PKOFKSSOIi O. TT. DAUWTX OX FlJjrSOTDAL ITAR.NFOXTO. AXALVSTS. 
f) I 7 
It is iiOAv necessary to evaluate the several definite integrals involved in tliis 
expression. 
It is well known tliat 
k(P,V) = .rd 
+ 1 ' / - 
It is easy to see that it is possible to express P' + -'' in tlie form 
p . -../, ^ pp._ pp,._^ + . . . , 
where A, B, (J . . . do not involve p. 
The value of xV may he found liy considering only the highest jiower of p on each 
side of the identitv. 
Now 
p. + _ u-pj- _ 1) 
20/1 w?p/ ' 
2/i 
_ ( _ + 1- -rt: i t 
' ' o; „ 2/,’ I ^ ^ ‘ ‘ ' 
2' . 1 ! ? — s 
and 
Therefore 
P' = ( _ Vi*-" • n> 4- 
' ^ 2''. t! t - .9! ^ ^ 
A =(-Tt 
I — S: 
1 - Si - 21-] 
Tlien, since the integral of the product of two P's of different orders vanislies, we 
have 
fjW’-a/f. = 1 3,, = (-)';d 
+ 1 i - Si - 2k \ 
+ 2 /. 
w e will next consider -where h is not zero. 
J J — 
+ 2 /.- 
p- 
The differential ecjuation gives 
f fi - + >■(<■ + i)P"” - P' 
tpL" ' </p j j - p- 
./ 
Jp 
(, _ p) i^p’I + p,-+ ,)p. 
/^j 
- p- 
P' = 0. 
Multiply the first of these hy P^ and the second ly P" 'and sulitract, and we 
have 
4/.(,s + A')r*P 
1 - 
+ 2 /' 
= P' 
21 
(1 
tlfi 
(' - 
l/p 
- P^ 
d 
dpi 
(1 - p2) 
d\ 
>.< + 2i' 
dpi 
