674 BELL SYSTEM TECHNICAL JOURNAL 



In three dimensions and in polar coordinates (those appropriate 

 to the boundary-conditions which we shall impose) the wave-equation 

 assumes the somewhat alarmingly intricate form: 



(131) 



The argument "^ can no longer be visualized as a displacement per- 

 pendicular to the equilibrium-position of the undistorted medium, 

 since all three dimensions are already used up. The reader may 

 visualize it, if he will, as a condensation or a rarefaction, after the 

 fashion of sound-waves. Perhaps not to visualize it at all would be 

 a better preparation for the study of wave-mechanics. 



In the familiar way, we essay a solution in the form of a product 

 of a function of time g{t), a function of radius /(r), a function $(0) of 

 the longitude-angle </> and a function Q{d) of the colatitude-angle 6. As 

 before, we find that the time-function is of the form: 



g(/) = A cos mut -(- B sin miit (132) 



and, as before, we shall find that the boundary-conditions confine the 

 coefficient m and the frequency mn/lir to certain "permitted" values. 

 The angle-functions and the radius-function are governed by the 

 differential equations: 



i(rf^] + nf-r'-f 



-r^ cosec 6 



;^(^°^""'^)+^^(^'"^^>l = ^' 



de)\ 



(133) 



in which Y stands for the product of 9 and $, and X for a constant 

 which seems to be arbitrary, but as a matter of fact is constrained by 

 the same circumstance as arose in the case of the membrane ; for, when- 

 ever cp is altered by 2t and 6 by tt, we are back at the same place as 

 before, and the function Y must have the same value as before; and 

 this will occur only if 



X = w(w + 1), n = 0, 1, 2, 3, •••, (134) 



these being the Eigenwerte for the differential equation in (133) for 



the angle-function.^" The corresponding Eigenfunktionen are spherical 



1" This and the following statements about the functions F„ are proved by writing 

 Y in the second of equations (133) as the product of a function ot d and a function 

 of 4>, and so dissolving the equation into two in the manner which I have already 



