electrodynamics in the Maxwell and in the Einstein forms 175 



We have quantities Mfj, Njj, for i,j= I, 2, 3, 4, such that 



M,, = iV, 

 matrices 



0, 3I,j = - M,,, 



m 



0, i¥,„ M- 



M 



21' 



M, 



i¥o„ M 



i¥„, M. 



i¥ 



325 



M, 



41' 



M.„ if^ 



^42' 



of which the second is determined from the former by an equation 

 given later, with the help of a certain quadric. The determinants 

 of m, n (supposed not zero) are respectively fi^, v^, where 



ya = M23M14 + M,^M,^ + M12M34, u = N^,N^^ + N^^N,^ + N,^N,,. 



It will appear later, from the equation connecting m, n, that 

 V = — fji. Connected with m is a certain other matrix 



0, 



M,„ -M. 



M. 

 M, 

 Mo 



34' 



^^34' 



0, 



M. 



M,^, M 



24' 



23 



M, 



31 



-M,„ -M, 



M, 







Multiplying this last with m we find that in the product every 

 element vanishes except those in the diagonal, each of which is — fj,. 

 Replacing the unit matrix by unity, as usual, we therefore say 

 that the last written matrix is equal to — fjbm~^. The corresponding 

 matrix derived from n is therefore equal to — vn~''-. 



We consider a certain quadric form in four variables, whose 

 coefficients a, h, ... will later be regarded as functions of variables 

 X, y, z, t, and put A for the matrix 



A = a, h, g, u 



h, b, /, V 



9, /' c, w 



u, V, IV, d 



The determinant of this matrix being denoted by 8, we put 



e = (- 8)*. 



Then the coefficients N^j are defined by the equations expressed by 



en = /iAw~^A 



which is the same as em = /aAw^A, 



so that Mij are the same functions of N^.j as are iV,,,- of M^j. At 

 greater length these equations are the same as 



