EQUATION AND ITS TRANSFORMATIONS. 
769 
These integrals form three pairs U and V, P and Q, B and S, either of the integrals 
in each pair being dedncible from the other by the substitution of —(p +1 ) for p: 
and, since the differential equation involves p only in the form p(p-\- 1), it is evident 
a priori that if in any expression satisfying the differential equation, p is replaced by 
— (_p + l), the new expression must still satisfy the differential equation. 
Also the pairs P and Q, It and S, are convertible the one into the other by changing 
the sign of a. 
4. If p is a positive integer the series in P and It terminate and the general integral 
of the differential equation is AP -j- Bit; and if p is a negative integer, the series in 
Q and S terminate and the general integral is AQ+BS. 
Thus, if p— 2, the general integral is 
u ■= Aar 2 {1 — ax -f \arx -} e ar ~b Bar 3 {1 -\- ax -f- la~x 2 }e~ ax ; 
and, if p— —2, the general integral is 
u =Acc -1 {1 —ax] e™ -f~ Bar 1 {1 + ax } e~ a ' x . 
5. As however we have six particular integrals, of which, for any given value of p, 
only two can be independent, it remains to investigate the relations between the 
particular integrals in the different cases that arise. 
(1°.) Suppose p unrestricted (except as mentioned in art. 1), but not equal to an 
integer. 
In this case all the series extend to infinity, and 
P=B=U, Q = S=V 
for, leaving out of consideration the factor x~ p that occurs in both P and IT, the 
coefficient of a n x 11 in P 
1 _ p i , Kp- 1 ) 1 . ■ / y p(p- i) • • • fp-fo-i)} 1 
n ! p (n— 1)! p(p—\) {n — 2)!2! . p{p — h)---{p— 2 ( n ~^)} n] - 
1 _p 2 p(p- 1) 2 3 _■ / y, p(p- 1) • • • 2 n 
n\ 2 p {n — f)! ' 2p(2p — 1) (n— 2)!2! ’ 2p(2p — f) . . . (2p — (n — 1)} n\ 
I \ 2p(2p — l) . . . {2p —(?i— -1)} 
= 2p(2p-l)...{2p-(n-l)}[ 
(2p—l)(2p — 2) . . . {2p-(n-l)} . p(p- 1) . . . {p-(n- 1)} 
_ TT r-2-••+(-) 2], 
and we see that the expression in brackets is equal to the coefficient of V in the 
expansion of 
MDCOCLXXXI. 5 G 
