Electromagnetic Waves. 159 



Combining (57) and (5*71) we find the results : 



(n + l)S w _i(»)-nS» +l (*) = (2n + l)S n '(«). 3 



The functions S», C„, E» are solutions of the differential 

 equation 



(Jr")(i + ;) s " (z)+s " w = 



found by writing n — 1 for n in equation (5*7), and then 

 eliminating S M _i from (5*71). 



This equation may be written in the form 



<g'+{ 1 _^^l)} S „ = ) . . . (5-73) 



which can be deduced directly from (3-42). 



Since E n is a second solution of (5*73), we see that 



E n (2)S»'(»)-E ll '(»)S»(a) = const., 



and supposing 2 to be small, we may replace S»(z), E»(«) by the 

 first terms of the formulae (5'5) and (5*61). Thus the value of 

 the constant is seen to be unity, so that 



V n (z)S n '(z)-.-E n '(z)8 n (z) = 1 (5-74) 



Again, by considering the form of (5-6), it is easy to see that 

 we can write 



\ tz (iz) 2 (iz) n J 



aud by substituting the last expression in the differential 

 equation (5*73) it will be found that 



Al =^+i), A3= (»-D(» + i> Ai , As Jj^Ki±z) K etc . 



These lead to the values 



_ n(n+l) _ («-!).. .(n + 2 ) __ (n-2). . .(n + 3) p .„ 

 Al 2~~ ' 2 27T~ "' Ag 274^6 ' etC ' 



. _ 1 . 2 . 3 . . . 2n 



2.4. 6. ;.2n 



. . . (5-75)* 



and it is easy to confirm the value of A» by direct differentiation 

 of (5-6). 



* The values (5-75) agree with those given by Sir George Stokes, 

 quoted in (3-492), above. 



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