784 
MR. J. W. L. GLA1SHER ON RICCATI'S 
and a similar expression derived from this by changing the sign of a, are particular 
integrals of the differential equation 
cPv n — ldv n 
-- 1 -«r=0. 
CtX/ X ctx 
Iii the case of Bjccati’s equation, 
~aW-*v=0, 
CLZ 
if q= the reciprocal of an uneven integer, the two particular integrals are 
g 3 — 1 / t \ , (g 3 —l)(3 s g 3 —1) / 1 \ 2 (g 2 - 1)(3Y-1)(5¥-1) / 1 \ 3 
^ H-Vfe) + 
g.2g \8 a# 
g.2g.3g 
8azi 
-f &c. \ei 
and a similar expression derived from this by changing the sign of a. 
These appear to be the best forms in which the integrals can be presented when 
the equations admit of solution in a finite form: but they do not suggest the solutions 
for the general cases when the letters are unrestricted. The series ultimately become 
divergent when they do not terminate. 
§ v. 
Evaluation of definite integrals satisfying the differential equations. Arts. 20-28. 
20. It was shown by Poisson* that the definite integral 
y= f e~ xm - h ^dx .. (5) 
J 0 
satisfies the Riccati’s equation 
. . ..( 0 ), 
so that the value of the integral must be of the form Athj+BV^, where U 3 , V 3 are the 
same as in art. 17, and q—\m ; it remains to determine the constants 
A and B, 
* ‘ Journal do l’Ecole Polyteclmique,’ Caliier xvi. (vol. ix., 1813), p. 23?. 
