OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
513 
point z. The part of the integral for P„ m (cos 0) which consists of integrations along 
the two finite circular axes is 
o t7r £) 
9 m 
n (- i) 
e mm sm m 
e f'{< 
J a 
t<l> + (m-£) (*-</>) t / _ - 
(— te 
or 
2“IT (— \) II (m — \) 
sm' 
_ e i«-m)*+(m- i) (»-++)* ( te ‘*)} (2 COS 9 — 2 COS <£)»-* d(f) 
9 f 2 cos {(m -j- |) 7r + (w + i) (2 cos 9 — 2 cos <b) m J c/(f>. 
J A 
The part of the integral which is taken along the circle of infinite radius is zero, 
and the part taken along the real axis is 
g-wr (m-i) 
2- ‘n(-i)n (m-i) 
e~ mm sm - ” 1 
m(n—m+l) + (n+i)v 
9 f {e~ l7T(n 
J 0 
- e ^-rn+l)+(n+i)^n-i)4nij ^ COS 9 + 2 COSh v) m ~* dv 
or 
2*n( 
— ^ - — sin m 9 f 2e (n+i)v cos (n -f- \ + m) n (2 cos 9 + 2 cosh v) m-i dv. 
27 Jo 
we thus obtain the formula 
P« m (cos 9) = 
^ I cos m7r . P/' ! (cos 9) -— sin mn . Q „ m (cos 9) l 
II (n -f- m) [ 7r J 
O TT . —~ 1 ( cos (n+k ^ 77 ) (2 cos 9 — 2 cos c?</> 
2™ II ( — 1) n (m — i) sm M! 0[Jo v 2r 1 2 /v r 
-f- j e ( n+i)v cos (n-\- ^ + m) u . (2 cos 9 -f- 2 cosh v) m ~* dv^ (109) 
which holds, provided the real part of n-\-m is negative, and that of m +-\ is positive. 
When the real part of n is between 0 and — 1, and the real part of m is between 
MDCCCXCVI.—A. 3 U 
