Solution of the Electromagnetic Relations. 261 



The terms involving h are seen to vanish, and therefore 



1 (r*a) + - -7, ^7j( sm ^7j f . 9/1 ^-j + « r-rt =0. b) 



Novv a, as a function of©, may be expanded in a Fourier series. 

 The coefficient u( m ) of the term involving cos (into + e (w) ) 

 satisfies 



"v () , ttW) + ^i a . ( siu , <g^\ _ |$g + , VttW=0 . 



^/- v y sm#d0\ d# / sin 2 



Writing M («) = XCPr(cos6>), 



where PJT denotes the associated harmonic of degree n and 

 order m. 



whose solution, finite (unless n = 0) at the origin, is 



— vl l = Ar-I J (kr) 

 »+i 



and the corresponding value of a becomes 



a = Ar-t .1 (kr) . P?'(e6s 0) (cos (may + e)). . (9) 



More generally, we must add, to the associated harmonic, an 

 arbitrary multiplier of the type Q™(cos 0), and to the Bessel 



function, a multiple of J (kr). For convenience, these are 



-»-§ 

 ignored, for since they have the same differential properties 

 as their primary types, their contribution to any formula 

 may be at once inserted. The electrical component x must, 

 by symmetry, satisfy the same differential equation as a, and 

 we thus write 



a = Ar-t J (kr) PIT (cos 6) cos (w© + e)" 



•+* \ 9 . (10) 



x = Br-i J (kr) P« (cos 0) cos (mro + 8) 



}•■ 



where (A, B) bear no necessary relation to one another. 

 They may be supposed to include a time factor exp. ik vt, 

 and a summation to be made for all integral values of 

 m and n. 



The remaining components now require calculation, and 

 are somewhat more diffieultto obtain. 



