electrodynamics in the Maxwell and in the Einstein forms 173 



where A, C, A', C are arbitrary functions of their respective 

 arguments. The expressions under the integral signs are perfect 

 differentials in virtue of the differential equation satisfied by 

 A 8, ^', S'. 



The equations are also equivalent to 



Ua = V^, RS'= Sy', 



V^' = - Va', Ry=- S8, 



together with those obtained by putting — e for e, where 



TJ - ~ -- V ~ (— ^-\ 



' dxj^ dti ' \ dzj_ dyj ' 



What is to be remarked here however is that the equations (I) 

 are evidently unaltered by replacing 



«, a, y, y', yi, h 

 respectively by aa, (j-^a , G-'^y, ay', ay^, (J-^^i, 



where a is an arbitrary constant. These are the equations of the 

 Lorenz transformation of the older relativity theory. If, putting 

 If = 0, we equate to zero the coefficients of i, j, k, i^, j^, k^, in the 

 identity (A) at the beginning of this article (§2), and write 



T = -4- — p.- , L = mP, M = mQ, N = mR, m =^ ~ ^c */ ^ , 



where i — V— 1, we obtain the six familiar equations 

 I V {X, Y, Z) = curl {P, Q, R), 



-f^S/iP, Q, R) = curl (Z, Y, Z), 



and the above results furnish a general solution of these, in terms 

 of two arbitrary functions satisfying the above potential equation. 



§ 3. We now pass to the equations described by M. Th. de 

 Bonder as the equations of electrodynamics in the Einstein field 

 (loc. cit., p. 93). Here instead of one system of two sets each of 

 three quantities {X, Y, Z), [L, M, N), which interchange, we have 

 two systems each of six, connected together in a way which shall 

 be explained, with coefficients which are quadratic functions of 

 the coefficients in a certain Absolute quadric. These two systems, 

 imitating the notation of the author, are denoted respectively by 

 (^23, M31, .Tfi2, if 14, M24, M34) and (iV23, N^i, N^^, N^^, N^^, N^), 

 and symbols ^33, M41, etc., are occasionally used where 



M32 = - M23, M41 = - il/i4, etc. 



