, we find 30~ = 0,52359......2 and multiplying by 6, 180~ = = 3,14159...... 45. If x = tang-, d - 1 1 = (I + 42).....Art. (423) the development may be mae as in the preceing atile; or and the development may be made as in the preceding article; or otherwise thus. Developing (1 + u2)-1 by the binomial formula, we have 56 DIFFERENTIAL CALCULUS. dx i 4( d 1 -_ 2 + U _6 + &C......(1); dzu and since by differentiation, the exponent of u in each term is diminished by unity, we must have = Au + B13 + Cus + &c.; whence dx A + 3B3t2 + 5C04 + &c.....(2). du Comparing the coefficients of the like powers of u in (1) and (2), 1 1 A = 1, 3B - 1, and B -; 5C 1, andC=-, &c.; whlence 3 5 7 x =tang-u= u- - - - - i + &c......(3). 3 5 7 If u -- I = tang 45~, this series becomes 1 1 1 x. 450 — = 7 q- -- + &c., 3 5 7 which is not sufficiently converging to enable us to determine the length of the arc with accuracy. To obviate this difficulty, we will make use of the principle that the arc 450 is equal to the arc whose tangent is I, plus the arc whose tangent is -3-.* * NOTE.To prove this principle, take the formula tang (a + b) tang a + tang b tang a tang b DIFFERENTIAL CALCULUS. 57 From equation (3), by the substitution of I and - for u, we have, I 1 1. 1 1 tang-' - + + &c., 2 2 8.23 5.2" 7.2,tan' i I 1 I I 1 tang - &c.; 3 3 3.3' 5.35.3' hence 45' = tang-' +- tang- - 1 1 1 1 1 1.- + &c. + - 3. - &c. - 0 78539.... 2 3.23 5.2" 3 3.3" 5.3" which being multiplied by 4 gives ir = 3,14159...... DIFFERENTIATION OF FUNCTIONS OF TWO OR MORE VARIABLES. 46. Heretofore our rules for differentiation have been limited to functions of a single variable; it is now proposed to extend them to functions of any number of independent variables. Let u =f (-x, y); x and y being entirely independent of each other. The second state of the function will evidently be obtained by giving to both Make tang a = - and a + b = 45~, then, tang 45~ = I = + tang b; whence tang b'; 1 - tang b hence 450 a + b = tang-'l + tang-l. 58 DIFFERENTIAL CALCULUS. x and y variable increments. First let x receive the increment h; f(X, y) then becomesf(x + Ih, y), which, (if y for a moment be regarded as constant), may be developed according to the ascending powers of h, by Taylor's formula; whence cdu d'2c it d3u lt3 f(x + h, Y) = u, L I- -- + &......(1), df? -'2 1.2 dZx' 1.2D3 cdu ciF, in which, c c.,a e2 ate te differential coefficients of u = f(x, y) cl, X, d.X2 taken imder the supposition that x alone is variable, and are evidently all functions of x and y. If in this development -we now put y +- J for y, we shall obtain in the first member f(x +- h, y k) which is the second state of the function u. The first term of the second member (u), being a function of x and y, will, when for y we put y -- k, become du 12u sJ72 d7P u k3 (,, + ) = u + k - _- + ~ 3 - __ + &C. d1y dy5 1.2 dy 1.2.3 In the salle mlanner -, when for y we put y 1- ic, may veloped, and will give, Art. (32), 1du,\ (duin (d _du cdx td ) cI 2 dx y=y+ - dx + dy d 2 + 1 or reducing ( iin c'u ciPin k'2 tyY+F = I~ I d xdy dzdy'2 1.2 Also DIFFERENTIAL CALCULUS. 59 dfcu d, ^ t k d4t k C., t y+ dZ dX dxcdy cdx2dly 1.2 dZ - d3u, dC14- k + &c. dz 3 y+ da? d(dx'Y These values being substituted in the second member of (1) give for the dlevelopment of the second state of a function of two variables du d" u kC2 Cdu t;3 f(x + h, y + + +-= } +. &c dyl (dy 1.2 y1.2. du~ d2+ tc d3u h7C22 - c h dxy + dxdys2 1.2 +......(2). d2u h2 cd3 Jh2k 4-_ - _' *- -4- c.) + dz 1- 1. 2 2d.2 d3u h3 + - -12. + &c. c" 1.2.3 In this development u is the original function; is its difierdy ential coefficient taken under the supposition that y alone varies, and is called, the partial differential coegfcient of the first order taken with r espect to y; &c. are successive cifferential coefficients taken under the same supposition, and are calledpartial differential coeficients of the second, third, &c. order taken with dzu d'u d3u respect to y. dx' cX are obtained from the original function dz dxa dx under the supposition that x alone varies, and are called partial differential coe7ficients of the first, second, &c., order taken with resC4ct u due pect to x; d is obtained by differentiating v with respect toy day d lt dx co and divicih. the result by dy, and is called a partial dc.erential co 60 DIFFERENTIAL CALCULUS. efficient of the second order taken, by differentiating first with respect to x and then with respect to y; and in general is a dx"dym partial differential coefficient of the v, + nt' order, and is obtained by differentiating first n times with respect to x, and then m times with respect to y. By an examination of these results we see that, from a function of two variables there are derived two partial differential coefficients of the first order, viz. du du and dx dy three of the second order, viz. d'u d'u d2u dx^ d.xdy dy2 four of the third order, &c. The expressions du i du d2u c 9 lcu d dx, dy, - dx' dxdy, &c., dx dy dx dxdy obtained by multiplying the several partial differential coefficients respectively by dx, dy, dx', dxdy, &c., are called partial cdiff reztials. 47. If instead of first increasing x by h we increase y by k, we shall obtain 7 du d'eu jI2 d3u P a f(xy + k) =" -- d +- T + - + &c. and if i t we pt h fo e shall evidently d ed 1.2. ce and if in this we put x + h for x, we shall evidently deduce DIFFERENTIAL CALCULUS. 61 du d2U 7J2 f(. + h, y + k) = t+ d h + d2 + &., d~x dxI1.2 du dSu 4- k 4- __ i+ osC", + dy + dydx d2u k2 dy 1.2+ &., which development must be identical with the one in the preceding article; hence the terms containing the like powers of h and k must be equal to each other, and we must have, d2u d2u d"u d3u dn+3uu dn+?"u dxdy dydx dxdy' dy2dx dx"dy'" dymdxn which shows that we shall obtain the same result whether we differentiate first with reference to x and then with reference to y, or the reverse. 48. Let it now be required to develope the second state of the expression u -= "x...........(1). Differentiating with reference to x and y, respectively, we obtain dlit in-ync dlu irx m-lyn...(2), = nxy... -...(3). dx dy Now differentiating (2), first with reference to x, and afterwards with reference to y, we obtain...... d - -..... m(t - 1)X2y2. (4), dxldy (5). dx22 m 62 DIFFERENTIAL CALCULUS. In the same manier by differentiating (3), first with reference to x, and then with reference to y, we obtain dc1 _ lu. l d2u d2u = -1 nY"-2 dydx- dxdy dy2 and by continuing the differentiation of (4), (5), and (6), d- u m.( - 1)(2 - 2)x-3y, d dy 72(7 - 1)2nxl"-y-1 &c. Substituting these values in the formula of article (46), we have (x + h)l"'fyk x += mxT "y + n(un - l)x~n~- + 1 c. 1.2 + mxZ-ly'lh + mnxm-y"-ll1k d- &e. -- m(nm - 1)x"-yn + &Ce 1.2 49. Resuming equation (2), Art. (46), and subtracting the pri — mitive state of the function from both members, we obtain fTx +h, y+ -f(xy)- d-7= + J- fk+- Y1. + 2d2c k )dx dy. x2^dx dy Extending the definition in Art. (8), to functions of two or more variables, we have, after placing for h and k the constants dx and dy, and taking the terms of the first degree with reference to tlhese constants; du du du - dxz + dy, dx dy that is, the differential of a function of two variables is equal to the sumn of thZe partial diferentials of the function. It is important DIFFERENTIAL CALCULUS. 63 to preserve the notation dt dlx and d dy, else the partial differendx dy tials might be confounded with the total differential (dl). Examples. 1. If u = ax2y3 d dx = 2ax.y3dx, dy = 3ax2y2dy; dx dy hence du = 2axyd3dx - 3ax2y2dy. 2. If b(a - d)2 2__ = _~ du_ - (a - 2) [2xydx (a -.x)d1y]. fy y3 3. If tang d ydx - xdy u = tang- du,2 +~gy y2 + x 4. Let u =.ay. 5. Let u xy. Vx22 + y2 50. Having obtained the first differential of a function of two variables, we may from this at once derive the successive differentials. Since d2 _- dx + du dy, dx dy (=dx ) 6(dL ) 64 DIFFERENTIAL CALCULUS. Differentiating ~d dx, first with reference to x, and then with redX ference to y, we have, -/duc - d'u d2u tr d __ax = __ d2 + _dxldy \dx J dx2 + dxdy and in the same way, d /d du d'u dyl +, d'u2 d _ -Z_-dydx + -dcy; d(d ) dydx dy2 d~u d2U whence, since d = _......Arl. (47), dxdy dydx 2 dhi7tt du cl 2u du - ~ dvx + ~2 dxady + ~- dy dx2 dxcly dy Differentiating this result, since /dct du 2I d3Ud Cx u X d ) ]= — + x d -dy, d3 f d tu wdxe) Tdx' + dxdy d'u da d 7 7 d{ dxdy, dXl3 dx 2dy +.dxf, \dxdy - xdy dxdy2 d(2t4dy2 d 3-udyT x + -dy, d )9 dy3dx dy3 we derive d-d dxdx +a dn 3&tt d X2 3 d'u 9d'u 9 dcu _= -— dx1 + dx2dy + ddxdyd + -dy dWx d d xdy 2 dxdy dy In the same way the differentials of a higher order may be derived. DIFFERENTIAL CALCULUS. 65 51. Let us now take the general case in which u is a function of any number of incependent variables; that is, let u =f/(,, Z &C.) It is plain that we may deduce the development of the second state of this function in precisely the same way as in article (46), by first increasing x and y, then in the result thus obtained increasing z, and in the new result increasing one of the other variables, and so on until each shall have received an increment; we shall thus find f( +h, +k,z+ &c.)=f(x, y, &c.) +d h+ t k + + &c.; dx dy dz whence du1 cdu d16 f(x h y + /, x + 1, &c.) -/ (x, y,, &c.) = C h + k + I &C. dx dy dz plus other terms, which will be of the second degree at least, with reference to the increments h, k,, &c.; we have then as in article (49), 71\ d du du dz = clf(x, y, z, &c.) d dx + dy + d dz + &c. dx cly dz that is, the difierential of a function of any number of variables is equal to the sum of the partial differentials of thefunction. Example. If u y= axyz3, dzu = ay23dx -- 2axyzddy + 3axy2zdz. 9 66 DIFFERENTIAL CALCULUS. 52. If in the development (2), article (46), we make both x and y equal to 0, the first member will become a -function of h and k; the first term of the second member, and the different coefficients of h and k, will under the same supposition become constants. Denoting by A what u or /(x, y) becomes when x and y are made 0; by B and B' what the partial differential coefficients of the first order; by C, C' and C" what those of the second order, and by D, D/' D1' and D"' what those of the third order, become under the same supposition; we obtain f(, k)= A + (Bh3 + B'i/) -- (CA _- 2C'Al + C",2) ~1 + 1. (DiP + 8D' 3 k + &c.); 1.2.3 or since we may change 7 and k into x and y, we have for the general development of any function of two variables, f(, A + (Bx + ) + 1. (Bx + B-) 2C'xy + Cly2) I~~ 12 +~ (Dx^m + 3D'zy + &c.). 1.2.3 If in development (2), above referred to, we make y and k each equal to 0, u becomes a function of x alone, and we have dzl d+' dl'u A3 f(x + )= dx dx2 1.2 + d' 1.2.3 which is Taylor's formula. In the same development, making x, y and k, each.equal to 0, ciu d'u and denoting by A, A', A", &c. what u, dT' d &c. reduce to under this supposition, we obtain DIFFERENTIAL CALCULUS. 67 f(h) A + A'i + A" l + A/",' - + &c., 1.2 1.2.3 or changing h into x, f(x) =A+A' + A" l + A"'/ -/ &Gc. 1.2 1.2.3 which is Maclaurin's formula. DIFFERENTIAL EQUATIONS. 53. Every equation containing two variables can, by transposing all the terms into the first member, be represented under the general form f(,y ) = 0............(1) in which y is an implicit function of x, the latter being usually taken as the independent variable: Or, since by the solution of the equation, the value of y rmal be found in terms of x, and substituted in (1), this function of x and y may be regarded as a function of x alone, and may thrf< ore be di{ferentiated as c function? of Ca single variable. Also, since the relation between y and x is such, that f(, y) must alays be equal to 0, its value is not variable, and can therefore have no difference betveen any two states. Its differential mnust then be 0, Art. (14); that is, cf(x, y) = 0. To obtain, then, from a given equation its dicferential equation, or the equation which expresses the relation. between the differentials of the f'unction and variable; transpose all the terms into one mermber, differentiate this as a function of a single variable, and place the result equal to 0: Or, if it be *not desirable to transpose all the terms into one mlember, each member, containing either x or 68 DIFFERENTIAL CALCULUS. y, or both, may be regarded as a function of x, anl the cliferential of the first will be equal to that of the second, Art. (13). Since every term of the differential equation thus derived will contain dx or dy, we may divide by dx, and then at once deduce the value of the differential coefficient d. dx 54. If an equation contain three variables, one will necessarily be a function of the other two, and all the terms being transposed into one member, this member may be regarded as a function of two independent variables, and may be differentiated as in article (49), and the result placed equal to O. In accordance with the same principles and in precisely the same manner, the differential equation of one containing any number of variables may be derived. If the differential equation derived by one differentiation be again differentiated, the new differential equation will be of the second order, and if this be differentiated we shall have one of the third order, and so on. Examples. 55. 1. If u = J'x, y) = X2 + 2 n= o............(1) du = 2.xdx + 2ydy = 0............(2), from which, after dividing by dx, we obtain Cy _...........(3). dx Y Dividing equation (2) by 2, and then differentiating; x, y, and dy, being variable, we have d(xdx) + d(ydy) = 0 DIFFERENTIAL CALCULUS. 69 or dx2 + ldy + yd2y = 0, whence d' _2 x2 d2y ( A "72 ) ( ) Y2 + t2 dxC2 y y Y' since — =........... equation (3). dx.2 f Equivalent results may be obtained by differentiating the value y = V112 - x, deduced from equation (1). 2. If u= y — 2mxy + - -x- a. = 0............(1) 2ydy 2mdy- 2mydx + 2xd 0...........(2); whence dy?m/y - x dx y — ~?mx )ifferentiating (2), and diviling by 2dx', we obtain &?Y dy dy (y -- m +) - d2m- 1- 0; dX2 dx2 dx from which after the substitution of the value of we may obdx tain the value of the second differential coefficient. 3. Let y 3axy - x O0 Equations derived as bove, immeliately from the primitive equation by differentiation, are named immediate dglerential equations. 70 DIFFER'ENTIAL CALCULUS. 56. By the differentiation of equations we may find others which will express the relation between the variables and their dif ferentials, for any values of either or all of the constants. Thus, if we take the equation of the right line y =ax +............(), differentiate and divide by dx, we have d - -.. (2), dx a result which is the same for all values of b. By the substitution of this value of a in equation (1), we have ydx = xdy + bdx, which is the same for all values of a. Differentiating (2) and dividing by dx, we obtain d 0, which is entirely independent of both ca and b. Take also the equation Y2 = mx + nx............(3) By two differentiations, we get 2ydy = mdx +- 2nrxdz dy' + yd-y = ndx2. By combining the three equations, m and n may readily be eliminated, and an equation obtained which will be entirely indepen dent of them. The result of this elimination is yfdx2 -'- x2dy2 -'- yx2d2y - 2yxdydx = 0. DIFFERENTIAL CALCULUS. 71 Again, by differeniatating the equation 3 - 2ax2 + a2 = 0, and eliminating a, we obtain 16yxSdx2 - 24x3dydx + 9y2dy2 0. And in general, all the constants of any equation may be eliminated by differentiating it as many times as there are constants. The differential equations thus obtained, with the given equation, make one more than the number of constants to be eliminated; an equation may therefore be derived which will be freed from these constants. Equations thus obtained are properly the differential equcations of the species of lines, one of which is represented by the given equation, since being independent of the constants they are evidently the same for all lines of the same kind referred to the same co-ordinate axes. 57. ]By differentiation we may free an equation of exponents, as in the example uG - Vn dze = nv~n-dv, ort vde = nv"dv, and finally vdu - nudv. Or thus, lu e= if'; whence lIt - nlv, du nadv - —, or vdu _ nZucdv u 2v 72 DIFFERENTIAL CA LCULUS. 58. The Differential Calculus enables us also to eliminate, from an equation containing three variables, an arbitrary function of either two, the form of which may be entirely unknown. Thus if ut = F (f [, ), the form of the function designated by the symlbol F being arbitrary, we can find a cifferential equation expressing a relation between x, y and the partial differential coefficients d, d which dx' dy will be the same, no matter what the form of the function F may be. Make f( ) =.........(1), then t = F(Z), ct = F'(z)dz......(2). Differentiating (1), first with reference to x and then with reference to y, and substituting the values of cd thus obtained in (2), we get du Jdx du dz d - PF ()...(3)'()......(4), dx ~ dA dy dy from which F'(z) may be eliminated, and the resulting equation, between x, y, -d anmd -, will be the differential equation required. dx dy Such equations are called pa. tial diferential equations. To illustrate, suppose 1. f(x, y) = Ge -+ by and Iu - F(ax + by). Place ax + by z —, then z, ncl b. dx' dy DIFFERENTIAL CALCULUS. T3 These values in equations (3) and (4), give d' dy F(z)b whence dx dy a b and finally du du a b - 0. dy dx 2. Let f(x, ) == x +y and?u F(x2 (r 2). Differentiating z, we find d% _, and dz _,7; d;x dy dz dz whence from which, by eliminating Fl'(), dx du - -Y = 0. 3. Let f(, y) = 10 I7 4 DI FFEREI'iTiAL CALCULUS. VANISHING FRACTIONS. 59. In the cliscussion of the lresulbt obtained by tile application of the Calculus, we often meet with expressions which, for a particular value of the variable, become i. This, although in general the algebraic symbol of an indeterminate quantity, does not indicate such a quantity in the particular cases referredl to. As in the example, ax ~;which becomes o when x - a; if we divide both numerator and denominator by the common factor a - x, we obtain a -- x and this, wnhen x = ca, reduces to —, which is the true value of the fraction in the particular case. Expressions of this klindi are called vanissing fractions, and reduce to o in consequen e of the existence of a factor common to both terms; wwhich factor becomes 0 under the particular supposition. All such fractions manvy be represented generally by the expression P (x - a)," in which P and Q are fulctions of x. There are three cases 1. When nm m= n'., the fi action becomes DIFFERENTIAL CALCULUS. 75 P ( -- a)'A P Q (x a)y" Q 2. When m > n, it may be put under the form P (x - a). Q m -- n being positive; and this, when x =a, becomes 0 O. 3. When 7 m, < n, the fraction may be put uuder the form P Q (x - a) g' ~,-?, being positive, and this, when x a=, becomes 0 60. Whenever the common factor is evident, the simplest method of obtaining the true value of the fraction is to strike it out, and then put for the variable its particular value. Biut as in moist cases it is not easy to detect this factor, other methods become necessary. Let r be a vanishing fraction, r and s being functions of x, and let a be the particular value which substituted for x reduces the fraction to o0. It is plain that, if we substitute a - h for x, and after reduction make ]I = O0 it will amount only to the substitution of a for x. Suppose this substitution made, and that in the result both nume 70 DIFFERENTIAL CALCULUS: rator and denominator are arranged so that the exponents of h shall increase from left to right, we then have (1 ~ A7- + B/17 + &c. S Jra-+h A'h" + B~"' + &c. in which A, A, B B',, n, &c. are constants. After reducing this fraction to its lowest terms, by dividing both numerator and denominator by that power of h which is indicated by the smallest exponent, we shall have one of three cases. 1. If m = n (5 ) ~A + B'7-' + &c. S Ah.' + Bv'h-" + &c. 2. If m > n /a AAhm- + &c. A/' d- &co 3. Ifn < n _q ~ A + &c. ks -a,, ^ A'-"'+- &c. Now making h 0, we have for the true value in the three cases, I, (')\ 4 A' 2. (-1 - 0 32. - l. O.~.= =X. - - ~ 0 Whence we derive the general rule. For the variable, substitute that value which causes the fraction to reduce to o, plus an increment; reduce the result to its simplest form, and then ma/ke the DIFFERENTIAL CALCULUS. 77 increment equal to 0. The final result will be the true value of the fraction for the particular value of the variable, and may be finite, zero, or infinite. Examples. 1. Take the fraction (X2_ a-2) (x - a) which becomes - when x = a. For x, put a + h, the primitive fraction then becomes (2ah + h2) Dividing both terms by h2, we obtain (2a + h)2, 3 which, when A- 0, becomes (2a)F, the true value. In this case the common factor (x - a)2 is evident; striking it out, we have (x + a) which becomes (2a)2, when x = a. 78 DIFFERENTIAL CALCULUS. 2. Take the fraction m sil-1 - a which. becomes -~ when x = 0. For x, put 0 + h, or 7, we then obtain 1 /h h3 7Th sin-' - + c. _ 6-a \= a 1.2.3a Art 4) h......Art. (44), or h. i m sin-' a _1 ~^~ ^?= ~~ + - _, 3 + &C., h a 1.2.3a 3c which, when h = 0, gives /m Sill-i __ _ m \ x' =o a The common factor in this case is x, as may be shown by de veloping m sin-', as in article (44). a 61. Another rule may be thus deduced. If the vanishing fraction, as in the preceding article, be u, — =; then r = 21S, DIFF'ERE'NTIAL CALCULUS. 79I cr = d- uds + sdu; in whicl, if we make x = a, we shall have (since s =a - 0), (d.r),=a = (uds)=,.; whence U' = s -- (d)=-........( \^s X (ds)x,' a for the true value of the fraction in the particularI case. If (dr) =a 0 this value is 0O If (ds)_. = 0 it is o. If both are 0 at the same time, the second member of (1) becormes o, and - is a new vanishing fraction; then, as above, we take the difeirentials of both its terms, put a for x, and thus obtain (d2r) - x = a~ = -(d's),, If this again becomes, we continue thle same process, and have _(d r)- t`= ff (d38)x =a and so on. The rule may then be thus enunciated. 2kce Je diflerentialcs of the numerator and denominator; in each, substitute that value of the variable which 9reduces the original fraction to o; if both do not redtce to 0 or infinity; what the formeri becomes divided by what the latter becomes, will be the true value of fhe fraction. If both redu ce to 0, take the second differentials, and O 3 DIFFERIENTIAL CALCULUS. make the. samze sutbstitution; or continue the difcerentia.tiaon, c. unt.?il:two differentials of the same order are obtained, both of which do 9not become 0 or i2ninity; u/hat onze becomes divided by what the oth/er becomzes, will be the true vcale of te fraction. It should be observed, that the effect of the application of this rule is, at each differeintiation, to diminiish by unity the exponent of the factor which causes the fraction to reduce to -, Art. (27). If the exponents of this factor in the numerator and denominator are friacnil, anCd not contained between the same twvo consecutive whole numbers, it is plain thalt the least one will be reduced to a negative ilimnber by a less number of difierentiations than will be required by the other. The differential of that term of the fraction whlich contains it, w-ill then, by the substitution of the particular value of the varable, reduce to infinity, while that of the other reduces to 0, and the true value of the fraction will be either O o o co 00 - ~- = CO or ~=0. 0 co If however, these exponents are contained between the same two consecutive whole nnum)bers, they will become negative by the same number of differentiations, and the differentials of both terms of the fraction; reduce to infinity at the same timne; as will the successive differentials,. In this the 02only failingy case of the rule, we shall not be able, by its application, to obtain the true value of the fraction, but must fall back upon the general rule, Art. (60). As an illustration of this, we may refer to example I, article (60), in Vwhich the second differentials, and all which follow, becomne in.finite when x = a. Exanmples If DIFFERENTIAL CALCULUS. 81 _ - S x-1 which becomes - when x = 1 dr -= -n"-'dx, ds = dx, ( =l (ds)-1 ( 1 -l 2. If r — sin x s cos X which becomes o when x - dr = — cos xd, ds = - sin xdx, (r) ( cos o. T J=2 sin x 2Zc r ax -- 2acx - ac2 s bx2 - 2bcx + bc2 dr (2cax - 2ac)dx, ds = (2x - 2bc)dx, both of which reduce to 0, when x- = c. Differentiating again, d2r = 2adx' d"s =- 2bdx, and trA a () b 82 DIFFERENTIAL CALCULUS. 4. Take a b when x = O. Aus. la lb. ~x ~ ~ ~ ~ x== 0.,, v m +ll + a,) a ax 1 sin x + cos x _7 sill +- cos 1 2 a- x-l ala + alx a - /2ax x2 8. _____,= 1 —x lx x - 2 sin x ____ x. -- x sill x 62. We sometimes meet withl the product of two factors, one of wihich becomes 0, and the other so, for a particular value of the variable. Let rt be such a product, in lwhich r becomes 0, and I infinite. It may be written t which, for the particular value, becomes o. Its value may then be determined as in the preceding articles. Ex ample. Let rt (1- ) tang ~z when x 1. 2' DIFFEREE TIAL CALCULUS. 83 Writing it under the proposed form, we have 1 — x 1 - - I X iX rt - - cot -,,rx 2 tang 2 2 the true value of which, when x = 1, is 2 63. The fraction - may become,in which case it may be written r ] s s which becomes oo x - = xo 0, and may then be treated as in GO the preceding article. 64. Sometimes also, we find expressions which become oo — o 1 1 Let -- r s be such an expression, r and s becoming 0. It may be written 1 1 s-r r s rs which will reduce to ~. For an example, take cot - 2 cos x 84 DIFFERENTIAL CALCULUS. which becomes oo - o, when x -. By reduction we obtain 2 2 sill x - 2 COS X cos %,te true value of which is, - 1, when x = - MAXIMA AND MINIMA. 65. A function is at a maximum state, or a maximum, when it is greater than the state which immediately precedes, and greate7' also than the slate which immediately follows it; and a mqinimum, 2when it is less than bot1t of tiese states. Thus, if u be a function of x, and x be decreased so as to give the next preceding state to u, denoted by u", and then increased, by the same quantity, so as to give the next succeeding state u'; if u be greater than both u"' and tu' it will be a maximum; if less, a minimum. 66. If u is a function of x, and x supposed to be increasing, it is evident that when passing from the preceding states to its maximum, u mutst increase as x increases, that is, be an increasing fanction of x; and when passing from its maximum to the succeeding states, it must decrease as x increases, that is, be a decreasingyfunction of x. in the first case, Art. (12), the sign of its first differential coefficient must be positive, and in the second, negative; therefore at the maximum state the first differential coefficient must change its sign from ptls to ginus. For a similar reason at a minimum state, the first differential coefficient mulst DIFFERENTIAL CALCULUS. 85 chanige its signfrom minus to plus. But as a quantity can change its sign only by becoming zero or infinity, it follows that no value of the variable will give a maximum or minimum value to the function, unless the same value reduces the first differential coeflicient to zero or infinity. The roots of the two equations d/-\ du dx ^ ^ du 0.....(1), and du_ o or......(2) dx dx dit will then give all the values of x, which can possibly make it a maximum or a minimum. After having obtained these roots, let each, first with an inrfinitely small decrement, andc then with an infinitely small increment, be substitutec in the given function; if both the results are less than the one obtained by substituting tlhe root, the latter will be a maximum; if both are greater, a minimum, Or if it be more convenient, let each of these roots, with an infinitely small decrement and increment, be successively substituted in the first differential coefficient; if the first result be positive, and the second negative, the root will make the function a maximum if the reverse, a minilmum. If the two results have the same sign, the root under consideration will give neither a maximium nor a imitimum. Since equations (1) and (2) may give several roots whinch,ill. fulfil the required conditions, there may be more than one max;imum or minimum state of the samne function. Examples. 1. If u a + - b)............(3 du dz I 2(v -- b) and d -— b dxz " du 2 (x - b) 86 DIFFERENTIAL CALCULUS. dzx Placing d -- 0 we have 2(x -) = 0; whence x - If in equation (3) we substitute first, b - hi for x, and then b + A7 we have u"l a c and u/' = a + 7t2 both of which for all values of 7 are greater thain u = a, the result obtained by substituting b for x; hence ze = a is a mzinimum. The only value of x which will reduce -to 0 is z Go; there dut is then no finite value of x which will satisfy this condition, hence x = b gives the only minimaum state, and there is no maximum. 2. If u = a- -( b)............