166 Professor Baker, On the transformation of the equations of 



On the transformation of the equations of electrodynamics in the 

 Maxwell and in the Einstein forms. By Professor H. F. Baker. 



[Read 9 February 1920.] 



The present note, which arose from reading the paper of 

 Mr W. J. Johnston in the Proceedings of the Royal Society, A, xcvi, 

 1919, 331, and the paper of M. Th. de Bonder, Archives du Musee 

 Teyler (Haarlem), iii, 1917, 80-179, has a very humble purpose. 

 (1) It is shown that the noncommutative "imaginaries" used by 

 Mr Johnston may be interpreted as aggregates of ordinary numbers, 

 much in the same way as the complex numbers of ordinary analysis 

 may be interpreted as aggregates of real numbers. It is shown 

 indeed how to interpret a system of any number of units obeying 

 the same laws of combination. (2) By this interpretation a form 

 of Maxwell's equations of electrodynamics is reached from which 

 the ordinary Lorenz transformation is obvious at sight. This form 

 of the equations is equivalent to a solution of the electrodynamic 

 equations in terms of arbitrary functions. (3) The equations given 

 by M. Th. de Bonder, and stated by him to be the equations of 

 electrodynamics in the Einstein field, are then considered, and 

 their invariance under a general transformation is established. The 

 purpose of this section is to show how simply this invariance follows 

 by the use of notation which is familiar in other applications. The 

 result of course includes the case of Maxwell's equations. 



§ 1. It is a familiar fact that the so-called complex numbers of 

 ordinary analysis are couplets of two real numbers {x, y), subject 

 to the laws 



(i) if m be a real number, m {x, y) = {mx, my), 



(ii) {x, y) + [x', y') = {x + x',y + y'), 



(iii) {x, y) {x', y') = {xx'- yy' , xy'+ x'y), 



those of the numbers which we ordinarily call real being couplets 

 {x, 0), and those which we call pure imaginaries being couplets 

 (0, ?/). By means of these rules any number (x, y) is expressible in 

 the form a; [1] + ?/ [i], where [1], [/] stand for (1, 0) and (0, 1) ; and 

 {%] [^] = — [1]. It is this last equation which we ordinarily write 

 i^ = — 1. 



We may similarly have systems of more than two numbers, 

 subject to laws of computation, addition (and subtraction), multi- 

 plication (and possibly division). In particular consider systems, 



