Field of a Particle on Einstein s Theory. 825 



Adding these three equations, we have the following 

 equation in X only : 



r 

 which on integration gives 



(8) 



-( i+ e 



where m is a constant of integration and a second constant 

 of integration has been chosen so that \— >0 as r->co . 



Substituting in (6), we obtain an equation in v which 

 readily integrates to give 



-£9 



m 

 \ 2 



(9) 



It is then necessary to show that (8) and (9) satisfy 

 either (5) or (6), and this presents no difficulty. 



The line element may therefore be written without 

 approximation 



ds 2 = -fl + ™X (dr 2 + r 2 dd 2 + r 2 s'm 2 Odft 



m 



+ 1 — \w- (io) 



The constant m is most readily identified with the mass 

 by considering the approximation of (10) when m/r is 

 small and comparing it with (2). In fact, (10) may be 

 obtained from (2) by means of the transformation 



'■ = '-i(i+^) 2 , (ii) 



and then dropping the suffix in r i . 



The advantage of the form (10) lies in the facility with 

 which it can be transformed from one set of co-ordinates 