(4) du -2 dx -3(x -b) = _________i? arnd 2= dx 3(x b)3 du 2 Placing u = 0 we obtain x -= co, which gives no finite soludx tion. dx Placing - 0, we have du 3(x b) == 0; whence x = b. If then in (4), we substitute first b - h, and then b -+ h, for x, we have 2. -- anu" == a h and u' = a- 7z, DIFFERENTIAL CALCULUS. 87 both of vwhich are less than u = a, the result of the substitution of b for; u =- a is then a maxzimumc and the only oze, and there is?zo mininmumn. If in the first cdiferential coefficients in the above examples we sabLstiLtte 6 -- h and b + 7i for x, we obtain in the first, for b - 7i a negative, and for b - 7 a positive result, and in the second the revorse, as it should be. 07. WT~lhen the states which immediately precede and follow the maximurn or minimum state of u, can be deduced from Taylor's for'mula, a more convenient rule may be applied. To demonstrate it; let? =f(x), then L' =fAx + h) 1i" =f(x - 7), and by Taylor's formula cld - d'2u h2 c3u h3 3 du d'u h' dcl' hP A "' u _ I h + - + &c. dx dx' 1.2 dx3 1.2.3 At. (34.) In order that u be a maximum it must be greater than both 6zZ dx~ 1.2 dPa 1.2.3 J In order that u be a lanmailll it must ve greater than both u' and ut", that is, the second members of the above equations, for an infinitely small value of 7i, must be negative; and for a minimum the reverse. But for any value of h less than the one referred to in article (II), (and of course when hi is infinitely small), the si1ns of the series will be the same as the signs of their first terms; but these terms have contrary signs, hence there can be neither m1aximum nor minimum unless the first term of each series be 0, which 88 DIFFERENTIAL CALCULUS. requires that d_ = 0. The roots of this equation will then, in tlhe dx case under consideration, give all the values of x which cai. possibly make u either a maximumn or minimum. Let a be one of these roots, and let it be substituted for: in the du two series, then, since (l-) 0, we h1ave T'he signs of ihle serles nor d pelnd upon thlat of U ~= 2', annld will f90t2U-1,/62SlC lct'. ]}_..i._ll]-l. if (- ) iO? _d3Tive; an the rears e if this is ___ti.e _ut if (d_____ -O, thG 7d~u\ both be neogajtive a a maximum, ii -- is neqa.tive; and the reverse if this is positive. Bt if 0, the signs of the series will again be contrvary acld there can be neither 11maximum n or mininmum lunless (d ) -- 0, in which case the sitgs will he the same as that o f (d ) And in general, if there be either a umaximum or miniimum, the first cifferential coefficient whichl does not reduce to 0 whern x = a,, must be of an even order, ne rative for a mqaximzum, and positive for a m??inimzum. Whence to determine the maxinum or miniimum states of a given fucteion. F.iid its first differential coebcicient cazd place it equal to 0; sulstitule eac/h of the real roots of the equclttion thus fo1rmed, in the second di;iffrential coeficient. Ectcc one which gives a negative result, will when substituted in the Jfunction.make it a,aximumzzm, anzd each ]which gives a positive result will make it a minimum. If either reduce DIFFERENTIAL CALCULUS. 89 the second differential coeficient to 0, substitute in the third, fourth, &c. until one be obtained which does not reduce to 0. If this be of an odd order, the root will correspond to neither a maximum nor msiinimum; if of an even order and negative, there will be a corresponding nmximum; if positive, a minifimum. Examples. X3 1. If u = + a. -- 3aZx, du du 2- x 2 2cx - 3a2, d 2_ 2 + 2a...........(1) du Placing the value of = 0, we have dx x2 4- 2ax - 3a2 = 0 the roots of which are x = a, and x -- 3a. The first substituted in (1) gives 4a, which being positive, indicates a miinim.mni. The second substituted in (1) gives - 4a, which indicates a maxinum. Substituting the roots in the given function, we have for 5aa the mininum =u ~ 5a, and for the maximum u = 9a3 2. If u = 24 +- ax, du 3 3 du 2 - 8 + a = 242...........(2), Placing the value of du 0, we have dx 8x + -a ==0; whence x _ a~. 12 90 DIFFERENTIAL CALCULUS. This value of x in (2) gives 6a2, and indicates a miiniBum, which 8a' is ^ -~ 8 68. Let v = Au, u being any function of x. By differentiation, &c., wae have dvA du d'2v Cd2u dx dx dx2 dx2 ddx will also e dv, and the reverse. Al that any of these -ATill also irnakle - O=.0 alLd te reverse. Also, that any of these dx values, when substitutec in the second differential coefficients, will give results affected with the same sign. Hence every value of x which will make u a maxilmlum or minimum will make eAu a maxinmum or minimumn. Therefore c constant positive factor may be oaited during tlhe search for those values of the variable corres-'pondiyo'r, to C,? zaximZ2um o'0 minimum.22. tVo il'lstrati, take. the example (2ax -- X2)......(1), a Omitting the constant factor, we may write Z - 2rX- Xs2, - 2a -- 2a, = -., dx dxi du _ Placing _t ~ 0O, we find x -= a, which in (1) gives the maxin alue ab. mu-m value ab. DIFFERENTIAL CALCULUS. 91 69. Let v - u' u and v beilng functions of x and n entire. Then dv n-cl du dx dx d2v -= nfl n- d'u -nn du_ dx2 dx' 2+ (n dxNow every value of x which will make du 0, will also make dx dv 0; and if the same value makes nu"-' positive, it will give to dsl cdv s du( dud d the same sign as d (since ~- ^ 0); that is, if it makes u dx"X d~dx2 dx' a maximum or minimum it will make v a maximum or minimum. If it makes nu"n- negative, it will give to a sign contrary to thoat of d; that is, if it makes u a maximum, it will make v a minimum dx and the reverse. All values of x, however, which will make v == u" a maximum or mininmum, will not necessarily make u a maximum or minimum, for the equation dv = nut - du= o d.x dx may be satisfied by making either nu- = 0 or du _ O0. dx Those values of x which satisfy the first, and not the second of these equations, will make u neither a maximum nor minimum, but may make v = a a maximum or minimum. As in the example, 92 DIFFERENTIAL CALCULUS. v = (03 3)2 -= U dv = du dv = 2udu = 2u dx dx dv We may make d = 0, by placing either 2u = 2(a3 -- 3) = 0; whence x - a or du _ X d= 3X2 0, X" 0. dx The value x = a evidently makes v a minimum, but as it does not reduce _ 3x2 to 0, it will make u neither a maxi dx mum nor minimum. The value x = 0 answers to neither a maximum nor a minimum. As the corresponding power of a radical expression is formed by omitting the radical sign, we may, in accordance with the above principles, omit it, and seek those values of the variable which will make the power a maximum or minimum. We are sure thus to get all the values which will make the root a maximum or minimum. Care should be taken, however, not to use any of those which belong only to the power. 70. In a manner similar to the above, it may be shown that any value of the variable which will render u a maximum or minlimum will also render log u and a" a maximnul or minimum. 71. It often happens that the first differential coefficient is composed of two or more variable factors, each of which, when placed DIFFERENTIAL CALCULUS. 93 equal to 0, may give values of the variable, corresponding to maximum or minimum states of the function. Let du - XX' dx be such a coefficient, X being 0 when x = a. Then d2u dX dXX' dxW dx dx or since X. 0 when x = a, (^b _ (dX dx,.= - d x__a That is, to obtain the corresponding value of the second differential coefficient; multiply the differential coegicient of that factor which is 0, by the other factors, and then substitute the particular value of the variable. To illustrate, let u = x2(x - a), d.n 2 = 2(Tx - a)5(4x - a), dx which is equal to 0, when 2x = 0; whence x = 0......(1). (x - a)5=; C = a......(2). (4 - a) = 0; = _......(3). 4 Taling the first factor 2x, and multiplying its differential coeflicient by the other factors, we obtain the expression 94 DIFFERENTIAL CALCULUS. 2(- a)5(4x - a); from which, by making x = 0, we obtain __2 - 2a6 which indicates a minimum. Multiplying the differential coefficient of the third factor 4:x - a, by the others, and making x = -, we obtain a negative result, which indicates a maximum. The second value of x reduces d to 0, but will make dt positive, and give a minimum, Art. (67). 72. If the function be implicit, we have only to find its differential coefficient as in article (17) or (53), and proceed as with an explicit function. To illustrate, take the example 2 - 2mxiy + 2 - a = 0......(1), and let it be required to find the value of x which will make y a maximun or minimum. By differentiating as in article (53), we obtain 2ydy - 2mxdy - 2mydx + 2xdx = 0; vwhence dy 7y - x dx y - mx Placing this equal to 0, we have my- x = 0; whence x = my, I IFFiliETIAL CALCULUS. 95 which, in equation (1), gives a ^ ma y - _ whence x = A/1 m2 1 - nzs2 Differentiating the factor my - x, equation (2), dividing by dz, and multiplying by, Art. (71), we obtain the expression y-mx d l (m-Y l), y -~ mx a which, by the substitution of the values of y and x, (since then d -- 0), becomes dx - 1 a-V/1 and indicates a maximum.'3. The only difficulty in the application of the preceding principles to the solution of problems, consists in obtaining a convenient algebraic expression for the function whose maximum or minimum state is required. No general rule can well be given by which this expression can be found. In order to indicate as clearly as possible the methods to be pursued, we will give the solution of several cases differing from each other. 1. Required the dimensions of the maximum cylinder, which can be inscribed in a given right cone. Suppose a cylinder inscribed, as represented in the figure. Let Ail DIFFERE'NTIAL CALCULUS. VA -- a, BA b, VC x, CO= y v \ then AC = a - x, and the solidity of the cylinder, which we denote by v, is equal to, 1...i........ /,' r y2(a-) )... (1). 7/',....,'''\, From the similar triangles VCO and VAB 9 -....-......' wJ e have the proportion bx a y::: ba 1; whence y =-. Substituling this value in (1), we have v - -a( - a)............ (2). (Omitting the constant factor, Art. (68), we may write U ==I at2 - ^3 wshence dx dx2 Placing d=, we find the roots x 0, anid x -2 a. The second value of x in (3) gives - 2a, and therefore will make v a. Arab2 Imaimum, wchich is ab 27 For the altitude of the maximum cylinder, we have a - = -a, and for the radius of the base y = -b. The first value of x in (3) gives 2a, which indicates a minimum, which is evidently v = 0. DIFFERENTIAL CALCULUS. 97 2. Required to draw a tangent to the given cquadrant ABD, so that the triangle CFG shall be a minimum. Let CB R, I B = x, BG y; then FG = x + y. The area of the triangle is equal to — CB x FG, which since -CB is constant, will be a minimum when FG is a minimum, Art. (68). In the right angled triangle CFG, since CB is per- C A F penclicular to FG, we have R2= — xy; whence y, x and FG = u - x R. x du R2 x2 - which, being placed equal to 0, gives x = R, and y = R. Hence the angle BOF - 45~. Obtaining the corresponding value of -, as in Art. (71), we find for a result -. dx~' 3. The whole surface of a right cylinder being given, it is required to find the radius of the base, and altitude, when the solidity is a maximum. Let m2 _ the surface, x -= the radius of the base, and z = the altitude, then v = qr7'X2Z. But 13 98 DIFFERENTIAL CALCULUS. m2 2xzz + 2sx2; whence z r - 2_.2 2tx' therefore V _ 3 2 and x \/, and = 2/, when v is a labinamum. 4. Required to divide a given quantity ct into two parts, such that the mlth power of one, multiplied by the nth power of the other, shall be a maximum. If x = one of the parts, theln x= +. m +?n1 5. In a given triangle, it is.required to inscribe a maximum rectangle. The altitude of the rectaangle =- ~ altitude of triangle. G. A certain quantity of water being given, it is required to find the relation bet-veen the radius of the base and altitude of a cylindrical vessel, open at the top, which shall just hold the water and bave its interior surface a nmniium. The radius - the altitude. 7. PRequired the maximum rectangle which can be inscribed in a circle. Each side = R/2. 8. Required the maximum cone which can be inscribed in a given sphere. DIFFERENTIAL CALCULUS. 99 9. Required the minimum triangle that can be circumscribed about a given portion of a semi-parabola. 10. Required the maximum cylinder that can be inscribed in a given ellipsoid of revolution. 11. Required the axis of the maximum para-bla that can'be cut from a given right cone. 12, Required tle minimum value of y in the equation y - ". MAXIMA AND MINIMA OF FUNCTIONS OF TWO OR MORE VARIABLES, 74. A function of two or more variables is a maximum when it is greater, and a minimum when it is less, than all of its consecutive states. Let u = f(, y), then u =f(x + h, y + 3), e' = h(p + p't) + - (q + 2q't + q"t') + &.....(), 1.2 after placing in the development of article (49), du due - t7, 1- -p, -:, dx dy d2u dz dtt dd.. q" &c. dx=, y dd d The sign of this series, when Ih is infinitely small, will depend upon the sign of its first term. Now we shall obtain all of the consecutive states of u, by giving to Ji and k proper infinitely small values, both positive anld nlegoative; and therefore, when u is either a maximum or a miniimum, the sign of u' - u for all these values of h and k must be the same: But the first term of the series (1) 100 DIFFERENTIAL CA CT-LUS evidently changes its sign when the sign of h changes; tllere can9 then, be neither a maximum nor a minimum, unless i(p - p't) = 0 or p + p't = 0, k and since this must be 0 for all values of t = k, we must have seh parately p = 0 and p' = 0, or 0......(2) 0.....(3). dTx dy The values of x and y, deduced from these equations and substituted in the second term of series (1), (h and k being infinitely small,) should mlake it negative for a maximum and positive for a minimum. This term may be put under the form 2'tq + z,+ r, which, if there be a maximum or minimum, must not change its sign for any value of t; but this requires that the roots of the equation + 2 qt + q - 0 q'l q11 be either imaginary or equal; that is, that q and q" have the same sign, and q'2 < qq" or qt2 = qq/. The conditions then are d'u < d'u d du o d da ddy < dx2 y or- dx X -; d2zr d'u and also that d~ and d have the same sign, after the values of d ede fdom e d x and.?y deduced from the equations = 0 and ~ 0 have dx dy DIFFERENTIAL CALCULUS. 101 been substituted: And since the sign of the second term will then dependc upon q", the sign of A- must be negative for a maxdy2 imum, and positive for a minimum. If the second term becomes 0, we must substitute the values of x and y in the third, which must also be 0, and the sign of the fourth negative for a maximum, and positive for a minimum; the discussion of the several conditions of which, although complicated, may be made in a manner similar to the above. Examples. 1. Required to divide a number a into three parts, such that the cube of the first, into the square of the second, into the first power of the third, shall be a maximum. Let x = the first part, and y = the second; then a - x - y the third, and u - Xyf (a -,x y) d - 2'(3a - 3y - 44, xy(2a - 3y 2x) Plancing these equal to 0, we have 3a - 3y- 4 == 0, 2a - 3?y-2xs 0; whence a a z =-7, y -5 2 3 We have also q = 2xy(3a - 3y - 6x), ctdx 102 DIFFERENlTIAL CALCULUS. f> =n =: x2Y(6a -9y 8x), dxdy q" X-. (2a - 6y- 2x), dy2 which for the particular values of x and y become a4 a4 a4 9' 12' 8 Henco 2 a a d2U a4.a ^ ~ ^ J d~u a4 q1 < qq" and ~= ~ a6'.~~ 1's dy( 8~ u is therefore a maxiimnSll when its value is 4. 432 2. Make the preceding proposition general, by putting for the cube, square, an.d f'rst power, the mth, nth, and rth powers. LThen u = Xmyl (a - x - y)"?zma na X _ __ _ y = O m +n+ r im + n +r 3. Required the shortest distance from a given point to a given p]ane. Let tlhe equation of the plane be placed under the form z = Ax + By + D, and the co-ordinates of the given point be x', y', and z'; then u (X - X')2 + (y - y')2 + (z -_ z)2 or putting for z its value, = (x - x')2 + (y - y')2 + (A + By + D- 1)2. DIFFERENTIAL CALCULUS. 1 0O Calling the radical, R, we shall have du _ y -' + (Az + - y + D - z')B dy R' du _ -' + (Ax + By + D - z)A dx R Placing these equal to 0, and solving the resulting equations, we may obtain the values of x and y; and thence, of z. Or otherwise, putting for Ax + By + D its value z, we have y- y' + B( -- z') = 0, and x + A(z -') 0, which are eviden:tly the equations of a perpendicular to the plane, and if combined with the equation of the plane will give the values of x, y, and z. 75. In order that a function of three or more variables be a maximnum or a minimum, we must have dt 0 du du - 0__ O, = 0, - = 0............&c., dl dy dz and the relation between the partial differential coefficients of the second order must be such, that the second term, in the development of the difference u' - u shall remain of the same sign, for all the consecutive values of the function. APPLICATION OF THE DIFFERENTIAL CALCULUS TO CURVES. 76. To x in the expression u =f(Z), 104 DIFFERENTIAL CALCULUS. assign a particular value, and deduce the corresponding value of u. These values, taken together, may be considered the co-ordinates of a point which may then be constructed. By assigning an infinite number of values to x, and deducing the corresponding values of u, an infinite number of points may be determined, which, being joined, will form a curve whose equation is u = f(x). Hience, we conclude that every function of a single variable mzay be regarded as the ordinate of a curve, of which the variable is the abscissa. 7'7. Let BMM' be a curve, the equation of which is y f(x); and M any point of this curve, the co-ordinates being x and y. Increase the abscissa AP or x, by the variable increment PP' =1 h; denote the corresponding ordinate ~ ~T A ~ B' l \ P'' by y'; and draw the secant M'MT'. Then M'Q = P'M' -- PM = y' - y Ph -- P'h......Art. (10). From the triangle M'MQ, we have tang I'YMQ MQ:= tang MT'X, ucnd placing for M'Q and MQ = PP', their values, this becomes Ph q- Plh2 tang MT'X -P = + Pr'h.......(1). Now if h be diniished, the point }1 approaches M, and the secant M'T' approaches the tangent MT, and finally when h i 0, the point M' coincides with M, and the secant with the tangent. If then in (1) we make h = O, we have DIFFERENTIAL CALCULUS. 105 tang ITX P = dy d' that is, the tangent of the angle which a tangent line at any point of a curve makes with the axis of X, is equal to the first difgerential coeficient of the ordinate of the curve. To show the application of this principle, let us take the equation of a circle x + y' = P; whence dy-.......... (2) dx y for the general value of tle tagent of tthe ang made by a tangent at any point of the circumference, with the axis of X. If the particular value at a point wlhose co-oOrdinates are x" and y" be required; for x and y, let x" andl y' bc substituted, then dy": 1; dx"' y Take also the equation iy -- mx + nx2; whlence dy nm + 2'nx im + 2nx. dx 2y 2Vmx -+ For the particular point y" and x", this expression becomes * NOTE.-The notation -, — ) &e., is used to indicate what the dx" d"2 frst, second, &e. differential coefficients become, when for the general variables x and y the particular values x" and y" are substituted. 14 1 06 DIFFERENTIAL CALCULUS. dy" _ n m + 2nx" d" 2 i' + nx'' 7'8. If it be required to find the point of a given curve, at whiich the ztan'ent line makes a given angle with the axis of X, we know that at this point the first differential coefficient must be equal to the tangent of the given angle. Calling this tangent a, we must then have cry dx and this combined withi the equation of the curve will give the particular values of x and y, for the required point. If the tangent line is to be parallel to the axis of X, then for the -ointi of tangency, d7y g O; and if perpen dicacuar, co. lx dx We will illustrate each of these cases by an example. 1. Let it be required to find the point on a given parabola, at which the tangent line makes an angle of 450 with the axis. The equation of the parabola is y2 = 2px by the differentiation of which, &c. we have dy _ dx y But as tang 45~ = 1, we have, for the required point, dx y an.id._ counbiuingo this witb. the equation y =- 2px, we find Y P. Ix _F y _ _.p. ~2 DIFFERENTIAL CALCULUS. 107 The tangent at the extremity of the ordinate passing through the focus, will then fulfil the required condition. 2. Let y = a+ (c - X)2.......(1) represent a curve; then dy -- 2(c - ) dx which is equal to 0, when x = c; and this value of x in (1) gives y = a. These are theln the co-ordinates of the point at which the tangent is parallel to the axis of X. 3. Let y = a + (c- x) represent a curve; then dy I 2(c - x) which is equal to infinity, when x = c.'x = c and y = a are then the co-ordinates of the point at which the tangent is perpendicular to the axis of X.'9. If x" and y" represent the co-ordinates of a given point on a given curve, whose equation is y = f(x); the equation of a straight line passing through this point will be y y I = a(x — / ") a being indeterminate. This will become the ecu.Uation of a tan 108 DIFFERENTIAL CALCULUS. dy"W gent line at the given point, if for a we put dy W thus ob dx" tain dy" Y - y,=" (- a"')..x......(.... () dxt' By differentiating the equation of an ellipse a2y + b2X2 = a2b2 we deduce dy b2x dy" Pb'xlf dy -_ b; whence d _ dz a2y dx" a2y/" and this value in (1) gives, for the equation of a tangent, to an ellipse at the point y", x", aiy" (x x") y l- _ (X, ) which, by reduction, becomes a2yy" + b'xx" = a"b. 80. If the equation of a tangent be required, which shall be parallel to a given line, or make a given angle with the axis of X; we may determine the co-ordinates of the point of contact as in article (78); and knowing these, the equation may be deduced as above. Thus, if a tangent to a circle be required to make with the axis of X an angle whose tangent is 2, we must have for the required point, equation (2), Art. (77), dy _ o dx y DIFFERENTIAL CALCULUS. 109 From this, we find = -_, which, combined with the equa2 tion of the circle, gives 2 R,I,t x: ~ -_ - x" y-= -- y" V-5 <5 and equation (1), Art. (79), becomes, when we use the upper signs, Y -1 ~- x ) or y 2x- R/5. 81. The general equation of a normal, deduced from equation (1), article (79), is evidently y _ y"~ = ~ (X - x"), dy" clx" dx"1 or ^Y -- -' (x- x"). dyl7 82. The right angled triangle MTP (Figure of Art. 77) gives PM PM = PT tang MTP; hence PT = tang MTP' or Subtangent = Y = y d. dy dy dx Also MT / —i2 -q +PT, 110 DIFFERENTIAL CALCULUS. or Tasngent y= y d2 1 + -dx2 d. ~dy The right-angled triangle PMR gives PR = MP tang PMR; but PMR = MTP; hence PR = MP tang MTP, or Subnormal = yd dx Also, MR = VMP + PR2; 1 enllce YVoraly = / + dy+ dY/1+ dx dxv2 To apply these formulas to a particular curve, it is only necessary to substitute in each the value of d, or d, deduced from the dy x' differential equation of the curve. The results will be general for all points of the curve. If the values for a given point be required, in these results let the co-ordinates of the point be substituted for x and y. For example, take the general equation of Conic Sections, y2 = mx + nx2; whence dy m + 2nx dx 2V/mx + nx2 d 2/mrx + nx2 dy m + 2n. These values substituted in the formulas, give DIFFERENTIAL CALCULUS. I11 PT _= 2(mx + nx2) PRi m + 2nx m -- 2nx- 2 jlyL(~ + 27z MT -/ mx + nI 2 + 4 (In t~ T' MR = / m + n2 + (nm + 22x)2. For the parabola n = 0, and these expressions become PT = 2x. PR = _ 2 MT _= m /~ + 4 42.= M mx + 83. If a curve be convex towards the axis of X, and the ordinate positive, as in the annexed figure, it is plain, that as the abscissas M / AP, AP', &c. increase, the tangents - of the angles MTX, M'T'X, &c., will also increase, and the reverse. Since' / 1 - T A F T" these tangents are represented by the corresponding values of the first differential co-efficient of thle ordinate dy ) i ust2 e an increcsing function of x, and its differential coefficient, i. e., d, must be positive, Art. (12). If the curve be still convex, and /S s the ordinate negative; the tangents A.... of the angles STX, S'T'X, &c. plainly decrease as x increases; d dx 112 I)I'FE'REN'TIAL CALCULUS. is a decreasin g function of x; and dy must be negative. dx3 If then a curve be convex towards the axis of abscissas, the ordi-nate and its second d'lgfercnical coefficient, taken at the different points, will have the same sign. / /.- If the curve be concave, and h-/1 + (P+P'))2, and z' -z < 7 1 + P2 P'h7 or he second embers of eac h expessions become+ the second bes of eac of hih expressions become yT D when h = 0. DIFFERENTIAL CALCULUS. 119 Therefore, in accordance with the principle of the preceding article, we must have dz= /VI + P2 = /l dy dx dZ= d x+dy2; that is, the diferential of an arc is equal to the square root of the sum of the squares of the differentials of the co-ordinates of its points. To illustrate, take the equation of a circle x2 + 2 =P 2; whence xdx xdx elyy -V - d and dz X2 2x2 Rdx 87. Since, also, by article (85), the limit of the ratio Z' -- z / % hV/1 + (P + P'A)2 MM' is unity, we prove that the limit of the ratio of a chord to its corresponding arc is unity. 88. Let 3BMP = s, be any area limited by a curve and the axis of X; it will evidently be a function of x. 120 DIFFERENTIAL CALCULUS. ^Y Let AP = x, PM y, PP' = h; ri' then P'M1' -y + PPh + P'1h2 and PMM'P' - s' s ~TA -p~~ "' 1the increment of the area s. The rectangles PMI and PQ being constructed, we have PQ yh, PM' ='P'W'h (y + Ph + P'7^2)h. But the area PMM'P' is always greater than the rectangle PQ; and less than PM'; whence s' - > y/, and s' s < (y + Ph -+ PQ'7)h, or S' -- s.' -S h- >/ - < < y + P/ + P'I2, both of wihich quantities become y, when - 0; hence, Art. (85), ds Y_ and ds ydx; dx that is, tle diTferential of the area is equal to the ordinate of the bounding curve into the di?'ererntial of the abscissa. The differential of the area included between the curve and. axis of Y, may be found in the same way to be ds =- xdy. If the axes of co-ordinates are oblique to each other, the rectangles PQ and PM' become pa rallelogam; the area of the first is yjhsino, and of the second (y -- Pht -I- P'h2)h sinc, DIFFERENTIAL CALCULUS. 121 w representing the angle made by the axes of co-orctinates; whence ds = y sin w dx. For an example, take the equation Y2 = R2 - 2; whence ds = ydx = ,(2y + Ph + P1'2) V1 + (P + P'h). Since the second members of both the above inequalities reduce to 2vy 7 1 + P' when h = O, we must have du du = 2,y dx' + dy; that is, the differential of a surface of revolution is equal to the circzmfrrence of a circlepe7pendicular to the axis, multiplied by the diZferential of the arc of the generating curve. If the curve revolve about the axis of Y, we may determine in the same way du = 2~xz/dx2 + dy2. If we suppose a parabola, whose equation is y2 = 2px, to revolve about its axis, we shall have DIFFERENTIAL CALCULUS. 123 dx ydy dz = — yd P du 2~y i/y dy 2 27rydy du 2 vy + c - 2~ydy <" + P2 p2'2 P 90. Let the area BMP revolve about the axis of X; it will generate a solid of revolution, which is a function of x, and which we denote by v. If x be increased by PP' = h, then the area PMMTP' will generate Y the increment (v' - v) of the solid. The rect- N angle PQ will generate a cylinder, which is al- M ways less than v' - v, and which is measured by yqh. The rectangle PM' will generate B another cylinder, which is always greater than A P'P v' — v, and is measured by u(y + Ph + PA'h)2h; hence we have v/ -v V/ -2 V hU > Ily 2 and h1n'- v < -r(y + Ph + P^h2)25 h ~ ry~h therefore dv _ - Y2 dv = ry2dxz; dx that is, the differential of a solid of revolution is equal to the area of a circle perpendicular to the axis, multiplied by the differential of the abscissa of the curve which generates the bounding surface. For the solid generated by the area included between the curve and axis of Y, we may find in the same way, dv = ~x2dy. If we take the particular case of the ellipsoid, the equation of the generating curve being 124 DIFFERENTIAL CALCULUS. y =_ (a2- x2), a2 we have b2 dv - = fy~dx -= _(a -- x2)dx. GENERAL REMARKS. 91. I-eretofore, in our treatment of the subject, we have regarded the differential of the independent variable merely as an arbitrary constant, Art. (7), without having fixed upon any particular value for it. All the demonstrations are then as true for one value, as for another. it is however of the greatest convenience, in the application of the Calculus to the higher branches of AMathematics and Physical Science, to regard this differential as infinitely small; that is, so small as to be contained in unity an infinite number of times; and hereafter it will be so regarded. The advantages of so regarding it will appear evident after a few illustrations. Let us tale first the simple function discussed in article (7), u =- axr, After x has been increased by dx, we have there found u' u = 2axd.v + adx2. Now, if the increment (dcx) of the variable be infinitely small, the two states ut and u' wnll plainly be consecutive, the expression, for their difference being 2axdx + adxz2........... (1). DIFFERENTIAL CALCULUS. 