﻿Quaterniotiic Form of Relativity. 793 



Then, developing the complete product of q, Q, by (3) and 

 (4), and by the fundamental rules of Hamilton's Calculus, 



^Q = Vrw + Iw -f sr - (rw) + si, 



and similarly , 



q' = Q ? Q = VwVrw - w (rw) + 2slw + s 2 v - 2s (we) + (s 2 - iv 2 ) I 

 = (w 2 + s 2 )y-2(vw)w + 2sIw+(s 2 --w 2 }1~2s(yv7), 



whence, splitting into the vector and scalar parts, 



r'=(M> 2 + s 2 )r — 2(rw)w + 2s/w ^ 



l' = ( s 2-w 2 )l-2s{rw) i ' ' ' ■■' ( 5) 



Comparing this with (2), we get at once, as the conditions 

 to be fulfilled by w, s, 



w 2 + s 2 =-\ ; s 2 — w 2 = y ; 2sw = i{3y 

 w = ivu. 



}• • • (6) 



Hence w= ±^/ (1—y) /2, s= ± x /(l + y)/2, where, to satisfy 

 the third of the conditions (6), we must take both square 

 roots with the upper or both with the lower sign ; therefore 



Q=+(y(i+7)/2+V(i- 7 )/2), 



and since in (1 a) the quaternion Q appears twice, the choice 

 of the + sign becomes indifferent. 



Thus, we obtain finally the required quaternionic expression 

 of the relativistic transformation 



q'=QqQ 



1 



with Q 



u being a unit vector in the direction of motion of S' relatively 

 to S. 



Observe that y = (l-^/c 2 )" 1 2 >1, so that the vector of Q 

 is imaginary, whilst its scalar is real. 



The tensor of Q is 1 ; thus denoting its angle by «, t. e., 

 writing 



Q = cos a-f u sin a = £ flU , (7) 



we have, by (L), 



cos«= \/(l + y)/2, sin u= s /(l-y)/2. 



