400 Dr. L. Silberstein on the 



geometrical relations o£ all the centres to remaining 

 the same) 



P=p'(sI2'+fl")i, 



and by adding to the last equation, 



that is, 



p=i(i+#yoii (3) 



where fl is the total operator, taking account o£ all centres, 

 as in (2) . Thus, in order to show that the resultant inter- 

 action is nil *, it is enough to prove that Xli = 0. 



Let us introduce as a reference system the four unit 

 vectors i, j, k, 1, drawn from a centre, say 0, to its four 

 nearest neighbours, 1, 2, 3, 4 (fig. 2). Then, first of all, 



i+j+k+l=0, (4) 



and ij = ik = il=jk, etc. Thus, multiplying (4) sealarly by i, 

 for instance, we have 



ij=ik=....= -i (5) 



which, by the way, is a very simple deduction of the cosine 

 of the equal angles 102, 103, etc. Using this, any vector R 

 can at once be represented by means of its four projections 

 upon i, etc., 



i^! = Ri, jR 2 = Rj, i? 3 = Rk, i2 4 = Rl. 



In fact, write 



'R = a 1 i + a 2 j+a 3 'k + a 4: l ; 



then B ± = % — J (a 2 + a 3 + a 4 ) , 



or R 1 = fa 1 — J 2 a;, etc. 



Of these four equations three only are independent, since 

 we have identically 



i? 1 -f-i? 2 + ^ 3 + ^4 = R(i+j+k + l) = 0. 



Thus one of the four coefficients U{ remains arbitrary. We 

 can put 2<2i = 0, so that Ri — ^a^ and therefore, the required 



* When E falls into i or 01, and therefore also when E has the 

 directions 02 or 03 or 04. This being the case, and p being at any 

 rate a symmetrical or self-conjugate linear vector function of E, the 

 proof that fii=0 will imply also that the iuteraction is nil for any 

 direction of E. 



