1038 Proceedings of Royal Society of Edinburgh. 
are proportional to a(a 2 -Q), (3((3 2 - Q), y(y 2 - Q), where Q = 
a 4 + /3 4 + y 4 . Inserting these quantities divided by the square root 
of the sum of their squares, in place of l, m, n in the factor outside 
the sign of summation, it becomes Ja 6 + /3 6 + y 6 -Q 2 , the proper 
sign having yet to be found for the cosines of the force. 
To determine the sign put in (a 3 l + (3 3 m + y 3 n), a=l-2£ 2 , 
(3 = rj, y — 0, where £ and r] are small positive quantities, and we get 
7] = 2£. Now assume l— - 2£, m= 1 - 2£ 2 , and (a 3 Z + (3 3 m + y 3 n) 
becomes - 2£ ; so that the force is outwards along ( - 2£, 1 - 2£ 2 ), 
and its component, in the direction indicated by l, is - 4£ 2 
multiplied by % This component must be identical with 
±a(a 2 + Q)/x/a 6 + /3 6 + y 6 — Q 2 multiplied by % Substitution of 
the values of a, (3, y gives the result ± 4£ 2 . Therefore the negative 
sign must be taken. 
Consequently the magnitude of the transverse force is 
and its direction cosines are a(Q — a 2 )/ J a 6 + (3 6 + y 6 - Q 2 , etc. 
13. Directions of Equilibrium in Zero Field . — The expression 
for the magnitude of the transverse force shows that the cones 
constant transverse force. The particular case, when c = 0, is of 
special interest since it gives the directions of equilibrium under 
the action of the internal field alone. If we put z= 1, the 
equation is a symmetrical cubic in x 2 and y 2 . If it be 
arranged in descending powers of x 2 , it is easy to prove that the 
term independent of x is positive or zero, and that the sum of the 
other three terms is also necessarily positive or zero. The two 
zero values occur together when x and y have the real values 
0 and ± 1. 
Thus the only directions of equilibrium are the 13 lines of 
cubic symmetry — i.e. the three quaternary axes, the four ternary 
axes, and the six binary axes. 
245 j2Ma 2 
2p 5 
r 2 (x 6 + y 6 + z 6 ) - (ic 4 + ?/ 4 + z 4 ) 2 = c 2 r 8 , 
where x 2 + y 2 + z 2 = r 2 , and c is an arbitrary constant, are loci of 
