Electromagnetic Waves. 151 



where U, V are any two solutions of the equation 



/,K^i[j_^U 1 B /■■ .BU\ _i_^U .,.„, 



c 2 dt 2 ~ Br 2+ 7^sin(9B(9V m ^B6'; + r 2 sin 2 6'd</> 2 



Although these very general solutions may be found 

 of value, yet the only applications made hitherto have been 

 obtained by expressing the general solutions U, V as sums 

 of terms of the special type 



F(r,t).Y(0,<t>) (3-4) 



If we substitute from (3*4) in (3*3), it will be found that 



r 2 f-d 2 F /*Kd 2 F-) 

 F I -dr 2 c 2 -dt 2 ) 



= ~Ylsln^PV Sm ^^) + sin 2 -^^j^ (3 ^ 



and since the two sides of (3*41) are respectively functions 

 of r, t and of 6 y <j>, it follows that their common value must 

 be a mere constant ; calling the value of this constant 

 w(n + l), it follows that Y is a spherical harmonic of 

 order n, say Y n (0, <£). 



If we restrict our problems to those in which all angular 

 space is used, the value of n must be a positive integer ; 

 for otherwise the value of Y (0, (f>) cannot be continuous 

 and single-valued in all angular space. 



The equation satisfied by F« then becomes 



where we now write 



c^ = c 2 IijK (3-43) 



The equation (3*42) can be solved very readily by a 

 transformation, due to Mr. R. R. Webb, which shows that 

 we may take 



F » = -(!-->- (3 - 44) 



In fact it will be found that 



and that 



\or rj \or r / or- r 



\dr r) \dr r ) or r~ 



