Quaternionic Relativity. 137 



with s, w as its scalar and vector parts, respectively, we 

 should have 



Gs-fGw=Gs + VGw— (Gw) = sG— VwG— (wG) ; 



writing here "dfdl instead of s and V instead of w, the 

 required definition of GD will be 



GD=|^-curlG-divG (3) 



ot 



Observe that DG could not be used at all, for G is right- 

 handed, so that DG would be torn asunder by the relativistic 

 transformation. 



Now to see the utility of GD let us compare it with DF, 

 which appears in (1). Since, by (VI. a), D C G=C C , we have 

 by the elementary rule, by which the conjugate of a product 

 of quaternions is the product of their conjugates in the 

 reversed order, C = G C D=-GD. But, by (VI.), C = DF. 

 Thus 



DF=-GD, (4) 



and using this in (1) we obtain the required expression for 

 the force-quaternion 



P.= — JG[D]F (XV.)* 



Thus, G[ ]F, when applied to D, or rather, when exposed 

 to the bilateral action of D, gives the force-quaternion ; 

 and this being the case, it has been easy to guess that the 

 same operator G[ ]F, when applied to a vector, say to the 

 unit surface-normal n, should give us the corresponding 

 stress and the component of the energy flux relativistically 

 associated with itf. This supposition proved, on trial, to 

 be correct, and having once verified it, the systematical 

 deduction from (XV.) of the energy, stress, &c, as shown in 

 the next section, has been a matter of course. 



2. Properties of the operator G[ ]F. Stress, electro- 

 magnetic momentum, flux and density of energy. — To see the 

 properties of the above operator, develop the right side of 

 (XV.), remembering that 



P. =/>{ -(pE) +E+ - VpJl]. = i(gp) +%, 



( C C j c 



% being the ponderomotive force (per unit volume), and 



* The Roman numerals, reserved for the more essential formulae, are 

 here continued from the first paper. 



f Especially as I have already remarked (Ann. der Physik, vol. xxii. 

 1907) that G[ ]F, when applied to a scalar, or simply GF, gives the 

 resultant flux and the density of energy. 



