WILSON AND LEWIS. — KELATIVITY. 481 



to have the velocity V, the expression becomes more compUcated. 

 Since W = (v + kO ' Vl — t)^ and d.s- = Vl — r- df, we have from (45), 



/ 



'^^ " V (T^^y ['^''f- (VXV) . (Vxv) 1 (V + k , ) r/<. ( ] 20) 



The two parts of this expression are precisely in the forjn obtained 

 by Heaviside and Abraham ^^ for the momentum and energy radiated 

 from an accelerated electron. 



53. When a singular vector field such as dg/dS is distributed 

 continuously over a hypercone and is of such a character that its 

 magnitude falls off along any element inversely as the square of the 

 interval of that (>lement measured from the apex (that is, inversely 

 as R-), or in other words, if it is of sucli character that the integral 

 of the vector over the surface of intersection of the hypercone with 

 any three dimensional spread is constant, then we may call such a field 

 a simple radiation field. (In three dimensional space the magnitude 

 would fall off inversely with 7?, »,nd in two dimensional space would 

 be constant.) The fact that the integral of dg/dS over the inter- 

 section of the hypercone with any two parallel planoids is constant 

 may be regarded as equivalent to the law of conservation of radiant 

 energy. 



While the discussion which we have given of the vector dg is in 

 complete accord with current theories of electromagnetic energy, there 

 is another singular 1-vector which is suggested by the geometry and 

 which may be of importance in case it is necessary to revise our ideas 

 of radiant energy. This vector also gives a simple radiation field, 

 in the sense just defined, and is likewise of the second order in M'; 

 but unlike the vectors dg and dg/dS it is continuously distril)uted 

 over a four dimensional field. This is the vector*^ (wM')'M' = b. 

 The vector b is along the element of tangency 1 by § 39. Indeed if 

 we take M' from (93) we have 



b= (wM').M'= ^,[c-c— (n.c)-](w — n) = |r,(cxn)21. (121) 



48 Abraham, Theoric der Eloktrizitat, 2, 116. 



49 To obtain a vector, of the second deg ee in M', out of M' itself is out of 

 the question; for the only two pn ducts ( f the soc nd deg "ee in M' which are 

 geometiically significant, namely M'-M' and M'xM', both vanish, since M' is 

 singular and uniplanar. The vector b involves not only M', the field of the 

 electron, but also w which expies.ses the state of motion of the electron itself. 



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