462 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
p = 
Q — mTTL 
it COS Hit 
{Q“ (ft) sin (n + m) v — Q_ n _i m (/x) sin (n — m) 77} . (18). 
This relation which has been proved to hold over the domain of the point 
ft = — 1, must hold over the whole plane. 
In the case m— 0, we have 
tan iiTT 
P» V)= V {Qn W — Q-»-l WI¬ 
TT 
If ft -f- m is a positive real integer, we have 
P-,T (/x) =- Q—mm cog mjT # Q_ n _* (p). 
7T 
If — m is a negative real integer, the relation (18) becomes 
IV" (/x) = - e r,m sin r/177 . Q” (p), 
7T 
we see therefore that in this case the two functions P,/" (p), Q n m (ft) are not distinct. 
Changing m into — m, in (18), we have 
PV m (ft) 
n (?t — m) 
7r cos ?i7r [II (it + ?/t) 
e~ m7ri U(n- m) 
7T COS 7177 II (w + m) 
Q, “ (/x) sin (71 — w) 
77 
II ( — 71 — m — 1 ) 
II (—n + m 
—~ Q_„_f"(/x) sin («+w)t 7 j 
sin (n — m) 77 (Q,“ (/x) — Q_*_r Ox)}, 
hence on substituting for Q_ M _ 1 “ (ft) its value given by (18), we have 
tv w = g - (bV) { p/ ‘ M ~ V” sil1 mn ■ q ’ M 
(19). 
Remembering the relation between P,™ (p), P_ n _!“ (p), we see that of the eight 
solutions Pv(fO> P—i*W, P,r m (ft), P-^r" (ft), Q,“ (/x), Q_*_r (ft), Q,-*W. 
Q_„_r w (ft), of the equation (l), six have been expressed in terms of the other two. 
Expressions for P* w (— /x), Q/ l (~ ft) m terms o/P M m (ft), Q,”'(/x). 
12. Since the differential equation (1) is unaltered by substituting — ft for ft, 
it follows that P» m (—ft), Q.f" (— ft) are particular integrals of the differential 
equation, and are therefore expressible in terms of P,™ (ft), Q,™ (ft). 
