170 Mr. G. H. Livens on the Initial Accelerated 



The conditions to be satisfied are : 



1. That the field in the sphere has to remain regular at 

 the origin r = 0. This requires 



<t>i(c't)+<j> 2 (c't)=0. 



2. That the field is continuous at the boundary of the 

 outgoing wave, which requires, just as before, 



/(0)=0, /(0)=0. 



3. The conditions at the surface of the sphere. These 

 are two in number : 



(i.) The tangential electrodynamic force is continuous. 

 In the case under discussion, in which the sphere starts 

 from initial rest with a small acceleration, this condition 

 is the same as that the tangential electric force is con- 

 tinuous to the first order. This gives in the usual way 



a 2 /" + af +/- «f= a 2 (</>i" + fc") + «(<£/ - fc') + fc + 0, 



at r = a. 

 (ii.) The discontinuity in the normal induction is 



4^ ; this giveS 



«/+/-«f=K[a(fc'-fc')+*i4 6] 



at r = a. 



The particular integral of these last two equations subject 

 to 



£i( c 'O+02<yO = O 



is 



es 

 <j> 2 (c't + a) = ^ 2 . ^—^ [(^ + a)H ?>a\c't + a) + 5a 3 ] . 



There is then the complementary integral of the equations 

 determined with f = 0. To obtain these we try solutions 



f=Be { a K 



fc=-We K a \ 

 which satisfy the condition 



