Mutual Induction. 369 
and the limits of <£, <£' are to 7r, and those of ft, p are 1 to 
taken twice over. 
n 2 (n + l) 2V y dr dr' \ d/i )^ \ V V=o 
Jo 
We have 
f 
T> , ^P» ^P n _i 
But P~— 3 — =nr n . 
dfju dfju 
.'. differentiating m — 2 times 
'-^F* + (m_2) ^^ — rf^^ =n *?i=i- (2) 
Now writing ^" i ~ 1 P» _-p ) " 
(2) becomes 
fC=^+(»-» + *)Itr • • • (3) 
.*. from (1) and (3) we have 
C m m 
+ mJ(l- l ^?- 1 [D^ 1 + : (n-i^)D!U 1 ]* ■ W 
Again, 
(1— /r 2 ) -j-2 =nP n -i— nfiP n . 
Differentiating m — 1 times 
i. e. 
(l-^)D;=«D:-_ 1 1+/ a(2m-2-n)D;_ 1 + (m-l)(m-2-n)D2. 2 . (5) 
