Quaternionic Relativity. 141 



Notice that n is a unit vector of quite arbitrary direction, 

 while <j is a real but otherwise arbitrary scalar. 



Resuming the results of the present and the foregoing- 

 section we can now say shortly that one and the same 

 operator 



G[]F 



furnishes us the force -quaternion, the corresponding stress and 

 the flux and density of energy, according as we till out the 

 vacant place [ ] by D, or by a simple vector or a simple 

 scalar, respectively. (Formulas XV. and XVI.a, XVI.fr or 

 their combination XVI.) 



Thus, our operator G[ ]F does all the offices of Minkowski's 

 matrix S of 4x4 elements * or Sommerfeld's and Laue's 

 " world-tensor/' the procedure however being in our case, 

 I daresay, a great deal simpler. Especially simple is the 

 structure of our operator, as compared with that of the 

 "world-tensor"; for, while in Gr[]F both of the bivectors 

 are ordinary quaternionic factors, Sommerfeld's "tensor" 

 {Ann. d. Physik, vol xxxii. p. 768) or, as written by Laue, 



is derived from the " Sechservector " $R in a very compli- 

 cated way. In fact, the definition of this symbol is actually 

 given by Lane's formula (80) (RelativiUltsprinzip, p. 74), 

 which contains for one only of the elements of the " tensor," 

 Tj k , not less than 16 products, while the whole " tensor " 

 consists (in the symmetrical case) of ten of such elements or 

 components. 



The utility of Gr[ ]P will appear also from what follows. 

 A similar investigation of the properties of the operator 

 F[ ]Gr, as applied, of course, to D C} W c , instead of D, k, is left 

 to the reader. 



3. The relatimstic transformation of stress and flux and 

 density of energy. — It is well known that, using the " world- 

 tensor," the stress components, &c, constituting its ten 

 elements, are transformed as the squares and products of the 

 coordinates of a point of the four-dimensional "world," 

 i. e.fn as x 2 , / 12 as xy, and so on. The corresponding for- 

 mulas are fairly complicated and rather hard to read and 

 more so to remember j\ The formulas of transformation 



* S is the " product " of two u alternating " matrices, eacli consisting 

 of 4x4 elements ; cfr. Minkowski, Gotting. Nachr. 1908, § 13. For the 

 " vacuum " Sommerfeld's S, being generally non-symmetrical, becomes 

 a symmetrical matrix. 



+ See for example Laue's (78), p. 74, loc. cit. 



