Electric Waves round a Large Sphere. 289 



by (131) and (132) if the series dependent on the limit e 

 can be ignored. We have seen that both these conditions 

 are satisfied for the series whose sum is intended, and we 

 therefore apply the formulae in question. The process of 

 calculation of Y and V 2 is as follows : — 



In the first place, the functions -^r at the zero point are 

 merely numerical. For 



and by the definition of t/t, since g)==k/ = at the zero point, 



2 



Accordingly, at the zero point, 



^ =1, ti = -i f 2 =Bi=t, ... . (135) 

 Secondly, it is sufficient, to the requisite order, to write 



and we require, for V 2 , the first two deriyates of U at the 

 zero point, denoted by U' and U" in (132). Now 



Uo' = "d/c^ (^ )= u'irQ + mv"^, 



U " = ii' / ^ + 2Lu / v"f> 1 + iuv'"^ 1 -uv //2 f 2 , 



differentiating the functions ijr according to their rule. 

 The derivates of Ui are not needed explicitly. Thus at 

 the zero point, where suffixes denote differentiations of 

 u and v, 



U -u, Uj = — l^j + t^««' 2 , 



JJ Q '=u l — \iuv 2 , U " = u 2 — m 1 v 2 ~ j;ttiv 3 — ^uv 2 2 . . (136) 



For the final result, two significant orders must be retained 

 in V of (132), but only one in Y 2 , for the function needed 



is V — j-^-. Thus in calculating V 2 , U may be identified 



with U , but this may not be done in calculating V . This 

 indicates that derivates of JJ 1 are not needed. Finally, 

 in (Y , V 2 ) we may write, in V , 



U = u ~z (i M i— A^aX 

 and in V 2 , 



U = w, U'= iti — \iuv 2 , 



