476 
DR. E. W. HOBSOX OX A TYPE OF SPHERICAL HARMOXICS 
In (A) or (B), n may be changed into — n — 1, and m into — m\ we thus have 
eight different forms which satisfy the differential equation, and as in each case two 
independent closed paths may be chosen, we obtain, on the whole, sixteen definite 
integrals which satisfy the differential equation (1). We shall proceed to express 
these definite integrals in terms of the functions P n m (/x), Q,™ (/a). 
Consider the integral (A), the path of integration consisting of a loop described 
positively round the point 1, followed by a loop positively round 0, another negatively 
round 1, and lastly a loop negatively round 0. When the loops are placed as in the 
figure, we shall suppose that the phases of u, 1 — u initially at A are zero, and that 
'LL 
the phase of 1 — — is zero at B ; when the loops are disjfiaced into any other position 
the proper phases will be obtained by the principle of continuity. We have then 
z n +m+ l f uH+m (l - - 4) du 
z 2 
- w 
yll + m + 
\bn r = cc TT ( m i r _ JL\ 1 f (1 + , 0+, 1-, 0-) 
f- S - 1 \ 4 - t** + " +r (1 - «)—* du ; 
1 r = o II (r) IT (m — ■§■) g 2r J v ' 
now 
' (l + i 0 + , 1 —, 0—) 
u !l+m+r (1 — u) m du = e (,l+r+ ’) " L € (n + to + r +1 , — to + ^) 
(n+, ' +?) " / lX) . (n + m + 1).. .(n + m + r) „ , , , , , 1X 
= e •(-!)' x.-n — , i r~ € ( n + m + ~ m + i) 
(■n + |)... ill + r + 1) 
II ( n + in)- 
ji-Oi+sn m + 1) • • • ( n + m + ’) . • / . \ 
= e ( -—;-— -7 T— • 4tt sm in + to) tt . „ . .. _ . 
(n + f) ,..(n + r + v ' IT (n + £) II {m — \) 
Hence 
/ <i 2 1 f( H-,0 + ,1-,0—) / % \ — i — m 
--Z - T 7 - I u» + m ( 1-4 du 
z n + m + l J \ / \ g J 
II (n+m) (/x 2 —Iff 
= — ie mu . 4 77 sin (n+m) tt . 
n n (m—i) 
yy/i + m + 1 
r F n+7n+ 1, n + f»- • 
Comparing this result with the formula (31), we have 
Q" W 
=Le {m ~ n) 7 r \ 2 r ‘ 
n (m—-i) n(-n ( At 3 -i)^ f< L+ -°- f _ x , ...c ttN - *”"* 
477 sin (n + m) ir z 
n + m + 
r( 
r J 
u 
u+m 
du (39). 