125 But since dx is infinitely small, its square will be infinitely small when compared with it: As may be shown by taking the identical equation d d_ dx2 dx dx2 dx3 fromn which, since dx is contained an infinite number of times in unity, it appears that dx2 will be contained an infinite number of times in dx; dxa in dx2, &c.: adx2 will then be infinitely small with reference to 2axdx, and may be omitted fiom expression (1) without materially affecting its value; hence in this case 2axdx vmzay be taken for, or is the measure of, the diferezce u' - u. This is true also in the general case, for all the terms of the difference, except the one which we have taken for the differential, will contain dx to a higher power than the first [see equation (3), Art, (7)]; they may then all be rejected, and the differential of the function taken, as the measure of the difference betwee two consecutive states of the function. It is plain, also, since du = pdx, dcu -= qdx2, d3u = rdx., &c.......Art. (26), that the second differential of a function is infinitely small when compared with the first, and the third when compared with the second, and so on. It is usual to call these, infinitely small quantities of the first, second, and third orders; and we see, from what precedes, that every infinitely small quantity may be omitted without error, when connected by the siqn -r with one of a lower order. In the application of the Calculus to curves these principles are of great use. Let BMM' be a curve; MP, M'P', any two consecutive ordinates; PP' = P'P" 1 - P"P"', &c., being each equal to dx; then M7 the difference between y and y', or y' - = y M'Q, is equal to dy; and z'- z = M'= dz: A B:P, ~I 12.6 DIFFERENTIAL CALCULUS. Or since z' - z may represent the difference MM', I M'M", M"M"', between any two consecutive states of the are, the different values of dz may in succession represent the infinitely small arcs M M', M'M, &c. the sum of all of which will be equal to the entire arc z. So the difference between the two areas BM1 P' and B3MP is equal to PMM'P' = ds; and the different values of ds may in succession represent the infinitely small areas PMM1'P', P'M'M"P", &c., the sum of all of which will equal the entire area s: Andc in general, if the variable be increased by its differential, the corrpesponding increment of the function may be represented by the diferential of the function, and the sum of all the diferent valzes of this differential will equal thefunction itself. In accordance with the above principles, the expressions in articles (86), (88), (89) and (90) are at once deduced. 1. The are MM' is equal to z' - = dz; and since the limit of the ratio of the are and chord is unity, they continually approach an equality as they decrease; and. when both are infinitely small, the one may be taken for the other. But the chord MM' = /MQ" + M'Q I V /dx2 + dy'; hence d = v/dx' + dy'. And if x, y and z denote the co-ordinates of the points of a curve w in space, we may find in a similar way dw = dx + dy' + dz". 2. The area PMM'P' = - s' - s- ds; and since the limit of the ratio P is unity, the area PMM'P', when infinitely PQ small, mav be taken for the rectangle PQ. But DIFFERENTIAL CALCULUS. 127 PQ PM x PP = ydx; hence ds = ydx. 3. The surface generated by the arc MM' is equal to u' - u du, and this will not differ from the surface of the frustrum generated by the infinitely small chord MM', which is equal to [2sy + 2t(y + dy)] 2M = S(2y + dy)dz = 2ydx, since dy may be rejected; hence du = 2jydz = 2,y /dx2 + dy-. 4. The solid generated by the area PMM'P' = v' - v = dv will not differ from the solid generated by the rectangle PQ which is equal to rMPx PP' = dy;dx; hence dv == ry"dx. SINGULAR POINTS. 92. A singular point of a curve is one at which there exists some remarkable property, not enjoyed by the other points. By a general discussion of the equation y = b + c(x - a)......(l), 128 DIFFERENTIAL CALCULUS. we shall meet with some particular curves, on which some of these points will be found. 1st. Let mn be an entire and even number. By the differentiation, &c. of (1), we have I mc(xY -_ m....(2), ) c (x - a)'......(3), =?/-..(x- l) ~.........().. - 1 ( )* a* 1CX" d'_:- rn m(m -- 1)......2.l.c. dx'" Placing - = 0, we obtain x = a. dxz This value of x, when substituted in (1), (2), (3), &c., gives y = b, and reduces the successive differential coefficients to 0, as far as the rmth, which, if c be positive, becomes a positive constant, and is of an even order; hence y = b is a minimum ordinate, Art. (67). Since for x = a, we have -y - 0; the tangent line at the excldx tremity of this minimum ordinate is parallel to the axis of X; and since (m and m - 2 being even) for all values of x except x = a, y and d2 _ ________L are positive, the curve at all of its points is -p dX2 convex towards the axis of X, Art. (83). If c be negative; the mrth differential coefficient will be negative; and x = a and y = b will be the coordinates of a point at which the ordinate is a maximum. Il this case, the second differential coefficient for all values of s, except x = a, is negativo, and the curve, for all positive values of y, DIFFERENTIAL CALCULUS, 129 concave, and for all negative values of y, convex, towards the axis of X, 2d. Let m, be an entire and odd number. When x = a, the first differential coefficient as before, is equal to 0, as also the second, third, &c. The mth, however, if c be po sitive, is a positive constant, and of an oddc order; there is then, in this case, neither a maximum nor a minimum, Art. (67). By examining the second differential coefficient, we see (since n - 2 is odd), that for every value of x < a, it is negative; that for x a, it is O; and when x > a, it is positive; hence for all values _| of x < a, which give y positive, the curve is concave towards the axis of X; and for all A ~ values of x > a it is convex, as in the figure. Therefore at the point whose co-ordinates are x = a and y - b, as x increases, the curve changes fiom being concave, and becomes convex, towards the axis of X. If c be negative; the reverse will be the case, and as in the second figure, at the point M, whose co-ordinates are x = a and y = b, there is a change from convexity to concavity towards the axis _ of X. Such points are singular, and are called points of inflexion. In both cases the _ \ tangent line at the point of inflexion is parallel to the axis of X, and also cuts the curve. 3d. Let mn be a fraction, the numerator and denominator of which are odd, as'. Then y = b + c(x- a), dy 3c dy 2 3c xdxa) s, &c.;d 5(x — a)`7 5 5(x - a)" 130 DIFFERENTIAL CALCULUS. x = a gives y b dy d2y 0 dx dxn If c be positive; dy for all values of x < a will be positive, and for all values of x > a, negative; hence for all values of x less than a which give y positive, the curve will be convex, and for all values of x greater than a it will be concave towards the. __ axis off X, as in the figure. If c be negative; the reverse is the case, as in the second figure. The point M, whose co-ordinates are x = a and y = b, is in both cases a point of inflexion at which the tangent line is perpendicular to the 14 axis of X. WAhence we may say: a point of inflexion is one at which, as the abscissa increases, hA -~t a curve changes from being concave towards any right line, not passing through the point, and becomes convex, or the reverse. If the right line be taken as the axis of abscissas, this point will always be characterized by a chaLnge of sign in the second dfferential coefficient of the ordinate. For, since the curve on one side of the point is concave, and on the other convex, the second differential coefficient in one case has a different sign from that of the ordinate, and in the other the same; hence at the point the sign must have changed. In order that this may be the case, the second differential coefficient must be equal to zero, or infinity. The roots of the two equiations d2y 0 and d c dx2 dx' DIFFERENTIAL CALCULUS. 131 will then give all the values of the variable which can belong to points of inflexion. It sometimes happens that a point of inflexion lies on the axis of X, as in the second case above discussed when b 0. In this case x == a gives y1 =- O~ allc dy = O. y O=-0 and _ 0, dx and the corresponding point M is a point of inflexion, at which both the second differential coefficient and ordinate change their signs. It is evident from the preceding discussion, A that if any right line be drawn tlhrough a point of inflexion, the curve on both sides of the point will either be eonvex towards the line, or concave. 4th. Let m be a fraction with an evven numerator, as 2-. Then y = b + c(( - a)3 dy 2c d2y 1 2c dx 3( - a)3 dx2 3 3(~0-a)3 1 x= a gives dy/ d'y yb -= d = dx dx2 c being first regarded as positive; if x < a, will be negative, dx and if x > a, it will be positive; hence at the point whose co-ordy dinates are x = a and y = b, _ must change its sign fron minus to plus, which change indicates a minimum ordinate, Art. (66). If c be negative; the reverse will be the case, there will be a 132 DIFFERENTIAL CALCULUS. change of sign from plus to minus, and the ordinate will be a max imum. In the first case, the second differential coeffi~ () cient for all values of x is negative, and the ordiM nate positive; the curve is therefore concave towards the axis of X, as represented in fig. (a). /A1~ ~ In the second case, dY is always positive. For dxl all positive values of y the curve will then be convex, and for all negative values of y concave, as in fig. (b). The tangent at the point M is in both cases perpen dicular to the axis of X. ~A — The point M is singular, and is called a cusp. It is a point at which the curve, when interrupted in its course in one direction, turns immediately into a contrary one. 5th. Let m be a fraction with an even denominCator, as 2. Since the denominator of the fraction indicates that the square root is to be taken, the double sign: must be placed before (x - a)2, and we then have y = b ~ c(x - a)2, dy 3' d2 3 d= ~ -c (x- a), dy 3c_ -X 2 da- 4 (x - a)' Every value of x < a gives y imaginary; x = a gives y -= b, and x > a gives two values, one greater and the other less than b. There is then no point on the left of that one whose co-ordinates are x = a and y = b; but on the right of this point the curve must extend indefinitely and consist of two branches. dy x =a gives = 0; the tangent at is th palel to te axis of the tangent at M, is then parallel to the axis of X. DIFFERENTIAL CALCULUS. 133 Each value of x > a gives two values for d, i dx' the one positive corresponding to the greater M value of y, and the other negative; hence the upper branch is convex, and the lower concave, A. as in the figure, and the point M is a cusp. 93. Let us now take the equation (y - 22 -= from which we deduce 5 y - xI i~ x2 dy 2 -5 C'y 25 3 I. -_ 2x z-~ 2 ) — 2~ -_^2 dx 2 dxt 2 2 When x = 0, we have y = 0. If x be negative, y is imaginary. For every positive value of x, there are two real values of y, both of which are positive as long as x2 >, or X < 1; after which, one is positive and the other negative. Whenl x = 0, dy 0; also when dx 10 2 ~- - 0 or x -, 2 25 hence the axis of X is tangent to the curve at the origin, aind the tangent to the lower branch, at the point whose abscissa is 16 251 is parallel to the axis of X. The first value of -d2 tbelongs to the upper branch, and is always dX2 134 DIFFERENTIAL CALCULUS. positive. The second value is also positive as long as 2 X>.^ x3 or x < -p; after which it is negative. The origin is then a cusp, at which both M/ \ branches lie on the same side of the AX f< =2 + ~common tangent, and is of the second species, those before discussed being of the first species. The point of the lower branch whose abscissa is 6- is a point of inflexion. 94. By differentiating the equation y ~ i (x- a) /t -c, we derive dy... q- = ~/- c i -- a dx 2 Vx,/ - c For every value of x < c, y is imaginary. For x =c, y b, and dy co dx For every value of x > c, there are two real values of y. For x = a, y b,and dy = - a c, and at the corresponding point M there are two tangents, one making an angle, the tangent of which is + yVa - c, and the other - Va- c. The point M is singular, and belongs to a class A called multiple points, or points at which two or more branches of a curve intersect. If but two DIFFERENTIAL CALCULUS. 135 intersect, the point is a double multiple point; if three, a triple, and so on. Since there will be a separate tangent to each branch, at one of these points, it will be characterised by two or more values of the first differential coefficient, for the same values of the variables. 95. From the equation ay2 - 3 + b2 _ 0, we derive fx(x - b) dy 3x - 2b a dx 2-Va(x -h) Since x = 0 gives y = 0, the origin A is a point of the curve. All negative values of x make y imaginary, as also all positive values less than b; hence A has no consecutive point. Such points, given by the equation of a curve, but having no consecutive pOints on either side, are singular, and are ( called isolated or conjugate points. At these points it is plain that no tangent can be drawn, and that therefore the corresponding vzalue of the first diferential coeilcient mnust be imaginary; as in the above example, x = 0 gives dy -b dx / - ab 96. We will close this branch of the subject by a discussion of the equation ay2 — X3 + (-C) - bcx -' 0; 136 DIFFERENTIAL CALCULUS. whence y _ /x(x - b)(x + c) dy _ 3x - 2x(b - c) - be a' d 2 V ax(x - b)( -+ c) Each of the values, = 0, x =, x= - c, gives y = 0. Every negative value of x > c gives y imaginary; while every such value less than c gives two equal values of y with contrary sigans: Every positive value of x < b gives y imaginary, and every such value greater than b, gives two equal values of y with contrary signs. The curve is then symmetrical with reference to the axis of X. Each of the values, = 0 z =b, - c reduces to oo dx hence at the three corresponding points the tangent is perpendicular to the axis of X. By solving the equation 3W2 _ 2x(b - c) - bc 0, we shall find two real values for x, and thus determine the points at which the tangent is parallel to the axis of ~C A ^ ~ X. The curve may then be drawn as in the figure, in which AC = - c and AB = b. If c = 0, the equation becomes ay2 _3 + b- x = 0, and the oval AC reduces to the conjugate point A, as in the precediingl article, if == 0, the equation becomes ay2- x2 -- cx2 = 0, DIFFERENTIAL CALCULUS. 137 and the curve takles the form indicated in figure (b), the origin being a double multiple point, since Y becomes equal to c ~ If b and c are both equal to 0, the equation becomes ay2 x = 0o; Vwhence y - -+- - and the curve will be as in figure (c), the point A being a cusp of the first species. OSCULATORY CURVES AND CURVATURE. 9 7. It is now proposed to examine the tendency which curves, with a common point, have to coincide with each other in the vicinity of this point; and also the use which may be made of this property of curves. Let there be the three curves BB', CCG' DD', having the point M common;, the co-ordinates of the first curve being j,/ representecd by andci y, those of the c / { second by x' and y', and those of the __ third by x" and y". Since the point M is common, for this we have AP= = a' = x" PMI = y y' y" Suppose the abscissa AP to be increased by the variable h, we shall then have cP'M',hF dyh' hs d3y h 3 PqI'M' f(zq-h)-y +A = + +h +x +&. dx dx' 1.2 x 1.2.3 18 138 DIFFERENTIAL CALCULUS. dy' ~cFy' h2 cp &c IP'M", rf' ( xI' + Ih) y',+ dy h + Y _ y -I L - + dx' dxl' 1.2 dx'^ 1.2.3 P/M/// jot __.cu(x,~//j y"j1/ +d/ y'' hd d3yll h d it) l.(, dxl"4 1.2 1.2. 13 in which dy d2y cl3y dx " dx2' dx " represent, what the first, secondc, &c., differential coefficients, obtained f.ioml the differential equations of the first curve, become by the substitution of the co-ordinates of the common point. -dy,' d~S, &c.d, are corresponding values for the second curve; dx' dx""-~'d y d'y" &c.......for the third. dx"J dxI By subtracting the second and third equations, each, member by member, from the first, and making dy d' A dy dy" _ dx d d' dxx d d'2y dy I d'y d'y" 3 ^ - ^ = A'd d-": B', &c., dx2 dx2' dx dx' we have MI'M" = Ah + A'l + Al" - + &c., 1.2 1.2.3 131 ]32 _a M " B= Bh - B' 7 -t B" - + &c. 1.2 1.2.3 Now, if 7t be made infinitely small, the points M', Ml", MP", iwill DIFFERENTIAL CALCULUS. 139 become consecutive with the point M, and it is plain that the second curve will approach nearer to a coincidence with the first, than the third does, if M'IMl is numerically less than M'M"', that is, if _2 0h2 7 Ah + A'- + A + &c. < BA B- +Bh + B — + &c. 1.2 1.2.3 1.2 1.2.3 This condition will necessarily be fulfilled if A is equal to 0, and B is not, as we shall have, after omitting the factor h, h is' is ith Al + A"12 3+ &c. < B + 1B- + B —' - &c., 1.2 1.2.3 1.2 1.2.3 a true inequality when h is infinitely small, as then the whole of the first member will be less than the finite quantity B. But A = gives dy dy' dx dx' that is, the first and second curves have a common tangent, or are tangent to each other at the common point. If A = 0 and B = 0, the three curves have a common tangent, and in order that M'MI < I'M"', we must have 7,2 I 73 7,2 h3 A/ + A." + &c. < B'h B" ^ - + &c., 1.2 1.2.3 1.2 1.2.3 which, it is proved as before, will necessarily be the case if A' = 0 and B' is not. We have thus in addition the condition d2y d__cy' dx2- dx,' dIy d2 /y l If B' = 0 or 2 also, then M'M" < M.MI" dx dx"i if in addition to the other conditions we have 140 DI''ERLEN'TIAL CALCULUS. d3y d3y' A/ 0 or d J;_ dx3 dx' and in general the second curve will have a greater tendency than the third to coincide with the first, if the first and second have more equal successive differential coefficients of the ordinate at the common point, than the first and third. Two curves which have a conmmon point, and the first differential coefficients of the ordinate taken at this point equal to each other, are said to have a contact of the first order, or are simplytangent to each other. If the first and second difierential coefficients of the ordinate ta — ken at this point are equal to each other respectively, the contact i r of thie second order. And, in general, if the first m. differential coefficients of the ordinate taken at this point are equal respectively, the contact is of the rmtl order. To illustrate, take the two equations y2 4x......(1), y- x+1........(2). By combining them we find a common point; the co-ordinates of which are x, " - 1 y" - 2. By differentiation, we find from (1), dy 2.... dy2. -.(3); whence d 1; dfx y./ dx" and from (2), dy =1.(4); dy dx dx" Differentiating again, we have from (3), d2y 4 d2y" 1 - 3 9 dwhence" I cx-2 y' dx"; DIFFERENTIAL CALTCULUS. 141 and from (4), 1d 0; whence d O; dx-2 -- The two lines having a point in common, and the first differential coefficients of the ordinate equal at this point, have a contact of the first order. Since the second differential coefficients are not eqlual, the order of contact is no higher than the first. 98. The constants which enter into the equation of a curve determine its extent and position with respect to the co-ordinate axes. If then one curve be given completely, and another in kind only, by its general equation, the constants in this equation being marbitrary, we can evidently assign such values to them as shall cause the curve to fulfil as many conditions as its equation contains constants; that is, we may maLe the co-ordinates of one point of the second curve equal to those of a given point of the first; and, in addition, as many differential coeficients of the ordinate taken at this point, for the second curve, equal to the corresponding ones of the first, as there are constants to be disposed of, less one; thus giving to the second curve an order of contact at a given point of the first, denoted by the number of constants less one. To ascertain the values which must be assigned to the arbitrary constants: Obtain first, the value of the ordinate from the equation of the second curve, (the abscissa being assumed equal to the abscissa of the given point,) and place it equal to the ordinate of the given point; or what amounts to the same thing, substitute the coordinates of the given point in the equation of the second curve; obtain then the first differential coefficients of the ordinate by differentiating the equation of each curve, substitute in these the co-ordinates of the given point, and place thie results equal; do the same with the successive differential coefficients, until as many equations are formed as there are arbitrary constants. Ty the solution of these 142 DIFFERENTIAL CALCULUS. equations we can find those values of the constants which will cause the conditions to be fulfilled. These, substituted in the equation of the second curve, will give an equation which will re-~ present the particular curve having the required order of contact. The curve, which at a given point of a given curve has a higher order of contact than any other of the same kind, is called an osculatrix. Thus, an osculatory circle is one which has a high er order of contact thanc any other circle. Since no more conditions can be assigned than there are constants; the highest order of contact which can be given to a curve, is denoted by the nzumber of constants less one, vwich enter its nmost general equation. Let these principles be applied: 1st. To find the equation of an osculatory right line. Let the equation of the given curve be y -f(-), and the co-ordinates of the given point, x" and y". For this point, we have yl =.f(x") The most general equation of the right line is y = ax + b.....(1), containing but two arbitrary constants. The first condition to be fulfilled is, that the value of y deduced from this equation, when -, = x",! shall be equal to y", that is y"' = - ax"t b.......(2) The first differential coefficient of the ordinate derived from the equation of the given curve is -i, which for the given point bedx DIFFERENTIAL CALCULUS. 14-3 dy" comes -. The first cdifferential coefficienlt lerived from equation (1) is aY = c; hence the second condition is dx dy" y a......(3). dcx" By the solution of equations (2) and (3), we find a dy"ll d a = /Y b= y dy " x cdx" dx" These values in (1), give the ecquation dy / it dy" or -y — = x dy -Y. - + Y"-dz"', or Y-Y = ( dx"IV' dx"' Cldx" This, as it should be, is the same eqrcltion as that deduced in Art. (79). 2d. To find the ecuation of the osculatory circle at any point of the curve whose equation is y =f(x). Denote the given point, or point of osculation, by x" and y". The most general equation of the circle is (x - a)' q- (y - = P......(1), containing three arbitrary constants. A contact of the second order may therefore be given to the circle. By differentiating the equation y =f(x), and substituting x" and y" in the first and second differential coefficients, we obtain dy" ady dx" dxl2 Differentiating equation (1) twice, we have (x - c)dx + (y - )dy = 0; wvhence y = dx Y —f . 44 DIFFERENTIAL CALCULUS. dy dx" -+ dy2 + (y - )dy2Y - 0; whence d2y _ ~ dx y - 13 ut the conditions that the circle be an osculatrix are, (x being assumed ecinal to x",) -,y diy J cy" d- y cdy" - Tdx dx" dcl2 dcx" We shall then have for thle three ecjuations of condition, ( / C)2a + (" -_ )2 = R-.....(2), d~I?" - I dty" cL xd 12"" 0 " x,, - (4). dz". y" -- fiP d''J" -- By the solution of these, we can find the values of R., a and [, wvhich substituted in (1) will give the equation of the osculatory circle. To illustrate, let us seek the equation of the circle osculatory to the parabola whose ecuation is y2 = 4x, at tihe point whose co-ordinates are x" 1, y" 2. Differentiating the given equation twice, and substituting the coordinates 1 and 2, we find d~y 2 _ _ dy 2- whence dy"; dxz y dx'" d'y 4 i d xy" 1 dx-2 y' dr""2 2 DIFFERENTIAL CALCULUS.' 14 These values, with the co-ordinates of the given point, placed in the equations of condition, give (1 - ) + (2 - 2)2 R2 1-~ 1 2 2- 2 2~whence o = c,i3 -- 2 R= /32, and the equation of the osculatory circle will then be (.- 5)2 + (y + 2)2 32. 99. Since in the three ecjuations of condition just considered, x" and y" may, in succession, he made to represent every point of the given curve, we may omit the cashes and write the equations thus (x- _)2 + (y -3)2 = R2.........(1), dqy............( dy x-~ = - - )..............(2), in which it must be recollected, x and y are the co-ordinates of the point of osculation, c and /3 the co-ordinates of the centre of the osculatory circle, and R its radius. Substituting in (1) the value of x- a, and reducing, we oltain 19 .l 4 DIFFERIENTIAL CALCULUS. 2 — (y - ))2 + d (y-(y (d.}dy2) (Y - PY' +:(y - P ( 2I whence, by the substitution of the value of y -, I (dx2 + dc- y) d' d:y which is a general value for the radius of the osculatory circle. If z denote the arc of the given curve, then dz = Vdi -- dy'...........Art. (86); hence the above expression for R becomes dz3 -_ \- -;,__ dx dxy 100. If p denote the angle made by the radius of the osculatory circle drawn to the point of osculation, with a fixed line as OP, M and Ml two consecutive points, and MC and TM'C the corresponding radii intersecting at C, then iMC- = IR, MiMM' = dz, 2Mn' = dp......Art. (91)..! +(~ ~Since MOGM' may be regarded as a triangle p ~ right-angled at M, we have MM' = MC tang MOM', and. since MOM' is infinitely small, the are which measures it may be taken for its tangent; hence dz = Rdp, and R =. dmp DIFFERENTIAL CALCULUS. 147 101. The first value of R in article (99) has been deduced under the supposition that x is the independent variable. It is sometimes desirable to change this independent variable, during the discussion of expressions of this kind, and to regard y or some other variable quantity in the expression as the independent one. A mlore general expression for R may be obtained without the particular supposition referred to, if we recollect that dy has been dedx dluced by the differentiation of dy regarding d(dx) = 0. If we differentiate this expression on the supposition that both dy and dx are variable, we have "cy _ cdxd2y - dydlx 1dx) d "~d^ dy') which must take the place of -, or for dxd2y we must pnut dxd2y - dyd2x. The value of R thus becomes (dX2+ dy )_ _y2) d T-t dxd2g _ - dyd - (1). dxd2y - ddx - dxzdy - dyd'x If in this, dx be regarded as constant, we shall have the value of R, as in article (99). If dy be constant, or y regarded as the independent variable, then (dxs + dye) 2 _A dydSx clydz If z be regarded as the independent variable, dz will be constant, and d(dz') = 0; whence 148 DIFFERENTIAL CALCULUS. dxd'x + dyd'y = 0. Adding the square of this, to the denominator of the value of R' taken from (1), we have -p-2 _ddz dz4 [(- (dy) +J- (d2x)'] (dy2 + dx2) ~ (dy)2 + (d2x) dz2 V(d2y)I + (d2x)' 102. Since the curve and osculatory circle at a given point have a tangent in common, they must also have the same normal; but the normal to the circle passes through its centre, the normal to the curve must then pass through this centre; or the radius of the osculatory circle, drawn to the point of osculation, is normal to the curv.e. 103. Let BB' be any curve, and CC an are of the osculatory circle. Then since dy = dy and dy_ d2y' ~ ~, -' dx d d' ddx dx'"" 1~~,i ~\;, vwe shall have, Art. (97), -M =A" + "' + + &C......(1). I- ~ ~ 1.2.3 1.2.3.4 W hen 71 is infinitely small, the sign of M'WM will depend upon that of the first term of the series, which will have the same sign as A" when h is positive, and a contrary one when h is negative; that is, M'M" and m'm" have contrary signs. If then M" is below the curve BB', m"v will be above it, and the reverse; and the circle CO must intersect the curve at M. It muay be shown in the same way, that any osculatrix of an even order intersects the curve; while one of an uneven order does DIFFE]RENTIAL CALCULUS. 149 not. As, when the order of contact is even, the first term of (1) will contain h with an odd exponent, and will therefore change its sign when h becomes - I. This will not be the case when the power of h in the first term of (1) is denoted by an even number. The osculatory circle, however, does not intersect at those points about which the curve is symmetrical with its normal. For, ordinates being drawn from the points of both, perpe ndcicular to the common normal, if the ordinate of the curve on one side is greater than the corresponding ordinate of the circle, it will be so on the other side; as may be seen in the figure, in which, if pn > po, then pn' > po';,; —..