﻿802 Dr. L. Silberstein on the 



Now, remembering that YVF + SVF = VF and using the 

 quaternionic differentiator D, explained by (11), the last two 

 coalesce at once into the single equation 



DF = C, (VI.) 



in which C is the current-quaternion, as defined by (10). 



Thus, the whole system of four equations (18), the funda- 

 mental equations of the electron theory, are represented by 

 one quaternionic equation, (VI.). 



This condensation is even more complete than in Min- 

 kowski's matrix-form, which consists of tivo equations, 

 lorf— —s, lorf* = (loc. cit., § 12), one for the first pair of 

 (18) and the other for the second pair, or in Sommerfeld's 

 equivalent four-dimensional vector form : ^u? / = P and 

 2to/*=0 (loc. cit., § 5). Here P is the " Vierervektor " 

 corresponding to the current-quaternion C, and /the " Sechs- 

 ervektor" corresponding to the bivector F, while f* is the 

 " supplement " (Erganzung) of/ which is another "Sechs- 

 ervektor/' though very nearly related to /. Minkowski's 

 / is an alternating matrix of 4 x 4 elements. But let us 

 return to our quaternionic differential equation (VI.). 



C is a (given) physical quaternion cov. q. The operator 

 D has also the structure of q. What is the relativistic 

 transformer of F ? By (V.) we see at once that it is 



Q.[]Q, 



or that F is transformed like a (scalarless) L-quaternion. 

 Thus, the answer is already contained in (V.). But to see 

 clearly the true meaning of the process implied in the 

 relativistic transformation, let us repeat again the whole 

 reasoning somewhat more explicitly. We have, in the 

 system S, as an expression of the laws of electromagnetic 

 phenomena, the equation 



DF=C ;■■;-.'.. '(S) 



Now, what the Principle of Relativity requires is the same 

 form of the law in the system S', i. e. 



D'F'=C (S') 



Suppose also that both of these equations have been fully 

 confirmed by experience. How are ¥ / and F correlated ? 

 To adopt language adapted to the general case, use in the 

 accented law or equation (S') the transformer already 

 known, i. e. in our present case Q[ ]Q for both D and C ; 

 then it becomes 



QDQF' = QCQ,orDQF' = CQ, 



