SCATTEEING OF PLANE ELECTRIC WAVES BY SPHERES. 195 



giving a corresponding approximation 



p; ( C08 e) = ^1 



sin 9 x/(2n7r sin d) 

 Thus 



where 



and we have replaced (n + ^) 2 /{n (n+l)} by unity, because n is large. 

 Thus, on putting x = 0, we find that (4 '8) gives the approximation 



- ir _inir 



/ /> i\ "1\/T / ** v> 



(64) M = cos <f> 2, 



KT "^ v /(2n?r sin 

 Similarly, we get the formula 



t*r ..inn 



(6-5) N = 



: - 

 7r sm 0) v 



With series of this type, the leading part is found by making the index of the 

 exponential stationary (regarded as a function of n). Now in both M and N there 

 is one index only, a- = 2\^ + n-7ro>, which can be stationary : and the condition is 



Now, from (6'l) 



I -,\^ da. 

 - - 



-~ = {z cos a (n + -j) } -j -- a = a, 

 an an 



and so the leading terms in (6 '4) and (6 '5) arise from taking 



2a = TT 6, or n + ^ = 2 sin ^9. 

 The corresponding value of the index a- is then 



To determine the form of the index near to this special value of n we take 



d'cr _ 2 da = 2 2_ 



dn 2 dn z sin a z cos \Q 



and then we find the approximate formulae 



| where 



