Electromagnetic Waves. 

 The formulae (3*5) then become 



157 



x = < n +Sl^ i(fCir) Y n{ } ^ 





Z = 



1 Own 



BY rt 



r d^ ' sin 6 <3<£ 



Co. = 0. 



r 



1 



c 7 



sin# B^> 



S> K< 



where the time-factor <? 6Krf is suppressed, and Y«(0, (/>) is a 

 surface-harmonic of order n. The solution for divergent 

 waves is obtained by writing E n (* 1 r) in place of S« («,?•). 

 The correlated solution, found by using V instead of U 

 in the solutions of § 3, is 



X=. 0, 



1 ~ ™ r ?i '* sin (9 d<£ ' 



Z = +^S n . 



1 BY. 

 9Y» 



a*' 



ca 



^t^S^^Y^^)^ 



c£ = 



ldS„ BY, 



1BS W 

 r or 



a*' 



1 dY„ 



>■ (5-4) 



sin d<£ ' 



As explained in § 2, the most general solution containing 

 e lKCt as a time-factor is obtained by superposing solutions of 

 the types (5*3) and (5*4). 



Solutions equivalent to (5*3) and (5*4) but differing 

 somewhat in notation and arrangement have been given by 

 Profs. J. W. Nicholson * and G. Mie f. 



Summary of properties of the functions S n (z), E (z). 

 By term-by-term differentiation we see that (5*22) gives 

 / 1 d\ n /amz\ 



~ 1.3.5...(2n+T) I 2(2n + 3) + 2 . 4(2n+3)(2rc+5) 

 Hence we can write also '."•(•• \ D D ) 



(5-51) 



S»(^=^/(f)j»+iW • • • 



in terms of Bessel's functions J. 



* Phil. Mag. ser. 6, vol. xiii. p. 259 (1907). 

 t Annalen der Physik, ser. 4, vol. xxv. p. 382 (1908). 

 % It is therefore easv to see that S n (z) is the function denoted bv u in 

 Prof. H. M. Macdonald's paper (Phil. Trans. A, vol. 210. 1910, p. 115). 



