WILSON AND LEWIS. — RELATIVITY. 479 



Now we may choose f/S perpendicular to k, and with jji-opcr sign, 

 then (/S* = k4 (IB. Hence, performing tiie nuihiphcation, 



'/g= ^'-J-|+e"-'k4)f/a. (116) 



Now if e' is interpreted as electric force in a radiation field, then we 

 are accustomed to regard c'- {= h'-) as the dcn.sifi/ of clrctromnqnciic 

 cnerfii/, and the vector c'- Ig/h, where I4.//4 is a unit \ector perpendicular 

 to e' and h', as the Poynting vector. Therefore dg becomes a vector 

 of extended momentum of which the components are the total energy 

 and the total momentum in the chosen volume rfS. The vector dg is 

 moreover independent of any choice of axes and is representative at 

 any point of the tube wliose cross section with any chosen space is 

 the volume JS- But the vector dgdB obtained by combining the 

 Poynting vector and a \ector along the k4 axis representing the 

 density of energy is by no means independent of the clioice of axes. 

 In fact we may state that no way can be found of representing the 

 densitv of energv bv a strictlv four dimensional vector. Thus we 

 have a vector of extended momentum for energy -quanta, but not for 

 energy density — an observation which is not without significance in 

 view of certain modern theories of light. 



52. It is interesting to note that the same energy vector dg may be 

 obtained from different 2-\-ectors M'. For anv two singular 2-\ectors 

 of the same magnitude and passing through the same element of the 

 hypercone determine the same vector 1' as above defined. If we 

 regard any singular 2-vector M' produced by an accelerated electron 

 as the extended electromagnetic field of the radiant energy which is 

 moving out along the space projection of the element 1 with the 

 velocity of light, then it is evident that, since there is an infinite num- 

 ber of such 2-vectors to which the element 1 is common, there is 

 something else necessary to characterize the light besides its energy. 

 In fact a l-vector such as 1' or dg upon which the condition is imposed 

 that it shall be singular has three degrees of freedom; a 2-vector such 

 as M' subject to the two conditions that it shall be singular and uni- 

 planar has four degrees of freedom. It is this additional degree of 

 freedom in M' which gi\-es rise to such phenomena as polarisation 

 which show a dissymmetry of light with respect to the direction of 

 propagation. 



If the vector dg represents radiant energy (moving out along the 

 hypercone with unity \elocity), then the integration of equation (114) 

 around the whole hypercone should give a vector representing the 



