electrodynamics in the Maxwell and in the Einstein forms 167 



built with the numbers of ordinary analysis (including complex 

 numbers), each system consisting of n^ numbers (w = 2, 3, ...). 

 These we may conveniently arrange in the form of a square; and 

 for purposes of explanation it will be sufficient to write down only 

 two rows and columns. Suppose then 



(a) = /«!!, a-^^\ , 



\^21' ^22' 



where a-^^, a-^^.^ ... are numbers of ordinary analysis, and introduce 

 the rules (i) when m is a number of ordinary analysis m (a) shall 

 mean the same as {a) with each element a„ replaced by ma„, 

 (ii) If {a), (b) be squares of the same number of rows and columns 

 («) + (b) shall mean the square of which the general element 

 is ttrs + Ks', thus (a) + (6) is the same as (6) + (o)- (iii) {C') (^) shall 

 mean the square whose general element c^s is formed by combining 

 the fth row of («) with the 5th column of b, in such a way that 



Cj-s = ^rl^ls ~r ^r2^2sJ 



thus (a) (b) is not generally equal to (6) {a), (iv) The zero square 

 shall mean that in which every element is the number ; denoting 

 this zero square by 0, we shall then have (a) + = (a), and 

 (a) = (a) = whatever (a) may be. (v) The unit square shall 

 mean that in which all the elements are zero except those, a,.„ in 

 the principal diagonal, each of these being unity. Denoting this 

 by E we shall then have E (a) = {a) E = (a), whatever (a) may be. 

 Addition of such squares is then not only associative, but is also 

 commutative. It is easy to prove that multipHcation is associative 

 also ; but in general it is not commutative. 



Of such squares, of two rows and columns, that given by 



Vy, x) 



i;^ easily seen to obey all the laws of the complex number {x, y), 

 the multiplication being commutati\"e. This square may then 

 equally be used to represent the numbers of ordinary analysis. 

 Putting 



(1) =fi' ov (0 = /o,-r 

 vo, i] \l, 0- 



this square \% x {\) + y (i). 



Now denote the two units of ordinary analysis by 1, e, where, 

 for convenience of notation, e is used instead of the ordinary i, 

 so that e^ = — 1. And consider the squares of two rows and 

 columns expressed by 



