General Relath'ity and Einstein's Theory. 401 



between the ^'s which satisfies the principle of general relativity, 

 i. e. it must have the same form in terms of the ^'s and .r's of one 

 system as in terms of the ^"s and x"s of any other system. 

 Secondly, it must reduce to Laplace's equation as a first approxi- 

 mation. As this equation is linear in the second derivatives of the 

 potential, it is to be expected that the exact law will contain no 

 derivatives higher than the second, and will be linear in the 

 latter. Moreover, as the exact law specifies the values of more 

 than one potential, the desired relations must be more than one 

 in number. Hence they may be expressed by equating to zero a 

 vector, or perhaps a dyadic. In this way four, or sixteen, scalar 

 equations will be obtained. The methods of building up an equa- 

 tion, or set of equations, of this character, which satisfy the re- 

 quirements of the principle of general relativity, will now be 

 considered. 



Consider two systems S and S'. Choose orthogonal coordi- 

 nates in each system. The coordinates x\, Xo, x.^ and time x^ 

 in S will be related to the corresponding quantities in S' by four 

 equations of the form 



Differentiating, 



dx =.2.^~^,dx ', (29] 



i^ 



and conversely 



dx^ 9xp 



where ^ is equal to the minor of ^ — } in the determinant formed 



9x^ 9x^ 



by giving /3 and v the values i, 2, 3, 4 respectively, divided by this 

 determinant. 



Moreover 9 _ yBx^' 3 



9xp~ vBxj^^xv" ^^^^ 



9 5> 9x 9 , 



^ — ; = -^0 . T — - (32) 



dx a dx dx ' 



Let kj, ko, kg, k^ be unit vectors in 6" parallel respectiveh^ to 

 the a\, .Vo. .n, .r^ axes. As these axes may be curvilinear, k^ etc. 



and 



