140 Dr. L. Silberstein on 



obviously, the well-known Maxwellian stress * [pressure 

 proper being considered positive, and tension proper 

 negative). 



But what we are here mainly concerned with, is another 

 form of this stress, showing- the utility of the operator G[ JF. 

 Now, recurring again to (10), we obtain at once, instead 

 of (11), 



/'n=iVGnF (12) 



which is the desired form. Moreover, we have for the 

 scalar part of the same triple product 



SGnF = SGYnF = - (G VnF) = (nVGF) , 



t. e., by (7) or (XVLa), 



1 SGnF=-($n). 



Consequently, the full quaternionic product will be 



iGnF=- ($n)+/n, . . . (XYI.b) 



which is the second property of our operator. 



Both of its properties, as expressed by the formulae (XVLa) 

 and (XVI.6), may be combined into one, by adding these 

 formulae together side by side, after having multiplied the 

 former by a quite arbitrary scalar s. Then we shall have 

 G[ ]F as applied to a full quaternion s + n. Take for this 

 purpose, say, a purely imaginary scalar s = icr, so that 



k = Lcr-\-ii (13) 



will be a special physical quaternion, cov. q. Then we shall 

 have 



iG&F=-{($n)-c™}+/n--$, . (XVI.) 



C G 



a formula which will be useful for our subsequent considera- 

 tions. This expression is, obviously, like h itself, a genuine 

 'physical quaternion, namely co variant with q |. 



* In fact, developing (11) we obtain at once 



/n = «n-E(En)-M(Mn), it = ±(E 2 +M 2 ), 



which is the usual form of Maxwell's stress. C/r. also (7). 



t For it has a real vector and an imaginary scalar part, like q=l-\-T, 

 and, since G' = QGQ C , F' = Q C FQ, &' = Q&Q, 



G'*'F' = QGAFQ. 



