OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
493 
P/ (p) — ~e mm sin mzr . Q„** (p) 
(p 2 - 1)*» 
II (n + to) 
\Un -~m) ¥» n(- i) n (to - i 
j- f (p + \/p a — 1 cos xp) n m sin 2 " 1 1 |/ d\fj (68). 
2) Jo 
This relation holds for all values of n and m, subject to the conditions that the 
real parts of m+ /x are positive ; the phase of p + vV 3 — 1 cos \p is the same as 
that of p when r Jj = br. 
In (68) change n into —n — 1, we then have, on using the relation (18), after 
some reduction 
P n m M ~ 
7T 
e mm sin mu . Q„ m (p) 
n (n + m) (p 2 - 
n (% — m) 2 "‘ n (- i) n (to 
sin 2 " 1 \Jf 
(p + \/p 2 — i COS \fr) 
71 + 7)1 + 
i d\\t . (69). 
34. From (68), (69) it is easy to find the corresponding formulae for the case in 
which the real part of p is negative ; in this case we have 
P/ (— p)-— e mm sin mu. Q„ m (— p) 
7T 
•> T mm 
IT (n + to) 
II \n—m ) 2'" II ( — ^) n (m— 
—- (p 2 — l) iw f e T(n m)irt (p + v/p 3 ~ 1 cosi^P m sin 2 " 1 \jjd\fj. 
— i) Jo 
The expression on the left-hand side is equal to 
e T,m p * /\ _ 2 si n(n+ja)_u ^_ mm , _j_ _2 & _ mn ^ mn>e ±n m - Q * / \ . 
7T 7T ' 
hence 
P„ OT (p)-- e -wm sin P7r.e 
7T 
±(R — 7>i) 7Tt ?>2 
Q»* (/*) 
II + to) 
n(?i—m) 2 ™ n (—^) n (??x—■§■) 
.i^ n / w ix (/r 2 - l) iw f (p+ v/p 2 - 1 cos t/f)« sin 2 " 1 1 fj dxp (70), 
where the upper or lower sign is to be taken according as the imaginary part of p is 
positive or negative ; (70) corresponds to (68). 
In a similar manner we find, corresponding to (69), 
— c T2,m P/' (p) d-e mm sin mu.e :f{n+m)m Q/* (p) 
u 
H(n + m) 
II (n—m) 2 m II( 
_ - _f- sm ~ m f _ dxl> (71) 
-4)11 (m-h) [l ' Jo (p + v/p 2 -! cos 1 |r)»+™+ 1 r K h 
