UJO Woola/td— Generalized Relativity and Gravitation. 



differential of are is unchanged, i. e., such that our equa- 

 tions shall be invariant. 



In ordinary Euclidean three-dimensional space, for two 

 points very close together, analytic geometry gives as the 

 differential of arc 



,/* J = dx* + dy- + dz> (3) 



In the four-dimensional space-time manifold of relativity, 

 the differential of arc may be shown to be ' 



ds 1 = c'dt' - dx* - d>f - dz 2 (4) 



If on any system of coordinates {x, y, z, t) we execute 

 any arbitrary transformation 



x ' =/, (*> y, 2 . a) 

 ;/' =/, (•''> y, z , O 



f =/< (*> y, b, *,) 



(5) 



where /,- is any function whatsoever, substitute in the 

 formula for the differential of arc, and collect, we shall 

 find that 



ds' = g lx dx" + 2<7 12 dxdy + 2# 13 ffocte + %g u dxdt ~) 



_1_ n r7u 2 -i-9n ditdx-i-9,n thidt ! 



+ y n ch f+ %&** dyd* + fy u dy di l 



+ U <& +%0 ti dzdt 



)- («) 



+ *„<»' J 



where g i} - denotes a coefficient which depends for its 

 value upon the values of the ith and jth coordinates ; this 

 may be concisely expressed as follows : 



ds*=^g if dx i dxj (7) 



8. In (7) there are ten coefficients, all of them 

 functions of the coordinates. Since this is a perfectly 

 general transformation, it includes all others, to any one 

 of which it may be reduced by a proper choice of 

 functions. We pointed out above that by a suitable 

 transformation, a gravitation field could be got rid of; 

 hence, these ten g ;J specify the gravitation field, since they 

 show how it can be abolished; but they also determine 

 the reference system of coordinates ; in the new theory 

 gravitation is inextricably mixed up with space and time, 

 the three forming an inseparable union. 



