EQUATION AND ITS TRANSFORMATIONS. 
779 
Therefore 
\J-gY=V'=H'-2g$, 
U+flrV=R / =P , +2flrQ. 
whence 
U=i(P'+R'), Q = S = V=i(R'-P'), 
which agree with the relations found for the case of a positive integer in art. 5. 
If r p is a negative integer =— i — 1, then ; we may therefore 
replace i by — i — I throughout in the integrals just obtained, and thus deduce the 
system of integrals considered in (3°) of art. 5. 
15. It maybe observed that, since the series for {1 — ^/(l — 4 t)} n and {l-py^l — ±t)} n 
in art. 11 terminate and recommence when n is respectively a negative or positive 
integer, it is evident that the solutions in series of the differential equation satisfied 
by them will present points of similarity to the solutions Q and P of (1). The former 
differential equation is 
rJZ V fill 
and its integration in series is considered in a paper “ Example Illustrative of a Point 
in the Solution of Differential Equations in Series ” (‘ Messenger of Mathematics,’ 
vol. viii., pp. 20-23). 
§ III. 
Transformations of the original differential equation. Riccati’s equation. 
Arts. 16, 17. 
16. If the differential equation 
cPu 
db? 
2 P(P + 1) 
a l u =-^—:—u 
( 1 ) 
is transformed by assuming u—x~Pv , it becomes 
dfv 
dx 3 
2 p dv a 
— —~a z v= 0 
x dx 
5 H 2 
(2) 
