WILSON AND LEWIS. — RELATIVITY. 475 



This may be expaiicU'd acoonliiig to (,72) when an axis of time has been 

 chosen. Then, noting that 1,^7 = (Ij — /.iv)xv, 



1 2 12 



M = — e ^^— rxv — € ,3 rxk4. (96) 



Where r is the vector 1 = 1.,— /4V from the contemporaneous position 

 of the eharuo to the point Q in the field, and / = /j — 1,,*V. Tiie 

 2-vector M is thus spht automatically into two 2-vectors, of which 

 one passes through the time-axis k.i, and the other lies in the planoid 

 kijs which constitutes ordinary space. These will be designated 

 respectively by the letters E and H. Thus 



M = H + E. • (97) 



This separation may in all cases be made whether the field is caused 

 by one or more electrons in constant or accelerated motion. We shall 

 thus see that the 2-vector M is precisely the "Vektor zweiter Art" 

 which Minkowski introduced to express the electric and magnetic 

 forces. 



Out of H and E spacial l-\ectors h and e may be obtained by the 

 equations 



h = H-k,,3, e = E-kj. (98) 



Then h is the three-dimensional complement of H, and e the inter- 

 section of E with three-dimensional space. Evidently 



(99) 



Referring now to (96) we see that in the case of a uniformly moving 

 electron 



e = e ^-=^' r, ■ h = - 6 ^^;^Vv) -1^123, (100) 



or ve = h«ki23 = H. 



Noting that (rxv)'ki-23 is that which in ordinary vector analysis is 

 known as the vector product of r and V, we see that these equation? 

 are precisely the equations for the electric and magnetic forces.^* 



It may seem surprising to one who is not fully convinced of the very 

 fundamental relationship between the four dimensional geometry of 

 relati\ity and the science of mechanics that we should thus be led 



44 See Abraham, Theorie der Elektrizitat, 2, p. 88. 



