522 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
or, 
(/ - 1) =(n-m+l)U(n+l, m) - (n + m + 1) (», m). 
Referring to the formulae (40), (50) for Q/ /l (/x), P /t m (/x), we see that by choosing 
specified closed paths for the integration in U (ft, m), each of the functions is of the 
form C,„ (/x 2 — 1 )-'"U (n, rn) ; we thus obtain the formulae 
(M* - 1) = (n - m + 1) P„ +1 - M - (» + 1) ,xP,- M ] 
(H?~ ]) ^2^ = (» - m + 1) Q, + M - (ft + 1) mQ„- (iU.) I 
(123). 
Next let Y (n, to) = U (— n — 1, ?n), we have then by changing n into — n — 1 
in the relation which has been found above for U, 
(/x 2 - = _ ( n + m ) V ( n - 1, m) + w/x Y (ft, to); 
special cases of this relation are 
rflY" (/x) 
(/x 3 - 1) 
d/j, 
n /xP^ (/x) — (ft + to) P ?( _f" (/x) 
f- • 
(j^ 3 — 1 ) = « /*Q* W (/a) - (w + w) Q„_ (/x) j 
from (123), (124), we have at once 
(2 n + 1) fiP* (/x) — (ft — to + 1) P* +1 * (/x) — (ft + to) P*_i* (/x) — 0 
(2ft + 1) /xQz (/x) — (ft — TO + 1) Q ;( + 1 w (jtx) — (ft -f ?ft) Q«_r (/x) = 0 
(124), 
(125), 
these recurrent relations between the functions for different values of ft hold for 
general complex values of to and ft. 
55. It has been remarked in Art. 1, that W, which is equivalent to U (ft, m), satisfies 
the differential equation 
/1 9\ d H (ft, 7ft) 0 / I i \ dU (77, 7ft) , / \ / i i i \ TT / \ A 
(1 — /x~) -——- — 2 (7ft + 1) /X---- + (ft — to) (77 + TO + 1) U (ft, 7ft) = 0, 
(IfP 
now 
rfU (n,m) 
dfi 
rf 2 U (ft, 777) 
= (2to + 1) U (/?, 7ft 4 - 1). -yi— — (2 to 4- 1) (2to 4- 3) U ( 77 , m 4- 2) 
thus 
(1 — /x 2 ) (2 to 4- 1) (2777 4- 3) U (ft, 7ft 4- 2) — 2 {in + 1)( 2ft7 4- l)/..U( 71, 777 4- 1) 
4" (ft — 7 ft) (ft 4 - 7 ft 4 - l) U ( 77 , 7ft) = 0 ; 
