528 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



Since both fields are propagated according to the equation 



and since any derivative of a solution is also a solution, we may 

 take 



- 



dydt' 



which satisfy the solenoidal condition and the equations (A) and 

 (B). If we assume </> to be a function only of r and , we have for 

 a diverging wave 



Differentiating < by the coordinates we obtain 



9< _d(f>dr _d(f> z 



dz~~drdz~'drr ' 



_ a/i d$\d_r = a_/i a^\^ 



~ * 



_ = _ 



dr (r dr ) fa dr \r dr) r 

 (3) 



8 2 <^> = _a_ /i a^\ 3r = _a_ /i a^\ ^ 



" dr\r dr) 'dy 'dr \r dr) r 



= . . 



8^ 2 ~ Z dr \r dr) dz r dr dr \r dr) r r dr' 



so that the forces are 



v ^ t^- ^$\ xz 

 *. z~ I "^ ] j 



dr \r drj r 



u) T -L(l*\yL 



~dr(rdr) r ' 



dr \r drj r r dr 

 The field is thus the resultant of two parts, the first, equal to 



drrdr' 



