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« = 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;
L. 1+ x2
and separating possible and impossible parts,

po cos ax
= nea;

(161) po sin ax,

(162)

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and consequently, as in the second case of the preceding example, 1.zm1 dx = "V-1 ( - 1)m-1-(1)-- 1

1+(-1)-1

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If in (163) and (165) m = 2p, and x is replaced by Vã, then

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po ap-1 dx

11** = cosec PT;.. ..(166)
po XP-1 dx
Jo

=
1-2
cot pr;

(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
- 1+co

J- 1+22
= ne-a;

(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

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(171)

. = A. The following are examples of this equation.

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the last integral being the same as (77), Art. 94, and as (107), Art. 100.

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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}
tee V-1;

(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

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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

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