482 
DR. E. W. HOBSON ON A TAPE OP SPHERICAL HARMONICS 
_ I)i-K r(i + . 0+, i—,o—) 
~n+m +1 
(1 — u) n ~ m 1 — — 
— il — Ml— 1 
du 
(/x 2 - l) im v n (n 4- to + r) 1_ j^. 0+ ’ l ~> °-> 
p‘ +m+ i n (r) n (?t + m) 
-If 
r+m—i ( I _ 
(I — u) n ~ m du. 
Now 
r(i + ,o+, l—, o—) 
U 
r+m- 
'* (1 — u) n m du = e ( ' l+, ' +f)t ’ r € (r -f m + n — m + l) 
..'■+■> 
+ !)■•• ( m ■*- r - i) , • / \ n (m - 1) n(/I - m) 
— e (+5;ir 7 --- 7 -:. 4 cos m tt sin (n — m) tt - =77 -———- 
(n + $)...(% + ?■ + 1 v ’ IT(>t + A 
hence the expression becomes 
e 0l+f) " r . 4 cos m tt sin (a — w)tt 
II (m — J) IT (n 
II (n + A) 
— {j l „n +m li . F (n + m + 1, m +i« + f,4 
Comparing this with the expression (31), for Q,“ (/x), we see that 
Qf" (/*) = te 
- , 0 {yn-n)ur 
9 hi 
n (u + m) n ( — |) (fj , 2 — i) 
4 cos m 7 t sin (n — to) tt H (n — to) II (m — |) 2 ’* +, " +1 
d+, o+, 1-, 0-) 
iN“=(l — u) ll - m l 1 
it 
o 
— a—m—1 
rfit . (54). 
26. We shall now consider the expression 
(/x 2 — i) im n l+ > j?+ > l -> j2 -> 
. w+?/i+l 
h _ -- 
— il— Hi— 1 
cZtt. 
Using the transformation u = z 2 — (z~ — l) v, the expression becomes 
- n rtn r 
- i)- ,m A- I 
n+m r(l + , z 2 + , I —,z 2 —) 
1 — 
| "I ill—J 
(tf — I) rt_m v -n~»i-l c l V} 
which can be expanded into the form 
2>i+»i TT (ryu _IT / 1 \r f(l + , 2 S +. l-j &—) 
-iiAU—U-/ — 1 V ( 1 — ) (v — \y- m v r ~ n ~ m ~ l dv. 
W L ) 92m ^n(r)n(m-r-i) 1 > ' - 2 1 1 ' ’ 
\ 
