474 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
p (1 — £)^ — m — (m + I) ^ — i (n — m) (n + m + 1) (1 — f) W = 0. 
Let W = £* <n+m+1) W'; we then find, on substitution, the following differential 
equation for W' : 
f (1 “ + i( n + 1) - (n + 2 m + f)£} ^ - (n + m + 1) (m + 1) W' = 0. 
Comparing this with the equation, 
f (i — O ” + {y - (« + /? + i)f} ~ - CC/ 3 W = o, 
which is satisfied by W' = F («, f3, y, £), we see that if a = n + m + 1, /3 = m 4- 1, 
y = n + 1, the equations are identical. It follows that our fundamental equation(i) 
is satisfied by 
V 1 = z -(n+m+l) ^2 _ Y ^1 _p n _|_ m _j_ n + |, ^ , 
or by 
V 2 = (p, 3 — 1)^ F f m — n , h ~ n tp) j 
where 2 denotes p + \/p 3 — b 
In z we suppose \/p 3 — 1 to be measured as hitherto, so that it has a single value 
at every point of the p-plane in which a cross-cut is made along the real axis from 
-j- 1 to — oo. 
It will be seen that mod z is greater than unity over the whole plane, the real 
part of -v/p- 3 — 1 having the same sign as the real part of p ; on the imaginary axis 
z is purely imaginary. 
In order to express the solutions V 1} V 2 in terms of P/' (p), Q u m (/x), it will be 
sufficient to compare these solutions for values of p whose modulus is very large, with 
the expressions (10), (22). 
These latter formula} show that for such values of p, the principal parts of Q,/" (/x), 
P** (/x) are, 
QtYlTTl 
2«+i 
n (n + m) n ( — |) 
n (n + d) 
(p 3 -1)* 
7JI .. —('/? + Vl+ 1) 
r* 
sin (n + m) 7r II (n + m) 
o»+i 
cos n v n (n + ■§■) n (— p 
(/x 2 - 1)*” IX 
—n—m —1 
4- 2" 
II (n—\) 
n(n-7»)n(-|) 
(p 2 - !)*’>*- 
respectively; for similar values of p we have 
