480 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
If we use the known transformation 
*(«• ft y. *) = n % -«- i) n I) - f| F & 1 + “ + * ~ * 1 ~ 
f 
n( « + 13 - 7 - 1 ) 11(7 - 1) 
TI(« - l)II(/3 - 1) 
(I — a:) Y_a p F (y — a, y - (3, y — a — (3 + 1, sc), 
we find for our expression 
2 t sin mu - 1 n(m-l)n(-i ) f U(n - j)Tl(-_ 2nO_ 
^ 1 ' ’ II (m - ft) |n(»-m)n(-m-i) 
F ( — n — m, 4 — m, ^ — ft, -) + 
n (- u - 1 ) n (- 2 m) 
n (— v. — m — 1 ) n (— 4 — ra) 
. — 2 / 1—1 
F in — m + 1 , - m + £, f + n, fi » 
which can be written 
TTi . g 
(a 1 
02m 
-j \-fm_ ^ f hi ( n 2) n+m _ _ 
n (m - i) [n (n - m) " 1 2 
sin (n + m) ir IT (n 4- m) 
+ 
cos uu II (n 4- 4) 
F ( — « — m, ^ — m, J — ft, — 
1 F (ft -f 1 — m, — m, ft + f, "A 
1 \ 
or, if we use the transformation 
F (a, /3, y, x) = (1 — x) y a * F (y — a, y — /3, y, x), 
it can be written 
n^ri) {f^rdo*— (F - l > i ” F (i + m ■ m “ »> 4-”> 
sin (n + mW II On + m) m „ , , „ i . , , , , 1 
+ co ,mr n <7Ti) 2 F- - 1)- I (i + *»,« + *» + l,» + f — 
on referring to (22), we see that it is equal to 
_ ^ 7rt n ( 2 ) J_ p ,«/ \ . 
n (m - i) 2™ * ’ 
we thus obtain the formula 
(^) 
_i_ on hi (m — ft) _ m _j / o , y> m f (1 
~2w z n(-si z w Ol. 
(!•>-, £ 2 -) 
U 
n+m 
U 
(1 i_ du (49); 
