﻿Quaternionic Form of Relativity. 795 



Similarly we could call our q and any covariant quaternion 

 a " world-quaternion " ; but possibly the less pretentious 

 name physical quaternion will do as well. Also, at least in 

 the beginning, no further specification of the " kind " is 

 needed. 



Thus « = Q[]Q, defined by (L), or by (7) and (8), is 

 what I should like to call the relativistic transformer of any 

 physical quaternion. 



To get the inverse transformer g> -1 , viz. that which turns 

 q' into q, apply to both sides of the equation q' = QqQ the 

 inverse quaternion Q" 1 as a pre- and a postf actor ; then, 

 remembering that Q _1 Q = QQ -1 = 1, the result will be 



?=q-vq-\ 



or o>-' = Q-'[]Q-', 



and since Q is a unit quaternion, its inverse is also its 

 conjugate, i. e. Hamilton's KQ, which may be more con- 

 veniently written Q c ; hence 



o,- = Q c []Q (1.6) 



where Q c =cosa— u sin«. Thus, we see that the inverse 

 transformer is got from the direct simply by changing the 

 sign of the angle a or by inverting the direction of u, — as it 

 must be. 



Observe that, since the product, of quaternions is dis- 

 tributive, the transformer co has also the distributive property, 

 i. e., A, B being any quaternions *, 



Q[A-f-B]Q = QAQ + QBQ, .... (9) 



and consequent!}^, if ^ be any scalar differentiator, also 



Q[j&A]Q=3QAQ, 



since Q, being constant, is not exposed to "d's action. Again, 

 by the associative property of quaternionic products, the dot 

 signifying a separator, 



A.QBQ = AQ.BQ, 



and so on. For our present purpose we scarcely need a full 

 enumeration of co's properties. 



* /. e. generally complete quaternions but also, more especially, pure 

 scalars or pure vectors, either simple- or bi-vectors, that is to say real 

 or complex. The heavy type (and this merely to suit the • general 

 custom) shall be henceforth used only for pttre vectors, both real and 

 complex. 



