500 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
or, on putting v = cosh w, this becomes 
o * (n) = —- e mn — n + (u* — 1 \i«* 
V ’ 2“ II (n - vi)' n (to - i) ^ ' 
[ {p + vV 2 — t cosh w;} w m 1 &\rdi~ m w div . (85), 
where the real parts ofm + i, w — m+l are positive. 
If m = 0, we liave 
Q w (^) = {/x + v //x' 2 — I cosh itf] dw . 
J o 
( 86 ) 
The particular case of (85), when m and n are real integers, is given by Heine.* 
When /x has a real value less than unity, we have, on using (31), 
/ m 1 IT (n + to) II (— A) . m n 
Q,r (cos 9)— - — ~-- —w sin 0 
2 m+l n (» — to) n (to — i) 
sinh 2 ™ to 
o (cos 0 + i sin 6 cosh to)' 14 ” 1 * 1 
cZir 
+ 
sinlr m ic 
J o (cos 9—i sin 9 cosh u’) a+m+1 
dw 
and from (30), 
P,/" (cos 6) — 
1 IT (n + to) IT (— -|~) 
. LIT " II (ft - TO) II (TO - 1) 
sin" 
(9 
sinh 2iil to 
o (cos 6 — i sin 6 cosh w) u+rn+1 
sinh 2m to 
f- 
Jn (C 
40. In the formula 
o (cos 6 + i sin 9 cosh to) ,1+5B+1 
dw 
dw 1 . 
TT (m 1 ) fl Jl n+m 
Q/ (p) = e w, H2'". n ^_ ^ cos mir (fi~ — l) J j* (1 + j^+h 
dh 
(43), 
which holds, provided the real parts of n + m + 1 , o — m are positive ; put 
n+1 
h — /X — y/pd — 1 cosh iv, then when h = 0, we have w = w Q = \ log,, —— , and when 
fl 1 
h = —, w = 0, thus since 1 — 2h[i + /r = (/x 2 "1) sinh 2 v, we have 
3 
Q,” M 
= —— cos mrr. — (p— \Z^— 1 cosh w )"*’"siuh Sm wdw (87), 
n (—i) Jo 
where w n = 4 log mod. ^ + ] , and the real parts of n + m -f 1, § — m are positive. 
U ^ O yx — 1 
* See 4 Kugelfunctiouen,’ vol. 1, p. 222. 
