﻿Quaternionic Form of Relativity. 791 



q whose components are the four coordinates of a space- 

 point, say 



q' = aqb, (1) 



where in the case of pure rotation a and b must of course be 

 either wm£ -quaternions or at least such that T 2 a.T 2 6 = l; 

 T denoting the tensor. 



On the other hand, it is widely known that the so-called 

 Lorentz-transformation of the union of ordinary space 

 (#, y, z) and time (t), which is the basis of the modern 

 theory of Relativity, corresponds precisely to a (hyperbolic) 

 rotation of the four-dimensional manifoldness (x, y, z, t), or 

 of what Minkowski called the "world." 



Hence the obvious idea of representing explicitly the 

 Lorentz-transformation in the quaternionic shape (1), — 

 which, together with some allied questions, will be the 

 subject of the present paper. 



To solve this simple problem we have only to write down 

 the well-known relativistic transformation, i. e., the formulae 

 of Einstein, then to develop the triple product in (1) and 

 to compare the two. 



For our purpose it will be most convenient to put 

 Einstein's formulae at once in vector form, eliminating thus 

 the quite unessential choice of the axes of coordinates. Let 

 the vector v = mi denote the uniform velocity of the system 

 S' (V, y\ z\ t') relatively to the system S (a?, y, z, t) *, Let 

 0, CK be a pair of points in S and S', respectively, which 

 coincide with one another for t = t' = 0. Call r (= ai -f ?/j 4- zk) 

 the vector drawn in S from as origin, and r' the corre- 

 sponding vector in S', drawn from 0' as origin. Then the 

 transformation in question may be stated as follows : — The 

 component of x' normal to the velocity v is equal to that 

 of r, i. e. 



r' — (r'u)u=r— (ru)u, (a) 



whilst the component of r' taken along the direction of 

 motion is altered according to the formula 



r'u== 7 [(ru)~-^], (13) 



where y=(l— /3 2 )"* 1 / 2 , ft=v/c<l, c = velocity of light f. 



* 11 being a unit-vector in the direction of motion of S' relatively to S 

 and v the absolute magnitude of its velocity. 



f In these and in all following formulae (ru), generally (AB), means 

 the modern scalar product of the vectors A, B, that is to say AB cos 

 (A, B):.; ^ hence (AB) is the negative scalar part of the complete 

 llarailtonian product, AB : 



(AB) = -SAB. 

 On the other hand, the modern vector product YAB is identical with 

 Hamilton's VAB. 



