EQUATION AND ITS TRANSFORMATIONS. 
807 
leading to the complete integral 
—fj—ll 3 ^ V’ l A 
— or. V -M or ?— ’ 
U — X 
x dx) x 
It will be observed that 
d r cos aP f 
J.(l Tm* d ^-Pl 
db) o(l + &| 2 ) 
£ 2 cos a% 
(1 + 5 P )? +1 
dl 
both integrals being finite for every positive integral value of p, and that the second 
integral when integrated with regard to a between the limits oo and a 
_ .f* g sin (cog ) r" g sin (ffg ) 1t 
~ v \ o (i+&pp +l f o (i+ n 2 y +1 d ^ 
The former of these two integrals is zero, as it can be shown that 
J n 
£ sin 
d£ 
, 0 (i+jn * +1 * 
diminishes as a increases, and, in the limit when a is infinite, vanishes, A similar 
remark applies to the second integration with regard to a. The above process does not 
therefore involve the assumptions, sin co = 0 , cos co = 0 . 
36. Poisson’s theorem quoted in § V., art. 20, viz. that the definite integral (5) 
satisfies the differential equation (6) shows that Riccati’s equation 
dhi 
— a~z 2q 2 a— 0 
( 4 ) 
is satisfied by the definite integral 
u 
> arz 2 ^ 
e ~ x ‘ q -± tf&'dx 
(31). 
dz~i 
Putting ", = a, and transforming the integral by taking x 2rjt =ax' 2 , we find that 
4 T 
which, if - — 1 = 2 i, 
1 if* i-i 
- = -oc 2 H x? 
9. Jo 
1 ,.*> 1 , J_ 
e~°*'~**dx, 
1 . v i/^vr 
\lx 
’ 2 [dec) Vu q K ’ 2 \du 
i+1 
o-2Va . 
so that the differential equation is satisfied by 
