﻿766 Dr. A. C. Crehore on the Construction of 



The Equilibrium of Turning Moments. 



It may be shown that the total sum of the turning moments 

 of all the atoms in the structure fig. 1 upon any selected 

 atom is zero, and there is no tendency to turn its axis. Any 

 two atoms tend to turn so that their axes are parallel and 

 the turning moment is a function of the angle a between 

 the axes. The moment of atom ' ; a, 9 ' fig. 12. upon A at the 

 centre is counter-clockwise in the plane aAtk, when viewed 

 from h. The line A/i is perpendicular to this plane, and the 

 turning moment of a upon A may be represented as the 

 vector Ah. The moment of the opposite atom b whose axis 

 is parallel with a's, being an atom of the same kind at the 

 same distance, is equal to that of a and in the same direction. 

 The moment of the pair a + b is then 2AA. The moment of 

 e on A is such that it is counter-clockwise viewed from ??, 

 and represented by Are, which is perpendicular to the plane 

 eAot. The axis of the opposite atom / is parallel to e at the 

 same distance, and this atom being of the same kind doubles 

 the moment, so that e+f gives 2Aw. The atom d gives a 

 counter-clockwise moment when viewed from s, the line As 

 being perpendicular to the plane Adtp, and the atom c 

 similarly doubles the moment, making that of c + d give 2 As. 

 The sum of the three vectors AJi + An + As therefore gives the 

 sum of the turning moments of the atoms a, b, c, d. e, and / 

 upon the central atom A. It is evident that this sum is zero, 

 for these three lines lie in the plane hlsjnq, being a hexagon 

 made by sectioning the large cube, and they are, moreover, 

 120° apart. As they are equal in magnitude their sum is zero. 



Now consider the atom k upon A. This is represented by 

 the vector Aj, since viewing from j the rotation is counter- 

 clockwise in the plane Akta. The four atoms h, i. j, and k 

 are similar in kind and have parallel axes. Hence the 

 turning moment of the four is 4A/. The effect of o on A is 

 represented by Al, perpendicular to plane Aote, and the four 

 similar atoms /, ire, n, and o give a moment 4AI. The effect 

 of p on A is represented by Aq, perpendicular to plane Aptd, 

 and the four atoms p, g, r, and s give lAq. Since Aj, A/, 

 and Aq lie also in the same hexagon and 120° apart, the sum 

 of the moments of h 9 i, j, k, I, m, n, o, p, q, r, and s is zero. 

 The only remaining atoms are those along a diagonal of the 

 cube. Since these, t, u, v, ic, #, y, z, and B, are all similar 

 atoms with axes in the opposite direction to that of A the 

 turning moment of them all is evidently zero, though for a 

 small displacement the moments of these would give in- 

 stability. The control of the position for stability lies with 

 the other atoms. 



