﻿Quaternionic Form of Relativity. 797 



To obtain the above representation or! the relativistie 

 formulae (2) we have introduced the quaternion q = l + r. 

 Now, for this purpose we might as well have used its con- 

 jugate, i. e., 



q c =l — r, (3 a). 



and the corresponding q c ' ' — V —x' *. It may often be con- 

 venient to recur to q c and it is therefore of some interest 

 to know how it transforms. Now, a glance at (2) suffices 

 to see that both of these formulae remain unchanged if, 

 having changed the signs of r, r' (and leaving /, V as before), 

 we change also the sign of u. Thus it is seen that 



q c ! = QcqcQc, sa y =c °cq c • • • ■ (Lc) ; 

 where Q e = cosa-~u sina = £~ nu . 



Now q e has precisely the same office as q, that is to say, 

 (I.) and (I. c) are but two expressions of one and the same 

 thing, namely, of the Lorentz-transformation. Hence q c and 

 any quaternion covariant with q c is certainly a physical 

 quaternion as well as q and its covariants. 



Thus, the conjugate of a 'physical quaternion will again 

 he a physical quaternion. If the original transformed as 

 </, its conjugate will transform as q c . If A is covariant 

 with </, then A c is covariant with q c , and vice ve?*sa. 

 Speaking of a physical quaternion we shall, when neces- 

 sary, add the explanation cov. q or cov. q c . But generally, 

 for the sake of shortness, this will be omitted, and any 

 letters, as A, B, a, h, &c, without the subscript c will be 

 used to denote quaternions covariant with q. Observe that, 

 with the above (formal) extension of our original definition, 

 two physical quaternions may be either covariant with 

 one another or not ; in the last case we may call them 

 antivariant, one being cov. q, and the other cov. q c . Thus, 

 by the above convention, A, B/or a, b c will denote pairs of 

 anti variant quaternions, the first in each pair transforming 

 as q, and the second as q c . 



The above transformer <w c = Q c [ ] Q c , which by (I. h) becomes 

 simply identical with w -1 , is, of course, distributive, quite in 

 the same way as g> = Q[ ]Q. Thus the sitm, or difference, of 

 two mutually covariant (but not of antivariant) physical 

 quaternions will again be a physical quaternion. 



* It can be proved immediately that (qc)' = (q')c Therefore both 

 may be written simply q c ' . 



Notice also that the invariance of q's tensor, Tq'='Tq } which follows 

 immediately from (I.) (since Q, is a unit quaternion), may be written : 



q'q c ' = qq e . 



