Dispersion of Diamond. 401 



representation of any vector, 



or written as a dyadic, 



i.i+j.j+k.k+l.l=4 (6') 



This may be a useful formula for various purposes *. 

 Returning to our problem, let us substitute in II 



u=}{M 1 i + ...+w 4 l} (6) 



In the first place, it may be interesting to notice that if 

 only the four nearest neighbours are taken into account, 

 then 12 = identically ; in fact, in this case we have, by (2), 



ft(i 2 34>=^2(3u.u-l)=^{3:i.i + ... + l.l)-4}, 



which vanishes identically by (6'). Next, passing to the 

 full operator, with the sum in (2) extended to the whole 

 unlimited assemblage, notice that II is a sym metrical or self- 

 conjugate linear vector operator and will thus have three 

 mutually perpendicular principal axes. If the three corre- 

 sponding principal values of such an operator are equal, the 

 operator degenerates into an ordinary scalar or numerical 

 factor. In that case, in fact, every direction is a principal 

 one. Now, turning to our operator 12, take i, that is, a unit 

 vector along 01, as operand. Then the result will obviously 

 be a vector along +i or — i, i.e. 



fii=I2 n .i, 



where I2 n is a scalar, viz. iOi. Again, remembering that 

 ij= -J, etc., 



12 =jI2i=I2 11 ji=:--lI2 11 , 



and similarly £l n = —l£L u , etc. Also, by the equivalence of 



i, J, i, 1, 



I2 n = 1222 = etc. 

 Thus, summarily, 



i2 n =n 22 =...=n aa =-3f2 a/3 , ... (7) 



where azfzfi—1, 2, 3, 4. Writing, for the moment, p — ., , 

 we have, by (2), 47rr " 



£l u = 2.p(3u 1 2 — l) ; 0, 12 = Xp(^u l u 2 ~ij) =%p (3^1*2 + J), 



and since I2 12 = — -JI2 n , 



%p(Zu x u 2 + i)~Zp(i— u\), etc. 



* Similarly, if i, j, k are three unit vectors drawn from the centre of 

 an equilateral triangle towards its corners, so that again i+j+k = 0, 

 ij=jk = ... = — \ ( = cos 120°), we have, for any operand, 



i.i+j.j+k.k=3/2. 

 Phil. Mag. S. 6. Vol. 37. No. 220. April 1919. 2 F 



