350 Radiation and Electromagnetic Theory. 



but XXX' — X«a' = XX 2 — X« 2 always, and therefore in this case 

 iXXX'^XX 2 -^* 2 , or integrating we have S = E e -E m . 

 But E^E. + E^, and therefore 



E e = S + iT, and E m = jT, . . . (68) 



i. e., the integral values of E e and E OT are separately deducible 

 from that of S. If we consider the current solution alone, 

 XXX' is not =0 at each point of the field unless %pi x = ; 

 but its integral vanishes because the divergence of (X Y Z) 

 for this solution is 



f — + A +r ^W—° and X— ° = 

 V dx * dy dz) dx ' dx 



With J2XX / rfr = goes ^Xua'dT=^iXoc 2 -iXX 2 )dT, i.e., 



E^S' + JT', and E;=*T'. . . . (69) 



Before proceeding to the general solution for any ellipsoid, 

 the simple case which corresponds to a sphere in statical 

 potentials may be briefly noticed. With i x = i y = i z = 



-x<=g (i-V), -x.(i-^*-«^-i^f 



, dyfr dyfr ^dX 



dy * dz . dx 



For -\Jr = C/R the surface R=R is an equipotential 

 surface, and may be treated as a conductor. We find 

 X = C#(1 — Xj» 2 )/R 3 , and if £ is the total charge, 



since the polar and equatorial semi-axes are R and 



Ro/^/l — S^ 2 respectively. Thus 



f.JLj, and then 8= | f s (l- V) -^=f# • (70) 



The potential is properly = ijr(l — Xjp 2 )==e(l — 2p 2 )/47rR, 

 which makes X'= r*. For a uniform volume distribution 



^ = p - (R 2 - iR 2 ) , <£ e = /?R 3 /3R ; and the integration 



\ipfrdri for S yields 3e 2 (l~X/)/20<7rR . 



Thus the typical elementary solution which corresponds to 

 that of the sphere in attractions here belongs to a particular 

 oblate spheroid. 



