486 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
and thence, after some reduction 
P,“ (/*) = ^ ^ + m) \ e~ {m ~ i)m 
7 r n (n + 2 ) 1 (f ~~ 1)* 
^m+u+l 
zi F (1 L+m, m, n+ 1, 
1- 
+ 
(2 2 -iy 
l(/F— I)' iB, F (4 + m, f—m, ?i+f, —-) f- . (58). 
This formula expresses P,/" (/x) in powers of ^-^ 7 ==^“ 
29. Let /x = cos d, then remembering that P/' (cos 6) = P/' (cos 0 + 0. 1 ), we 
have 
P,“ (cos #) 
If 
7r n (?i+ 4 ) 
2’”n( 2 ) n (n + m) w . ^ f e jr/i 1 rr i_ m ?1 1 3_ 
-ifl 
+ 
(2e* l7r sin 0)" i+i 
/pi 
sin 0 
( 2e ur/2 sin 0) 
n + f, 
sin 0 
Hence 
P,/" (cos 6) 
2 U(n + m) - cos ( ?7T i 0 - ~ + -T ) 12 _ , JJ|S cos I ^ 
- V^r n (n + 1) — i -- D + 
/_ 
(2 sin 0)* 
2.2» + 3 
_4 
(2 sin 0)* 
+ 0 
l 2 - 4 ) 7 i 2 .3 2 - 4m 2 cos ( n + A 0 - _r + - t 
2.4.2 n + 3 . 2n + 5 
75 a 5?r , m7 M 
“_1Y 1 IaI + ■■■ ■ < 59 ): 
(2 sin 0)i _ 
this series represents P„“ (cos 6 ) for unrestricted values of n and m, provided it is con- 
7 r 
vergent, which is the case when - < 6 
57r 
6 ' ' ' 6 
To find the corresponding expression for Q/' (cos 6), we have from (57), 
Q" (cos 9 + 0.l) 
_ i2>i n(-i)n (71 + m) e ( “—)‘ 9 («*" sin 0)“ 
II (7i + 4) g + (2e t7r/2 sin #)"' + i 
i F (£ + w > i - m > ™ + 
2f 17r ' 2 sin 0 
- ..mm 14 ( 
£) IT (71 + 777 ) 0 -(ni + I)0i--4 r _ l2 _ 
n (71 + i) (9 sin mi I 1 2 2 
(2 sin 0)i 1 
- 4m 2 e~^ s + 
. 2n +3 2 sin 0 
l 2 - 4m 2 .3 2 - 4m 2 e ~ + ^ 2) 
+ 2.4.2/7 + 3.2/7 + 5 2 sin 0 
• * * ' 1 
Similarly we find 
Q,“ (cos 6 — 0 . t) = e” wi . 11 ^ ±_E>. i 
t (n + i) 0 + wr/^ 
IT (71 -f ■§•) (2 sin 0)* 
_ F - 4)/i 2 F< 9 + ;r2 ) | 
2.2?i + 3 ’ 2 sin ^ I 
thence using the relation 
