526 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
n(p 0 ) 
n (p 0 + £ - m) n ( - i) 
sinfficr. e (p0+i m)tr ] 1 + 
(i + m) (p 0 + h - m ) _ 2(r 
1-J>0 
(£ + to) (| 4- to) (p 0 - TO + 1) (p 0 - TO - 1) 
^ l-2.Po^o-l e 
(4 + m )...(£ + to + p 0 - l)(p 0 - TO + £) . (- TO + £) ^ 
l-2...p 0 .p 0 (p 0 - !)•••! 
which we shall denote by S 2 ; and second the undetermined form 
2“ * in ~ + 7 7 • -rf7 to'tt sinh^o-. e (p +m+ V« F (m-f £, p+f+m,_p + 2, e 2<r ) 
cos (^ + i) 7T II (_p+1) II (— v 1 2 5 1 ' 
d-2 w 
n (p) 
(£ + »»)• • • (| + TO+ffo) (p -TO + 1) . Mj; -TO + 1 -p 0 ) 
n (_?+£—»») n (-i) 1.2... (p 0 +i) .p<jj-i)... (p—p 0 ) 
e(jJ+i -m- 2 ^- 2 ), f x + i + m+j?0+l.j?-7n, + ^-ff 0 - e 
^o + 2 .p-p 0 -l 
+ • 
•I 
where in the limit, p — p 0 . 
The numerical coefficient of the second series is equal to 
2 ,„ _ n ( p) __ n(j3 0 + m + b) u(p - to + i) n(p-p 0 - 1 ) 1 
" n (p + i — to) n (— T)' ri (to — b) ' n (p — p 0 — m — p,' n (p) ' n(p 0 +1)’ 
which is equal to 
2’" II (p 0 + TO + I) n (p 0 — P + rn — ^) sin (p 0 — p + to + it 
n (j3 0 + 1) ■ II (TO - 1) n (-1) * n(p 0 -p) ' sin (p 0 - p + l) tt 
Now the limiting value of the ratio 
sin (p + \ + m) ir / sin (p 0 — p + m + J) tt 
cos (p + l)ir / sin (p 0 — p + 1)tt 
whenp — p Q , is easily seen to be — 1, thus the coefficients of the two series are equal 
and opposite infinities. 
Evaluating the indeterminate form according to the known rule, we obtain first an 
expression, which we shall denote by S 3 ; this is 
S„ = 2” 
n (p 0 + to + |) 
n ( -1) n (to - n (p 0 + i) 
. sinh” 
L n= 
V ~ Po 
p Po sin (p — p 0 ) 
■‘P=Po 
d_ 
clp 
n (Po ~P + m - J) 
sin (p 0 — p + m + £) 7r . e 
n(p 0 -v) 
T I I + ™ + p 0 .p - Po - rn - b ^_ 2 
-()» + !) <r+ (p—2p 0 ) a- 
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