« = 0, when x = + 00, and when y = 00; also the element-function = 0, when x = +V-1; of which values only the upper one is to be taken, because y, which is the coefficient of V -1, is capable of only positive values. In this case by (153), B = 0(*) <, when x = v=1, = V=; :|27) dx = 1(1) (160) In illustration of this formula take the following examples ; (1) Let $(x) = axv-1 = cos ax+V–1 sin ax; in which case it is to be observed that sinar) f'(x+yv-l-e ay(cos ax +V 1+x_y2 + 2xyVI which vanishes when x = +00, whatever is the value of y; and, when y = +00, whatever is the value of a; thus, (V-1) = ena; and consequently po cos ax +V-1 sin ax, - dx = nea; po cos ax (161) po sin ax, (162) and consequently, as in the second case of the preceding example, 1.zm1 dx = "V-1 ( - 1)m-1-(1)-- 1 1+(-1)-1 If in (163) and (165) m = 2p, and x is replaced by Vã, then po ap-1 dx 11** = cosec PT;.. ..(166) = (167) which are results of the same form as (72) and (73), Art. 94. In this case however p may be any positive proper fraction. 107.] If the form of f'(x) is such that all the corrections for infinity and discontinuity vanish; then a = 0; and from (159) we have f'(x) dx = 0. .. (168) All these conditions are fulfilled when earvi-e-a cos ax + V-1 sin ax-e-a f'(x) = 1+x2 for (1) the element-function = 0, when x = + oo, whatever is the value of y, and = 0 when y = 0o, whatever is the value of x. (2) A=10(V-1)=0; since earr-i-e-a = 0, when x = V 1. . [ cos ax +V-1 sin ax -e-a dx = 0; 1 + a2 .. so that equating possible and impossible parts, . . p® cos ax dx J- 1+22 (169) po sin ax dic T2 = 0; (170) J- 1 +22 the same results as (161) and (162). 108.] The limits of the x-integration in the preceding Articles are oo and –00; they may however be changed to co and 0 by the following process. Since (171) . = A. The following are examples of this equation. the same result as (120), Art. 101. 109.) M. Cauchy has also made another application of the general principle of the inversion of order of integration in a double integral which it is expedient to insert, as it exhibits the applicability of the principle to another form of function. Let f'(z) be the element-function of the required definite integral, wherein z is a variable whose modulus is r and whose argument is 8; so that z = r {cos 8+ V 1 sin 0} (185) and consequently the element-function is f'(reov); and let us suppose this element-function, as also its first derived function, to be finite and continuous for all values of its modulus less than R. Now, since and Let the two members of this identity be multiplied by dr de ; and let the limits of r be r and 0; and of 0, 25 +a and a; then (kole some vidr = Talde rentydd.(187) Let us moreover suppose f'(reov-1) to be such that, when 2n +0 is substituted for 0, its value is unaltered; then the righthand member of (187) is manifestly zero; and since |