EQUATION AND ITS TRANSFORMATIONS. 793 
Tins result may be readily obtained by differentiating both members of the equation 
r COS bx 7 7 T 7 
1 - dor - "~ al 
J 0 (ct~ + x~) n 2 a 
n— 1 times with regard to « 2 : see infra, art. 31. I now proceed to investigate the 
value of the integral when n is unrestricted : it is to be observed, however, that n 
must be positive and greater than unity, for otherwise the integral is infinite in 
value. 
It is easy to prove that the integral 
satisfies the differential equation 
for, by actual differentiation, 
„ cos ap 
cPu „ p(p — 1 ) 
~——a z u= - - — u ; 
dxr x* 
and by a double integration by parts we find that 
xp 
Thus the value of the integral 
where U and Y are as defined in art. 3, and A and B are constants to be determined. 
It is however more convenient to avoid the determination of the constants by 
deducing the value of the integral from the formula (8) of art. 21. 
In the ‘Journal cle l’Ecole .Polytechnique,’ Cah. xvi. (vol. ix.), p. 241, Poisson has 
proved a formula which, after some unimportant transformations, may be written 
mus ^ °f tl ie form x p ] (AU + BY), 
x^e 
dx= 
r(% + l)f°° cos 2 bx 
*fir J 0 (l+a; 2 ) 
dx ; 
Poisson’s demonstration holds good for all values of n such that the integral upon the 
right-hand side of the equation is finite. Putting n —1 for n and transforming the 
vd 
right-hand integral by assuming x— this equation becomes 
* ‘ Quarterly Journal of Mathematics,’ vol. xii., p. 130. 
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