﻿796 Dr. L. Silberstein on the 



In the above we have been concerned with q as an 

 example, or in fact the very prototype, of a physical 

 quaternion. Another example, which will be needed in 

 the sequel, is the quaternionic equivalent of Sommerfeld's 

 '* Viererdichte," or Laue's " Viererstrom," say 



G 



=4+~p) (10) 



which we may accordingly call the current-quaternion. 

 Here p means the volume- density of electricity and p the 

 velocity of its motion relatively to the system S. To prove 

 that C is a jyhysical quaternion, write "p=dr/dt, and con- 

 sequently 



C=vJ, ...... (10a) 



and notice that, the charges of corresponding volumes in S 

 and S' being equal (by a fundamental postulate), dl/p is 

 itself an invariant of the Lorentz-transformation. 



The transformer (I. a) may, of course, be applied not only 

 to quaternionic magnitudes, but also to operators, as, for 

 example, to differentiators, which have the structure of a 

 quaternion. If 12 be an operator of this kind, in the system 

 (§, and O' its relativistic correspondent in S', and if XI' = QX1Q, 

 we shall say that the operator 12 has the diameter of a 

 physical quaternion. 



As a chief example of such an operator, which also will 

 be needed for what follows, we shall consider here our 

 quaternionic equivalent of Minkowski's matrix called by him 

 " lor " to the honour of Lorentz. This w T ill simply be the 

 Hamiltonian V pins the scalar differentiator 'd/'dl. Let us 

 denote it by D, 



D =| +V ( U > 



=B/dZ+id/d^+JB/By+kB/d*. 



Comparing this with 



we see at once that the operator D will transform precisely 

 as q did, i. e. 



D' = QDQ (12) 



Thus D has the character of a physical quaternion. 



