508 DR. E. W. HOBSON ON A TYPE OE SPHERICAL HARMONICS 
(z -j- ax + fiy) H = r n [p + \/p 2 — 1 cos (<p — xp dz m)}” ; 
we should, therefore, expect that if {p + vp 2 — 1 cos (<f> — xp d: tu)}" is expanded in 
cosines and sines of multiples of (p, say 2 (w, a cos vup + v m sin mcp), the coefficients 
u m , v m would be linear functions of the functions V n m (p), Q/' (/a). 
Let io — v/p 3 — 1 e ±L{<i> ~* ±LU) , we then find that 
{/a + V p 3 — 1 cos (<£ — ± (,«)}” = (2w)~ u (p + w — i)' J (p + w + 1)“. 
If w < log mod a/ ^ + | , one of the expressions (fx iv — l)", (p + w -f- 1)" can 
be expanded in positive powers, and the other in negative powers of iv ; if, however, 
u > log mod /\J d - ~t . j } both expressions can be expanded in positive powers, or both 
in negative powers, according to the sign taken in ± lu. 
Case I. — u< log mod a / C + \ . 
° v p — 1 
In this case all the powers of w in the expansion are of positive or negative 
integral degree, thus 
{p + \/p 3 — 1 cos (<p — xp ± ai)} n — t u m cos m<p + v m sin m<j5>, 
i?i=0 
where m has all positive integral values 
We have 
\ f2jr 
'V'm — 
7T 
{p + v / 'p- — 1 cos (<p — ip ± m)}“ cos nup 
except that 
also 
v n 
= 2P„* (p) - v n( - ° cos m (i// T i«)» 
v ‘ II (a + ?/t) v 
U 0 = P n (p)> 
pir _ 
= — (p + i/p 3 — 1 cos (<p — xp i iu)} n sin nup 
7T Jq 
= 2P ”“ w n (« ( +m) sin m 
hence 
(p + \ // p r ” 1 cos (<£ — ^ i tw)}’ 
= P H (p) + 2 S 
n (w) 
= i n (» + m) 
by (81), 
P,™ (p) cos m ((p — xp i im) (98), 
