electrodynamics in the Maxwell and in the Einstein forms 169 



The squares e, i, j, h thus obey the laws of combination of the 

 symbols occurring in Mr Johnston's paper above referred to, de- 

 noted by him by o, i, j, k. 



If we consider a square of which every element is e times the 

 corresponding element in eijk, and denote this by m, we find at once 



m^ — — u, em = — me, im = — mi, jm — — mj, km = — mk, 



and e, i, j, k, m are a system of five squares obeying the same 

 laws of combination as the three quaternion squares I, J , K. We 

 have eijkm = — eu, and save for multiplication with — 1 or e, or 

 both, there are sixteen squares arising by multiplications of 

 u, e, i, j, k, m, namely these six and those obtainable by the ten 

 products of two of e, i, j, k, m. 



As was remarked to me by Dr W. Burnside*, F.R.S., the above 

 work is capable of generalisation. 



If e^, e^, ..., figr-i be squares each of m rows and columns obeying 

 the laws e-^ = — u, ejej = — e_,e^, where u is the unit square of 

 m rows and columns, then the 2r+ 1 squares of each 2m rows and 

 columns given by 



E=/0,-u\, E,= fO, eA, {i=l,2, ..., {2r-l)), F= fiEEj_...E^r-i, 



where fju, = (— 1)^^^ + '"^' \ is 1 or e according as r is odd or even, 

 obey exactly the same laws — namely if U be the unit square of 2m 

 rows and columns 



^2 _ ^.2 =-U, EEi=- EiE, E,Ej = - EjEi. 

 For instance for r = 0, m = 1, this gives 



ikiy {i~i) 



as two squares of which the square of the latter is the negative of 

 the former. These are the two units (usually denoted by 1, *) of 

 ordinary analysis. Or if, denoting these, as we have done, by 1, e, 

 we take r = 1, m = 1, the theorem gives the binary squares 



/O, - 1\ , /O, e\ , /- 6, 0\ , with /I, 0\ , 

 U, 0/ U, O) \ 0, e) \0, l) 



say, I, J, K, U, as units satisfying 



P^J2 = K^ ^-U, JK = - KJ = I, 

 KI = - IK = J, IJ = -JI ^ K, 



which are the laws of Hamilton's quaternions. 



* Dr Burnside writes: "In group notation it really comes to this: — The opera- 

 tions »§j, (§2, ..., S , such that 



8^^ = S^^ = ... = 8^^ = T, T' = E, 8 ^8 J = 5/,.T 

 generate a group G„ of order 2^^*+^ In this group E, T are the only invariant 

 operations if n is even; when n is odd, the factor group G^^ \ \S182 -"Sj^} is identical 

 in type with G^_i.'" 



