800 
MR. J. W. L. GLAISHER OX RICCATI’S 
and, if p is a negative integer = — i— 1 , so that 7 =— then, from the second and 
fourth forms, 
rz= ( 2 _ 2 ?+ iy +i ^ 2,+ ^"^)’ 
or 
v— f 
7\ s / - zi --*« N 
ay / n^e s +c 2 e e \ 
& 
)■ 
31. These formulae may be readily connected with the series-integrals found in § I., 
for, comparing (20) with the series P in arts. 3 and 5, we see that 
x 
i+1 
1 ^Y +1 -nr \ -if ! ^ T — 1) a ~ x ^ afr 8 
i *) 6 =■ A * j 1 - V7- ^ T _P_i) "sT 
+ &c. \e ax , 
where A is a constant. Putting i — 1 in place of i, this equation becomes 
1 d 
x dx 
i -1 . (i-l)(i-2) «¥ (i-l)(i-2)(i-3) afr 3 
e ax =A.x 21+1 \ 1 ——ax , 
z-1 fr-l)fr-|) 2! (i 
-l p2 i-3 a 3 P , c 1 „ 
-ix<-*x<-2) ^ +&c -r : 
and, observing that the coefficient of x { e? x is a\ it is evident that 
_/ \ z -_i fr'P + 1 ) • • • (2i— 2 ) 
A=(-)'■ 
9J -1 
a. 
Writing the terms in the reverse order, as in § IV., we find 
2 ) A-fe rt- • (21) : 
2.4 
that is, on replacing a? by pfr, 
d 
dx 
e a-Jx — 
1 (< + iW-l X <- 2 ) JL_ &C . 
2y/xJ [ 2 apa; 2.4 a-a; 
which is a known formula (see, for example, Schlomilch’s ‘ Analytische Studien ’ 
(1848), p. 86 ). 
The formulae which result from comparing the solutions of PiCCATi's equation (4) in 
arts. 17 and 30 are 
/ (1 \ * + 1 a 
4 _%+l f) * = 
(~)‘ 4 1 ~ PA^+r PAl ( !rhV 1 £* ( 22 ), 
(2 — 1 X 3 ? — 1 ) . . . {( 2 i — 1)2 — 1 } [ A ? — 1 ) (?(? — 1 ) 22(22 — 1 ) 