-. or if pn < po, then pn' < 2o'; hence, in this case, in the vicinity of the point M, the circle lies entirely within or entirely without the curve. In these cases it will be found that the order of contact of the circle is odd, and higher than the second, fur- unless A" -- O; the circle must intersect, as shown by the preceding dem-onstration. Since the osculatory circle has a more intimate contact with a curve at a given point than any other circle, it will necessarily separate those circles which are tangent without the curve firom those which are tangent within. 104. The curvacture of a curve at a given point is its tendency to dej.rt fironm its toangennt at t/hat point. Thus, of the two curves AC and AB, having the common tan- A gent AD, the former has a greater tendency to de- \. part from the tangent, and has the greatest curvature. The curvature of the circumference of a circle, is evidently th.e same at all of its points, but of two different circumferences, that one curves the most which has the least radius; as in the figure, 150 DIFFERtENTIAL CALCULUS. a the tendency of abd to depart from the ta.ag ent is greater than that of ab'd', and this l )b t tendency plainly increases as the radius decreases, and the reverse; that is, the curvatlre in tzwo different circles varies inversely as their radii. This being the case, the expression ~ may be taken as the measure of the curvature of a circle whose radius is R. Since the contact of the osculatory circle with a curve is so intimate, its curvature may be taken for the curvature of the curve at the point of osculation; and the two in the immediate vicinity of this point, may be regarded as one and the same curve; hence, to compare the curvatures at dif__J~.~ —'-`\' ferent points of a curve, we have I y"' ~ / \', only to compare the curvatures of (\^ )~ \ J ^the osculatory circles drawn at these points. Thus in the curve MM', 1 1 curvature at A: czurvCature Ct Al":' - 1_. 105. The radius of the osculatory circle at a given point of a curve is called the Radius of Curvature, at that point. The general value of this radius is given in article (99), and it may be found for any particular curve, by differentiating the equation of the curve, and substituting the derived values of dy and d.y in the formula, (dx2 + dy2)2 If the alue at any ptic p of the cve e reuied If the value at any particular point of the curve be required. DIFFERENTIAL CALCULUS. 151 for x and y in the value just deduced, substitute the co-ordinates of the particular point. As only the absolute length of the radius of curvature is requirec in determining the curvature of curves, we may use either the plus or minus sign of the formula. It is best, in general, to use that which taklen with the sign resulting from the expression, will make R essentially positive. Let it now be required to find the general expression for the radius of curvature of Conic Sections. Their equation is yF mx+ nx"; whence dy ( + 2nx)dx 2y 4d2 dzy 2nydx' - (n +~ 2nx)dxly [4ny2 - (nn + 2nx)~2]dx 2y2 4y3 These values substituted in the formula, give _ [4(mn + 2nx2) + (7n + 2nx)]2, 2mn2 and this, after dividing both terms of the fraction by 8, may be put under the form 2=(Vnx + nx, + n (9n + 2nx)) ( 4 the numerator of which is the cube of the normal, Art. (82): Hence the radius of curvature at any point of a conic section, is the cube of the norial divided by the squcare of half the para 152 DIFFERENTIAL CALCULUS. zmeltr, and the radii at different points are to each other as the cubes of the corresponding normals. Tf in (1) we make x = 0, we have, at the principal vertex, R, - = one half the pc'ameter, 2 B2 which for the ellipse and hyperbola is t. A Tie radius of curvature at the vertex of the conjugate axis of the ellipse is obtained by substituting in (1), 2B2 B32 M -- ~A n ~ 2__ and x = A. The result is A2 R A: one half the 2arameter of the conjugate axis, B It may be readily shown'that - is the least value which R ad" 2 mits of; therefore the curvature at the principal vertex of a coA2 nic section is greater than at any other point. Likewise, ~ is the greatest value of R in the ellipse; hence the curvature of the ellipse is least, at the vertex of the conjugate axis. The curvature of the other two curves diminishes as we recede from the vertex. For the parabola n = 0; we then have Rp (2 + 42mx) (m2 - 4mnx) 2niY DIFFERtENTIAL CALCULUS. 153 EVOLUTES. 106. If at the different points of a given curve osculatory circles be drawn, and a second curve traced through their centres, the latter is called the Evolute of the formler,, which is the ilnvolute. Thus CC"' is' the evolute of the involute MM"'. \ Points of the evolute may always be \ constructed by drawing normals at the =' different points of the involute, and on each of these normals laying off the corresponding value of P1, deduced as in article (105). 107. If C and /3, the co-ordinates of the centre of the osculatory circle, be regarded as variables, they will determine all the points of the evolute; but u, /, and R, are functions of x and y, the coordinates of the points of osculation; and the relation between these five variables is expressed by the lthree equations of Art. (99), which may be written thus, (x - ) + (y -G)2...........(1), (x- a)dx + (y - 3)dy = 0........(2), (y - g)d2y + d' + 2= (........) If we differentiate (1) and (2), regarding all the quantities, except dx, as variables, we obtain (x - a)dx + (y - 3)dy - (x - a)da - (y - /)d/l= dUR,, di + dy --- (y - /)dy - dxd. - dydf-3 0,'20 154 DIFFERENTIAL CALCULUS. and these, by means of equations (2) and (3), are reduced to -( x -.)da - (y -R)d R.......(4), - dxda - dydf 0..............(5). Equation (5) gives x d,3...(6). dv ydci CX is the tanglent of the angle which a normal to the involute dy at the point (X,?) makes with1 the axis of X, Art. (8 1), Mld - da the tangent of the angle which a tangent to the evolute at the point (oC, ) makes with the same axis; hence these angles are equal. But the normal at the point (x, y) passes through the point (ra /), Art. (102); therefore the normal and tangent form one and the samey line; that is, the radius of curvature is normal to the -invlctie, and tangent to the evolute. The evolute mlay therefore be constructed, by drawing a curve tang-lent o t he normals at the different points of the involute. From what precedes, it is plain that the evolute may be regarded as formed by the intersections of the consecutive normals to the involute, and that the point of intersection of any two consecutive normals may be taken as the centre of the osculatory circle, which passes through the two consecutive points of the involute at which the normals are dravwn. 108. Equation (6) of the preceding article, combined with (2), gives x - a- -(d - 3 *l _ 3dD ) DIFFERENTIAL CALCULUS. 15 Substituting this value in (1), we have, after reduction, (y_ )( + ) R..........^). - (eda 2 +. d2(7). df3 Substituting the same value in (4), reducing and squaring both members, we obtain (y - S)~(dt + d -2)S = R2d. (Y - t ~d/'dR. Dividing this by (7), member by member, and taking the root, Vd2 + d/2 = dR. But if z represent the arc of the evolute, we have dz = V/da + d.......... t. (86); hence dR dz, dR - dz = d-0- (R-) = 0; whence R - z must be a constant, Art. (14), or:R = - +c. 109. If any two radii of curvature be drawn, as one at M and the other at M'; the first being denoted by R, the second by t', and the corresponding arcs BC and BC' by z and z', we have R = z + c R' z' + c; whence R - R ='- z; 156 DIFFERENTIAL CALCULUS. that is, the diference between any two radii of curvature is equal to the arc of the evolute intercepted between them. If in the equation R = z + c, we make z 0, and denote by r, the corresponding value of R, we shall have r =-0 - c =c; that is, the constant c is always equal to the radius of curvature which passes through the point of the evolute, from which its arc is estimated. M' If we estimate the evolute of the ellipse from the point C, we have MC ~ /~C" R2 c = MC = -...........Art. (105). hence A Also, since M'C' A, MR'C MI -- = C C'. B A If the evolute and one point of the involute be given, and a thread be wound upon the evolute and drawn tight, passing through the given point M, fig. (a); when the thread is unwound. or evolved, the point of a pencil first placed at M, will describe the involute; for, by the nature of the operation, CC' is always equal to M'C' - MC. 110. The equation of the evolute of any curve may be ound DIFFERENTIAL CALCULUS. 157 thus: Differentiate the equation of the involute twice; deduce the expressions for dy and d2y, and substitute in the equations Y - - "- +........() l d2y. Art. (99); du Z a = i - d:;Y-,"-)-~.....(2). combine the results, which will contain the four variables a, 13, x, and y, with the equation of the involute, and eliminate x and y; the final equation will contain only a, 3, and constants, and will therefore be the required equation. As an example; let it be required to find the equation of the evolute of the common parabola. The equation of the involute is y2 _ 2px; whence dy p dx y 2y 2dx pddx3 Pydx-, d y... yp2d2 p Substituting these values in (1) and (2), and reducing, we have P\ P y-/=l- i; whenc, e - ( 1......(3); -- = -Y _. p...........(4); p'and putting for y, in (3) and (4), its value V'2px = (2p)Ax2, we have 3 3 22x2 —, - "- _a= - 2-x —p. p2 ~ = 1." " ~K 2x ~ p P: 158 DEDIFFERENTIAL CALCULUS. The value of x z-_( - p) taken from the last equation, and substituted in the preceding, gives /2 = 8 (a _ 3), P- (a - p3, 2 7p which is the required equation. If we make 3 = 0, we have a = p, and laying off AC = p, C will be the point at which the evolute meets the axis of X. If we trans-,^/ X fer the origin of co-ordinates to this point, we C"vJ^ have A ce ca' =C - p, 2' = {; hence /2' = 8 a'3. 27p Since every value of a' gives two values of /', equal with contrary signs, the curve is symmetrical with the axis of X. If a' be negative, 1' is imaginary, and the curve does not extend to the left of C. The branch CC' belongs to AM, and CC' to AM'. TRANSCENDENTAL CURVES. 111. The most general division of curves is into the classes, Algebraic and Transcendental. When the relation between the ordinate and abscissa of a curve can be expressed entirely in algebraic terms [see Art. (5)], it belongs to the first class; and when such relation can not be ex DIFFERE!NTIAL CALCULUS. 159 pressed without the aid of transcendental quantities, it belongs to the second class. 112. One of the most important of the latter class is THE LOGARITHMIC CURVE, so named, because it may always be referred to a set of co-ordinate axes, such that one co-ordinate will be the logarithm of the other. Its ecquation is usually written Y =- log x, or, if c be the base of the system of logarithms,,x - ay. The curve is given when a is known, and - i' may be constructed by laying off on the axis \ of X the different numbers, and on the cor- responding perpendiculars, the logarithms of these numbers: Or it may be constructed / ~ xI from the equation x = ay, by makiing y =:, / 3, ~, &c.; whence the correspolding values ~ of x are Y M x V- =i /a, 2 -- a/ su = V7a, V c. When y = 0, x - 1. This being the case for all systems of logarithms, shows that all logarithmic curves, when referred to the same axes, cut the axis of X, or axis of numbers, at a distance from the origin equal to unity. If a > 1,and x > 1, y is positive and increases as x increases; if x <' 1, is negative and increases numerically as x decreases ]'4 DIFiERl;ENTIAL CALCULUS. until x -- 0, vwhen y - o. If x be negative, there will be no corresponding value of y. The curve will then be of the form indicated by the full line in the figure. If a < 1, the reverse will be the case, and the curve will be represented by the dotted line. 113. If now we differentiate the equation y = log x, M being the modulus, we deduce dy M d2y M_ dx x dx2 X' When x, 0 dy = ~=; dx 0 hence the tangent at the corresponding point is the axis of Y; and since for x - O, y =- o, this tangent is an asymptote. When x dy c 0. dx co But x = co gives y co; hence there is no tangent parallel to the axis of X, at a finite distance from it. The value for the subtangent on the axis of X is PT yJ = log x. If the subtangent be taken on the axis of Y, we have SS'- x dy - M. d. that is, the subtangent on the axis of loyarithms is constant, and equal to the modulus of the system in which the logarithms are taken. DIFFERENTIAL CALCULUS. 161 If M 1, SS' = 1 AB. Since, when a > I, d2y is negative for all values of x, the part dx2 BM is concave towards the axis of X, and BM' convex. When a < 1, M is negative, ds will be positive, the part B3nl dx2 convex and Bm concave. 114. Another remarkable transcendental curve is, THE CYCLOID, which is generated by a point in the circumference of a circle, when the circle is rolled in the same plane, along a given straight line. Let AB be the given line, and suppose the circle to have been placed upon it, so that the generating point was at A, and then to have been rolled to the position RME. The generating point now at M, has generated the arc AM. T, A I' c B Take the origin of co-ordinates at A, and let AP x, PMi = y and RE, the diameter of the generating circle = 2r; then AP -AR P- P............(l). But since every point of the circumference from M I to R, as the circle was rolled, came in contact with AR, we have AR = are iR -- ver-sin-'RN = ver-sin-'y. 21 162 DIFFERENTIAL CALCULUS. Also, PPr = MN = VlN x NE Vy(2r - y)= V2ry - y Substituting the values of AP, AR and PR in (1), we have x = ver-sin-l - v-2ry -.............(2), which is the equation of the Cycloid. After the circle has been rolled over once, every point of the circumference will have been in contact with AB, and the generating point will have arrived at B; wvve have then AB = circumference of cyengcratig circle -- 2ir. The given line is called the base of the Cycloid, and the line CD = 2r perpendicular to AB at its middle point, is the axis. If the rolling of the circle be continued beyond the point B, an infinite number of arcs, each equal to ADB, will be generated. Every negative value of y in equation (2) makes x imaginary; hence there is no point of the curve below the axis of X. y 2= r, gives x - ver-sin-2r r - -- AC. Every value of y > 2r makes x imaginary; hence the greatest ordinate of the curve is equal to the diameter of the genera.ting circle. By differentiating (2) we have, Art. (42), dx - rdy rdy - cdly V/2ry - iy V2ry - y2 or reducing dx == yd -................. (3) d yd -y (), V/2,ry -2 DIFFEIRENTIAL CALCULUS. 163 which is the differential equation of the Cycloid. 1.15. Substituting the preceding value of dx in the formulas of article (82), and reducing, we have Subtanyent, PT = _ 2/2ry -y Tangent, MIT y 27y,/2ry- y2 Subnormcd, PR = /2ry - yy. NTorcal, MR V- /2y. Since the subnormal PR = -V2ry - y2 = MN the diameter ER and normal MR intersect the base at the same point. Hence, to construct the normal at a given point, join it with the point at which the corresponding position of the generating circle is tangent to the base: Or, upon the greatest ordinate CD as a diameter, describe a circle, and, through the given point M, draw a line parallel to the base, from the point F in which it cuts the circle, draw the two chords CF and DF to the extremities of the diameter; a line through the given point parallel to CF will be the normal, and one parallel to DF the tangent. If it be required to draw a tangent parallel to a given line as TT'T"; draw the chord DF parallel to the given line, from F draw FM parallel to the base; the point AM is the point of contact, through which draw a line parallel to T'T". 116. From equation (3), article (114), we have dy y- _ = /..........2. () dx y y 104 DIFFERENTIAL CALCULUS. which becomes 0 when y = 2r, and oo when y = 0; hence at the extremity of the greatest ordinate, the tangent is parallel to the base; and at the points A, B, &c., where the curve meets the base, it is perpendicular. If we square both members of equation (1), we have dy2 2r dx' y Differentiating both members of this, we have 2dyd2y 2rdy d2y r 7-, or dx2 y dx y This second differential coefficient being negative for all values of y, the curve is concave towards the axis of X, Art. (83). 11 7. Substituting the values of dy and d2y in the expression (dx + dye)2 dxd2y we obtain 2ryd j -2?/2-2 3 _ - 2 3y2.1 2-V2ry rdxc y2 or since /2ryis the expression for the normal, Art. (115), the Radcius of Curvature is equal to twice the normal at thepoint of osculation. If y = 0, R = 0; and if y = 2r, R =4r; hence the radius of curvature at A, (see figure in next article) DIFFERENTIAL CALCULUS. 165 is equal to 0; and at D is 4r; therefore, Art. (109), the arc AA' - 4r. 118. To obtain the equation of the evolute let us substitute the values of dy and d'y in equations (1) and (2) of article (110). After reduction, we find y — 2y, x — == 2 V2ry - y'; whence y =- (, x = - 2V/- 2r- -. These values, in the equation of the involute, Art. (114), give v= ver-sin' -P + 1/ - 2r - - 2.....(1), for the required equation. If we produce DC to A' making CA' = C, and then transfer the origin to A', the new axes being D A'X' and A'1D, and the new co-ordi- nates a' and /3', we shall have for' any point, as M', B AG-, G-M' =-/3, A'P' - a', P'M' = /3' Since AC ~r, and CG = A'P', -c =' ^r cc^: and since G P' 2r, GM' = 2r-', or -/ 21 -r Substituting these values in (1), we have 166 DIFFERENTIAL CALCJLU LS. r a' - ver-si'-(2r ~ /3') + -r' ~3 2 whence a' = ^r - ver-sin-'(2r - f') - /2r' - /3/2; But frr - ver-sill-n(2r --') = ver-siin-/3'; hence the last equation becomes' = ver-sin-w/' V/2r'' - /3'2 which is the equation of the evolute referred to the new axes, and is of the same form and contains the same constants as the equation of the involute, therefore the two curves are equal. Since arc AA' = 4r, its equal AD = 4r, and ADB = 4.2r that is, equal to four times tie diameter of the generating circle, THE SPIRALS. 119. If a right i-ne be revolved uniformly, in the same plane, about one of its points; a second point of the line continually approaching, or receding from the fixed point, in accordance with some prescribed law, will generate a curve called a spiral. The fixed point is called the pole or eye of the spiral. The portion of the spiral generated while the line makes one revolution, is called a spire; and since there is no limit to the number of revolutions, the number of spires is infinite, and any straight line /' / \ dadrawn throughl the pole of the spiral will i 1 A- jl intersect it in an infinite number of points. A\ \ 1 / The system of polar co-ordinates being \~. // ~ used to determine the different points of a'^,i' spiral, its equation will, in general, be of the form DIFFERENTIAL CALCULUS, 1 07 u = f(t), in which u denotes the radius vector, an d t the variable angle. 120. Before discussing the particular spirals, it will be necessary to determine general expressions for the subtangent, &c., and the differentials of the arc and area, in terms of polar co-ordinates. The subtangent, in such case, is the part of the perpendicular to the radius vector of the point of contact, intercepted between the pole and the point where the tangent meets this perpendicular. Thus, if A be the pole, and MT the tangent, AT perpendicular to AM is the subtangent. To find the expression for it; let the arc t receive the increment PP', (AP beingo 1); de- scribe MC with the radius AM 1 =; /~ \ draw the chords MC and MM', and the.-X'h! \ line AT' parallel to MC, and produce,,',IMM' to T'. From the similar triangles MIM'C and M'AT', we have M/': MC Mi'A T'A; T'A MC x A..) M'C C Also from the similar sectors APP' and AMC, i: PP':: AM: arc MC; arce MC AiM PP'i Now suppose the increment PPI = clt, then MI'C clu, Art. (91), M' becomes consecutive with MI, the secant M'T' coincides with the tangent MT, T'A = AT, AM' = AM = u and chord MC arc MC - udlt. Maling these substitutions in (1), we have AT - subtcngen zt _-__.......(2). du 38 DIFFERENTIAL CALCULUS. From this we deduce AT AT vudt....= __ = -taBng AMT. u AM du The tangent MT = I/AM + AT' == u /l +1 jd, Thle similar triangles AMT and AMR, give ~ d~t, AT: iu::: AR; AR iL =- i _ -= subnormal, AT dt When M' is consecutive with M, M'C may be regarded as a triangle, right-angled at C; hence MMI' /-'C2 + MC''. But MM' is the diffeiential of the arc; therefore dz = -V/du- + utdt. If ADM be any segment, AMMA' will be its increment when t is increasec by dt. Calling the segment s, AMM' will then be ds, and may be measured by the sector AMIC. But the area of the sector AMC = -AM x arc MC -- dt 2 2' Tt2dt ds - uc 2 It should be observed, that all of these expressions may be found precisely as in the corresponding cases in rectilinear co-ordi DIFFERENTIAL CALCULUS. 169 nates, but it is better to avail ourselves of the more simple process indicated in the general remark, Art. (91). 121. An equation, from which the particular equations of most of the spirals may be deduced by assigning particular values to a and n, is U at" If n be positive, t = 0 will give u = 0, and the spirals represented by the equation have their origin at the pole. If n be negative, t = 0 will give u = co, and the spirals have' their origin at an infinite distance, continually approach the pole, and uz becomes equal to 0 only when 122. Let n = 1, then u = at, and if u' and t', u" and t", represent the co-ordinates of any two points of the spiral, we shall have u'' = at', u" = at"; whence U': ":: t: t i", or the law in accordance with which the generating point must move is, that the radius vectors shall be proportional to the corresponding angles. The curve thus generated is the Spiral of Archimedes. 22 170 DIFFERENTIAL CALCULUS. If we take foi the unit of distance, the length of the radius vector after one revolution; then u = 1, t = 2%, and the equation gives 1 = a.2%r, a = 2~ and the primitive equation becomes t dt u =; wVhence du -. 2,r 2~ This spiral may be constructed by dividing a circumference into any number of equal parts, as 8, and the radius AB into the same number of equal parts. On the radius AC lay off one of these parts; on AD two, AE three, &c.; on AB Tf) —-T"" 1eight, then again on AC nine, &c. The /', I. /"'.,\\, distances thus laid off will be proportional to the angles BAC, BAD, &c., and the curve through their extremities the re\ I red/ spiquired spiral. " -,.,....- Substituting the values of uz and du in equation. (2), Art. (120), we have AT = subtangent _ 2,, If t = 2, that is, if the tangent be drawn at the extremity of the arc generated in one revolution, we have AT = 2q. = circumference of neasuring circle. If t = m.2r, or the tangent be drawn at the extremity of the are generated in nm revolutions, AT = — m.2 =- q.2'nmr; DIFFERENTIAL CALCULUS. 171 that is, equal to m times the circumference described with the radius vector of the point of contact. For the subnormal we find AR du 1 dt 2,' 123. If n = ~, the general equation becomes u at2, 01r 2 = a2t. This equation being of the same form as that of the parabola, the curve given by it is called the Parabolic Spiral. It may be constructed by first constructing the parabola whose equation is y2 = a2x, and then laying off from P to B, C, D, &c., along the circumference, any assumed ^ abscissas, and from A to M, M', &c., the cor/'\ { J2 responding ordinates; the points M, M, &c., [,, \ will be points of the spiral, since for each we have'-a__ ^ _y, =- a2x, or u2 a2t. 2u3 The subtangent at any point is AT = ~j. a 124. If n - 1I, t == at' becomes u - at -l, or utt - a, t and the spiral thus given is called the Hyperbolic Spiral. If u' and t', u" and t", be the co-ordinates of any two points of 172 DIFFERENTIAL CALCULUS. the spiral, we have u' and ~'; whence tl I I 1.1 u: ZC": _:, t' t" or the radius vectors are inversely proportional to the angles. If M be any point of the spiral, AM = u, MAP = t. The right-angled triangle MAP, gives MP sin t Substituting this value of u in the equation ut = a, we find M in t t As t is diminished, this value approaches nearer to a, and since (sin t) 1; when t = 0, we have MP - a. If then at a distance AC a, a line be drawn parallel to AP, it will continually approach the curve and touch it at an infinite distance. The subtangent AT _ = d a. du It is then constant and equal to AC. Also, -dt tangAMT - t du that is, the tangent of the angle made by the tangent and radius DIFFERENTIAL CALCULUS. 173 vector is equal to the arc which measures the angle made by the radius vector and fixed line. We may apply these properties to the construction of the curve by points, thus: With A as a centre and radius = a, de- T=-....-~-...scribe a circle; join any point / \. T with A, draw the indefinite / x' radius vector AM perpendic- A P ular to AT. Make AD = \ are PN; join D and N, and - draw TM parallel to DN, M will be a point of the curve; for by the construction AD = tang AND- tang AMT = arc NP. 125. The spiral represented by the equation t = log u is called the Logarithmic Spiral. Differentiating, we find Mdu dt -; whence tang AMT =-= dt; du that is, the angle formed by the radius vector and tangent is con stant, and the tangent of this angle is equal to the modulus of the system of logarithms used. If the Naperian system be chosen, M = 1, and AMT = 45~. 174 DIFFERlENTIAL CALCULUS. Since t is the logarithm of u, if it be increased uniformly, so that the different arcs t, t', t", &c., shall be in arithmetical progression, then u, u', u'", &c., " must be in geometrical progression, and the / 7" curve may be constructed thus. With i\ A =AO = 1 describe a circle, and divide the \ / circumference into any number of equal.^... —-. parts, and draw the lines AO, Ap, Ap', &c. The distances laid off on these lines are to be in geometrical progression, since the arcs Op, Op', Op", &c., increase by the constant difference Op. To find the ratio of this progression let t = 0, then u = AO = 1. Now make t = the arc Op, and find the corresponding value of u in the system of logarithms used, which lay off to m, then Am the ratio. AO0 On Ap', A", &c., lay off Am', Am", so that AO: Am: Am': Am": Am': &c., mn, im' m", &c., will be points of the curve. PART IL INTEGRAL CALCULUS. 126. THE object of the Integral Calculus is to explain how to pass from differentials to the functions from which they may be derived: Or in any particular case, to find cn expression which, if it be differentiated, will produce the given differential. This expression is called the integral of the differential. The symbol f when prefixed to a differential, denotes that its integral is required, thus f du = t, and this integral (du being infinitely small) is plainly the same as the sum referred to in article (91). 127. ~Ve have found, article (15), dAu = Adu; therefore fAdu = f dAu = Au - Af du. From which we see that a constantfactor may be placed without the sign of integration, without affecting the value of the integral; thus, 176 INTEGRAL CALCULUS. x3dw 1 J'b(a - x')dx = bf (a - xc)dx, f -fx'3dxo C C Also in article (18), we have d(u + v I &c.) = du + dv ~ &c.; hence f (du + dv ~L &c.) =fd(u + v ~ &c.) = u- v ~, &C. =fdu + -fdv ~c &c.; that is, the integral of the sum or ditference of any ncumber of differentials, is equal to the sum or difference of their respective integrals. Also in article (14), we have d( +- C) -- du, no matter what the value of the constant C may be; hence an infinite number of expressions differing from each other in a constant term, when differentiated will produce the same differential. For this reason, to the integral immnediately found we always add a consta.nt; thus, fdu = u -- C. INTEGRATION OF MONOMIAL DIFFERENTIALS; &C. 128. By article (22), we have cd.m+l = c(m + 1)x'dx; and from this, cdzx+l.'m+l cxLdx ed cdx m+l m -1 INTEGRAL CALCULUS. 177 hence m+l m+l fcxmdx - fcd C. Sm q - 1 ~ -. Therefore, to obtain the integral of a nmonorial diierential: if[ultiply the variable with its p2rimgitive exponent increased by ulnity, by the constant factor, if there is one, and divide the result by the new eponent. Examp les. 1. If du = xdx, f du = Sxdx -- C. 2 2. If d =, fdu= - xldx - + C. c c 4c 5 L 2 bxl' 3bx~ 3. If du = bx dx, u = - += C.?n n-1 x dx nx n 4. If du =, Z = e e(n- n-) 2adxc 3x 3dx c2x4dx 5. If du= +, b e adx 3x dx C24d.A (X uc=f-'3-f -J... Art. (127). Vx e The application of the above rule does not give the proper integral when mn = - 1, as in this case we have 23 178 INTEGRAL CALCULUS. f -l dx =, = 00_ — 1+1 whereas f x-'dx = lx + C......AArt. (37). This result was to be expected, sincef or lx cail not be exx pressed in algebraic terms, Art. (5). a dx b x af df-x a l C, b X b or u =log x - C, the logarithm being taken in the system whose modulus is a. b 129. Many expressions, by the introduction of an auxiliary variable, may be transformed into monomials, and then integrated as in the preceding article. I. Let du = (a + bx")'c'x"-'dx. Place a + bx"= z, then nb6"'dx = dz xadx = d, bn IN'TEGRAL CALCIULUS. 