EQUATION AND ITS TRANSFORMATIONS. 
781 
(3 n .) n — a negative uneven integer, 
P 1 =R 1 =U 1 =^(S 1 '-Q 1 '), Q^S^V^-MQ.' + S,'); 
where 
i 
9 1 = (-) 4 
K»+1) 
1 2 .3 2 .5 3 . . . n~ 
o a. 
This is perhaps the simplest form in which the six integrals can be exhibited ; and, 
having regard merely to the simplicity of the series and to the expression of the 
manner in which they are related to one another, (3) should be preferred as the 
standard form of the differential equation both to the original form (1) and to 
Riccati’s equation (4), which is considered in the next article. The form (3) is that 
adopted by Bacii in his memoir (see iv. of § VIIL). 
It may be observed that if p=i, a positive integer, the differential equation (2) is 
satisfied by v=- X coefficient of Zf 41 in the expansion of e ax ' /a+1 ‘\ and if r p= — i, by 
v=xX coefficient of li l+l in the expansion of e a ^ +K> \ these results follow from §11. 
i 
17. Transforming the equation (3) by assuming x=nz~\ it becomes 
or, putting n — 
s- 2 dH * n 
2 'd*- aH=0 - 
d~v o n _o A 
— — z v—0 
dz i 
Riccati’s equation in its original form is 
d l+bf=cz m ; 
it may without loss of generality be written 
|+r=*”, 
1 dv 
and, putting y = - —, it becomes 
dh 
dz 2 " 
■z m v— 0, 
(4). 
Thus (4) is the equation derived from 
dy t o o o,, ra i 
; 4 - 
dz J 
by assuming y = , and it is convenient to regard it as the standard form of Riccati’s 
J & J v dz ® 
equation. 
