dx2 dx dy dx dy This is the general equation to 2 + dx (da) * = 0. dy surfaces generated by the motion of a line which constantly rests on two given lines while it remains parallel to a fixed plane. while (22) Eliminate the arbitrary functions from ≈ = 4 (ay + bx). ↓ (ay – bx). Taking the logarithm we have log = log (ay + bx) + log ↓ (ay – bx), 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, Multiplying by a, b and subtracting, we obtain as the result of the elimination of the functions (23) Eliminate the arbitrary functions from (1) x ƒ (a) + y $ (a) + ≈ † (a) = 1, where a is a function of x, y, and ≈ given by the equation (2) x ƒ' (a) + y p' (a) + ≈ f' (a) = 0 ; f', ', ' being the differential coefficients of f, p, Y. Differentiating (1) with respect to x, da {x f'(a) + y p′ (a) + ≈ y' (a) } + f (a) + † (a) da which by the condition (2) is reduced to In the same way, differentiating with respect to y, we have dz $ (a) + 4 (a) = 0. dy dx dy dz Since from these two equations it appears that and dx are both functions of a, the one may be supposed to be a function of the other, and we may write dz 'dz dy CHAPTER V. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE SECT. 1. Taylor's Theorem. 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 a receive an increment h, then 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 f(x) by accents affixed to the characteristic of the function. In this way Taylor's Theorem is f(x + h) = f (x) + f'(x) h + ƒ" (x) written. If we stop at any term, as the nth, which is f(n-1) (x) ha-1 1 .2 (n − 1) ... maining terms lies between the greatest and least values which h" f(x) (x + Oh) 1.2 n ... can receive; where is less than 1. Also De Mor This is Lagrange's Theorem of the limits of Taylor's Theorem. See Lagrange, Calcul des Fonctions, p. 88. gan's Differential Calculus, p. 70. Ex. (1) Let f (x) = (a + x)". (a+x+h)"=(a+x)"+n (a+x)"−1h + 1.2 Then n (n-1) (a+x)"−2h2+ &c. If we stop at the nth term the error lies between the h" greatest and least values of a*+0h) (log a)" least value is found by making = 0, and the greatest by making 0 = 1, and therefore the error lies between (3) Let f(x) = log a. Then since by Chap. 11. Sec. 1, and the error of stopping at the nth term lies between |
