Quaternionic Relativity . 139 



Now, the vector part o£ the product of any three vectors 

 A, B, C * is easily proved to be 



VABC = B(CA)-C(BA)-(CB)A-) ^ 



= B(CA) -(AB)C-A(BC) J ' 



which is here written in a twofold manner on purpose. 

 Taking G, F, V instead of A, C, B and using, with due 

 caution, the first and the second form of (10), respectively, 

 we obtain 



VGV • F= V(G . F)- F . divG-(F . V)G, 

 VG. VF = V(F.G)-G.divF— (G.V)F, 



where none of the separating dots is superfluous : in the first 

 formula it is G only which is exposed to the action of the 

 differential operators as if F were constant, and vice versa in 

 the second formula. Taking their sum, the first terms of 

 the right-hand sides give us the full gradient of the scalar 

 product (GF), i. e. V(G-F) without dot, so that 



VG[V]F= V(GF) -F . div G-G . div F-(F . V)G-(G.V)F 



= i{^(GF)-div(FG 1 + GF 1 )}^j{...}+k{...} 



=idiv{(GF)i-F(Gi)-G^Fi)}+ .... 

 Thus (9) takes the form 



$Mxw = — idivf i~J divf 2~~ kdivf 3> • * ( 9a ^ 



where the vectors f x , f 2 , f 3 , constituting a self-conjugate 

 st?*ess, are given by 



fi or/i = i{(GF)i-F(Gi)-G(Fi)},&c, 



or, n being any (say unit-) vector, by 



f«=/n=KOF)n-iF(Gn)-iG(Fn), . . (11) 



or, finally, by the dyadic f 



/"=KGF)-iF(G -iG(F . . . (11a) 

 This stress, represented by the linear vector operator /, is, 



* Which may be ordinary- or bivectors. 



t Here, and in what follows,/ 1 is a dyadic or a linear vector operator, 

 which may be called the stress-operator, and which, when applied to 

 the unit &urface-normal n, gives the corresponding pressure-vector f n . 