179 Substituting in the given expression, and integrating, we have fC'Z'm C' /c' eZ'm+l f du _ f c'z dz d = m bbn bn bn m J+ I and replacing the value of z, we have, finally, c'(a - bx) + C; (m + 1)nb that is, to integrate a binomial differential when the exponent of the variable without the parenthesis is one less than that within: lAultfidply the binomial with its primitive exponent increased by unity, by the constant factor, if there is one, then divide this result by the product of the new exponent, the coeficient and the exponent of the variable within the parenthesis. Examples. 1. If du = (a + bxS)eexdx, u_ ( b C. 2. If du = (2 - 3x5) —'3x4dxC, u_- (2 - 3S5)2+ C. ~ ~ -' - (a- bx) 3. If du= (a- bx)3 ) d, u=x 3(a C in p, i 4. Let du = a(b cz- i) i -1 c7 IL Let a axn-'dx du _ b: xn 80 INTEGRAL CALCULUS. Place b 4 x" = z; then - nx"-dx = dz x'-dx = ~i d n and adz a a 2 = - f ~ a -z _~ - l(b ~) xG) + C. nz n2 n In the same way we may find the integrals of the following expressions. a( b + 2cx)dx 1. Let du =- (b - 2cx)dx a + bx + cx2 Place a + - + cx2 =z, thlen (b + 2c)dx = dz, dZ j =: mf 71Zz = Mn(a -+ bx - cx2) + C. 2. If du = 2dy 2, (a - y) + C. a —y 3. If du = (2 + 2x)d 2-it (2x + x2) + C. 2x + x2 2z 2 dz 4. Let du 1-z2 Since in general f adu we see that in all cases where the numerator of an expression is the product of a constant and the differential of the denominator, INTEGRAL CALCULUS. 181 its integral will be the product of the constant and the Naperian logarithm of the denominator. 130. If we have an expression of the form du = (a +- bx + cx2 + &c.)x"dXnd in which m is a positive whole number; the integral may be found by raising the quantity within the parenthesis to the nrth power, multiplying each term by x"dx, and then integrating it as in article (128). Elxamples. 1. Let du = (a + x')2x3dx, or du, = (a2 + 2aC2 4-+x)x3dx; then fu =f(a 2xdx + 2axSdx - x7dx) + ax4 2a x 0 2. Let du = (b - ) xa)2dx. 3. Let du =- (b- cx2)2 dx. 131. Every expression of the form du = Ax'"(a -- bx)"dx, can be integrated, when either m or n is a positive whole numoer. If n be positive and entire, we may integrate as in the preceding article. 182 INTEGRAL CALCULUS. If m be positive and entire, n being either fractional or negative, place a - bx = z, then x = b dx. d du AQ A j __ " b' b A (z a)m/ndZ which may be integrated as in the preceding article. The value of z being then replaced, the integral will be expressed in terms of x. Examples. 1. Let du == bxZ(a - x)2dx. Place a -x= z, then x = a — z, dx = dz, iu f - b(a - z)2zdz ba2 = 2- b-a5 + 4- Z" 3 5 7 and finally, by replacing the value of z, 2 6 4 s 2 -1 it ba(a - B(a -)x + b(a - x)2 + C0 3 5 2. If du_ _rl (1- 3)2' it may be placed under the form INTEGRAL CALCULUS. 183 - 2 1 d = 2'(1 3x) 2-dx; whence u - f(1 f z)z dz, and finally, u - 1 X(1 - )+ (1- 3 X)- 0. 3. Let d d. 4. Let d = ydy 1-x (3- 2y)' If 1du = (Ax2? + BxP + Cx + &c~,) (ax + b) we may place it under the form - Ax"'dx + Bx'dx du z= +- + C., (ax + b)n (ax + b) ) and may then integrate each fraction as above, if m, p, q, &c., are entire and positive. 132. To complete each integral as determined by the preceding rules, we have added a constant quantity C. If in the particular case under consideration, we happen to know what the integral must be for a particular value of the variable, this constant can be determined. Thus, if Xdx = X' C............(1), X' representing the function of x obtained at once by the application of the rules for integration; and we know the integral must reduce to N when x = a, we have N - X'= - C=, C C - - X'.=a. In general, however, this constant is entirely arbitrary, since 1 84 INTEGRAL CALCULUS. rwhatever value be assigned to it, it will disappear by differentiation, Art. (14). This arbitrary nature of the constant enables us to cause the integral to fulfil any reasonable condition. Thus if in equation (1), it be required that the integral reduce to the particular expression M, when x -= a; we may determine the value which must be assigned to C, by writing M for fXdx, and substituting a for x in the function X. Calling the result of this substitution A, the equation reduces to M - A + C; whence = M- A, and fXdx X' + M -............(2), which will fulfil the required condition. If TM = 0 C -- A and,fXdx X'-A. The integral fXdx = X' - C before any particular value has been assigned to C, is called a complete, or indefinite integral. After a particular value has been assigned to C, as in equation (2), it is called a particular integral; and if in this particular inte'gral, a particular value be given to x, the result is called a definite integral. We should thus have, when x = b, fXdx = B + M- A............(3), B representing X',=b. That value of the variable which causes the integral to reduce to 0 is called the origin of the integral; and in every particular integral this origin may be determined by placing the integral equal to 0, and deducing the value of the variable from the resulting equation. If in (1) we make x = a, and then x = b, we have INTEGRAL CALCULUS. 185 S(Xzd)o = A + C, f(Xdx).,b = + C, whence by subtraction, J (Xdx), - J(Xdx)=, = B A. This is the integral taken between the limits a and b, and is usually written a Xdx == B - A, the limit corresponding to the subtractive integral being placed below. Tf a, b c........., 1, be several increasing values of x, and we have J Xdx = Al' Xdx = B'........ Xdx = K'; then evidently dx- A' + B' C'...... K Ix ample. f6 2dx = 2x +- C; ts a complete or indefinite integral. If it be required that this reduce to 4, when x 1, we have 4- = 2+ C, C 2, and f 6xdx = 2x3 2, the particular integral. 24 186 INTEGRAL CALCULUS. For the integral between the limits x = 0 and x 3, f(62dx),o = 2, f(6x2d)3 56; hence f 6x'dx = 54. The origin of the particular integral is obtained by placing 2x3- 2 0; whence x3 =- 1, x= --. INTEGRATION OF THE DIFFERENTIALS OF CIRCULAR ARCS. 133. 1. In article (42), we have found du dx =- d in which u = sin x, the radius of the circle being unity; then x= f.= sin-u ~t C. Expressions of a similar form my be readily integrated by the Expressions of a similar forlm may be oradily integzated by the aid of an auxiliary variable. 1. Let d = d......(. 0). Va2 -u2 MtalH u = az then du = adz,.\/a2 _ 2- al2 -' Sub itituting these values in (1), we have INTEGRAL CALCULUS. 187 dx; x sin-lz = sin-,l - s C. 1-z2 a 3dx 2. Let dy. V/2 - 2 This may be integrated directly, by placing x == /.z, as in the last example, or by a simple comparison with it, by placing V2 for a. Thus y dx - 3 sin-Lx + C. 2 /=X2 2dx 3. Let dy == ~9 /9 - 38 This should first be placed under the form dy -= _ V/33 - 3 2 II. In article (42), we have also dx = d -~ =- dcos-'u; whence C di X -= cos- C.2 2du dx d - ~-, x = cos — + GC. Va _ it a if dx _ 2du v4 -' 188 INTEGRAL CALCULUS. x du = x- 2- / v u_ = 2 cos-'- + C, V4 - 2 by placing 4 for a2. III. We have also dx _-d _ d ver-siin'u /2u ~ u2 whence x =-. = ver-sin-'u + C. du 1. If dx -- 2au~ - U2 place u az, then du =adz, and ^du d I dz X V, dJ V= g -~z = ver-sin-s *V2au U2 l _~ z-u = ver-sin-l - + C. a 2. If dx 3du_ x = 3 ver-sin-' + 0'v4iu- - te-2 2 IV. We have also dx = du d tang-;'u 1+-u'2 'INTEGRAL CALCULUS. 189 whence x- f -- _ tang-'u + C. J1 i+ u 1. If dx -d a2 -- U2 a2 +.' make u = ar, then du =adz, and d F1 dz 1 1 z x - d = tang-l 2= tang — +C a2+ u J +Z2 a a a d'u 3 - U 2. If dx = x- - tang + C. 2+u V2 2 ~-~ 7~ 2dx 3 Let dy= 2 2 -3x2 du If we multiply and divide the faction 2 e a 1 a id - a2 a:2 + u2 whence a =2 a2du 12 a2 l Js -T ~2 -1tang^ 4- GC Art. (42), the radius being a; and in a similar way, all the above expressions may be transformed. INTEGRATION OF RATIONAL FRACTIONS. i34. Every rational fraction which is the differential of a 190 INTEGRAL CALCULUS. function of x, will appear as a particular case of the general form,, (Ax' + Bx'- q- xZm-2 + &c.)dx A'x" d- B'xn-1 + C'x"-I2 + &c. in which m and n are whole numbers and positive. If m be greater than n, the numerator may be divided by the denominator, and the division continued until the greatest exponent of x in the remainder is one less than in the denominator; the quotient will then consist of an entire and rational part, plus the remainder divided by the denominator, and may be written Axm + Bxm-t +~ &c.)dx _ d + (A"xn-l + B/ixn-2 + &c.)dx Alx" - B'x"-' + &c. A'/x + Bn]-1 + &c. and the integral of the primitive fraction will be the sum of tle integrals of the two parts. It will be necessary then to explain only the manner of integrating the second part, or those rational fractions in which the greatest exponent of the variable in the numerator is at least one less than in the denominator. First, suppose the denominator to be divided into its simple factors of the first degree, and let them be represented by x-a, x-b, -c, &c. There will be four different cases, each of which will require a:eparate discussion. 1. When thefactors are real and unequal: 2. When they are real and equal: 3. When they are imaginary and no two alike 4. Whfaen they are imaginary and alike, two and two, INTEGRAL CALCULUS. 19. 135. 1. As an example of the first case, let us take the firaction (cix + c)dx zX2 b2 The two factors of the denominator are x + b, and x b; -then (ax + c)dx (ax + c)dx w2- b' (x + b)(x- b) Place ax -t c A A' x2 - b2 -- b x- b....... A and A' being constants to be determined. For the purpose of determining them, clear the equation of its denominators; then ax + c = Ax -Ab + A' + A'b. By placing the coefficients of the like powers of x, in the two members, equal to each other, we have a = A -- A' c - A'b-Ab A ab-c' ab -- c 2b 2b Substituting these values in (1), multiplying by dx, and prefixing the signf, we have I(ax- - c)dx ab - c dx ab c+- c dx x2 - b2 2b x - b 2b J - b ab - c ( )+ ab -f- c 2 —~ I (x + b) + — C0. 2b 2b 192 INTEGRAL CALCULUS. The method pursued above indicates the following rule for all similar expressions. Place the primitive fraction (omitting the cdiferential of the variable), equal to the sum of as many partial fractions as there are factors of the first degree in its denominator; the numerators of these fractions being constants to be determnined, tand the denominators the several factors of the original denominator; clear the resulting equation of denozmio ators, eqzuate the coeicient-s of the like powers of the variable in the two members, and thence determine the conslants; then multiply each partial fraction by the dif erential of the variable, and take the sum of their integrals as in case II., article (129). 2. Integrate the expression (OX -1) dz X3 _ X The factors of the denominator are, x + 1, - 1, and x; then 3x -1 A A' A" + + _x-~x x+1 x - - x Clearing of denominators, 38" - I = Ax2 -Ax + A'lx + A'x + A"x -l A; whence 3 = A + A' + A", 0 A + A 1 A and A = I = A' = A" INTEGRAL CALCULUS. 193 Then d" (3^~i) r dx - + X X d$=<+j. dS-.-1= (x -+ 1) + 1(x - 1) + 1x = (X - x) + C, as may be seen at once, since the numerator of the given differential is the exact differential of the denominator. 3. Integrate the expression (1 - y)d 2 2y -2 Placing the cenominator equal to 0, we have Y - 2y - 2 0; whence y = I V/3, and the corresponding factors are y- (1 — / 3), y (1- -1/3), or y- n and y — n. Finally, f (1 - y)dy ) n) -i. Jy -_ 2y 2 n- m 3) -nM^ 4. integrate (2x + 3)dx;3 _- 2 - 2X 5. Integrate (3- 1)dx 2 -- 4 25 194 INTEGRAL CALCULUS. 136 II. In the second case it may be remarked, that if all the factors of the denominator are equal, the fraction will take the form (Axn-1 + Bx"-2 + &c.,) ~ - a)~ dx, which may be integrated as in article (131). Ae need then only consider the case where a portion of the factors are equal. The rule of the preceding article is not applicable here, as will be seen by taking the expression adx (x - b)( - c)' in which two of the factors are equal to x - b. By an application of the rule referred to, we should have a A A/ A (x - b^)2(x,_ ) - b x - b x c A A+A A" B A" = + = -1- -- - x b x —c x -b x-c since A + A' must be regardec as a single constant. If this equation be cleared of denominators, and the coefficients of the like powers of x in the two members placed equal to each other, we shall evidently form three independent equations, with only two unknown quantities, B and A". We obviate this difficulty by writing, for the equal factors, the B B' two fractions B B, and thus have (.x- b)y x - b aC B B' A - --- X -'-~ (x - b)2(x - ) (x - b)2 x - b x - c which, being cleared of denominators, gives INTEGRAL CALCULUS. 195 a = B(x - c) + B'(x - b)(x - c) + A( - b)2; whence B' + A = 0, B -B'c-'b - Ab= 0 B'bc - Bc+Ab2 =a, three equations with three unknown quantities, which can then be determined. And. in general if there be n equal factors, we should write n partial fractions of the form B B' B(~-1^' (x - b)" (X-b) -- b' the numerators of which are constants, and the denominators the different powers of the equal faetor from the nth down to the first power. After B, B', &c. are determined, each partial fraction, being first multiplied by the differential of the variable, will be integrated as in article (129). Examples. (2 -+ x)dx 1. Integrate ( - (x - I)(~x- 2) Place 2 + x B B' A (x - )( 2) (x - 1) x - I x — 2 Clearing of denominators, and equating the coefficients of the like powers of x, we have 0 =- B +A, 1 = - B- 3B' - 2A, 2 =-2B + 2B' + A, B - 3, B' -4, A = 4; and finally 196 INTEGRAL CALCULUS. (2 +x)dx 3 - -4 (X-1) + 41 ( — 2) + G (x - 1) - 2) - - -1 xdx 2. Integrate xd X _ + _ X ^ i If there are different sets of equal factors, partial fractions must be written for each set; thus, 2 A A' B B' (x - 1)2(X +1)2 (x + ) (X + 1)2 +1 137. III. We know from the general theory of equations, that imaginary roots are found only in pairs, and that for each pair we must have a factor of the second degree, of such a value, that when placed equal to 0, it will give the imaginary roots. Each pair of roots will always appear as a particular case of the general form x, = a - b............(1), and the corresponding factor of the second degree will be,z -_ 2ax + +b = [x-(a + -b')] [ x-(a — b)]. By a comparison of the imaginary factors, in any given case, with these general values, we determine the corresponding values of a and b. Thus, if the factor of the second degree be - 2x + 5, we place it equal to 0, and find the two roots x- 1I V — 4; INTEGRAL CALCULUS. 197 whence, by comparison, a = 1, b2 = 4, b = 2. Now, in the third case, for each pair of imaginary factors, let a partial fraction be written, of the form Mx + N _ Mx + N x2-2ax +a + b2 ( - a)2 + ^b By clearing of denominators, &c., as in the preceding articles, M and N may be determined. We shall have then to integrate the expression (Mx + N)dx (x - a)2 + b2 For this purpose, make x - a = z, then x = z+- a, dx = dz. Substituting these, the original expression becomes (Mz + Ma + N)dZ. z2 + b2 or by making Ma + N = P, and dividing the expression into two parts, Mzdz Pdz z2 + b,2 - b2 The first part may be integrated as in case II., Art. (129)o Thus, C Mzdz M + I - 1(+ b2) =MZV/2 + b2 MIz(-a)2+b2 The integral of the second part is 198 INTEGRAL CALCULUS. pP dz tang-~ z PJ;2 --- = tang-......Art. (133), case IV., or by substituting the values of P and z, - N +A- M tanog- L b J2 + b b; and finally r (Mx + N)dx (r g)c+ b2 (x a)2 + bb x + a, - a)' + tag a) +......() b b Take the particular example (X - 1)dx x2 + x2 + 2x The factors of the denominator are x and x2 + x + 2, the last being the product of the two factors corresponding to the imaginary roots X - — 7 ) -k 2 4 which compared with (1), give a = -, b, b =;/-. Place x- I A Mx+ N x3 - -x + 2 X X 2 x + 2+ 2 Clearing of denominators &c., we find INTEGRAL CALCULUS. 199 1- 1 __1 3 A ^ -, N2 2 2 Substituting these values of M, N, a and b, in formula (2), obAdx 1 dx I serving that f = f -- l - - 2lx and redux 2 x 2 cing, we have (X - I)dx Ix+l V +x I JG + 2 2 2 5 x+ 2 _. t+ tang-1 + 2) + C. 2/7 ^2 V7 138. IV. In the fourth case, where there are several imaginary factors, alile two and two; those of each pair multiplied together will give the same factor of the second degree, and if there be p such pairs, the denominator will contain a factor of the form (' - 2ax + a2 + 62) For this, we write p partial fractions; thus Mx + 1 ~'llx + N' M()'x- N(p-' [( - a) +- b ]f' [(a +- ~a) + b' ( - + Clearing of denominators &c. the values of M, N, M', N', &c. may be determined as before, and since the several partial fractions, after multiplying by dx, are all of the same form, we have only to explain the mode of integrating any one of them except the last, which is to be integrated as in the preceding article. Take the first (Mx N) dx [(X - a)2 *.+ b+' 200 INTEGRAL CALCULUS. and ma.ke x - a = z; the fraction then becomes (Mz + Ma + N)dz (z2 + ^2)p or placing RMa + N = P, Mzdz Pdz (Z + 2). + (Za + b2)P The first part is integrated as in case I., Art. (129). Thus f Mclz __ M(z2 + + _) — _ M__ J ( + ^)P (- p + 1)2 2(1 - p) (2 + b2)-' By mieans of a formula hereafter to be determined, [Formula D, Art. (151)], we shall find " Pdz dz C' z 9 (z + Y)P ) C + b2 =f(az) + tang-; then (Mz+ P)d M C' x J ~ )P^ ^'2b +^f(z) + b tang ~+ C, ( -2- b2)P 2(1 -p)(z2+ b2)P-L () + b tang - b+ C, after which, substituting the value of z, we shall obtain the complete integral of the primitive expression. 139. By a review of the preceding discussion, it will be seen that all differentials which are rational fractions can be integrated; provided the factors of the denominator can be discovered; and that the integrals will depend upon one or more of the four forms. d-r Id.2( d a d a' 1 x + a J td J (x2+a2)P J / 2 + 2 INTEGRAL CALCULUS. 201 INTEGRATION BY PARTS. 140. In article (19), we have found duv = tudv - vdu; whence uv = f udv +fvdu, and fJdv = uv -f v- u d.....(1); from which we see, that the integral of udv can be obtained, whenever we are able to integrate vdu. This method of integrating udv is called, cIntegration by parts. Examples. 1. Integrate the expression xadxV/a - x. This may be divided into the two factors, x2 and a' and xdxa-/a - x Place x = u and xdx Va - x = dv; then _ (G 2a - du = 2xdx, v =fxdxV/ a- x Substituting these in formula (1), we have fi. (- X) 2 x (a -'I { ) 2 263 -ud + j- xLcdy; 26 G02 INTEGRAL CALCULUS. and finally f x2dx (a- - 2 - (a - 2 + C. 3 5 (1 -- x2)2dx 2. Integrate (I) - dx Place (1.- 2)2 = u and 2 = dv; then (l _-2) l - sin-'x + C. J x_2_ x 3. Integrate dxVl -. Place V1 -- x = u, and dx dv; we then have by formula (1), X dx fdx V I-. -2 1 - 2 X+ J 2.......-(2). If we multiply dx V1 - l2 by -- x -e ~/1 x may write dx v I - oxdx fdx Vl s2 -~ ~- fd ~ f xd...........(2 J /1 - J Vq/1 - Adding equations (2) and (3), we have INTEGRAL CALCULUS. 203 __________ ________ -dx 2fdx /1 2= V 1- + ~ 7 2 2 Jb~tz 3/1. $2 25,8/ -1- 2 _l_ Sill- l 1T } + + 4. Integrate Xd. (a 2 _ X)2j INTEGRATION OF CERTAIN IRRATIONAL DIFFERENTIALS. 141. In the preceding articles, rules have been given, by which every rational differential may be integrated, except the case referred to in article (139). It may then be taken for granted, that, in general, every irrational dierential which can be made rational in terms of a new variable, can also be integrated. Let ax_ dx nm p bx la + cxF be a differential, the irrational parts of which are monomials. Make h x z kq; then Zx zh m P n = zmkg, nP dx = knpknq-l dr x z'k- zn"', dx = knqz5~q1 dz. These values substituted in the given expression, evidently make it rational in terms of z and dz. It may then be integrated, after 204 INTEGRAL CALCULUS. which the value of z in terms of x must be substituted. We may then enunciate the following rule for the integration of expressions of this kind. For the variable, substitute a new one, with an exponent equal to the least common multiple of the indices of the radicals; then integrate by the known rules, and substitute in the result the value of the new variable in terms of the primitive. Examples. 2 2x2 3x2 i. Let du 2 3 d^.....(I) 5xo The least common multiple of the denominators or indices being 6, we place x = zs, then dx = 6zSdz, z = x, Substituting in (1), we have du = (2z- - 3 Z)6 zd_ = 2 z7dz _ ~18 dz 5z 5 5 and integrating, 12 18is 3 A 2 3 u __ z =_x3 _ ~x~ + C. 40 45 10 5 2. Let du - 3. 3Let du = ade 2 x - x% b - cx 2xa-S3 bcvx 142. If the irrational parts are all of the form (a + bx) the expression may be made rational in terms of z, by placing INTEGRAL CALCULUS. 205 a + bx = Zr, r being the least common multiple of the indices of the radicals. We shall thus have z a rzg-dz X dx, b b which substituted in the primitive expression, with the value of a + bx, will evidently give a rational result. Take the examples; 1. du dx (1 + ) + (1 + x)~ Place + x = z; then dx = 2zd, z=(1 + x). These values substituted in (1), give 2zdz 2dz u.+ Z - + 2; whence u = 2f z = 2 tang-z = 2 tang-' (1 + x) + C. 2. Integrate the expression xdx du (1 - x) + ( - ) 143. Differentials of the form Xdxa + bx in' + bZ' X being a rational function of x may be made rational' by 306 IINTEGRAL CALCULUS. placing a'- =,z deducing the values of x and dx, and a/ -I- b substituting them. For example, let &u = - x& k -h............ (1)' Place + then x dx 6z llac e 21 x — " Z3, ten X - 3 -— ~=(z3 + 1)2 These values in(1), give (3 1)6z4dz du^ — = (3 + 1)3 which is rational. 144. Every radical of the form -Va + bx ~:f cx2 can be written thus, \/~ - + ~Xq2=i ^ba+3~xX after making a - a, and f. c c To render rational a differential, the only irrational part of which is a radical of the above form, it will then only be necessary to find rational values for x, dx, and V/o + 3x ~ x, in terms of a new variable and its differential. I. Take the case in which the sign of x2 is +, and place INTEGRAL CALCULUS. 20'7 aU + fx -+i = z(- X...........(1). Squaring both members, we have a - /3 = Z2 - 2zx; whence Z2 ~ x=............ (2)' y iereating his value of, e otin By differentiating this value of x, we obtain 2(z2 + 3Z + I )dz d( + )2)............(3) and by substituting the value of x in the second member of (1), w2 -t Z + cc + e - x = -- +__............(4). These values of x, dx, and -c/a + 3x -- x2, substituted in the primitive differential, will evidently give a rational expression in z and dz. After integrating this, the value of z, taken from (1), must be substituted..Examples. dx dx!. Let du - [ Example. Let du = xdx(a +- bxz)3 216 INTEGRAL CALCULUS. m -- = 1, n 3, p 1, q 3, m + P 1. n g These values in equation (4) give xdx(a + bx)3 _ z3dz (3 b a V a ) in which z3 = a-3 + b. From what precedes, we see that every binomial differential of the proposed form can be integrated: if the ezponent of the parenthesis is a whole number; if the exponent of the variable without the parenthesis plus unity, divided by the exponent of the variable within, is a whole number; or if this quotient, plus the exponent of the parenthesis, is a whole number. 148. Let us now write p for P, and then divide the expression q x:-ldx(a + bxz)P =x xn"xddx(a + bx ), into the two factors xm- t= u and xn-ldx(a +- bxn)P - dv; whence du = (m- n)x.-. l'dx, v = ( + ^)b.......Art. (129). (p + l)nb Substituting these values in the formula f ud = uv -fvdu............ Art. (140), INTEGRAL CALCULUrS. 2 1 and making (a -+- bx") = X, we have X-nX X+ (n _- n) fi (.. ) (p + l)nb (p + 1)nb But since XP+- = XX XP(a + bf" ) = aXP + bx"X:, f:.-n-l'dxX'1 af xZ'?-.-l.xXP +4 b1f x-ldxX". Substituting this value in (1), and clearing of denominators, (p + l)nbfx-mldxL:P = 1nXp-~-l - (m - n) [afx -x-'dxX + bxf x"'dxX3j transposing, &c., we obtain /m-p _ X"rn-nnl - a (72) - 22) f xm-_"-' d,-XP -~dxX =x ".. + - a- n)x -...... b(pn + nm) By a single application of this formula we cause f'lldxXP to depend upon f t.-ldzP~ in which the exponent of the variable w ithout the parenthesis isdiminished by the exponent of the variable within. By an application of the same formula to f xt". XldxX, it may be made to depend upon f x21-2-l'dxX, and finally, by repeated applications, f Jxn-dxXP. will depend upon the expreesion a (m - in) f ~ )x."r-ldxXP, in which - represents the number of times m will contain n. Lf m. is an exact multiple of n, then m - rn = 0, the term containing 28 2183 INTEGRAL CALCULUS. the expression to be integrated disappears, and the integration is complete. If pn a + m = 0, the second member of the formula becomes infinite, and it fails to answer the purpose; but in this case + - 0, which, substituted in equation (4) of article (147), n gives an expression which may at once be integrated. 149. We may also write l -;fdxXp fx1-dxX X^ afx^-ldxXP-l + bfxi^-X., If now in formula A we change m into n +- n, and p into p - 1, we have,f Xvtn-'dxXP =X- -- amf xl"-'dxXP-' b(zp + n) Substituting this value in the preceding equation, and reducing, we obtain fxni-_Ix, _ v'"XP + praf x'-ldxXP-.......... 7pn + -? by which the primitive expression is made to depend upon another, in which the exponent of the parenthesis is one less than before. By repeated applications, this exponent may be reduced to a fraction less than unity, either positive or negative. 150. The use of the preceding formulas may be illustrated by the example fxl2d;x(a + bxl)!. INTEGRAL CALCULUS. 219 Place a + bx" = X, m = 3, n = 2, p, then from formula A^ 5 3 JfxdxX'3 ZX2 - acf dXXv 6b Applying formula M to the expression JdxX~ after making n = 1, n =2, p = 3, we have A W j dxX- =.*X2- + 3af dxX2 f' dxX4 and by another application xX2 + aj~I f dxX- - Substituting these values, we have finally 5 3 1 xX axX2 a-x X a Xr d fxdxXX ~- 6b 24b 16b 16b dx dx The expression: may be integrated as in Art. (144). 151. If in the primitive expressions, mn and p are negative, the effect of the application of formulas A and Wa would evidently be to increase them numerically. Other formulas are then required. 1. From i by transposition and reduction, we find 220 INTEGRAL CALCULUS. Xm-n —dZ' =iX- p+' h -( + ~np)f.l-'d.xX (m?- 2) If in this we change m into - m +- n, we have f n —'dXP; -'nXp+l lZ 2, -n+ p)fx-m+nl-1dXrX' am " by the application of which, -- m will be numerically diminished by the number of units in n. 2. From S, by transposition and reduction, we find f sl-1'dxX' -- xX_ X + (12 + n).f x/-ld,'dXP If in this we change p into - p + 1, we obtain /f X _....' IDx X-, ~+' aX- ~ xX- X+l — (m ~ - npfx".'d.X.. a.(p) - I) in which, the exponent of X is numerically one less than in the primitive expression. If p - = 0, the second member becomes infinite, but in this case p I, and the primitive expression reduces to a rational fractiono 152. Let us illustrate the use of these formulas by the example x-2dx(2 - x)- 2 Making in, =, a= 2, b = -, na 2, p = - 2, we have _ X-X- 23 f aT^x(2 -x'- = b= dX- 2T -............(1), INTEGRAL CALCULUS. 221 By formula e, after making m = 1, n = 2, a = 2, b - 1, p 2 -, we have f dxX2- - - X Making the proper substitutions in (1), we obtain finally 2d(2 _ 2) - 2 + - + C in which X = 2 x2. 153. By the aid of formula ZD we are now able to integrate the expression dz d dz(z2 + b2)-P....r.... t. (18). (Z + b2)p By making m 1, x=z, a = b2, b = 1, n -2, we cause J(2 d_2)_ to depend upon the integration of another expression in which the exponent is one less, and by repeated applications, we shall find that the integral will depend upon tiexpression f b2 1 tang- z + C. J W + T & 6 154. For the expression x xqd J 2cx - x 222 INTEGRAL CALCULUS. we may write /xdx(2cx - x2) = fx2-'dxc(2c x), to which applying formula &, after making m = + -q - -, a = 2c, b = -1, p --, n= 1, 2 2 2 and recollecting that x-2 ^x~l x~, and x q- x- x 2, we obtain f xqdx _ xq-' V'2cx - x q 2 +a /2 -( xy l^f Z 1..... By repeated applications of this formula, when q is a whole number, we make the primitive expression depend upon / 2 = _ver-sin1 — + C......Art. (133). -J2cx-X c INTEGRATION BY SERIES. 155. If it be required to integrate the expression Xdx, X being any function of -x; it is often convenient and useful to develope X into a series by any of the known methods, generally by the binomial formula; and then, after multiplying by dx, to integrate each term separately. This is called integrating by series; since we thus obtain a series equal to the integral of the given expression, from which, when the series is converging, we can for particular values of the variable deduce the approximate value of the integral, INTEGRAL CALCULUS. 223 i. Let us take the example d d1 L_ dx(l + x)-l By the binomial formula, we have (1 + x)-' = i -- + x2 - x3 + &c. Multiplying by dx, and prefixing the signf, J d_. = f (dx - xd + x2dx -xdx + &c.); whence I __ - x ~ + _ t + &c. - C, J + x 2 3.4 or since &:i(l +x), I +x 2 X3 X4 I (l -4 - + - + &c. 2 3 4 But when x: 0 the first member becomes 1 (1) 0; hence C 0 and X2 $3 94 (. 2 + X4 = x - + &c A......Art. (38). 2 3 4 2. Let du - x(1 -- xI)2dx. By the binomial formula we have 2.54,. 4 INTEGRAL CALCULUS. (1- ~1 -22 2 4 X6 2 8 16 Mifultiplymg each term by x-adx, &c. f ~x(1 - -)2dx 2 / 1 &C G.. 2 8 w/hence.?