498 
DR. E. W. HOBSON" ON" A TYPE OF SPHERICAL HARMONICS 
Suppose the real part of p, to be positive, and the path of integration to be a circle 
with centre at the point p,, and of radius greater than mod (p, — 1) and less than 
mod (p- + 1). On this circle take a point C such that the angle between [xz and p.C 
is \jj, and let (/> be the angular distance of any point of the circle from C. If we put 
t — [x -f \/p a — 1 T “, the point t represents, for different values of </>, points on a 
circle of centre /x and radius e™ mod (v/pv — I); we must thus take u to be such 
that mod (y/p., a — l) > mod (p, — 1), and < mod (p, -f l), or m < | log mod . 
Take tire circle commencing at C to be the path of integration; we have 
t~ — 1 = 2\Z[x z — 1 . e‘ 0 * ,- ' w Tu [p, -f \/jX 3 - 1 cos (</> — i/j T: iw)]. 
Hence we have 
P>1 >1 (p) — , v n 0j) J o [p + v/p 2 — 1 cos (c/j — i/; ± cl(f>, 
or 
^ f2jr __ 
— {p, -+• \/p T — 1 eos (</> — *p i iu)} >l (cos mxf) — l sin m$>) d(f) 
* 0 
- P w ^ O J— 
” " n o + m) 6 
On changing m into — m, and remembering that 
we have 
:tt j o 
T . / \ IT (n — m) „ , . 
p,r m p = fpy-- P,r (p 
' H [n + m) v 7 
(p, -f '/p' — 1 cos (<j) — \Jj di tw)}" (cos m<f> + t sin m</>) d<f> 
= p„*M - 1 - 
IJ -+- m) 
we thus obtain the formulae 
