OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
485 
change from h to w as independent variable in the integral, where h = — (I — iv ); 
we then have 
q,r (h-) = - . 2 ™. n - (m ; )n( . l ) 
^ 477 sm (n + m) 7 r 
(z 2 -!)*+» 
(/—l)i-[ 
(OH , 1+, 0-, 1-) 
u~ m ~ i+r (1 -u) n+m ( 1 + - 
u 
du 
II (rn — \) II_( —_ 4 ) 
477 sin + 77 
_—- („2_ 1 Um V (_1 V 11 ('"*' +2+0- 1 _ 
(s 8 -!) m+i ^ ; r fo v '' n(r)n(m-i) (2 2 -iy 
(0 + , 1 + , 0-, 1-) 
W 
—m—J-H-r 
(1— «) n+,B dw. 
On evaluating the definite integrals we find 
Qu M 
2 W . n( n (/-l (t x ' 2 - 1 ) imF ( m +i w +i 
1 — 
( 57 )> 
_ 2 ^ 
which gives an expression for (p) in powers of (— - - f 1 ~ h-, which is convergent 
2 vp 2 — 1 
for the part of the plane over which this expression has its modulus less than unity. 
Using the formula 
— mm 
P '™ M - W* (^) sin ( n + w) 77 - (p) sin (n - m) 77}, 
77" bUo /< 71 
we find from (57) 
p.- w 
2~n (— i) (n (» + >») sin (» + m) 7r z*~“ , , .1 II II 1 
= eos^ -»+*,»+*, izs 
no-i) y y (^_pi .f / m+ 1 -m+i, 
n (?i—?ft) o 2 —i)" ,+i v ’ \ 2 ^ 25 1 — 2 s / 
Now by the known formula for the transformation of a hypergeometric series 
whose fourth element is 1 — x, into a linear function of series whose fourth element 
is x, we find 
F(m + 4 -m + 1 + I, 
1 - z 3 
— n ( n 2) n ( n + 2) -p / m 1 1 
14 (7t — 7ft) II (ft + 7ft) ' 2 ’ 
m + & 1 
1 - z 2 
+ 
n (- n - I) n (n + i) 1 
n(771 -i)n(-m-|)(i- z *) n+i V i - z 2 ; 
F (1 + m, - m, w + I, 
1 - 
