OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
495 
where, as before, the phase of cos 9 dt i sin 9 cos \p is ± it, when \p = according 
as the upper or lower signs are taken in the exponentials. 
36. In the important case in which m is a positive integer, we find, from (68) and 
(70), that 
II (n + to) (p 3 — 1)*" 
II (n — to) 2 m II (— |) II (to 
( (p + x/p 3 — 1 cos xp) /l m sin 2 "* i// chp 
~ 2> JO 
is equal to 
P H m (p), oi' P«" (p) — — e ±nm sin rnr. Q/ 2 (p) . . . . (77), 
according as the real part of p is positive or negative. 
From (69), (71), we find in this case 
P(/*) = 
( jj ~ — l) im 
II (n + to) 
n (n — to) 2 w n ( — 4) n (to 
rut 
Sill 2 * 1 -yfr 
(p + \/p 2 — 1 cos yp)“ 
+ in + 
! d\ji, 
when the real part of p is positive, and 
e T2mu V„ M (p) -f — sin mr . Q n m (p) 
7T 
(p 3 - 1)1“ 
riiC. 
sin 2 ’" yp 
II ( n + to) 
II (n — to) 2“ II ( — |) II (to — 4) Jo (p + \/p 3 — 1 cos yp) n 
1 + hl + 
. (78), 
when the real part of p is negative, the upper or lower sign in the exponential being 
taken according as the imaginary part of p is positive or negative. 
When p = cos 9, we have in the case in which m is a positive integer, 
P/‘(cos0) = ( — iy 
sin " 1 6 
TL(n + m) 
n (n—m) 2 “ n (— 4 ) n (to— 4 ) 
—— ( (cos 9 i 1 sin 9 cos *p) n m sin 3 "* \p clip. (79), 
2 ) J 0 
when 9 may have any value between 0 and n. 
Also 
II ( n + to) sin'" 6 
P/*(cos 9) = (— 1)" 
sin 3 '" yp 
n (n — to) 2“ n (— |) n (to — 4) 
where 0 lies between 0 and hn, and 
— e Tim P/ 1 (cos #) (e Tilm + i sin W7r) + — e Tim sin «7r . Q/' (cos 9) 
. , u II (n + to) sin'" 6 f 7r sin 2 '" ^ 
1 1 rr z™ _ ™ \ hiUrr / _ i\ tt J 
0 (cos 6 + l sin 6 cos yp) n 
+ 1)1 + 
i dip) 
II (n — to) 2'" II (— J) II (to — |) J 0 (cos 9 + t sin 0 cos yp)' 1 
+ M + 
jC lip . (80), 
where 0 lies between ^77 and 77 . 
