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XIII. On a Type of Spherical Harmonics of Unrestricted Degree , Order , 
and Argument. 
By E. W. Hobson, Sc.D., F.RS. 
Received December 23, 1895,—Read January 16, 1896. 
Introduction. 
The ordinary system of Spherical Harmonics or Laplace’s functions is obtained 
from Laplace’s equation 
V 2 V = 
dx~ 
+ 
5 
by choosing special values of V which satisfy this differential equation, and are of the 
forms 
cos . / \ i cos , , , 
r n . m<h . u n m (a), or r n ~ 1 . m<p . up (y), 
sm 7 sin r 
where n and m are real positive integers, x, y, z being expressed in terms of r, y, (f> 
by means of the relations 
x = r (1 — cos y = r (1 — y?f sin </>, 2 = ry ; 
the function u n m (y) is a particular integral of a certain ordinary linear differential 
equation of the second order, and is known as Legendre’s associated function of 
degree n and order m; these solutions, in which y is restricted to be real and to lie 
between the values dz L and in which m is restricted to be less than or equal to n, 
are the solutions of Laplace’s equation which are required in the very important 
class of potential problems in which the boundary of the space considered consists of 
either one or two complete spheres, or of surfaces which differ only slightly from 
spheres. 
It appears, however, that the functions m<£ . u n m ( y ) are required for the solution 
of certain potential problems in which the boundaries are of forms other than 
complete spheres, and in some of these cases the values of n, m, and y are not 
subject to the restrictions which hold in the case of the primary potential problems 
in which the boundaries are complete spheres. In the case in which the boundary 
is a spheroid or two confocal spheroids, the functions up (y ) of both kinds are 
3 L 2 25.9.96 
