808 
MR. J. W. L. GLAISHER ON R1CCATPS 
Similarly, by transforming (31) to the form, 
If® --l -**-4 7 
u—-\ x q e x ax, 
SJ o 
we find that the differential equation is satisfied by 
u 
=(£T°- 
2Va 
(33), 
if'-1=— 2i—2 that is, if q= — 
q 1 2t + l 
The formulae (32) and (33), on substituting for a its value in terms of z, lead at once 
to the solutions given in art. 29. 
This method was applied by Poisson* to show that the equation (4) is integrable 
when q=±^- i . 
The formula (25) of art. 32 may be easily deduced from the equation 
for we have 
and also 
so that 
whence t 
qlx= • 
2 s/a 
dir! d 
r 
J o 
r a -a*'--! d71 
x~’e x dx———. 7 , . 
J n 2, \ da) a 
r dh 
‘ 00 
wm=- w£r< 
5 \i t >— 2 V ( ab ) 
\/a 
d \V~2Vlai) 
—2 V(ci) 
(34). 
* ‘Journal de l’Ecole Poly technique,’ vol. ix., pp. 236, 237. Poisson’s investigation is reproduced in. 
De Morgan’s ‘ Differential and Integral Calculus,’ pp. 703, 704. The formulas (32) and (33) are 
obtained in the same manner as in the text from the integral (31), and the solutions in art. 30 are deduced 
from them, in a paper “ On Riccati’s Equation ” (‘ Quai'terly Journal of Mathematics,’ vol. xi., 1871, 
pp. 267-273). 
f In the paper “ On Riccati’s equation,” referred to in the preceding note, the following two formula) 
occur 
«n*d*y _ Vsw)= (-yyyj e -^w>. 
These are inaccurate owing to the omission of the factor dft in both, and a wrong sign in the latter; 
when these corrections are made, both become identical with (34). The formula (34) is given, and (35) 
is deduced from it, in a paper “ Sur une propriety de la fonction e '•'■ v ” (‘Nouvelle Correspondance 
Mathematique,’ vol. ii. (1876), p. 240). 
