482 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If a k4 axis has been chosen, b maj^ be obtained in terms of e', or of 

 e' and h'. For instance with M' taken from (108), 



/ V + ki Isxe' + /4k4xe'\ Lxe' + /4k4xe' 



D = 



V Vl — f2 h J h 



When we perform the reductions, remembering that l^-e' = 0, we 

 find simply 



e- [^ _ L.v\ fh ^ ^\ (122) 



Vl — v^ ^ h J \U 



If we use M' in the form M' = E' + H', we find ^o 

 1 



b = , [e'xh' + ve'e' + vh'h' - ir-v + (t-'^ - v.e'xh')k4], 



Vl — x~ (123) 



where e'xh' has been used to denote the 1-vector (e'xh')'kio3, which 

 is the three dimensional complement of the 2-vector e'xh'. Another 

 equivalent form is 



. /4-L-v 



Vl — tHi 



(e'xh' + ?'%). (124) 



The coefficient (/4 — ls'V)//4'^l — 'v^ is unity when r is negligible 

 compared with the velocity of light, and therefore in all such cases b 

 is the sum of two \'ectors one of which is the Poynting vector and the 

 other along k4 equal in magnitude to the density of energy. Since 

 the vector b comes so near to being the extended vector of energy 

 density, the possibility is suggested that the energy of an electro- 

 magnetic field may not depend solely upon the field itself but to some 



50 For rapid calculation a rule for obtaining the three dimensional form of 

 some products is useful. The most important of these rules is that if 



A = a 'kiss — bxk4 and c = Cs + cds.i, 



where a, b are three dimensional vectors, then 



c«A = c.,xa + cjb + (c^«b)k4. 

 Thus we have here 



b = (w.MO'M' = — =L=:[(v + kj).(h'.km - e'xk4)].(h'.ki23 - e'xk4) 

 Vl — V- 



^ [^^<h' + e' + (ve')k4].(h'.ki23 - e'xk4) 



Vl - f2 

 1 



VI - v"- 

 whicli is identical with the form given 



[vxh'xh'+e'xh'+(v.e')e'+(vxh'.e'+e'.e')k4]. 



