WILSON AND LEWIS. — RELATIVITY. 459 



ami 

 I f I I (/i^-*"2^^-''3^-'"-» + f-i (l-^'sdxidxi + f:i<lxi(l.r->(!.ri — fidxiiLvtcl^s) 



= /'/77Tf +f +f +f>-^-''--''-^*- 



J J .7 J Loxi dx-y dx3 dx.\J 



The theorems may be used to (l(Miionstrate in a vectorial manner 

 such an equation as (52), 0*(0*^) ~ ^^- ^^r 



As the final integral extends over the boundary of the closed three 

 dimensional spread which bounds the given region of four dimensions, 

 the final integral vanishes, since the closed spread has no boundary. 



Geometric Vector Fields. 



43. The idea of a vector field is ordinarily associated with concepts 

 such as those of force or momentum, which are not wholly geometri- 

 cal in character; but it is per- 

 fectly possible to construct ^^* 

 vector fields which are purely /^ ^ — -^q 

 geometrical. Thus in ordinary 



geometry we may derive a /?/ .^^ 



vector field, when a single 

 point is given, by constructing 

 at every other point the vector "'^ 

 from that point to the given 

 point, or that vector multiplied ^ 



by any function of the dis- '' ""^•^.^^ 



tance. "^ 



In our non-Euclidean four Figure 21. 



dimensional space we may as- 

 sociate with any (5)-curve a vector field derived from that curve in 

 the following way. At each point of the (5)-curve construct the 

 forward unit tangent w, and the forward hypercone.^^ At each point 

 Q of these hj-percones construct the vector w/7?, parallel to the vector 



38 That half of the hypercone Ij'ing above the origin, enclosing points which 

 will represent later times than the time of the origin, will be called the forwai'd 

 hypercone. 



