[420 
11 
420] 
ON RICCATl’s EQUATION. 
op so soon as there 
n if the particular 
tions to be satisfied 
foregoing values of 
and the subsequent 
- 1 ; the coefficients 
e for a the finite 
q , thereby obtaining 
Reverting to the equation in z, we have next a particular solution of the form 
z = Ax + Bx q+1 + Cx 2q+1 + Dx? q+1 + &c., 
giving between the coefficients the relation 
(2 + l)H+(ç + l) q B = 0, 
(3q + 1)B + (2q + 1) 2q G = 0, 
(5q + 1 )C + (3 q + 1) 3 q D = 0, 
(7 q + 1) D + (4>q + 1) 4g E = 0, 
If A = 1, we have 
A = 1, 
(g + l)(3y+l) 
ri _ , K'i T ~ *-) 
{q + 1) q (2q + l)2q' 
(g + l)(3g + l)(5g + 1) 
(q+l)q(2q + l)2q(3q+l)3q’ 
&c., 
where, as in the former case, the series is considered to terminate as soon as there 
is an evanescent factor in the numerator, without any regard to the subsequent 
coefficients which contain in the denominators the same evanescent factor. [In particular, 
q = — 1, we have the solution ^ = #.] 
Hence if we have (2i +1) q = — 1, the series terminates, and we have for u the 
finite particular solution, 
from which, changing the sign of x q , we deduce the other finite particular solution, 
op so soon as there 
n if the particular 
tions to be satisfied 
foregoing values of 
and the subsequent 
- 1 ; the coefficients 
e for a the finite 
q , thereby obtaining 
where q (2i+ 1) = + 1, we have (writing D = 1) 
—XI 
> q 
= CP'+_Q' 
y CP + Q 
2—2