EQUATION AND ITS TRANSFORMATIONS. 
815 
and, by integration by parts, 
t sin xt 
2p{ 
J n 
,dt= 
sin xt 
OO /.CO 
, . X cos xt 7 
+ 1 . dt 
Io (a 2 + t 2 y +l L (« 2 + ^) ? Jo Jo (a* + t 2 ) p 
whence the right-hand member of (44) 
„. , cos xt f cos xt , 7 
=x~P I \ Ult 
05 ), 
J I 
0 [<> 2 + G> p (er-M 2 F +1 
cos xt 
dt—cAu. 
o (a? + t 2 y +1 
If the limits were a, /3 the differential equation would still be satisfied if the 
cpiantity in square brackets in (45) vanished between these limits. This is not the 
case for any other values of a and /3 besides 0 and co , but if in (43) p-\-1 is replaced 
by — p, so that the integral is 
u=x p+l \ (<x 3 +£ 3 )^ cos xtdt. 
then the quantity in square brackets =—(a 3 -f- t z ) p+1 sin xt, which vanishes when 
t=±ai', and therefore the differential equation is satisfied by the integral 
rai' 
'z=xP +l \ (a~-\-t 2 ) p cos xtdt. 
— ni' 
Since in this case the quantity in square brackets vanishes in virtue of the factor 
(a 3 +F)^ +] , we may replace cos xt by cos (xtp-a), a being any constant, so that the 
solution of the differential equation may be written 
fai' 
u — Cx p+1 (a 2 + t 2 )p cos (xt -f a)dt 
J —ad! 
. . . (46). 
If in the differential equation a 3 be replaced by —a 3 , this integral becomes 
u—Cx p+1 i (t 2 —-apy cos (xt-\~a)dt .(47), 
J — a 
which is Mr. Gaskin’s formula (38). 
40. This is not however, as stated by Mr. Gaskin, the general integral of the 
differential equation, as if in fact contains only one arbitrary constant. For, evidently, 
