Quaternionic Relativity. 143 



take first <7 = 0, and then <r = l, and remember that, by (16), 



o- '=-/3y(in); a 1 , -(r t =y; N/-N ' = — /3yi; 



N ' =n + (y-l)i(in) = y-n, 



where e is a linear vector operator, stretching every vector 

 normal to the direction o£ motion in the ratio y : 1 and not 

 affecting vectors parallel to that direction, or in dyadic form 



e=i(i + yj(j + «yk(k (18) 



Of the four relations, obtained in this way from (IT a), 

 (17 &), one, namely that which contains ty n — c/3/ wl , is a con- 

 sequence of the three others (one of which contains the 

 resultant 9$). 



These three relations, after a slight rearrangement of terms 

 and without splitting in Cartesians, give us the required 

 relativistic transformation of the density and flux of energy and 

 of the stress in the remarkable form 







(XVII.) 



e being the operator explained in (18). 



The three formulae in (XVIL), one being scalar, one 

 vectorial, and one dyadic, are completely equivalent to 

 Laue's ten ( = 1 + 3 + 6) transformational formulae (102)*, 

 which the reader may verify at once, by expanding the 

 second and third of (XVII.) and remembering that /', 

 like /, being a linear operator, f'v = vfi = vfi,(vf'v) = v 2 f iu 

 and so on. 



To obtain u f , *$', f in terms of u, ^}, /, we have only to 

 transfer the dashes to the symbols without dashes and to 

 write —v instead of v, leaving the coefficient 7 and the 

 dyadic e unchanged. 



Of particular interest, especially as regards its application 

 to relativistic dynamics, is the case in which the flux of 

 energy ', and consequently also the electromagnetic momentum, 

 vanishes for one of the two systems S f S', we are comparing; 



* Laue, he. cit. p. 87. 



