of Four-Dimensional Vector Analysis. 403 



If S is the current four-vector and equal to 



p u x ii + p u y i 2 + p u z i^ + pi 14, 



where p is the density of charge and the u'§ components of 

 velocity o£ the charge, the general equations ma}' then be 

 written : 



LorP 2 = $, 



Lor Q 2 = 0. 



and for the case when the permeability and dielectric con- 

 stant are both unity Q 2 = P 2 '. 



Thus from (11'4) and (11*5) we may summarize the theory 

 by stating that the normal component of the six-vector P 2 

 over the surface of a three-dimensional volume is numeri- 

 cally equal to the amount of S within the volume, while the 

 total tangential component vanishes over the surface. By 

 the amount of S is meant the numerical value of 



It is the value of the component of S along the four-vector Idr. 



§ 13. The four- vectors of Minkowski's Calculus are limited 

 in that they are subject to a linear transformation. The 

 vector p becomes a new vector a where 



(j> is a linear vector operator and can be expressed as : 



cPp^-i^Vo^p + /3 1 V u /3 2/3 + 7l V o72 p + 8{V 8 2 p). (13-1) 



Thus (/> is a dyadic and associated with it are certain 

 invariants just as in the case of three dimensions, and 

 whereas in the simpler case $ satisfies a certain cubic 

 relation so here cj) satisfies a quartic. 



In three dimensions the ratio of the two scalar products 

 ScfrXcfrficfrv to SXyitv is independent of A, /u, and z> ; to this 

 corresponds the invariance of the ratio TV 4 cf>\cj)/jb(j)P(p7T to 

 TV 4 \//,v7r, which means that the ratios of the four-dimen- 

 sional volumes after and before the operation of <f> are a 

 constant. When cf> denotes the Einstein transformations this 

 ratio is unity. 



The linear transformation referred to is analogous to that 

 occurring in the theory of elasticity in the case of homo- 

 geneous strain, and since a further condition to be satisfied 

 by the four-vector is that of unchanging tensor such a trans- 

 formation is analogous to a rotation. 



