Propagation of Lie/ Jit in a Gravitational Field. 587 



Maxwell's electromagnetic equations in vacuo retain, in any 

 gravitational field, their familiar form 



^+curlE = 0, div3)? = 0, 



a? 



-^ -curl M = 0,div (£=/>. 



c, . nt 



In absence of gravitation (£ = E, and 53? = M. In any 

 gravitational field & and 53? are certain linear vector functions 

 of E and of M respectively. The relations between the two 

 pairs of vectors are given, in the usual notation, by the six 

 reiativistic equations 



F*=0^F ajB , ...... (1) 



due to Kottler, taken over by Einstein. Now, in the system 

 in which the determinant 



\g lK \ is equal —1, and g u =g 2i = g u = 0, 



and which can be employed without loss to generality, we 

 have, by (1), and writing g tK = r /i K , 



M- 1 = F 23 = I9?i(722Y33 — 723 2 ) + ^2(7237.31 — 721733) 



+ ^3(721732 —722731), • (2) 



and two similar expressions for the second and the third 

 component of M, and 



-(y 1 = F u = 7 4 ,(7 1] F u + 7 12 F 24 + 7 1 3F 3 4) 



= 744(7ll E L + 7l2F2 + 7l3E 3 ), .... (3) 



and two simitar equations by cyclic permutation. Thus, if &> 

 be the three-dimensional linear vector operator 



7n 7i2 7i3 



721 722 723 

 731 732 733 



or the matrix which belongs to <y u as submatrix of the 

 <jontravariant tensor y lK , we have simply 



S= — 74i • SE. 



Next, the coefficients of 5)?! &c. in (2) are the minors of S 

 belonging to y n , y l2 , 7^, respectively. Thus, inverting (2) 

 and writing [a>| for the determinant of co } we have 





2Q2 



