2 Differentiating with respect to y, dx dx dx d'% w = 0. dyl daadw dy dx dy' (da) dy This is the general equation to surfaces generated by the motion of a line which constantly rests on two given lines while it remains parallel to a fixed plane. (22) Eliminate the arbitrary functions from %= (ay + bx).4 (ay – bw). Taking the logarithm we have log x = log o (ay + bx) + log y (ay – bæ), and as the functions are arbitrary their logarithms are also arbitrary functions, and we may replace them by the general characteristics F and f. Therefore, differentiating with respect to x and y successively, 1 43 = 6F" (ay + bx) – 6f" (ay – ba), Differentiating again, -) = bF"(ay + bx) + b2 f"(ay – bx), = a' F"(ay + bx) + aʼf" (ay – bx). Multiplying by aʼ, h? and subtracting, we obtain as the result of the elimination of the functions fd’z 1 dz) 22 dl x 1 (dz)) Idx? \dx) (1) 6f (a) + y Q (a) + x4, (a) = 1, (2) u f'(a) + y P'(a) + xy' (a) = 0; f', , being the differential coefficients of f, 0, y. Differentiating (1) with respect to X, {«f'(a) + y $'(a) + x \'(a)} * * + f (a) + y (a) ** = 0; which by the condition (2) is reduced to f(a) + y (a) ** = 0. In the same way, differentiating with respect to y, we have ♡ (a) + y(a) * = 0. dz Since from these two equations it appears that and dc are both functions of a, the one may be supposed to be a function of the other, and we may write drid Eliminating the function F from results this equation there (2) G ) - ( ) - 0. This is the differential equation to developable surfaces. ELIMINATION O CONSTANTS AND FUNCTIONS. ELIMINATION OF CONSTANTS AND FUNCTIONS. 51 (24) Since was l _ ) is a homogeneous function of m dimensions, we know that CHAPTER V. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE DEVELOPMENT OF FUNCTIONS. This theorem, the most important in the Differential Calculus, and the foundation of the other theorems for the development of Functions, was first given by Brook Taylor in his Methodus Incrementorum, p. 23. He introduces it merely as a corollary to the corresponding theorem in Finite Differences, and makes no application of it, or remark on its importance. The following is the statement of the theorem : If u = f (x) and x receive an increment h, then . žu n du h f (x + 1) = + + de 1.2* dix 1.2.3* If we avail ourselves of the method of the separation of the symbols of operation from those of quantity, this theorem may be expressed in a very convenient form, which is useful in various parts of the Integral Calculus: viz. It is frequently convenient to use Lagrange's notation, and to represent the successive differential coefficients of if (x) by accents affixed to the characteristic of the function. In this way Taylor's Theorem is written If we stop at any term, as the nth, which is f(n-1) (0) ha-1 1.2... (n − 1)' i, the error committed by neglecting the remaining terms lies between the greatest and least values which Oh" fla) (x +01), - can receive; where 1.2...n is less than 1. can This is Lagrange's Theorem of the limits of Taylor's Theorem. See Lagrange, Culcul des Fonctions, p. 88. Also De Morgan's Differential Calculus, p. 70. Ex. (1) Let f(x) = (a + 2)". Then (a+&+h)*= (a +x)"+n(a +a)=="h+ n(n-1) (a+2)»-2K+&c. as 12 (2) Let f(x) = a*. Then as a * = (log a)" a", a*+= «® {1 + (log a) h + (log a) + (log a) +...}. If we stop at the nth term the error lies between the greatest and least values of al* +0l) (log a)" — The 1.2 ...n least value is found by making 0 = 0, and the greatest by making 0 = 1, and therefore the error lies between h 1 h 1 h log (x + h) = log x +-- 2 +- - &c., |