WILSON AND I.KW IS. — REI.ATI\ 11 V. 483 



extent upon the \ rlocity of the cinittiiif,^ electron. It is interesting 

 further to note that by tlu- applieution of ruhvs already gixcu we may 

 evaluate O'b «in(l sliow tiiat it \anishes. Hence 



0-b= Vb, + ^j'^^ = 0, (125) 



where hg is the vector which we have just shown to be approximately 

 equal to the Poynting vector, and hi is approximately equal to the 

 density of ener<iy. This equation is therefore entirely analo<i;ous 

 to the familiar theorem of Poynting. If we integrate over a tlu-ee- 

 dimensional volume, 



/ / j V'hsdx idiylr-i = — ^^ j j j bidvidiylxs, 

 or 



J fbsndS = -^^J j fb,cLv,dx,dxs. (126) 



Thus the induction of hg through any closed surface is equal to the 

 rate of loss of 64 in the enclosed volume.^ ^ 



If in the vector field b we cut the hypercone by any planoid, it 

 will be e\ident that the integral of hdS over the surface of intersection 

 will be independent of the position and direction of the planoid; for 

 the surface dS always lies in a tangent plane and b varies inversely as 

 R- and hence as dS. The vector hdS bears a simple relation to dg 

 which we have studied. For dg = {dS*'M')'M', where f/S is deter- 

 mined by any planoitl. We may therefore choose r/S perpendicular 

 to r/s, that is, to w. Then dS* is Wf/3 and dB = d!Sds, and since b by 

 definition is (wM')'M', the integral of dg is the product of ds and 

 the integral of hdS. We might therefore by a consideration of b 

 alone ha\e obtained the same vector of extended momentum for the 

 total energy emitted by an electron in the interval ds. 



We shall not pursue further the study of this interesting vector b, 

 but it may be well to point out that the two fields M' and b cannot 

 both be additive. For since b is qiKidratic in M', we obtain a differ- 



51 In general if a 1-vector field in four dimensions is of such a character 

 that its four-dimensional divergence vanishes, we may obtain in three dimen- 

 sions an equation of the type just found, wherein the surface integral over a 

 closed surface of the space component of the vector is equal to the negative 

 time deiivative of the integral of the time-component of the vector over the 

 enclosed volume. Such an equation may be interpreted as a continuity ()r 

 conservation equation whenever the space component appears as a velocity 

 multiplied by the quantity defined by the time-component. 



