1 74 Professor Baker, On the transformation of the equations of 



The differential equations in question, writing |, 77, ^, t for 



^ d^ d d 

 dx' dy' 8s' dt' 

 respectively, are 



m^ -iMu-rM^i = pyy, 



^^23 + 7^^31 +^Mi2 = P, 



^N^, - CN,, - tN,, = 0, 



- ^iV24 + ^^14 - ^^12 = 0, 



If herein we replace M23, M^-^, M^2> ^u^ ^245 ^34 respectively 

 by X, Y,Z,-L,~M,-N and N^^, iVgi, iVjg, N^^, N^^, N^ by any 

 constant multiples respectively of — Z, — M, — N, X, Y, Z the 

 ♦ functions on the left become, save for W, the coefficients of 



in the identity (4) of § 2. 



Our object is to show that with any transformation of x, y, z, t 

 leaving a certain quadric differential form in dx, dy, dz, dt unaltered, 

 and appropriate corresponding transformations of the elements 

 Mij, N ij of the two systems, and of V x, ■•■, p, the equations change 

 into others of the same form. In the Maxwell form of the equations 

 the differential form is simply — {dx^ + dy'^ + dz^ + df^). 



In the paper referred to M. Th. de Donder bases his work, after 

 Hargreaves, Trans. Camh. Phil. Soc, xxi, 25 Aug. 1908, upon the 

 transformation of differential (integral) forms, which he compounds 

 together by rules which appear to have a certain artificiality. In 

 what follows this is replaced by the use of Grassman's alternate 

 units*. For the actual transformation here considered M. Th. de 

 Donder does not give the proof in the paper in question {lac. cit., 

 p. 147), but refers to Mem. Acad. roy. de Belgique, i, 1904. But 

 more, in the present note, full use is made of the theory of matrices, 

 which not only brings out the identity of much of the algebra with 

 what is familiar in other work, but is also very much briefer, as 

 the reader may easily see by comparison with the original paper. 

 A short account of the use of matrices, under the name of squares, 

 is given in § 1 of this note. 



* Mr Bateman, Proceedings of the Lond. Math. Soc., vm, 1909, p. 245, to whom 

 M. Th. de Donder refers, has abeady suggested that Grassman's units might be used. 



