490 
DU. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
To obtain a closer approximation for large values of n , we use the theorem 
II (n) = \Z'2tth . e "n u ( 1 + + . . . 
we have 
now 
hence 
or 
n (n) 
\/n . c u . n u 1 + 
^ \/ n + ^ . e (>i+i) (n + ^) n+i (l +77 
12 ii + 6 
, approximately 
= 77, ( J + in) 1 e * • (' + "> ne g lectin g terms in jj , 
/ 1 \ —n 
lo g ( 1 + 2^i 
— — n ( y — n =>) — — i ~h "q ; 
\ In 8 n~ ~ 8 n 
1 
1 + 7^1 = e M 1 + hT)’ approximately, 
8n 
n 
= -±)(i 4 . ±\ = A = ( l _ A 
II (n + |) s/n\ 2 n ) \ 8 n ) \Ai \ 8n 
when terms in . . . are neglected. We thus find as an approximation to 
n (n) 
II (n + m) 
V„ m (cos 0), by taking the first two terms in (59), 
V ^s?( 1 -£){ sin ( ,, ' + i^+f + T) -L 4? i d : ne 
cos(n+f9+j +Y 
or 
IT (n) 
IT (n + m) 
P/ 1 (cos 6) = fsj 
2 
mr sin 6 
1 — 2m-\ . /— r - . n . 7 T VlTT 
1 -W“) sm ( ,i+ = S + T + T 
1 8 p cot 0 cos (p + | 0 + —■ + y 
(63). 
Similarly we find 
n (n) 
II (n -f m) 
Qr (cos 0) = sj 
7T 
2 n sin 0 | 4 n 
1 - 4m s 
, 1 + 2m S \ /- T/1 7T fW7T 
1-7-) cos (n + Id 4- — + — 
+ 
8 n 
cot 6 sin [n + \Q + [■ . (64) 
In (63), (64), n is large but not necessarily integral, and m is not necessarily integral. 
32. When p is real and greater than unity, let it be denoted by cosh ifj ; in Art. 
28, P u m (p) has been expressed in terms of two hypergeometric series, in both of 