~(l -x =' ) x_ ix ~,- &c...... G 3'7 44 3. Let du = axdx. In article (36), we have found kx kx c C3 kX3 a = +- ~ ++ 1 4 &C. - 1 1,2 1.2.3 hence fka. k2x k13XA /f ax == x + -t- + -- + &c.......+ C, 2 0 24 n which = lat. If a e, then k = le==1, and X2 X3 X4 2 6 24 /fed - x + _ _ + _ + x. + de......+ 0. dx clx 4. Let du __ _ V/^- - Vxvl-x Make V/x. = u; then dx = 2 /xdu, and dx 2du xl X. ~ INTEGRAL CALCULUS. 225 which may be readily integrated, and we shall obtain dx x 3x2 __dr__- 2sin 2 V/(1+ - +. ~ &c....)+C. I - x/2.3 2.4.5 5. Let du = dx -/~2ax - X; dx --- ~ x2 6. Let du -- /1 -- -x2 Developiung V /1 (1 e l 2N 2), we have /1 l _ _ ele, _ 2 11e4' eV XIi e'Ix1/_1 2 e1 --- &,.; 2 2 4 hence fdxVi~e/^:= 1-,-!f'2~e - 1 1ie,4x4 0.) Ja/ ~ —~ ~1/ ~~-e'. _ — e'V x-&c. -_/ l1-: ^\ 2 2 4 J Vi- _X2 After the multiplication, each term of the second member will be of the form A ~, which by formula as _1 -' may be miade to depend upon' d- = SilIl 7. Let du d ) 2z- d b V(2cx: )b - x) V2cx- VTb -- x 1 ( If we develope (b x)- and multiply b-x 29 22 6 INTEGRAL CALCULUS. dx Axqdx by d, each term will be of the form Axd v2cx~ - x V'2 cx - whiich may be reduced and integrated as in the preceding article. 156. By the application of the formula for integration by parts, Art. (140), to the expression Xdx, we obtain f Xdx - Xx -f xdX............(1), and then to xdX, &c. Pr dX, ~2 dX f X d2X fxdX =..dx - ~.........(2), f 2 dc2X fd2X 2Xc2;x x3 d2 s3 d3X dx 2 dx J 2 dx' d J 2 dx J dx2 2 2.3 d2 J 2.3 Od'........... &C. Substituting in successionl tile values above deduced, eqluation (1.) will become d,2 d72X x3 fXdx I _x d5, - &C. f Xdz = X__z _ - - _ —_ -_ &c., dxz 1.2 c 2 1.2.3 a series, expressing the integral of Xd in terms of X, and its differential coefficients; which has received the name of its distinguished discoverer, John Bernouilli. 157. If in the integral f Xdc = f(x) = u, we male x = x + h, we have INTEGRAL CALCULUS. 22 (f Xdx) =, h =f(+ ~ h) - U; and by Taylor's formula,,'-u =du dc'u hW u'~ u = __A + + &c............. (1) dx dx' 1.2 But since f Xdxv = u, Xdx = dzc, = X, dx du dX d'u d'X dx dx dx' dx These values substituted in (1) give,iY h, dX1~ h a tu' u = Xh + dX + 2X L + &c. dx 1.2 d,2 1.2.3 If in this series we make x = a, h = b - a, and denote by A, A' A &c, A"hat X, ~, ~ &c. hedx dX &C. become under this supposition, it is plain that what u becomes wilt represent the value of the integral when x = a; what iu' becomes, its value when x -a + -a = b; then what u' - u becomes, will be the value of the integral between the limits x =a, and x = b; whence Xd= A( - a) + 1 - b -)2 _ (b a)3 - &c., 1.2 1.2.3 a series from which the approximate value of a definite integral may be obtained. If b - a is so large, that the series does not converge, or does not converge rapidly enough, then let it be divided into n equal parts, so that 228 INTEG-RAL CALCULUS. b - a = n, and talke the value, first between the limits aan and a, then between a +- a and a + 2a, &c., and suppose the results to be Bo+B' ~ B+ + &c., 1.2 1.2.3 C +- C' + + &c.,.(2), ~t.,~ 1.2.3 Da + D'2 D &c., 1.2 1.2.3 &c.; then by article (132) we have Xd - (B + C + D + &c.)a (B + C(' + &c.) _L +.la 1.2 &c...........(3), and as a is arbitrary, the separate series (2) [and of course the final series (3)] may be made to converge as rapidly as we please. INTEGRATION OF DIFFERENTIALS CONTAINING TRANSCENDENTAL QUANTITIES. 158. But few of these differentials admit of exact integrals. We can, however, by the aid of formulas previously deduced, obtain, by series, their approximate integrals. By the examination of a few expressions, we will endeavour, as far as possible, to indicate to the pupil the general method to be pursued, and then leave to his ingenuity and industry, its application to the different cases with which he may meet. INTEGRAL CALCULUS. 229 159. Take first the expression Xa'dx, in which X is an algebraic function of z. If we divide it into the two factors X and a"dx, and recollect that axladx = da............Art. (36), whence da," -, r a ad.x __a and f adx -- ~; la la we shall have from the formula for integration by parts f Xadx X = - fadx - XX............ la J cla If now we take the successive differenltials of X, andl place dX = X'dx, dX' = X'dx, dX" = X"'dx, &c., we obtain adlX _ X' a ra f dX J la ) (l)a J (la)2 radX, x'i r _fa dX"f J (1a ~)2 J (lay) &c. These values in equation (1) give JXa-dx 2 aO X _......__ r (2) ( te la)+ {(la) tl) ( a) i. If the function X is of such a nature that one of its differential 2,80 IrNTEGRAL CALCULUS. coefficients X", X ct&. is constant, the differential of this will be 0, and the corresponding term a rdXn' J (a)+' The integral will then be exact. The expression xna'dx, admits of an exact integral when n is entire and positive. If n be fractional or negative, we write for ca its development, Art. (36), and then integrate as in Art. (155). 160. Take now the expression X(x)"dx. If we divide it into the two factors Xdx = dv, and (x)"= u; whence fXdzx = v = X', du = nl(Z)n-lI z, and then substitute in the formula of Art. (140), we have fX(Zlx).)dx X )(l) - n fX'(lx)n-ldx -....(1). By this the integral of the primitive expression is made to depend upon the integral of another similar one, in which the exponent of (Ix) is one less than at first. INTEG-RAL CALCULUS. 231 If the n be entire and positive, after repeated applications of the formula,'the exponent of (1x) will become 0, and the expression upon which the integral depends, algebraic. For a particular case, let mz+1 X - Inl then /xSmdZ = X m -r I and this in (1) will give fx'!(L)ndc - ( (_x) - x'-(f )"-' x)......(2). r n —I-i r.~1 If in this we substitute for 2, in succession n - 1, n 2, n- 3, &c., we have ______ - ( - 1 ) —............& c. These values in (2) will give a general formula, in which, if n be positive and entire, the last term will be n(n - 1)...2.1 fx(l)od (n (m + 1)"1 (On -+ 1)21 We shall therefore have .232 s XINTEGRAL CALCULUS. J'x,(Zi7.)ndx-_ Rx_[(ixv)1K D- ] C...(3). -' z'" ~(Z)u -' i "- -.W in-+ 1 nr -+ I (ni + 1)" The sign of the last term will be plus when n is even, and minus when n is odd. If nm =1 and n 1= 17 we have f xlxdx' 1 Lx + C. 2 If m 0O and n 1, we have flxdx =. a:(l x 1) - C. If m - 1, the second member of (3) becomes infinite. In this case the differential becomes (Ix)n dx Making lx = z, we have _ - dz, and. X dx z'+ (lx)n+L f(x) - = z"d fz = - - C, x n- +1 I 1 which is true for all values of n, except when n - 1. In ths case the expression becomes dx xlx dr. MayHing lx = z, we have - - d, andl Mk(l dz a xd I xlx J 7 INTEGRAL CALCULUS. 233 161. Take now the expression Xdx sin-'x. Place Xdx - dv, anc sin-x = u, then f Xdx = v = X and du = -. (1 - ) Substi-tuting in tlhe formula of Art. (140), we have JfXdx sin-x = X' sin-l -f X'd (1 - 2) Thus the integral of the primitive expression is made to depend X'dx upon the integral of the algebraic expression X.. (1 - X2) Let X X",'thlen f Xdx = fx"dx - 1- X', n - n+ I and we have fxSdz sin-'x --. y' i X C1x&dx lf "dx sinx Sil= ~_ s-x d n+-1 n+ 1 - By thle application of formula'1 or C, when n is entire, the last term may be reduced, and then integrated; except when n = -~ 1, in which case the expression becomes dsix ll-I, ~30Cx 30 234 INTEGRAL CALCULUS. which can only be integrated by series. In the same way, like expressions may be found for JfXdx cos-'x, fXdx tang-x, &c. 162. By article (41) we have d sin nx = ndx cos nx, d cos nx= - ndx sin nx; hence f cos nx sin nx dz sin nx -- ~ os 7 _, f dx cosnx = o n n In the expression dx sin2r, 1 cos 2x we can place for sin'x, its value,, and then have 2 2 f.g dx cos 2xdx x I f dx sin" x =- I sin 2x + C J2 J 2 2 4 and in general the integral of similar expressions containing any power of either the sine or cosine of x, can be obtained by first substituting the value of the power in terms of the double, triple, &c. arc, as determined in trigonometry. The expressions dx sinmx, dx cosmx, when gm is entire, may also be integrated as follows. Make sin x = z, then x si-'z dx = d ( I - (1 - e)2 INTEGRAL CALCULUS. 235 whence f dx sinx -- f Zdz (1 -2) This expression, by repeated applications of formula a or d, may be made to depend upon dz r zdz f (I-, or - (1- _ ") (1 -- z2) In the expression dx tang"x, place tang x z= then A = dz dx = +z fdx tang' - /f I which is a rational fraction. Examples. Integrate 1. du dx sin3x. 2. du= d a cosax 8. du dx 4. du = dx tang2x. sin x 163. In the general expression 236 INTEGRAL CALCULUSo dx sin"' x cos"x, we may place sillx = Z then cos x = (1- d2) dz e (1-Z ) and finally n~L fdx sinmx cos"x = S z"dz(l 2)-, which may be reduced by formulas ~i, ), Q, and ), and in some cases integrated,, as in the example du = dx sinsx cos2x; whence u =fS dz(l - Z) INTEGRATION OF DIFFERENTIALS OF THE HIGHER ORDERS. 164. By an application of the rules previously demonstrated, we may readily obtain the primitive function, from which differentials, containing a single variable, and of a higher order than the first, may have been derived. Let there be-the differential dn=t f(x)dx", Dividing by dx"', we have d - d2' (Z), A INTEGRAL CALCULUS. 2.37 and since dx-' is a constant, this may be written, Art. (24), d (du —) =f()dx. Integrating both members, we have dn-1 d - - f ff(x)dx = f'(x) + C. After multiplying both members of this equation by dx, it may be written d f () J)dx + Cdx; and integrating as before, dn-2U,d_ (.) =f,(x) + Cx + C'; which by another transformation and integration, may be reduced one degree lower, and finally after n integrations, we shall obtain Cx"-n C'X"- it = F(x) + C n _ — +,(,_l. ) 1.2...(n - ) 1.2...(n - 2) The above operation may be indicated thus, u = f(x)dx"; the symbol f" indicating that n successive integrations are re" quireeL 238 IN'i'E(lRAL CALCULUS. Examples. i. Let d2u = ax2dxZ The required operation is indicated thus, u =f 2ax2dx2, and may be read, the double integral of ax2dx2. Let the expression, after dividing by dx, be written d=d _ d(d ax2dx whence by integration du ax 3 ax3 dz- + J+, du dx + CdxG ax 3 3 Integrating again, we obtain 41X4 u = - + Cx + C'. 12 2. If d3u = bdx, u = J bdxd, which is called a triple integral. We may write d3u f d( u b _ d_ ~ d )Z bdx; whence INTEGRAL CALCULUS. 239 cnu d 2 - bx + C, dx? and finally as in the last example 3 Let d'u d- - 4. Let d3u = /dx2 INTEGRATION OF PARTIAL DIFFERENTIALS. 165. HIitherto, we have explained the mode of integrating only the differentials of functions of a single variable. It yet remains to extend our rules to the integration of those which contain more than one variable. These differentials are either partial or total, Art. (49). When pa.rtial, they belong to one of tzo classes. I. Those obtained from the primitive function by differentiating with reference to one variable only. II. Those obtained by differentiating first with reference to one variable, and then with reference to another, &c., Art. (46). 166. The differentials of the first class may be expressed generally thus, d"%t f (x, y,, &c.) dx" du = f'(x,, z, &c.)dyn, &c., in which 2u is a function of x, y,, &c., and may evidently be obtained by successive integrations, precisely as in article (164); all the variables, except the one with reference to which the differentiation was made, being regarded as constant, and care being taken 2 4t) 0 INTEGR.AL CA LCULUS. ~to add, instead of constants, arbitrary functions of those variables which are regarded as constant during the integration. Examples. 1. Let d2u = bx2ydx', which, after dividing by dx, may be written d = bxclydx; whence J blx-yd.x -bx y 3 du = lbydxd + Ydx, 3 an d, f~2b yx2 -b = y + Yx + Y, 12 in which Y and Y' are arbitrary functions of y. 2. Let d3u = cxS2yz2dy3. 167. The differentials of the second class may be written generally thus, dm+l-'.u = f(x, y, z, &c.)dx'n dyn dz...... and the mode of integrating is plainly to integrate first, m times INTEGRAL CALCULUS. 241 with reference to x, then n times with reference to y, and so on until all the required integrations are made. To illustrate, let d2u = c)(x, y)dxdy, which may be written d2u du d y=,(x, y)dy, or d() )-(xy y)dy; whence by integration with reference to y, du - =I p(x,? y)dy + X, du dZxf p(x, y)dy + Xdx, dx and u =f dxzf (x, y)dy +f Xdx + Y, or u =f2cp(X, y)d.yd +/fXd. + Y, there being no necessity of indicating with reference to which variable the integration is first to be made, Art. (47). Examples. 1. Let d3u = axyzdy2dx. This may be written d ax3 _ ydx, or (d ydx. d 31 816 242 INTEGRAL CALCULUS. Integrating with reference to x, d2u ax3y + Y dy2 3 which may now be integrated as in the preceding article. 2. Let d3u = axzdxdydz. 3. Let d4u = (x +- y)2dx dy2. INTEGRATION OF TOTAL DIFFERENTTIALS OF THE FIRST ORDER. 168. If u -f(x, y), we have found, Art. (49), du du X du dx dy du da in which, ~dx and udy are the partial differentials dx dy of f(, y); and also, Art. (47), d (d) d(^ d2 u d2Zs \or dx dy....(1). dxdy dydx dy dx If then an expression of the form Pdx + Qdy......(2) be the total differential of a function of x and y; Pdx and Qdy INTEGRAL- CALCULUS. 243 must be the two partial differentials of the function, and by the integration of either, we shall obtain the function itself. To ascertain, in any given expression of the above form, whether Pdx and Qdy are such partial differentials, we have simply to see if the condition (1), or dP dQ dy dzx is fulfilled. If so, the given expression is the differential of a function of x and y, and we have u =fJPdx + Y......(3), Y being a function of y, which is to be determined so as to satisfy the condition d = Q. dy To determine this value of Y, let equation (3) be differentiated with reference to y. Then du _ dfPdx dY dy dy dy or representingf Pdx by v, du dv dY Q dy = y dy whence dY dv _f(ddv' dy dy f dly and finally u =fPd +f(,, - (v dYf 244 INTEGRAL CALCULUS. Examples. 1. Let dut = (2axy - 3bxy)dx + (ax2 - bx3)dy, which compared with equation (2), gives P 2axy - 3bx2y, Q = az2 bx3, dP 2 - 3bx2 = dQ dy dx This condition being fulfilled, we then have u = f(2axy - 3bx2y)dx = axCy - byAt + Y. To determine Y, we have v -= Pdx = ax2y - byX3, and dv a - bx3; whence Q -, Y C dy dy 2e If du= dz + 2?. Y, u= Jdx _ X y. 1Y 3J Since v -=fPdx = -, we have y dv x dy y; INTEGRAL CALCULUS. 245 hence Y (Q d)dy - f2ydy y2 + C, and, = i + y2 + C. 3. If du yd. +, ldxy- x 8. If U - r, u = tang- x 4- C. 4. Let da - (6zy - y,)dax + (3 e 2oxy)dy. 169. If a function of two variables, composed of entire termrs, is homogeneous with reference to them, its differential will also be homogeneous; and such a relation will exist between the function and its partial differential coefficients, as will enable us at once to obtain the function, when the differential is given. To explain this relation, let t =Af(, y), and mn denote the sum of the exponents of x and y in each term, For x and y substitute tx and ty respectively, the primitive function then becomes t..u. In this expression, for t put (1 + s); then tmu = (I + S)mu. Under these suppositions, x and y, in the primitive function, have become, respectively, x + sx, and y + sy. 246 INTEGRAL CALCULUS. Developing this new state of the primitive function, as in article (46), we have U + d -sx}- dsy +-i d' +2 du2sxy2 + &c. \\dx dy} 2\dx2 dxdy m(n -~ 1)us2 p = (1 + 1S)m == U + US + + &C.... 1.2 Equating the coefficients of the first powers of the indeterminate s, we have du du,x + y = mu......(1). Hence in the differential du = Pdx + Qdy, if P and Q are homogeneous of the (m -- 1)th degree, we shall have, by comparison with equation (1), Px + Qy = mu; U = Px+ QY For example, let du = 4xy2dx + ay3dx + 4x2ydy + 3zaxydy, in which, n - 1= 3, m- =4, 4xy q ay3 = P, 4x2y + 3ay2 = Q; whence u + Qy 2xyX2 + axy2 3 4 INTEGRAL CALCULUS. 247 170. The method of obtaining the integral of a differential, containing several variables, is readily deduced from what precedes. Let dU = Pdx -+ Qdy + Rdz = df(, y, )............(l). If for a moment we regard cz as a constant, and then in succession y and x, it is plain that we shall have the three expressions Pdxl - Qdy, Pdx + Rdz, Qdy + Rdz......(2), which, taken separately, are differentials of functions of two variables, if the primitive expression is a differential of a function of three, and the reverse. But the conditions that these be each an exact differential, are dP dQ dP _dR dQ dR ---- 7 --- 2...(3); dy dx d dz dx dz dy hence if we have given an expression of the form Pdx +- Qdy + Rdz, and the conditions (3) are fulfilled, it will be the differential of a function of three variables, and we can obtain the function by integrating' either of the expressions (2), as in Art. (168), taking care to add to the integral a function of that variable which is regarded as constant. Thus, denoting the integral of Pdx + Qdy by v, we have f (Pdx + Qdy + Rdz) v + Z.......(4), Z being independent of x and y, and a function of z alone. If now we differentiate equation (4) with reference to z, we find 248 INTEGRAL CALCULUS. T dv dZ d dz whence dZ dzf d- z dv ^ L - -^; Z ==J L ~z-' ) \ z0 and finally =f (Pdx + Qdy + Rdz) = v +df(R d\)dz + C By a similar course of reasoning, we may deduce the integral of the differential of a function of any number of variables. 171. In article (168) we have denotedf Pdz by v; whence dv =P. dx Differentiating this with reference to the variable y, we find d() dPdY),~d dP -- T ~/~.. Art. (168); dy dy d. whence tgai d y ntegrati w ith rerence to the dPiable, we hax. Integrating with reference to the variable x, we have INTEGRAL CALCULUS. 249 dv dP f _dx, dy J dly or since (dP)dx - d(Pdx), df Pdx d(Pdx) dy dy By which we see that 6we ay dienate with fe rene It ce d asnother variable, the indicated integral of a partlial diferential, by,imnpy dis(erenStiatyn thr quantity under the sign. INTEGRATION OF DIFFERENTIAL EQUATIONOS. 1:'2. These equations when of the first ordei, and when derived from equations conitanini but two vaQ iba.!es, lill appear as particular cases of the general fol Idx -+ Qdy - 0, and may of course be inteorated as in article (16'8), when dP dQ dy dx and give fPdx + Y = C. In practice, however, it will in general be found, that in consequence of the disappeabance of a fi ctor, commonlo to both terms of the difierertial equation, or when the difihrential equation has been obtailned by the elimination of a constant from the primitive and its imm.ediate differential equation, Art. (56), this condition is not fulfilled; hence other means of obtaining the integral must be so1ugh for. 5 ) 250 INTEGRAL CALCULUS. In the first place, it is evident that, if by any transformation the equation can be placed under the form Xdx + Ydy = 0, X being a function of x and Y of y, the integral can be found by taking the sum of the integrals of the two terms; thus f XdSx +f Ydy C. 173. Among the most simple forms with which we meet, are I. Ydx + Xdy = 0. II XYdx + X'Y'dy = 0. The variables may be separated, in I. by dividing by YX, and in II. by dividing by YX'. The results dx dy. 0 VY and X Yd X ddx + - dy = O, are under the proposed form. In general, if the value of d, dedz we have dy Xc and f y fX dx. INTEGRAL CALCULUS. 251 Examples. 1. Let ydx - xdy = 0. Dividing by yx, we have dx dy _x 1 y C, x y or making C =l C', we have ~X X _ C:='t - - -=0C', C' Oy. Y Y 2. Let xy2dx + dy = 0. Dividing by y, dx + dy 0; y integrating, and reducing 2y - 2 = 2Cy. 3. Let (I - )2ydx - (1 + y)x2y 0; whence ( )Z - 1 dy = o, Y Y and 21x - x - ly - y = C. 4. Let (1 + x2)dy- -V/ dx = 0. 25 2 INTEGRAL CALCULUS. 5. Let ydx- (3y + 1) Vdy 0. 174. III. In all cases where the equation is homogeneous with reference to the variables, they can be separated, and the equation placed under the proposed form. Suppose the general form of the given differential to be Ax"y'"dx -+- Bxt'dy d = 0, in which n + mn = h + c = g. Make y =, and substitute; we thus obtain Axgz"'dx + Bxzdy -- 0; dividing by x', and putting for dy its value, zdx -+ xdz; we 1 ave Az"'dx -+ B3(zdx + xdz) = 0; dividing by (Az' + BBz+')x, we have dx Bz"dz x Azn2 + 3 3z~+ whilch is under the proposed form. EIxamnplcs. 1. Let x2dy -?ydx x-ydx - 0. Make y = zx, then dy zdx + xdz. Substituting in the given equation, we have xzdx - x2dz - ~c'dx ~ xz2dx - z 0; INTEGRAL CALCULUS. 253 reducing and integrating, xdz - z2dx = 0, - lx = C. z Putting for z its value, we have finally lx (C + -). 2. If ~ Y dy = ydx, = --- 1 + -. x — y 2y x 3. Let xdy - yd = dx t/,2 + y2. 175. IV. The equation (a + hb -I- cy)d + (a' + b'x + cy)dy = 0, may be so transformed, that the variables can be separated and the integral found. For this purpose let us make x = t + - y = uI +'; whence dx dt, dy = du. These values in the primitive equation, give (a J b - + c' --- bt -" cu)dt + (a' +- b' +- c'5' + b't - c'u)du = 0. By placing a + bh -- cJ5' =0, a'' a -'' + = 0, we can determine proper values for the arbitrary quantities 6 and v', and our equation reduces to 254 INTEGRAL CALCULUS. (bt + cu)dt + (b't + cu)du = 0; which being homogeneous with reference to t and u may be treated as in the preceding article. This transformation is always possible, save when the values of a and 6' become infinite, which will be the case only when bc'- cb' - 0; whence C1 z 6; ~b'x +- c'y (bx +- cy). b' The primitive equation thus becomes adx + a'dy + (bx + cy)(dx + bdy) = 0, in which the variables may be separated by making bx + cy = z. Substituting this, and the resulting value of dy, the equation reduces to dx + _(a'b + b'z)dz abe - a'b2a + (be - b)')z.If be - bb'- =0, we have at once the integral 2a'bz +- b' 2(abe - a'b) in which the value of z is to be substituted. INTEGRAL CALCULUS. 255 176. V. In the equation dy d- Pydx = Q............(1),: P and Q being functions of x, the variables may be readily separated by making =............(2), X being a function of x, for which a proper value is to be deter mined. By differentiating equation (2), we have dy = zdX + Xdz, and by substitution in (1), zdX - X(dz + Pzdcx) = Qdx............(3). Suppose X to have such a value that zdX = Qd............(4); equation (3) then becomes X(dz - Pzda) = 0; whence dz ^=_-P dx; lz f Pdx, or taking the numbers o t k efr tNOTE.-En, uationr of this kind being of the first degree with refere. to y and dy, are sometimes improperly called lincar equations. 25( IlNTEG-RAL CALCULUS. From equation (4), we have dX = Qdx _ Qe f'd.- 9 -whence X =f Qe Pdzdx. he]se values of z and X, in equation (2), give y =- e-~SQeJ'r.dx. 177. Equations of the form ay-.'dxdx + bzPc x +- cxzdy = 0, may sometimes be rendered homogeneous by making Y = Z? k being a constant to be determined. From this, we have dy = kzz-ldz, yn = Z:. These values in the primitive equation give azr"mx"dx + bxPdx + ckxz-'dz = 0O wiich will be homogeneous, if kjm -- n- =p -q + k - 1, that is, when P - f - I -- q - Is 9n INTEGRAL CAL(!ULUS. 257 1 8. It has been remarked, article (1 2), that differential equna,tions sometimes fail to fulfil the condition of integrability, in consequence of the disappearance of a common factor. Whenever this factor can be discovered, by trial or otherwise, the integral can at once befound, as il article (168). Let Pdx + Qdy = 0 be a difierential equation, in which the condition is not fulfilled, and suppose that Z =f(x, V) is the factor by the disappearance of which the given equation has resulted. The immediate differential equation will then be Pzdxz -- Qzcly = 0, from which we have the condition dPz dQz dy dx or performing the differentiation zdP + Pdz zgQ Qdz dy ldy dx dx or (fp)' () 0dz...(l).. dy dx I\dy dx) This equation expresses a relation between z, z, and y, brit its 33 258 INTEGRAL CALCULUS. solution in the general case is so difficult, that nothing will be gained by attempting it. In the particular case, however, where z is a function of x only, its value can be determined, as we shall then have dz dy and equation (1) will reduce to Qdz dP_ dQz 0 dx \dy dx } or dz I dP ~Qdx. z Q dy dx But by hypothesis z is a function of x, therefore 1 /tdP d X; then d = Xdx; whence 1z =fXdx, z = e.fx Let this be illustrated by the example dx + 2xydy +- 2y2dx 0, Im which INTEGRAL CALCULUS. 259 P = + sy2 Q = 2xy; whence l(dP dQ 1 Q dy dx/' and z efX — efdx Z = x, x being the common factor, the immediate differential equation must be xdx + 2x2ydy + 2xy2dx. 0, which can be integrated as in article (168). In a similar way, if z -f'(y), its value may be determined. 179. Differential equations of the first order, containing the higher powers of dy, may arise, as in the last case of article (56), from the elimination of the higher powers of a constant. Such equations, after division by dx", may be put under the form (dyn. + U, ()........U........(1). The determination of the primitive equation will then depend upon the solution of equation (1), or upon the division of the first member into its factors of the' first degree. There are n such factors, and it is plain that each, when placed equal to zero and integrated, will give an equation between y and x which may be regarded as a primitive equation. 260 INTEGRAL CALCULUS. If, then, the values of y be denoted by V, Y', V", &c., dx equation (1) may be written dx j dx ) dixN ) 0, which may be satisfied by placing V - o,,0 V O &c...........(2); dx dx and if the integrals of these equations be denoted by P, P', P, &c., respectively, we shall have PP'P"&c. 0 o..........(3), for the most general primitive equation, particular cases of which may be obtained by placing P = 0,' = 0, or the product of any of these factors taken two and two, or three and three, &c. It would appear, since a constant is to be added in the integration of each of the equations (2), that (3) ought to contain n arbitrary constants; but equation (1) can only be deduced from its primitive equation by the elimination of the nth power of a constant: [Or by raising (i -~ V) to the nth power, in which case dx the primitive equation must be y =f Vdz]. It is plain then that the constants added ought to be equal, or that the same should be added in each integration. The n differential equations of the first degree which are factors of (1) are readily accounted for, by supposing the primitive equation to be solved with reference to C, which will have n values, each of which differentiated will give one of the equations referred to. INTEGIIAL CALCULUS. 261 As there will be difficulty in the solution of equation (1), when the degree is higher than the second, it will be well to discuss some particular cases which admit of integration by other means. 180. 1. If the proposed equation does not contain y, and it be easier to solve it with reference to x than with reference to dy which we will denote byp, we can then obtain / ()............(1). But dy = cdx, and by parts, article (140), yJ -= 1- Xp -~ - f px f(i) dp - C; whence, if f(p)dp can be integrated, p may be eliminated by -the aid of equation (1), and the primitive equation between x, y and C, deduced. II. If the proposed equation does not contain x, and may be solved with reference to y we shall have:, = f/()............ (3), dy- df(p) or pdx df p); whence dx X —f(p)= fdf() c 45 ty) 262 INTEGRAL CALCULUS. Combining this with equation (3), and eliminating p, a primitive equation will result between x, y and C. III. When both variables enter, but y enters only to the first power, we may take its value in terms of p and x, differentiate it, and thus obtain dy = Rclx + Sdp; or, since dy _ pdx, (R - )dx + Sdp = 0. If this can be integrated, the result may be combined with the proposed equation, p eliminated, and a primitive equation between y and x determined. Suppose the deduced value of y to be y = px + P............(4), in which P = fp). By differentiation, we obtain dy = pdx + xdp + P dp; dp or (x + dp )d o, which may be satisfied by making dP + 0......(5), or dp = 0......(6). dp Equation (6) gives p = C; whence by substitution in (4), INTEGRAL CALCULUS, 263 y = Cx + C', C' being what P becomes when p = C. Equation (5) expresses a relation between x and p, and if it be combined with (4), and p be eliminated, an equation between x and y wvill result, which will contain no arbitrary constant. Let there be for a particular example ydx - xdy = n/cdx + dyS; whence y = _px + n/ 4- 2...... (7) dy =pdz + xdp + n dp /1 + +2s dp4 ( + v2__p 2 0; whence + -nP = O, dp= or p 0 v1 + F This value ofp in (7) gives y = Cx + nV + C. From the other factor we have.wP = _ 7 n in () gives2 which in (7), gives 264 INTEGRAL CALCULUS. yX2 - n2 a result containing no arbitrary constant which will be further explained in the following article. SINGULAR SOLUTIONS. 131. It has been seen, that many differential equations of the first order result from the elimination of a constant, from the primitive equation and its immediate differentidl. Thus, let f(x, y, C) 0......(1), be the primitive equation containing the variables x and y, and the constant (C Pdx + qf- y - 0......