506 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
where n — m is a real integer, and u > \ log,, mod 
P + 1 
A* - 1 
It has been shown in 
Art. 11 that the expression in (93) is zero when n — mis a negative integer. 
When m and n are both integers 
1 [ 2 * _ 
^ j. {/* + vV ~ 1 cos (<f> — xfj db w)Y 
n (n) 
iill( ^ w±ra “ d(f) = - yj\ P/‘ (p) . (94), 
n (n + m) 
the right-hand side is zero when n and m are positive, and n < m, since in this case 
P/ (p) = 0. 
Next change m into — m, in the formula (93), the expression on the right-hand side 
then becomes 
II (n) 2 _ . „ _ . , 
n (* - m) l p ’ r "' M “ V c ~"” Sln • Q ““"M 
or 
II (n) f 2 
frv--r 1 P/ 4 (p) — — e~ mm sin mn . Q/ 2 (p), . 
n (71 + m) [ vr/ 7T vr/ J n (n + m) t r 
n pa 2 
— sin mr. e~ (n+2m)m ,Q n m (p), 
XI (72/^ 
which reduces to ^ ^ ^ ^ ^ P/* (p), since n -f m is a real integer; we thus have the 
formula 
^ j o + vV ~ 1 cos w)Y l e ” u( * wTm “ d( i> = (p) (95), 
which holds for all values of m and n such that m -j- n is a real integer; when m and 
n are positive integers such that m > n, we have P,/“ (/a) = 0, and the integral in 
(95) vanishes. 
44. In (93) change n into — n — 1, we have then 
JL C” 
27r J 
b— mi ±»iit 
{/X + \/p 2 — 1 COS ((f) — + m)} 
)i+l 
d(f> 
n 11 y A - 1 ) { P "" M + V c ”‘ sin w w 
where m — n is a real integer; now, suppose m and n are both real integers, it is 
then necessary to evaluate the undetermined form on the right-hand side ; to do this, 
suppose the modulus of p is greater than unity, and substitute the series in powers of 
— for the functions P/ z (p), Q_„_p ! (p); the expression then becomes 
