596 
Proceedings of the Pioycd Society 
where Qi is as in § 1, and P* is a function of fx and p. The 
equation (10) becomes successively 
d (PeQi) 
dy (0-~P 2 ) ( ff) + 1 — y 2 df ' + *'(*+l)I’«Q.'=0, 
1 d*(P<Qi) 
(TP, f n dPidQi „ <2 2 P 
dy + ^ ~ r*) l 2 
0 
d[x dfx 
+ Q i 
1 
) 
Q { eZ 2 Pi 
i -- n 
+ 1 - f 2 dp 2 ~ u > 
2 ^* + rfP, 
fx 2 Qi djx 
d 2 ?j\ Qj d 2 Fj 
-t e\ ^ *-». \J « 
d/x 
df 
) 
1 - /x 2 dp 
n „ 0 
+ 
Qi d 2 Pi 
dPi ' 1-y 2 da 2 
= o, 
and, finally, 
0-/*•)« -/*■) a ^ 
2 ^ ^2 dPi \ . _ 4 c? 2 P,- 
+ Qi =0. 
If we put, for a moment, 
d{x 
(T^VjQ? 
dv (which has a real meaning, see § 11), 
and suppose Qi to he expressed in terms of v instead of /x, calling 
it the equation may be written 
d 2 Vj 
dv 2 
(24). 
Hence it appears at once that Pi cannot contain p except in the 
form of factors, such as cos. sp, sin. sp, in the several terms of 
which (as an integral of a linear equation) it must he composed. 
Hence, as before, 
(s) 
P i = 2o A 5 ©i cos. [sp + a), 
and, keeping to one value of s, 
<2 8 0i (,) 
dv 2 