(2) its immediate differential equation, and P'dx - Q' 0......(3), the result obtained by the, elimination of C efron (1) and (2). It may now be asked; may not such a function of x and y be substituted. for C, that the result of the combination of equation (1) under this supposition, and its irmmediate differential, shall be the same as before? To answer this, let equation (1) be differentiated,., y, and C being regarcded as variables, we thus obtain Pdx -r Qdy + C'dC 0......(4). Now if C'dC 0, it is plain that equation (4) will be the same as equation (2), and the result of the elimlination of C between it and (1), -will then be the sanme as elquation (3). INTEGRAL CALCULUS. 265 If then for C in equation (1), we substitute the variable value deduced from the equation CldC = 0, that equlatiou -will cotnain no arbitrary constant, and yet will be a mutlc a primie equation, asn one contaiing tlhe arbitrary cous ta t. Suclin rsult s ar termed sing ri ul ar s ol utiozs, inasmuch as they cian not poss.ibly ie. otatm, ecd f rom the comp llete integ rl, Art. (132), by assigninl to Jtho arbi'iaryi const',W a p 7articular value; the latter results beinr g called piarticulccr itzgcrs. The equa -ton C ~' -- 0 can be satisfied, by rnma-ing dC - 0 o C' - 0. The filt g i 0ves C - a constant, the p, art;icular values of which whein su bst'tulte d in cqu.lation (1) give particular integ'ral. Te Ivlt s, o C o d frc O C - 0' 0, if vadriable, will then gi.e e the only 0si.gula,'ol"tioJ o, 11ns. To illus;trat, let us re mine the comnplete integral of equationl (7), in the preceding article, y =- C + nil 4 O.....(5). DifferenLatating ith reference to C, we have 0 z ndC nd whence 34 26 6 INTEGRAL CALCULUS. + - 0......(G), 1 + C2 and,2 + x2C2 n2C2 or C.= -; n2 - 2 the negative value of C being plainly the only one which will sat" isfy equation (6). Its substitution in (5), gives y = -- ~- V 1/1^ " x2 I'" y V= n'2 X or y2 -2- x2= n', the singular solution found in the preceding article. ITNTEGRSAT'ION OF DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 182. Of these equations, which in their most greneral form contamin d_ i,, y, x, and constants, we shall only discuss thoso particular cases which admit of integration. dhy 1. The proposed equation may contain only d 7 -,, ad cond2x stants; in vhich case, solving it with reference to d y- we have ~dx~2y~ d2 ~dxa~2~dx a Y_ -(a3) INTEGRAL CALCULUS. 267 which may be integrated as in article (164). 183. II. It maycontain only d y, yand constants. oolying the equation as before, we obtain d2y Y dx2 Multiplying by 2dy, 2dy d2y 2Ydy, dx dx and integrating, 2 f2Ydy + C or 1 whence dx ___ 2JYdy+ = Y +,'Yd d/2Y~ + 2 f /Ydy + 0 Examples. 1. If a2d y + ycd: 0, c2Py y 2dy d-y 2ydy X2 2 x a d2 dy a2 dx dz Ya2 d.- - 0 -+ C) d ^ ^ d^' d~~~x~~V^^.i 268 INTEGRIAL CALCULUS.,_ dy a2 which may be integrated as in case I, article (133). 2. Let d'y Vay=- dx2. 184. III. The equation may contain only d, d- andcon dcl d and con stants, being expressed generally thus, F(dd) d 0............ (l) Make dy =d P; then d -Y d L and (1) becolmes dx d, dx F(d.pLj) 0, which i3 of the first order with reference to dp, and may be solved with reference to dx; whence dx - F'(p)dp......(2), x =-fF'(p)dp + C.....(3). Multiplying (2) by p, we have pdx - dy = pF'(p)dlp; y f= F'(p)dp + C......(4). Eliminating p from (3) and (4), we;hve the primitive equation between x, y, and the two arbitrary constants C and C' For an example, let dzdy dp ~dx ~-p~ INTEGRAL CALCULUS. 289 whence,dz = adp ap dp 37dx j~pdx = dy -_ (1 + 2)- (1 +)2) Integrating the last two expressions, we have x= C + ap Y a,-G+~^___, ^G' — ^a V$ +- P v+ and eliminating p, (x - C)2 + (y - C')= a-, as was to be expected, since the proposed equation expresses a constant radius of curvature. 185. IV. If the given equation does not contain y, it may be cxpressed F(d, fy) o0, or F,, ) ) 0, Ta^ _ d(^ dx J which is of the first order with reference to dp. Its integral will give an equation of the form f/(P, ) - 0, in which, p being replaced by dy, and the result integrated, we shall have /'(, x) = 0, with two arbitrary constants. 270 INTEGRAL CALCULUS. For an example, let d2y dy 1 dx2 dx x or dp_ dp dx dx x p x p = lx + C, P = C'x, dye, and C.y + C. dx 2 186. V. If the given equation does not contain x, it may be expressed F( dy. dyY..0... (1). Since dy = pdx, dx dy d2y dp __ pdp -dx ~. ^ - = - p' d2 dx dy' and equation (1) may be written YF (P^dr =) 0, which is of the first order with reference to dp and dy. Its integral will then be expressed F'(p,y)= 0, or F, dy )=O dx ) INTEGRAL CALCULUS. 271 aid thils may be t reated as in case II. Art. (180). 187. VI. If the equation be of tihe form, d'y dy + + X y - X 0,............(1). dx6 dx then dy fud c d,2y x d'\ dx dx' dx) These values in (1) give, (since the common factor e-' disappears,) dlu C7 + it2+ Xt + X'I o, dx: whlic is of the first order with reference to du. After integration, the value of u being determined and substituted in (2), will give the required primitive equation, y e.Oxd INTEG-RATIONi OF DIFFERENTIAL EQUATIONfS OF IIGHIERi ORDERS THAN T IE SECOND. 188. Of these, it -will also be suflcient for our purpose to discusa a few of the most simple cases. 2' 2 INTEGRAL CALCULUS. I. ipp-ose the equation to contain only -d ~y, d' ld s'ants, it may then be expressed, tny d-'l? Fd d,, 7.) - 0............ (1). M'ake d7-Y d-nY d? dz - xu; te ten dxt7-l' Cdx" dx These values in (1) give F ( 0, which is of the first order, and its integral will give u in terms Olf d, or = X - 1J-_ X,x iand finally, y - f-'(X + C)Cdn-'. 189. II. Suppose the equation expressed thus, F ( dy............(1). Make dn- y d"ny du d-~- u, then - dx d^z-2 _, dx" - dx' INTEGRAL CALCULUS. 273 and equation (1) will become F(T/ u ) 0, which may be integrated as in article (183), and the value of u t= (x) determined; we shall then have d "-2 () and y?- ) 190. III. Suppose tlhe equation to be of the form dPy + Adycdx + Cdydxh + l D 3 0............(1). Make y-........... (2), u being an'arbi;trary function of x; then dy -- eCiu2G d^y = eV (d2'u + dZ62) d3y = c(de u + o3dud'2t + du3). These va,.lucs in (1) give d'u + 3ciducld - - dzi +'A(du -f- dWC)dx -- Edudx2" -- Ddx3 -: 0........... (3). Since u in equation (2) is arbitrary, let such a value be assigned to it, that its differential shall be constant, in which case d = m-cldx, d d =0 0. Equation (3), under this supposition, reduces to,3. — Am~ + B + 1) — 0............(4). 274 INTEGRAL CALCULUS. Fromi this equation we may determine the value of the constant m. Denoting the three roots by nz, nm, iM " we have for du the three values du - md, d = m'dx, du == mZdxu; whence u -' mIx -+ C, u - 1 mx + C', ^ 7m" + C", and. Mx ~ Cm'C' ~mx-C<. y e +, y C — e. c or calling e -C, C cC — * C/," y C G., y- = G.., y- c e G'. But since these values of y each contain but one arbitrary const:ant, they must be particular cases of the general value of y, vwhich must be of such a form that. either of the above particular values can be deduced from it; that is, y = Cc' -It CG'.x + C"ell"., from which the first particular value is deduced by making C' and C" == O; and in a similar way, the others. If two of the roots m, m', m", are equal, that is, if -- m', we should have the equation y = (C -+ C)eC7n + C"e/"lt -= Ce..'. _- ClG"c"1..,^ containing but two arbitrary constants, C d- C' being denoted INTEGIRAL CALCULUS. 275 by 0. It is not then general. But in this case, y Ce"' being a particular value, y= e........... (5) will be another; for, difierentiating it, we have dy C"'e'(l1 + mx)dx, dJy = C'eT(22m d- m2x)dx2, d\J = O'e'"'(3.'2 + 3.,)dc.z3 and these substituted in equation (1), give (in3 + Amt + B2 - + D)x + 2Am -- B) = 0...0......(6). But the coefficient of x is the same as the first member of equation (4), which has two roots equal to m; and 3m2 +- 2Am +- B is its first derived polynomial, which, when placed equal to 0, must have one root equal to m (see Algcbra); hence both terms of (6) are 0, and y = C'xe"'' satisfies the given differential equation, and must therefore be a particular value of the general one, y = Ce""m CI'xec7 + Cl "em"I If m == m' =m", it may be shown also by trial, as above, that y = J/fxn: is a particular value; whence the general value must be y = enr(C + Cx + Cl"x). Two of the roots may be imaginary, but as the discussion in this case is quite complicated, and of little value to the student, we omit it. 23 I76 INTEGRAL CALCULUS. To illustrate the above, let dy -+ 2d2ydx - dydx2 - 2ydx3 - 0. Comparing this with equation (1), we have A =2, 1 --, D= —2; aund equation (4) becomes m -n 2m2 -- 2 0; whence n - 2, 1, and -1, and the general value of y is y Ce-2x + C'e - + C"e —. 191. It is plain that the preceding principles can readily be extended to the general equation d" - + d"-ydn, - Bd"-ydz........+ Mydx" =r- 0, and that the general value of y will be y = Ce"x -t C'em'" 4- C"e.. + &c....... 192. If the equation be d3y + - Xdydz - X'dydxz + X"ydz3 0......(1), iu which X, &c. are functions of x, the difficulty of integration is much increased. If, however, we know three particular values of INTEGRAL CALCULUS. 27 yp Cy', C'y", C"y"', each of wlich will satisfy the given equation, then the general value of y will equal their sum, that is y = Cy, + C'y" + C"y"'......(2). To verify this, let equation (2) be differentiated three times and the proper values substituted in (1), Awe shall thus obtain C(d3y' + yXdy'dx. - /'cy'dW d- X"/''d:1 ) + C'(diy" + Xd^y'"dix - I.d, + " y'l"dx2) 0, + C"(dI3y"' -I- Xd2y/'dx -+ X/'dx"d — 1 XI'lJ"d. ) which is satisfied, since each of the three terms is by LhylJothe:ai equal to 0. 193. The above demonstration can be generalized, and a similar result obtained for the equation d"y A- Xd —-...'ydx -...X ( 0. This, and the equations discussed in the three preceding articles, belong to the class termed linacr. See note to article (176). INTEGRkATION OF PARTIAL DIFFERENTIAL EQUATIONS OF THEI FIRST OIRDER. 194. A partial differential equation of the frst order, derived. from an equation between the three variables z, y and x, z beino regarded as a functeion of x and y, contains in its most general form, the three variables, the two partial differential coefficients, 278 INTEGRAL CA.LCULUS. -c and i-, and constants. VWithout attempting to discuss the dx dy most general, we will confine ourselves to a few of the most simple cases. I. If the equation contains but one partial differential coefficient and the two independent variables, that is, if r p nd r P being a function of x and y; we integrate at once as in article (166). For example, if dz x _..~= Jr 2 ) = 2=2 + 2/ + Y. dx V + 2T 195. II. Let the equation be dz p A, dx R being a function of the three variables. Since the partial differential coefficient has been obtained under the supposition that y is constant, the proposed equation may be regarded as a differential equation between Z and x, and may be integrated as in article (172), taking care to add an arbitrary function of y. Exacmples. 1. Let dX -- dx z INTEGRAL CALCULUS. 2 9 By the separation of the variables, we have d zdz bzd, x i —,oz2 and by integration -_' - Vy' - ~' +

4^ ^4 48 For the hyperbolic spiral n =- 1, and the general value of si becomes 2t which is infinite when t 0. For the integral between the limits t = b and t- c, we have I1TECGRAL CALCULUS. 299 S a2l 1l)'2 b c J In the logarithmic spiral, when M 1, t = 1, dt -, u2dt fudu c. C r e2 m 2 4 or estimating fromn the pole where u = O and C = O; +we have 4s' that is, equal one-fourth the square described upon the radius vector of the extreme point of the curve. AREA OF SURFACES OF REVOLUTION. 211. In areticle (89), we have found for the differential of the area of a surface of revolution du = 2ryv/cldx2 + dy2; whence for the indefinite area, we have u =f 2ry Vd + d......... (1), the axis of X being the axis of revolution, and 1c/dx2 + dy2 the differential of the arc of the generating curve. The indefinite area of any particular surface will tlyen be obtained, by deducing from the equation and diferential equation of the meridian or generating curve, the values of y and dy in terms of x and dx; or of dx in terms of y and dy, and substituting in 300 INTEGRAL CALCULUS. formula'(1). The result of the integration will be the area required. 212. Let the line AC, by its revolution about AB, generate the surface of a rilht cone. The origin of co-ordinates being at A, the equation of AC is A B y = ax; whence dy = adx, and u =f 2- axdx V/a + 1 =- azx\/a + 1 + C. Estimating the area from the vertex, where -- 0, we have C = 0, and U' =- axz2v/c + 1. Making x = AB = hA, we have the area of the cone whose altitude is h, and the radius of the base BC - b, u"= r-ah2Va ~- 1, b or since a =, = 27rb /b 2AC. 2 2' that is, the circumference of the base into hay the side. 213. From the equation of the circle, we have / ~,~ ^ (r - x)dx y = V2rx -xy = - The surface of the sphere is then INTEGRAL CALCULUS. 301 u = 2y x/d2 + (r - ))d / 2,rd u = 2rrx +- C. Taking the area between the limits x = 0, and x = 2r, we have u" = 4sr2 = four great circles. 214. From the equation of the ellipse, we have b bx Y — /a 2_ x, dy=- dx; a a2y whence for the area of the ellipsoid of revolution,:P 2,rb u = ^/-da42 - (a2 - 2) 2wrb aa- _2 a4 or placing 2/- = C', and - _, - 2 a,6 6bLYI-ja2 - b2 u C'f dxV'2R - 2. But fSdxczV'/' - x2 = area of a circular segment whose radius is R', and abscissa x, Art. (88). Integrating this between the limits x = 0, and x = CB a, a and calling the segment CBFG -D, / we have / I u" = C'D - area of Ellipsoid. E 2 302 INTEGRAL CALCULUS. If a == b in the primitive value of u, we shall have u = J'21rfadx = 2rax +- C, for the surface of the circumscribing sphere. Let the area of a paraboloid of revolution be determined. 214' By the substitution of the value of dx, Art. (201), in the general expression for u, we have for the surface generated by the revolution of a cycloid about its base, U = 2, V-2Wrfydy(2r - y)- Placing 2r - y = z, and integrating as in Art. (131), we have 22(2r — Y)-'u= 2V'-/2r(- 4r(2r y)~ _ (2? y)- - C. Taking the area between the limits y 0, and y = 2r, we have = 32r' 3 for one-half the surface. The whole is - the area of the generating circle. CUBATURE OF SOLIDS OF REVOLUTION. 215. The operation by which the solid content, or solidity of a solid, is determined, is called its cubature. For the differential of a solid of revolution we have found, Art. (90), INTE GRAL CArLCULUS. 303 dv -' i dx or v = dx...........(1); in which y and x represent the co-ordinates of the curve which generates the bounding surface, the axis of X being the axis of revolution. For the cubature of any particular solid; we find, from the equation of its meridian curve, the value of y2 in terms of x; or from the diferential equation of the curve, the value of dx in terms of y and dy, and substitute in the above formula (1); the result of the integration will be ca exp2ression. or an) indefinite por'tion of the solid. 216. Let thle rectangle ABCD revolve about AB and generate a right cylinder. The origin of co-ordinates being at A., the equa.tion of DC will be, y A.D = b then I A B v - f,,r'y2d = f'^rbdx = xbez + C. Takldig' this bet.ween the limits x = 0, and x = AB = h, we V'" =-'h b2=- the base into the altit'ude. 217. The equation of the ellipse gives 2 b7 2 D Yp= /(a 2-v2); f af (a A 4 3 whhence for the ellipsoid of revolution b bf2 7 X 2y f 3' v -f -(a -- xcdx (a"x ) + C. a CZ2 3 a ~~~ ar t. -iri''i IC I.NTOAL CALCULUS..E;stimat_'il the solidity from the plane thlrovuglh the centre, per-?rindicular to the transverse axis, we have x 0, C 0, a nd v' - (2 ~ax~- ). Making x a, we obtain for one half the solid, 7rb2 a3 2 v" - ~(a _) - _^ c 3 3 and for the whole 4 2 =: -a 2~bt" X 2a; 3 3 or, equal to two-thirds of the circumsciibing cylindero If the s-ame ellipse revolve about its conjugate axis, we have, zf x'dy f b2 (- y) dy, which between the limits y = b and y = b, gives v" _4 la2b 2 aC x 2b = - ra X 2b. 3 3 The latter solid is called the oblate spheroid, and the former the pr)iolcate spheroid, and we have the proportion the prolate: the oblate:: 4 b a: 4ca'b: b a. 3 3 If un either expression a =b, we have 4- spe' a3=: solidity of a sphere. INT'EGRAL CALCULUS. 305 Let the origin be now taken at A, when fY -' (ax - x), a and the solidity be determined. Give also'the cubature of a sphere directly, by using the equation y2 + $_ x r2. 218. Give also the cubatures of the following solids of revolution: 1. The right cone, v" base x - of altitude. 2. The paraboloid, V" -. circumscribing cylinder. 3. The solid generated by a given portion of the common parabola revolving about the tangent at its vertex, v" =-' cylindcer with same base and altitude. 4. The solid, the bounding surface of which is generated by the curve whose equation is j -= 5. The solid, the bounding surface of which is generated by one bianch of the cycloid revolving about its base. APPLICATION OF THE CALCULUS TO SURFACES. 219. Since the equation of every surface expresses the relation between the co-ordinates of its points, it must contain three variables, and may be generally written 39 3 i; (: INTJEGI AL CALCULUS. U = F (, Y, z) = 0......(1); or since either two of these variables may be assumed at pleasure, aind the remaining one determined from the equation, the latter,may be regarded as a function of the other two, they being entirely independent of each other, and the equation of th'e surface be thus otherwise expressedl z — 7(x )...... (2). In the equation of every surface considered, z will be regarded as a function of x and y; and the co-ordinate planes will be taken at 1right angles to eacha other. The differential equation of a surface may then be obtained, either by differentiating equation (1), as in article (54), or by differentiatilg equation (2), as in article (4 9). By the latter method we obtain dz dz dz =- dx+ d............(3). dx dy 220. Let MA be any point of a surface, a portion of which is re)pr.esented in the annexed figulre d r~7 -\ Tl.The co-ordinates of this point are N, ^' \,/ \ / If, then, in the equation of the d''i' —~T" surface, we make x = x"',a nd INTEGRAtL CALCULUS. 30T suppose z a.n y to vary, they can only belong to points in the curve dMd', the intersection of the plane and surface; and if we suppose y to receive the increment CC' = c, we shall have, by Taylor's formula, CIZ d -z t2 N'Q' =f(", y + ik) = + 1 ~.+ 2" -- ( ) dy+ 1.2 + in wihich xt is regarded c cs onstant and equal to x". In the same way, if y = y" in the equation of the surface, and z and x vary, we shall have the curve eMIN, and if x receive the increment bb' = h, dz dtdz h2 NQ) =.f(x3 + h7 y1) = Z - + + d2 12 + &c..... If now x and y at the same time receive the increments h and k respectively, we have, Art. (46), M'P z' = f(x + h, y + I) - z + 7 + d - + CIZ d2Z d2z I +- -C - T - +&C. y ddy 1, - z =ph p+'I +- (qh' + 2qIk +~q"Is') + &c.; by making dz dz d2z d. ^ " dy dld PI 303 INTEGRAL CALCULUS.'When x -- x:", equation (3) gives dz dz dz -- dy ='Id y or dy ='.......(4) equations which evidently belong' only to the section dMd' parallel to YZ. If y = y", the corresponding equations for the section parallel to XZ are dz dz ad = dz -,z= pdx or ~= p 5......(5). dzx h1e value of A, equation (4), is the tangent of the angle which a tangent to the section dMd', at any point, makes with dz the axis of Y, or with the plane XY; and d-, equation (5), the corresponding expression for the section eMN; and since these angles are the same as those made by the curves at the point of contact, with XY, they give the inclination or slojpe of the surface in the direction of these curves. 221. If it be required to find the slope of the surface at any point, as M, along the section MM' made by the plane MM'PP', we takl the equation of this plane y = c1x 4- /3......(1), z inzdeterminate; cc being the tangent of the angle made wiith the axis of X by the trace PP', and equal to - K Now in order that z shall represent only the ordinates of points INTEGRAL CALCULUS. 309 in the section MMT', the relation expressed in equation (1) must exist between the variables x and y, and we must have dy = acdx, which in equation (3) of article (219), gives dz _ (p +'i)cldx. The, l op -- 3,'A'1 The limit of the ratio P P, is evidently tire tangent of PP1 the angle (S) which the tangent, and consequently the curve at the point M, makes with:P', or with the plane XY. But since PP', =/2 + PQ' -h /V + a, we have M'P' - MP z' - z the limit of which is 1I dz Vp + ap, -_ X - - tang S. V1 V,' -/ + ^ To find the direction in which the section MM' must be made in order that the slope at a given point lM, along the curve cut out, be greater than along any other, it is only necessary to obtain tlhatl value of a which will render the expression p +- ap' /1 + c,? a maximum, the values of p and p' being taken at the given point the- vein-u~ i-h at — the ivenpoin 310 T INTEGRAL CALCULUS. M. Differentiatingl th-e expression with reference to a, and placing the result equal to 0, we have P' - p' (1 + Ct22 whence p ~ p- a = 0, a — p This value of a substituted in equation (1), (13 being first deterineid by the condition that the line PP' shall pass through P), will give an equation, which, conibined with that of the surface, will determine the line of greatest slope. 222. The co-ordinates of a given point M, being x", y", and z" the equations of a tangent to the section parallel to XZ at this point, will be z -z" m(x - x"), y y"; and to the section parallel to YZ, z z" nz(y - y'), x = x" in which um and n represent what d and d equations (5) and (4) of article (220), become, when x", y" and z" are substituted for x, y and z. The line, of which the equations are i Z// - ( I/ ~ Z - ~I (x - x"), " ~ is perpendicular to both of these tangents, and, of course, to their plane, which is tangent to the surface. This line is then a nor INTEGRAL CALCULUS. 311 mal to.the surface at the given point. The equation of a plane passing through this point is A(x - x.") + B (y-y") + C(z -,") -. To maLe this plane tangent to the surface, it is necessary to introduce into its equation th.e conditions that it be perpendicular to the no;ral, which are A - mC, B - C; whence - m(nx - x") -' n( - ") -t ( - z') = 0, dz" dz" or substitutiu.ig for Zm tad in their values, d- and d-,, and d dlx" cly" reducino 7"(" -. d') + (- - 1) - 0......(l).,(x, - ") + F(y- y") (- ")= 0(1). The equation and differen'lial equation of the Ellipsoid are Mz' Ny2 + + L - P = 0, ancd M.zdz + iNydy + Lxdx O0; whence dz" Lxx" dz" Ny" dx" Mz"' dy" -- M"' wlvich in equation (1), after reduction, give.2L"(x -- x") + Ty"(y -- y") + - Xt/(I-.11) - 0, or since 312 INTEGRAL CALCULUS. _ L/2 _ -N/ _,2- Mz/2 P _- AMzz" + Nyy" -+ Lxx" - P 0, for the tangent plane to the ellipsoid at a given point. If M == N L, we have, for the tangent plane to the sphere, Zr" + yy" +.. R2 0. The distance from any point of the normal to the point of contact is, = -v/ (x-:s,)2 + (y- _,,)2 q (Z -Z/)2 (z" - )/1 -- In + (- 2, 2 If z - 0, vwe have D -\/ 1 -1- w m" — "'2, for the distance from the point of the normal in the plane XY, to the point of contact. 223. One surface is osculatory to another, when it has with it fa muore intimate contact than any other surface of the same kind' and the conditions which must exist in oirder that a surface, given in kind only, shall be osculatory to a given surface at a given point, can be determined by a method similar to that pursued in article (90). But from the nature of the case these conditio-ns are more numerous and complicated, and their determination more difficult; so much so as to render osculatory surfaces of little use in the mneasur of curvature; hence another methocd has been devised which will now be explained. INTEL -L CALCULUS. 81 Let i be any point of a surface, at which it is proposed to ex amine the curvature. Let this point be taken as the origin of co-ordinates, z and let the normal at this point be the 0 axis of Z, the axes of X and Y having any position in the tangent plane M XMY. The equation of the surface, / X Art. (219), will be z =f(, y)........ ().Y Through the normal let any plane ZMX', making an angle p with the plane ZX, be passed; it will cut from the surface a curve 1MO, For any point of this curve, as 0, denoting the abscissa iMXl by N', we shall have = x / cos p, y = cx' sin............(2), and these values, substituted in equation (1), will evidently give the equation of the curve referred to the two axes MZ and MX'. 1ow by varying the angle?p, all the normal sections at the point M may be obtained, and by examining the curvatures of these different'sections at the given point, an accurate idea of the curvatm'e of the surface may be formed. The general expression for the radius of curvature of one of these sections, Art. (105), may be put under the form R = +d,. (3). dx"2 Differentiating equations (2), we have dx = dz' cos p, dy = dx' sin qp............(4); 314 INTEGRAL CALCULUS. and substituting these values in equation (3), Art. (219), dz dz = p cos +dZ' - p' sin cdx,', or p cos p +-F p' sin p......(); Differentiating again, recollecting thait p and p' zar inpiiEit faictions of x', we have Art. (220), __5_ ~ cos ( Q_ -!- q' d ) -r- sin (n f ^d - __ sq~ p,; dx^'^xz dx'x' dy d d''d' or since equations (4) give d- - cos and _ sin, If these values of Z- and z be subsituted epesin expre ( dx' c dx' we shall have the general value for the sradius of curvature of cany one of the normal sections. But as we only desire this value for the point Mi, we may first substitute the co-ordinates of this point, w:hich are xI't -- 0, y" 0 z 0; and since the normal at this point coincides with the axis of Z, vwe must also have, Art. (222) -0, 0 -i O 0, or p = — 00. dx"' dy/ substituiting these values in equations (5) and (6), and the resIult in equation (3), rwe obtain p_ - __ q C__OS_........... Si q cosq -p" 2qi' cosO sinp -1- q" single INTEGRAL CALCULUS. 310 in which q, q' and q" are what the ipartial didirenitial coefficients of the secondc oder of the function z become, vwhlen 0 is substituted for x, y and z. Dividing by cos2 p9 and recollecting' tht -- tang (, cos c this value inay be put under the form p: _ 1 -1- tang2 9.. q + 2g taL - ta q tang2 9 We have taken the positive value of R, Art. (105), since, as the surface is represented in the figulre the sections are above the axis d2% of X' and. convex towards it; d~^ must therefore be positive, Art. (83), and the value of R positive, as it should be when laid off from M above the plane XY. If the section at the point M lies below the plane XY, it must still be convex towards this tangent dsz plahne; will be negative, and iP negative, and must therefore be laid off from M below XY. By assigning all values to (p from 0 to 360~ in equation (8), we shall obtain a value of Pi for each normal section. Among these values there must be one which is greater, and another which is less thlan all the others. The values of (p which will give these principal values of R will be obtained as in Art. (66). Iifferentiating equation (8), we have dRl 2(c' tang2 o + (q - q") tang p') d tang p (q 42' ta +' an " tang'2 p)2 If the denominator be placed equal to 0, we shall obtain values of the tang (p, which, vwhen real, will reduce the value of rI to infinity. The curvature of the corresponding section will then be zero, and the section itself a right line, or the point M a singular point, Art. (92), cases which do not occur in all surfaces. Let us then place the numerator equal to 0, we thus have 316 INTEGRAL CALCULUS. tang' p + - tang q - 1p 0............(9). This being either of the first or second form of equations of tho second degree, the roots will always be real and their product equal to - 1, that is, denoting them by tan.g p' and tang p" tang p' tang p" -t ]. = 0; hence the normal planes in which the greatest and least radii of curvature are found, must be perpendicular to each other. Their exact position will be determined by solving equation (9). The values of tang p' and tang s" being determlined and the ~~z ttraces of the normal planes constructed as in the figure; let us talke MX" as a new axis of X, and MY" as a new axis of Y, and suppo;se the surface to be re//'.'.X ferred to them with MZ as an axis of Z. Then we must have for these new axes tang' =0, tang ",c tang q/ tang " co, which requires in equation (9) that q' = O. Substituting this value of q' in equation (7), we have =R - ~............(10). q cos2 (p - q" sin' q) Substituting in this the values of p, corresponding to the maximum and minimum radii as above determined, viz. = 0 and qp = 900, and denoting the values of the principal radii thus determined by B' and RT", we have INTEGRAL CALCULUS. 