﻿Quaternionic Form of Relativity, 805 



where C c is the conjugate current-quaternion. f>{t>— p/c) and 

 D c the conjugate differential operator "dfdl — V> as already 

 explained. 



We now see, by (V.a), that G is transformed like an 

 R-quaternion, i. e. 



G' = QGQ C . ..... (VILa) 



Again we may write, similarly to (20), j 



G = YR = Yde c , ..... (20a) 



d, e c being a pair of physical quaternions covariant with q 

 and q c respectively. And since Gr is a 'physical bivector, 

 just as much as F, we may again call R = de c a (right) 

 physical quaternion of the second kind. 



Notice that, at least for the time being, we have no need 

 of both F and G, since we require either F only or G only. 

 (Possibly for the further development of Quaternionic 

 Relativity the simultaneous use of F, G may turn out to be 

 convenient or even necessary.) 



As regards the relation of (20a) to (20), observe that 

 generally we cannot write d=a, e = b ; in fact, the reader 

 will easily prove for himself that this would require (EM) =0, 

 i. e. E-LM, and would not, consequently, be sufficiently 

 general. The only essential thing here is that in (20) 

 it is the first and in (20 a) the second factor which has the 

 subscript c. This is shown also by the symbols L (left), 

 R (right) . 



Let us return to the quaternionic differential equation for 

 the vacuum, in its first form, i. e. 



DF = C (VI.) 



Remember that DD C =(TD) 2 = J^ 2 + J^ 2 + |^ + J^ 2 



is the four-dimensional Laplacian, or Cauchy's □ , 



dd,= d (IX.) 



Hence, if <E> be an auxiliary quaternion and if we put 

 F=— VD C <I> (since F is scalarless), or more simply if we 

 write 



F = -D c <£> (X.) 



demanding at the same time that 



SD fl <S> = 0, (XI.) 



