136 Dr. L. Silberstein 



on 



some short expressions for the ponderomotive force combined 

 with its activity, which union I called the force- quaternion 

 and denoted by F e . One of these forms has been : 



P tf = Jr{DF.F-G.DF}, (1) 



loc. cit. (28), p. 808. I hoped then to be able to prove that 

 this particular form is apt to show the properties of the 

 corresponding stress, and of the density and flux of electro- 

 magnetic energy. In fact, I found since, that the above 

 formula gives all of these in an unexpectedly simple way, if 

 subjected to a somewhat peculiar but slight transformation, 

 of a purely formal character. The formula (1) so transcribed 

 proved then to be very convenient for further application, 

 inasmuch as it led to very simple formulas for the relativistic 

 transformation of stress and of density and flux of energy*. 

 To show this, along with some allied matter, will be the 

 subject of the present paper. 



1. Simplified form of the force-quaternion. — In order to 

 give to the second term of the right side of the above 

 formula (1) a form similar to the first term and guided by 

 the principle of alternation t (as explained in my first paper), 

 I recurred, after several trials of other forms, to the peculiar 

 form 



G[D]F, 



in which the differential operator D is intended to act both 

 forward and backward, and which I define explicitly by 



G[D]F = GD.F + G.DF, .... (2) 



the dots symbolizing always separators, i. e. stopping D's 

 differentiating power. What is still to be explained in this 

 symbolism, to make it entirely clear, is only the meaning 

 of GD, which is unusual inasmuch as the operator D follows 

 the entity operated on J. The way how GD is to be defined 

 naturally suggests itself : If D were an ordinary quaternion, 



* The corresponding set of formulae lias been communicated by the 

 author to the Societas Scientiarum Varsoviensis in December 1911. 



t Remember that the electromagnetic bivector F is a (scalarless) left 

 quaternion and the complementary G a (scalarless) right quaternion. 



X The reader must not be afraid of this departure from convention. 

 Oliver Heaviside says, in a similar situation, simply : " A cart may be 

 pulled or pushed,'' Electromagnetic Theory, vol. ii. p. 218. Besides, 

 J. W. Gibbs taught us to employ linear vector operators as pref actors 

 and as postf actors. 



