﻿806 Dr. L. Silberstein on the 



then we get at once from (VI.) 



□ *=-C, (XII.) 



which is the well-known equation, obtained by Sommerfeld 

 for his " Viererpotential." But here, I daresay, it follows 

 from (VI.) more immediately, than by the use of four- 

 dimensional divergences and curls or "Rotations." 



The above <!>, which may be called the potential-quaternion, 

 is easily proved to be a physical quaternion, namely, cov. q. 

 For by its definition, (X.), and remembering that F is cov. L, 

 we have immediately 



<I> cov. D c _1 F cov. DF cov. DZ, 

 i. e., by (IV.), <I> cov. g, — q. e. d.* 



Writing the potential-quaternion 



<£ = *</> + A, (23) 



where </> is a real scalar and A a real vector, it is seen at once 

 that <fi is the ordinary " scalar potential "" and A the ordinary 

 " vector potential. " In fact, developing (X.) we have 



r\A 



F = VVA- §-r -f iV£ = M-£E, 



whence the usual formulae 



M= WA=curlA, 



E=— V<£ ^r 



c ot . 



Also the condition (XL) is expanded immediately into the 



usual equation 



i|t+divA = 0. 



6* Qt 



Finally, notice that the " equation of continuity," as it is 

 commonly called, L e. 



|f+div(>p)=0 3 



assumes the quaternionic form 



SD c C = (XIII.) 



The scalar of D c is, in fact, the same thing as Sommerfeld's 

 four-dimensional divergence Div. 

 Or we may write, equivalently, 



SDC c = (XHIa.) 



* This is seen even more immediately from (XIL). For, since 

 □ =(TD) 2 is an invariant, <I> is transformed like C and, consequently, 

 like q. 



