Electromagnetic Waves, 



153 



o£ / with respect to its argument, while fo=f (so as to 

 fall in with the general sequence of suffixes). 



The formula corresponding to (3*492), with the function g, 

 differs by powers o£ — 1 ; the sequence of powers is ( — 1)*, 

 ( — l) n_1 , . . . , ( — 1)°, each index being the same as the order 

 of the differential coefficient involved in the formula. 



For instance, with n=l, 2, the two formulae are respectively 



/i+J/i+?/o 



fh + 7.$o> 



9i- 



3 3 



"Z9i + 39o< 



The proof of formula (3*492) is not hard by induction ; but a 

 more direct method is to note that the result is clearly of the 

 right form and to determine the numerical coefficients by using a 

 s pecial function. A suitable function is given by taking 



/ 



,«i(e 1 *--r) 



as in § 5 below. The corresponding function was found by 

 Sir George Stokes * in the form 



K3*i(«i*-r) f 1 n(n+l) (n-l)...Q + 2) 

 [ 2 . iK-^r 2.4. (iK x r)' 2 



the numerical coefficients being the same as in (3-492). Now 

 here 



A = ("l)Yo> fn-i = (tK 1 ) n - l U etc., 



and so the result is established by taking ~K = (tK 1 ) n . 



The fact that the coefficient of f n in (3*492) is unity follows 

 by direct differentiation of (3*49). 



If we substitute the form (3*4) in (3:1) and (3*2) we 

 obtain the special solutions : 



n(n + l) 



\2 



x = 



Y = 

 Z = 





P Y 



l BY, 



sin d d<£' 



cot = 0, 



KBF, 



6/3 = 



C7 



1 3Y„ 



re B^ " sin # d</> ? 



_K^ BY« 

 re ~dt ' ~b6 ' 



-A }> (3-5) 



; 



where for brevity the terms arising from V have been 

 omitted, and ¥ n is defined by (3*49), while Y„ is a 

 surface harmonic of order n. To include the V-solution 

 is quite easy, but it is hardly necessary here. 



* Phil. Trans, vol. 15*. p. 447 (18(58) ; Mathematical and Physical 

 Papers, vol. iv. p. 300 ; Rayleigh's * Sound/ vol. ii. ch. 17. 



