468 
DR.. E. W. HOBSON ON A TTPE OF SPHERICAL HARMONICS 
U . f LL \ ~ n ~> n ~ 1 
that of t is 6, so that the phase of 1 — ~ is — 2tt, thus (1 -) is equal to 
gaoi+w,)™ ti mes the value given by the Binomial expansion ; we have therefore 
p-{n+i)Ln i TT (r> 4- m 4- 95 f(- 1 + . 1_ ) 
Q.- M = i--• e~ 2niri (/- - l)i" $ ( rr \ } (*-!)" t~ n ~ m ~ r ~ l dt, 
1 ' 2 4 sin nir 2 n H (n) Vl ' IT (r) ) c v ' 
r (-1 + . 1 -) 
The value of (t° — l) n t? dt may be found as before by first considering the 
J C 
case in which the real parts of n and p are greater than unity ; in the present case, 
o 
-/ 
o 
o 
+/ 
the phases of t are tt in the integral from 0 to — 1 , and from —1 to 0, and zero in 
the integrals from 0 to 1 , and I to 0 , hence the integral is equal to 
fjl — v zy v p(e™ l e- 2nm eP rim —e nm . 1 . e pTl " + e nm . 1 . l — e~ ,im . 1 . 1 ) dv, 
ip - 1\ 
or 
(e nm — e~ nm ) (l + e pm ). 1 
H (n) n 
8 n((« + S±l 
The extension of this result to the case in which one or both of the quantities n, p 
have their real parts greater than — 1 , can be made as before. 
We thus find after reduction, as in the preceding case, 
Q,"M -- 5 — .2". 
,'n + m — 1\ 
n(-X— ) n(- I) 
IT 
n — m 
(/r 3 - l)* ra F 
n + m + 1 m — n . 0 
O-’ -’ 2> 
n + m\ 
n —s- n(-i) 
_|_ c (i«, + l»). 2 ». / _ _ V — (/X 3 — l) iW >F 
n 
m — 94+1 944 + 71+2 
|,/X 2 • ( 26 ), 
which is the formula that corresponds to (25). 
