814 
MR. J. W. L. GLAISHER ON RICCATI’S 
p being supposed to be any positive quantity ; and the process of verifying that this 
is a solution of the equation is as follows. By actual differentiation, we have, as in 
art. 26, 
“ iK-f {^-(2p+3)jr 2 } ■ 
dx 2 
(41), 
and, by a double integration by parts, 
cos a% 
1 sin of 2(p + l)| : cos a% 
\< i (x‘+py*' d £ _a(x'+i-y* 1 a? (y+py* 2 
2(p+1)r {^-(2p+s)e}~f^de. . . ( 42 ). 
The integral (40) therefore satisfies the differential equation, since the quantity in. 
square brackets vanishes between the limits of integration. 
If these limits had been any quantities a, /3 independent of x, instead of 0, oo , we 
should have obtained a result corresponding to (41), but the quantity in square 
brackets in (42) would not have vanished. Replacing y> + l by —p, it is clear that 
u 
—x (x 2 -\-£~) p cos a£d£ 
* a 
will satisfy the differential equation if a, /3 can be so chosen that 
^(jc 2 +P) ? sin 1 cos 
is zero. This would be the case if'" /3=xi, a= — xi', but these values of a, /3 are 
inadmissible as they are not independent of x. 
Transforming now the integral in (40) by the substitution a£=xt, w r e have 
x p+ if cos a%> +1 x~ p f C0S xt dt 
J o (.r 2 +J 0 ( a s+ ty + 1 ’ 
and therefore 
u = x~ p ( 
cos xt 
J 0 (a 2 + ^ +1 
dt 
also satisfies the differential equation. 
To verify this, we find by differentiation 
(43) 
d~u pO + 1) n 
•---- - -u— 2px p 1 
7 ° 
I • 
X* 
t sin xt i r cos xt , 
pdt—:—-—~dt . 
o (a^ + t 2 )^ 
J 0 (^ + < 8 )' +1 
( 44 ) 
* As i denotes a positive integer in this memoir, i is used to denote V ( — 1). 
