﻿Quaternionic Form of Relativity. 803 



or, by the non-accented equation (S), 

 DQF' = DFQ. 



Hence, rejecting an additive function of obvious properties, 

 i. e. requiring that F' shall vanish together with F, 



QF' = FQ, 

 or finally, Q being a unit-quaternion, 



F'=Q„PQ, ..... (VII.) 



which is the required correlation, identical with the above *. 

 Henceforth we shall have to admit, in the name of Relativity, 

 bivectors transforming like this calling them, say, physical 

 bivectors (or in Minkowski's way, " world "- bivectors). Or 

 we can make the L-quaternion (of which F is the vector part) 

 the master, calling it, say, a (left) physical quaternion of the 

 II. kind, and writing F as its special case 



'F^VL^Yacb (-20) 



(The supplementary scalar, Sa c &, necessary to convert F into 

 a full quaternion, would present no difficulties, since it has 

 been proved to be an invariant.) The short name physical 

 quaternion might then continue to stand for physical quaternion 

 of the first kind, of which q is the standard. 



But leave aside questions of nomenclature and return to 

 (VII.). To verify this short formula remember that, by (I.), 



Q- V(i + 7 )/2 + uV(l-7)/2, Q c = V(l + y)/2-W(l-7)# 

 and expand the right side of (VII.). Then 



P'=(l-7)(Pu)u + 7F + ^7VFu, . . (21) 

 or splitting into the real and imaginary parts and remem- 

 bering (19), 



E / = (l-7)(Eu)u4-7E4-yS7VuM l 



M / -(l- 7 )(Mu)u + 7M-/57VuEr * * (21rt ) 

 * Our quaternionic formula (VII.). resembles entirely Minkowski's 



/'-a"Ya, 



in which A is a matrix: of 4x4 elements, and A" 1 its reciprocal ; he. cit. 

 § 11. The reason of this analogy will easily be seen to depend on 

 the circumstance that both the product of quaternions and the product 

 of matrices have the associative property. But at any rate the multipli- 

 cation by a quaternion, like Q or Q c , is actually done in a much more 

 simple way than the application of a matrix of 4x4 elements. 



Observe also that the above analogy does not extend to the trans- 

 formation of Minkowski's vectors of the I. kind and our physical quater- 

 nions ; in fact, here the matrix»form is 



s = sA , with s = | s u So, s S) s 4 |, 

 whereas the quaternionic form is 



3 G 2 