317 1 - 1 l — 5, R"'J-, q q and finally from equation (10), 1 I + 1 -- C os'C2 g- q sin2 p n coS2 ( + sin2, Rt R' R, w-hich expresses the reciprocal of the radius of curvature of any normal section, in terms of the principal radii and the angle p. If I' and R" are both positive, all values of R will be positive, and the greatest of the two will be a maximum, and tlhe least a minimum, and all the normal sections at the point M will lie above the plane XY. If R/ and R" are both negative, the sections will lie below XY. If one is positive and the other negative, a part of the values of R will be positive and a part negative, and a part of the sections will be above and a part below the plane XY, and this plane will cut the suxeace at the point M, giving a point analogous to the point of inflexion, Art. (92). If R' I R", all the values of R become equal to R' or R", and the curvature of all the sections will be the same; as at any point of a sphere, or at the vertex of a surface of revolution. 224. To determine a general expression for the solidity of any solid; denote the solid AbPc-MZ, included by the surface, the co-ordinate planes and / the parallels ece' and dbd', by v. Since, by the equation of the bounding surface, z will always be given in terms of x and y, / the solid may be regarded as a function of'' x and y. Let x be increased by h, y re- /' / maining -the samle, ve shall have the solid' InH' 31 8 INTEGRAL CALCULUS. dv d'2v h2 bb'QP-Nd - v' - v - / +h - 2 -1- &c. If y be increased by k, and x remain unchanged, we shall have the solid dv d'v ks + cPO'c'-N'e =v v dv + ^ d- k2 + &c. dy dy 1.2 If now x and y be increased at the same time by the variables h and k respectively, we shall have the solid whose base is cPbb'P'c', or d2v k2 d dv v:"'~+ ~7- 72 + &C. solid PQP'Q' - MMA' = — dy + - hk ~+ &c. dxdy 1.2dx-2cly If through M and M', planes be passed parallel to XY, two parallelopipedons will be formed, having the common base PQP'Q' and the altitudes MP and M'P'; the limit of the ratio of these solids will evidently be equal to unity, and since the solid PQP'Q'-MN is always less than one and greater than the other, the limit of its ratio to either will also be unity, Art. (85). The solidity of the first parallelopipedon being 7k.MP, we have the ratio INTEGI-IA L CALCULUS. 319 dYv d3v hI2 d'v d3v 7h dIk +id - + &c. - - - + &e1. dU-dy cd(iicy 1.2 dxdy. dx+ cy 1.2 h/k.MP z and passing to the limit d2v dx^dy d2V L =.; - ^ L z1 l; dxdy; or d~Y- ZJ d (v Zdx. dxj dy integrating with respect to x, d'v — zdx -- Y. dy From this dv = dyf z'dx + Yd/y. Integrating both members with reference to y, v =clfdyfzdx +-I-Ycy + X, Go Art. (164), v =Jfzdydx + fYdyd +y X. Since the integral'zdx -1- Y is evidently the area of one of the parallel sections as efge'; to obtain the whole solidity represented in the figure, we miust first taLe the integral between the limits 3930 INTEGRAL CALCULUS. x- O and xl = c', and lthen the second integral between the limits y - 0 and y AY. To illustrate, let us determine the solidity of the pyranmid A'D - C; the equation of the plane BDC, being + 2y +- 3z - 2 = 0; whence 2 - 2y - x / ^^ —--— ^< The equation of DC is x + 2y = 2, or x- 2 - 2y, AD 1, AC = 2, AB- 3,v =. f2zdxdy =f (dyf dx (2 - y - ) 3 integrating with respect to x, v=fdy( f2x - 2yx C Y ( 3 6 + or taking the integral between the limits x= 0 and x= ce'-2 — 2y, f (4 - 8y + 4y2) r6tn t Integrating now with referLence to y, between the limits INTEG RA L CALCULUS. = 0 and y AD - I, we obtain for the solidity 4 1 2 2 1 1 v - - x - x 1 x - - AB x AD x - AC 18 2 3 3 2 3 BAD x -i AC. 3 225. Let BMMi,' be any curve in space, and B'PP' its projection on the co-ordinato plane XY. Let the plane of the curve MM' -~~ malke an an loe / with the plane / B- /i X Y, and let its intersection w-ith / that plane be taken for the axisx /A Q of X. Then, if the ordinate oTQ be denoted by y', tihe area of the curve'MM' will be s = yl:_ da o..........Art. (203). But any ordinate PQ of the projection is plainly equal to the corresponding ordinate M:Q of the curve multiplied by cos MQP = cos, or y = y' cos f; hence the area of the projection B'PP', denoted by S, is S =f ydlx ='fy'os / dx cos ffy'dx cos 3 s; that is, the projection of any plane area is equal to tohe area ~multiplied by the cosine of the angle included between its plane and the plane of projectio-. 022 INTEGRAL CALCULUS. 226. N]ow, let u denote the area of any curved surface. It will be a function of x and y. By a process identical with that of Art. (224), we shall find the surfaco d'u d3u hik 3TNIAS'N'/ - +yh _L -d + &......(). dxay dx'dy 1.2 If a tangen't plane be drawn atl M, and the four planes PN, QM3', &e. be produced, they will form on the IA-= s tanoenit plane thie plt-'lleloglram MORS' "- -........ Tlhe lilmit of he ratio of thisd parallelogram - l / a|nd the surfae M^N' will be unity, as I \ 1 lay hbe j]roved biy a process. similar to that P.; _. —-... —-Q p'ursu d in article (S89). T/ e area of t e pa al. illog.ram is equal to Q " ^ its projection PQP''Q divided by cos /3c; being the angle vhich'the tangent plane makes ewith XY. But 13 is also the anigle which lthe noirmal nt' miakes witlih MP or the axis of Z; hence cos P / 1 — I- __ - n'1 - m and -- n representing the tahgcents (- d- and - 7 of the \ x dy angles which the projections of AR' make witlh tl axis of Z, Art. (222); hence PQP' Q h. _ to area MORS -- AQ'?'- -_ cos 1 1 /1l +m'~ +^ 9',2 Dividing equation (1) by this, we have INTEG RAL CALCULUS. 1 3 C}'iNz d3dt hM-lTM/\ cl Ldxy! d-ly 1. 2 x MOLtS'\/1. -t- m2 +- n+ Passing to t'he limit, we hlave ud2,l d _ L d — __ 1; whence dt dx - mxdy,/ + V fs-+2', and t =fct7xdy17/l + P2 n2.'or the sphere, we have 2 ~ Y2 + Z _ P2; whence dz y -- d ~ / _ _ f/dy z22 2/2 ~v + wn2 +4 2zn _ ~, and Rdxdy 2 - y_ " '32 4 INTEGRAL CALCULUS. Maling V/.R2- y2 = II, and integrating with reference to aX we h1ave:=J"Rdy d f S__cdy sin-l - - Taking the integral between the limits, x 0 o,d ce' = -/ 2 — ~, we have -t SRdy 2 "A~~~ X~~~~2 Integrating again, with reference to y, we A ^ I- have C^" —-— 3^ y -[- u y+OC, 2 and between the limits y =, y = R, 2 for one-eighth of the s,:rtace. The entire surface is then 4^,,fR' P ART II. CALCULUS OF VARIATIONS. FIRtST PRINCIPLES. 227. A function may be regarded as given-, when the forml of the algebraic expression, which determi nes the relation between it and the variable or variables, is given, and the conltant' -which enter this expression are known. In this ease, the only change which the function. can be:made to undergo, is that wihich arises from a chanoge in the variables. Wheon these variables-, receive infinitely small increment',% the correspond'ing infinitely small increment or chan'e of t.e function is taken for the di-frcnzticl of the functioz, Art. (91). All our previols applications of the Calculus have been made to functions of the kind above referred to, and the term differential can, with propriety, be applied to no other change. It will at once be seen, that if a function be not given a(s abo-ve described, but mCerely subjectecl to ce'tain cocditiGons, it maly lbe made to undergo a chang'e by altering the relation which exists between it and the variables; and this may be cone by chano'ino'g either the form of the elxpression or the consltants; whe ic e-nter it, in any way consistent with the given conditions. Now if such a, change be made as to give anotler fu.mnction2 cor..srctf7ive.,im lt.ho 326 CALOCLUS O' VA.IATiONS. first, the7 in finitelj Knall - ch mgVjol l-ich tho fi c ll e i; 1 undergOil s' is called its vartiation anid -1.e co (r 0:spon'ding chalo.es of the variables are their varications. The difference l,twe the tiellrm ds l"difier ential" a Ud variation," will be mace more plain by geometrical illustration. Let BC be any cuirve a furnctieon of i and y, of whlich M and M' are any two consecutive points, the co-ordinates of M being x and y. Now if the constants which deteri mine the curve be changed in any way c / \ J so as to,give a different curve B'C/, in-,,)-^"^ fin,',itely near to BC(, and so that the o'/[ pitsM and 4' shall take the posijtio:s?,I and n', Pp2 ill be the variation A f h P < 0of x ancd i l tlhe variation of y, while PP' is the dififrenti al of x ad M'Q the difffbrential of y, Art. (01).'The conditions under Avlich the va-ia'tioll is maide, may be such that one of the variables vill bave n o varia.tion; and when this is the aehe he operacitions to be pel rformiled vill be much simplified' Thu:, if it be requ-ired that the points 3 I 1 —g —--'_n'f m?:i' n/ i:&hall e fund in lines parallel / 1 ^ — ~ ^to -th ^ of Y at'n and nm', Mm will be the variation of y, ]while x has no vai'. n'ril-ation; the difflren-tials of x and y being A. - - P' and I'TQ as be-l'e. As the differential ^ dIn otedc by the I:symbol d, the Greek character 6 is used to denote the variation, and fiom the illustrations just given, it appears tlhat while the forimer symbol denotes the changes which take place in passing' from one point to anzother of the same czurve thie latter is used for a very different purpose, tc denote the changes in passin; from points of one curve to the corresponding points of another ilnoiely inear to it. CALCULUS OF VARIATIONS. 327 223. From the naitre of thev term as above explained, we see th.at to obtain thle varia tion of any function of x, y, &c., we have only to put for ty z, &c., cx - 6', y q+ 4y, &c., and then take, as in the Di sfeUt-, ia+l Calulus, Art. (49), those er tms of te developmenlt;:'_iclh are of the first degree with reference to the vanriations of tie.iarbl.es: Or, since the development may be made precisely at i an lArt. (51), by substituting Jx, Sy, &e., for,, k &c., it is pl) in t hat,re slhall have du, clg duu d "au = __,y -po -6 ~z -H- the. (d-. 6dy dz It is also plain thatL the principles contained in articles (13) and (15), as also thel particular rules demonstrated in articles (18)... (24), e re eanllul'Jy app1, caiblo to var.ations. )22 9. In thle - nc rio bi /'(j.f)...........l), let us; su,.bstituilte -u1- s Y for.t aid denote the new function by,/'(:x); then by t otie doe-i i on, iArt. (22 7), 8' -/ ) - )........... (2), and since from tile rllation expressed in equation (i), x is a function of.7 tlhe second member of equation (2) will be a function oi t, and lwye nmay 1 twrite aJ -- (,)......... (3). If in this equation woe ut, fr fi, z -l- dv --', we shall have,' - " ((''); subtracting equation (3) 32 CALCULUS OF VARIATIONS. Su' - p(r qu') - p(u) - dip(u) = dSu, But u' ~- G = du9, and takiing the varia.tions, Arts. (13) and (18), we have c' -~ u =J cdiu; hence ldu- dS u............ (4), That is; thle variation of the differential of a function, is equal to the diCt,'cnt'ial of its vartiaion: Or when both of the symbols d and 8 are prefixed to a function, the order in which they are wlritten, or in which the operatiions indicatedc are performed, can be chang'ed at pleasur-e without aftietn il te resut. The principle above enulnciated is true for any order of the differential; for if in equation (4) vwe put du for 2 7, we have da(alL) = dSdu or d_ u: ddJu -- d"u. If in the last equation we put du for u, we have S61(du) =C dS6duI, or 6d3', == d3^z, and so on; hence e wmay conclude tehat, Sd"z l- d"Su,. 280. Let v be any differential of a function of x, and place f = v', then d-vt = v Jdv' = - v, or de v' = 6-v CALCULUS OF VARIATIONS. 329 and by integration,v' =f Jv, or f/v =f -iv. The principles demonstrated in this and the preceding article; are evidently true for functions of any number of variables; since the varlation of the differential of such a function is but the sum of the partial variations, and the converse. 231. In order to consider the subject of variations in its most general sense, when applied to differential expressions, we must regard the differentials of all the variables a.s variable, as well as the variables themselves. In this sense, if u be a function containing x, y, and their successive differentials, we shall have, Art. (228), flu M lx -- hM'ldc -I - M"cds'x +- &c..........(i). +-Nfy -+- N'6dy + N"ld2y + &c. in which M,,' M/ " &T. are the partial differential coefficients of iC taken with respect to x, dx, c2x, &c.; and i, N', N" &c., the corresponding ones taken with respect to y, dy, d2y, &c.; and if this expression be first extended to any number of variables, by adding for each an expression of the form Mzx - +M'ldx + M"l"dx + &C. it may then be made to give every particular case which can arise, by making the particular suppositions upon dx, d'z, dy, d"y, &c,, which the case requires. We often meet with differential expressions contaiinng only the variables x,, dq, -, d q, &c. If we denote such expression by v, we shall have, as in Art. (228), 330 CALCULUS OF VARIATIONS. Sv tx + -NSy 4+ N'Sp3 - N, -I- &c........(2). And if this expression be taken in its most general sense, dx must be regarded as variable, in which case we put for Orp,, c,, th.eirvalues obtained as in Art. (24), viz., dy sdfdy -- d'yS6dx dc/ - pdoex dx dcx' d dp Sdxc'Sdp - dpSx dlp - qdzx dx Cdx" dx:If dx be regarded as constant, equation (2) is under its most simple form. 232. If zu be still regarded in its most general sense, we have, Art. (230), f u fSSu-; and by the preceding article,.ftu -= (MA S + M'x -r M"d2x -- &c)......(1). +f(Ny -+- N'S/y 4- N'"sd'y + &c.) By the application of the rule for integrating by parts, we find fM'lSdx —.J''drx: = M'Sx -fdSM''x; fJM"sIds =fXfM"d'x A- M"lSdx -- fdM"dlSx - M"'dx -- d'"6x +f df'ci'"1Sx; f"g/, d~3x J': f,"d MX; — I P4clf/d",dx -= W'"d d clM"'dS. -- fd 2,l"'"dSx = i"'d"x - dM..."'cldx J,- (]5{/"'0i -fd3/ / S CALCULUS OF VARIATIONS. 331 Also JN'cdy =N'y -SfiN'Sy; fN"d2y -= I"dSy dN"y — I2fc1W'G"y; fJ'!s'N"' =- 1-"'d2Sy dNi"dy + d2N"'dy — fd3N"'Sy. Observing that the second member of equation (1) is equal to the sum of the integrals of the terms taken separately, and substituting the above values, we obtain, fu = (MI - dM" + d2M' - &c.) 6x + (M"' - dM"' + &c.) dcx - (M" - &c......) d'2x + &c. +(N' - dN" + dN'- &c.)Jy + (N" - dN"' + &c.) dOy + (N1" — &c......) d2Jy + &c. I-(M.- dM' + d'M" - dM'1, + &c.) 6x................... (2). -+f(N - dN' + d'N" - d3N' + &c.) y. By examining the above expression, it will be seen that there is no term under the sign f which contains the symbols d and 6 applied the one to the other; and also that the parts containing 6x are exactly similar to those containing 5y. The formula may therefore be extended to any number of "variables, by adding for each new variable similar parts containing its variation. 233. It should be remarked, that if the multipliers of Sx and Jy following the signf, in equation (2) of the preceding article, are both equal to zero; fJu will be complete, or Su will be the differential of some function. But in the expression SftU f itG 332 CALCULUS OF VARIATIONS. it is evident that if fu contain any terms which can not be freed from the signf, sfu must contain the variations of these terms still under the sign, and fSu can not be complete. Hence if Ju is a differential, u itself must be so. And conversely; for if ft, is entirely freed from the signf, then lfu can not contain this sign, and its equal f&u must be complete, or &S be a difierential. Hence if the conditions M - dM' + d2M" - &c. 0 - dN' + d2N" - &c. 0, are satisfied, u will be the differential of some function, which may be obtained by integration. 234. Let us now take the expression fvdx, in which v, as in Art. (231), is a function of x, y, p, q, &c., we have, Arts. (19) and (127), S'vdx ==fJF(vdx) =fvsdx + fdx.v. But, Art. (140), Jfvdx = fvdSx = vdx - Jdlvox; hence S/vdx = vSx + f (dxv - dvx)............ (1). Substituting in that part of the second member which follows the signf, the values of dv and 3v, Arts. (51) and (231), dv = Mdx + Ndy + N'dp + N"dq + &c.; 6v = Mx + Nay N+' +J N"Sq 4- &c.; we have CALCULUS OF VARIATIONS. 333 dxSv - dvSx = N(dx6y - dy~x) + N'(dxSp - clpx) + N"(dxSq - dqlx) + &c............ (2). Since dy =s pdx, we have dx6y - dySx = dx(Sy - p~x) = udx, by making oy - p96x = w. Also, if for Sp we put its value, Art. (231), we have dxSp - dpfx = dSy - pdSx - dpSx = d(Sy - p~x) = dw. If in this last expression we put p for y, and q for p, and recollectthat q,=, we have dx dxSq - dqdx = d(p - q'x) = d _dx _ ),_ __ d \dx Substituting these values in equation (2), and prefixing the sign ~ we have f(dxzv - dvfx) =/Nwdx + fN'd + fN"d ) + &C.....(3). Again by Art. (140), dw/dl [ d7 dx dJ dx dx dx J dx = Ndu _ dNW W + f ldN."wdx. dx dx J dx dx d Now substituting these expressions in (3), and the result in (1), we obtain 334 CALOULUS OF VARIATIONS. dN" du fvdx vx N+' - + &C. ) + (N" - &c.) d- +&e (_ dx & c.>. +f(N - d I d dN"~ - &c.)dx. dx dx dx ) If we now put for w its value Sy pdx, the part affected with the signf will become f!(N dN &c.)dxSy - f(N - dN &c.)pdz x. d+( dd1 From which we see that, in this case, the coefficients of 6x and Sy have such a relation that if one becomes equal to zero the other will. 235. The principal, and far the most important application of variations, is to the determination of the maxima and m2inima of indeterminate integrals, that is, of integral expressions of the form /f Vdx + dy, J/ yd...... &c. containing x, y, &c., and their differentials, in which the relation between the variables is entirely unknown. Thus, if it be required. to determine the relation between x and y, in order that firydx taken under certain conditions, shall be a maximum or minimum, the problem is one not capable of solution by the ordinary method of article (66), since the principles there developed require the form of the function to which they are to be applied, and the constants which enter it, to be given; whereas the object now proposed, is to ascertain what this form and these constants must be, in order that the expression, when subjected to the given conditions, shall be a maximum or minimum. Questions of this kind are readily solved by the aid of variations. CALCUTLUS OF VARIATIONS. 335 23C. Lec- u be a function of the nature discussed in Art. (231), and suppose cl: x, y, dy, &c., to be increased by their variations, and let the diffirence between the corresponding function u' and u be developed, which is done at once, by putting 6x,,, dx, &c., for h, 2, k,,&c in the development of Art. (51), we shall thus obtain -- nvi M. +- ~ N6y + MlSzd + N'Jdy + &c., plus a term of the second degree with respect to 6z, Sy, &c.; plus other terms. By the same course of reasoning as that contained in Art. (74), vwo see that u can be neither greater nor less than u' for? all values of 6z7, y, &c., unless the term, of the first degree with reference to these variations, is equal to zero. But this term, Art. (231), is the vari2ation of su Hence in oride;r that ui be a mazzxiamnu or mint-;''uM 6 e z must s be equal to zero. ff tihe conditions which make the variat'ion of e equal to zero, nmakel the term of the second degree, in the above developmaenut, posit"ive, for all values of dx, Cy/, &c., u will be a minimum; if negative, u will be a lmaxinmum. The discussion of the various circumstancei s in whicih this term will not change its sign, is of too complica:ed a naturee, and likely to lead too far, for an elementary treati.e. NIeither is it necessary, in gen'eral, as vie shall be able, from the naiture of nearly every case, to determine without a referenc to this second term, vwlethelr we ha've a naximu. m or miinimum. 237. In the application of the foregoing principles to the indeterminate integrals referred to in Art. (235), it may at first be remarked, that if the integral be indefiznite, Art. (132), from its nature it canL have no maximum nor mi1inimum. The application can then only be made to definite integrals, or those which are taken between some well defined limits. If then, it be required thantft be a maximum or minimum, we may write the variation off f, Art. (232), thus, 336 CALCULUS OF VARIATIONS. cJu o =f u mSx ~ 2Jr J + -v'dx + r'Sdy + c., + ( + 1'y)............(1), a.nd this when taken between the prescribed limits must be equal 1o zero. We have seen, Art. (233), that this expression can not be integr'ated unless the quantity following the sign f is equal to zero: That is, there can be no integral to be taken between limits, and of course, no maximum nor minimum. We must then have for the first condition AX. + lc'Jy - 0.....(2), and since, in general, this must be so for all values of Jx and dy, which are iindependent of each other, we must also have k = 0 and k' - 0 or, Art (232), M- dM' +- dM"' - &c. = 0............ (3). N - dN' - d'N" - &c.= 0 Again; if -we denote by I and 1' the results obtained by substituting the limits in succession, in the remaining part of equation (1). we must have for a second condition, I' - = o............ (4). Should there be more than two variables in the function m, the quantity following the sign f in equation (1), will consist of as -many terms as there are variables, each of which, if the variations are independent of each other, must be placed equal to zero, and will thus give an equation expressing a relation between these variables and their differentials. CALCULUS OF VARIATIONS. 337 If, however, the conditions under wlich the variations are made are such as to render these variations in any way dependent, we shall be able, by means of the equations which express these conditions, to eliminate from equation (1), one or more of these variations; then by placing the coefficients of those which remain under the sign f, equal to zero, we shall have a system of equations from which we may determine the nature and extent of the required function. The system of equations (3) will, in every case, express the relation which must exist between the variables and their differentials, in order that the function shall be a Inaximum or minimum, but they must be subjected to the conditions deduced from the equation - I' 1 0, which can, of course, contain no variables except those which belong exclusivelv to the limits. Where u is under the form vdx, it has been seen, Art. (234), that the two equations (3) will both be satisfied, if one is. They will therefore give but one independent equation. The solution and discussion of the following problems will serve to illustrate and more fully develope the preceding principles. 238. Problenm 1. —Rtequired the nature of the shortest line joining tvwo given points in a plane. Let x', y', and x", y", be the co-ordinates of the points. The general expression for the length of the line, Art. (197), is = fV/dx2 + dy'. Taking the variation of this, we hatve 43 J( ds jd+ ) 43 0338 & CALCULUS OF VARIATIONS. which upon comparison with ecquation (1), Art. (231), gives M-=O N —O. - dz Nd and all the other terms equal to zero. He-n.ce equations (3), fc the preceding article, become d ( -) 0 and - 0, whence by integration, dx C. dz dz Eliminating dz and integrating againl, we have c, dy -= dix = cdx, y = ax -)- Z.....(1), which gives the required relation between y and x, and indicates that the line must be straight. The first part of equation (2), Art. (232), becomes M/'x + N-yo. Since in this case the limits x' y' and x" y" are absolutely fixed, we must have Fx', by', &c., equal to zero, which being' substituted in the above expression give M'x+' -t N'y' = 0 Mx" -+ N'y" = 0, whence results the fulfilment of the second condition 1' -- I=, CALCU'LUS OF VARLATIONS. 339 and it remains only to determine the constants a and b, in equation (1), on condition that the line shall pass through the two given points. 239. Problem 2. Piequired the shortest line that can be drawn from one given curve to another. Let y =/(.) a nd y f'(x) be the equations of the curves, their differential equations being dy -= p'dx dy ~- 1"dx......(1). As in the preceding problem, we have /=f Vx ~+ dfy, ~f f df - i dyd), from which is deduced, precisely as before, the equation of the required line y = ax + b......(2). But since the ends of this line must be in the given curves, the variations of z and y, at the limits, must be confined to these curves, that is, by', 6x', by", 5x" must be the same as dy and dx in equations (1), whence b' = p'S' Sy" = p"S". Substituting these, in succession, in the first part of equation (2), Art. (232), and subtracting the results, we must have 340 CALCULUS OF VARIATIONS. (zI' ~ I +' + +SX-(d-t C p/)/ 0, Al'' d^ J ^h''z" ~ dxl'~) and since this contains two independent variations, it can only be satisfied by malilng the coefficients separately equal to zero; hence dx' + dy'p' = 0 dx' - dy"p" = 0, whence dy' 1 dy" 1 dx' p' dx" p But these are the equations of condition that the required line shall be normal to both curves at the points (x' y'), (x" y"), respectively, Art. (81). In order to determine the constants a and b in equation (2), we must first find the values of x', y'? x", y", on condition that the normal to the first curve at the point (x', y') shall also be normal to the second at the point (x", y"), and then cause the line to pass through these points. 240. Problem 3. —Required the shortest line, on the surface of a sphere, joining two given points of the surface. Let the equation of the sphere be x2 + y2 + - z= _ 2...........(. The general expression for the length of a line joining the two points will be, Art. (91), w = f/Vd + dy' + dz, the variation of which is CALCULUS OF VARIATIONS. 341 dx 6dx + f ^ dy ~y 6fdz\; Jdw dw cdw whence, by adding an equation containing Jz'to those of Art, (233), and comparing, we find M:dx' N' dy p, dz dw dui dw' and thence the first condition required in Art. (237), c( 5x + -d( 7-)]y -I — dd+ j z 0......(2). dw ) dw J di But in this case the variations must be confined to the sirface of the sphere, that is, taking the variation of equation (1), we must have 2xzc + 2yyj + 2zfz: 0. Combining this with equation (2), and eliminating 6z, we obtain d dx+ d._ d _ ( x - ) d ~ \ ~ fda \ Oy -=-.~0, d Td) dW z dw \dw) I which, containing two independent variations, gives zdd-_ _ xd ( \dz 0O zd( d\ yd dz 0. {dwt) dwzo dw Kdw rNow if we regard dw as constant, these equations become zd'x2 xdz- =d 0 zdy - ydz =O 0, from which we dceduce xd2y - yd'x 0. 342 CALCULUS OF VARIATION':;. Integrating' the last tlhree equations, we have zdxr - xz z — a, dy - yz -- b, xdy - yx = c. Multiplying the first by y, the second blly - x, the third by:z and adding, we obtain cay- bx -- cz 0= 0........... (3), which is the equation of a plane passing through the centre of the sphere. The required curvle muast lie in this plane, and therefore is the arc of a great circle. The limits in this case, as in problem 1, being absolutely fixed, we have at once, as in that problem, the fulfilment of the second condition' -- I = 0. Equation (3), may be put under the form — y -- - k -' 0, or a'y - b'x + - 0, C C and the constants a' and b' detcermmmined, by causing the plane to pass through the given points. 241. In many cases where there are conlditions confining the variations, whether at the limits or not, the method of reducing the number of independent variations explained in Art. (237) and pursued, in Arts. (239-40), will be found of very difficult application. In all these cases the following less direct, but very elegant method may be used. Let r — 0 s8- 0 &c. be the equations between x, Y, &c., expressing the conditions CALCULUS OF VARI1ATiONS. 343 to which the variations are subject; then at the same time that we have 6fu - 0, we must also Iave S' =0 s -== 0, &c., or denoting' by c, c', &c., arbitrary constants, we must have tho equation (f J + cfr +- c'&s + — &c. 0......(1) for all values of the variations of f:< y, j&c. Placing the coefficients of these variations separately equal to zero, we obtain equations from which ve can eliminate thle cnstnts c, c', &c. and tlhus deduce an eq uatjuio or euaios ihich Aiill express the propel relation betiween x y, &,..iAs n illuist-ration let us. take, P>robleam 4:. -4 equired he natue of the line, of a given lengtlh, joining two points, which h with,he ordinates of the points and axis of X, will inclose the g'reatest area. In this case we have, Art. (203) and since the length of the are between the limits is to be const ant, the variations must be subject to the condition Jfd/dz —. d' + dy2 a hence (uatvdv -2- di 0, Equation (1) will then becomei 344 CALCULUS OF VARIATIONS. Sfydx + cS/fv/dx2 + d y= 0, or putting for the variations their values, we have f* Q d.^ J~ ^d~ cdxdx -- cdycdy 0 Comparing' this with equation (1), Art. (231), ve see that dx_ dy M = O, M' _ y + c, N - d' N= c -, d -z dz and these being substituted in equations (3), of Art. (237),. give -d y -x 0 d c^ -d (dy 0 and by integrating dxc dy__' y $c~=5 x - -cJ- b'. Eliminating c from these two equations, we obtain dy x - b dx y - which is evidently the differential elquation of a circle whose equation is, (Art. 98), (y - b2 + ("x- b -- 9,)2 b, b' and R being arbitrary constants, which must be determined on condition that the circle pass through the two given points, and that the included arc be of the given length.