Undamped Vibrations in an Unbounded Dielectric. 671 

 force are of the type 



a = dRldij-d(^ldz 



+ tt {dRn, Jdij - ^G„. Jdz) E„ («r) ^-v'. 



At a great distance from the centre, and selecting the real 

 parts, these last take the forms 



a =2S/f(?/H„,K— eG„, k) cos (/cV^ — a„^^ — /cr + n7r/2)(/cr)~"~^, 



the approximations being to the order 1/r. Evidently the right- 

 hand members of these equations cannot vanish for all values 

 of the time unless the coefficient of each cosine vanishes 

 separately ; therefore, if there is to be no radiation, for each 

 value of K we must have three equations of the type 



the summation extending over all the values of n which occur. 

 It is readily seen then that the only harmonics which can 

 occur are of the first order, and that for each value of k 



Fi,K = ^^^, Gi,« = Xy, Hi,^ = X-, 

 where \ is a constant. Accordingly the values of the mag- 

 netic and electric forces outside the sphere vanish, not merely 

 to the order l/r, but absolutely. Again, in periodic motion 

 these forces and all their spacial differential coefficients are 

 continuous throughout all space except at sources. Hence 

 we see that at any time they must absolutely vanish in any 

 space which can at that time be reached from the outside 

 without passing through a source. Accordingly, in order 

 that there may be no radiation in periodic motion, the sources 

 must include a continuous distribution over a closed surface 

 or surfaces, though there may be other sources inside them ; 

 and the distribution must be such that the dielectric outside 

 these surfaces is absolutely free from disturbance ; in other 

 words, the vibrating medium must be finite and bounded 

 externally. This seems indeed to be the proof indicated, too 

 briefly, by Pocklington *. 



It thus appears that, if either Poynting-'s expression for the 

 rate of radiation, or Macdonald^s modification of it, is to be 

 accepted, it is impossible for any combination of electron- 

 sources executing periodic motions to constitute a molecule 

 which, when isolated, will not lose energy by radiation. This 

 does not contradict Larmor^s f result that if the period of the 

 slowest constituent vibration is small compared with the time 

 which radiation would take to travel across the molecule, the 

 arrangement may be such that the radiation is very small 

 compared with the sum of the radiations from each electron 

 ♦ >'ature, March 26, 1903. f '^ther and Matter,' p. 232. 



2 Y2 



