466 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
where X, s are positive integers which we can so choose that the real parts of 
n + 'X + 1 , p + 2s -j- 1 are both positive, in which case 
r p -f 2s — 1 
c . 
r(-i+, i-) 
( t 3 — l)' !+A t p+2s dt — t sin ( n + X)tt . (l — e i ,+1+2 *" r ) 
whence we find, as before, 
, i \ II (n) n 
r(—i+i i-) . v ' \ 2 
j ( t 2 — l)“ tP dt — l sin 7177.(1 + e~ pmL ) 
XI (+ A, + 
'P ~ 1 
p + 2s + 1 
n [n + 
p + 1 
We have now, letting = — n — m — r — 1, 
. TUn)H( - n+m+r -l 
Q ™u ) = e ~ {n+l)m £ (1 _ e ( ii+w+ .-M n ( ft +!!L±i1> __ X _ 2 _ , 
4,1 [IX) 44 sin 7477 ^ 2*11 ' II (r) fn-m-r' * 
u( — - — - — -l)u(n + m + 2s) 
c -(»+l)ur oo l 9 / V ' 
- __ /I _ Rfc+m+O _> _ ^_ L _ 
“ 9»+2 V 1 ° / + f n — m — 2S’ 
s=0 n (2s) n -———- 
2s 
o —(71+ l)t*r 
TT , — % — m — 2s — 1 \ -,—r 
n (-^-— i j n (tv + vi -f- 2s 1) 
+ “^TF" (1 + 2 
3 = 0 
IT (2s + 1) n 
?i — m — 2s — 1 
2s+ 1 
By the known transformation theorem II ( — +) IT (cc — 1) = 77 cosec xtt, we have 
II n ~ m ~~ ^ s -d 
2~ 
„ (m — n + 2s \ fn + m + 2s 
II ( -—-— _ l cosec ( - ■ „ - h 1 ] 77 
n 
n — m — 2s 
2 
n 
n + m + 2s 
I m — n + 2s^i 
cosec [ - 177 
[m — n — 2 \ . vi — n 
II I---1 S Sill - ; -77 
„ !m + 74 \ . 714 + 74 
n I — + s sin — +— 77 
also 
n ■- n - m — 2.s - 1 - 1 
IT Im — 74 + 2s — 1\ ( 714 + 74 + 2s + 1 . 
II ( - | cosec [ -p 1 77 
n 
74 — 7/4 — 2s — 1 
74 + 7/4 + 2s + 1 
II 
0/4 — 74 + 2s + 1' 
cosec I-- ] 7 r 
rr /7/4 — 74 — 1 \ 7/4 — 74 
n (- h S COS -r - 77 
„ ,'7/4 + 74 4- 1 \ 774 + 74 
n I-p S COS -77 
