232 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Again, if we write a 2 = 1/3 — £ /5 2 = 1/3 — rj, y 2 = l/3 + ^+> 7 , where £ and 
>7 are small, we find A T / 2 = P£ fF ' 2 = l ) r], v T' 2 = P(f +>?)■ In these expressions 
P is a positive quantity. Hence the transverse force acts so as to turn the 
system of co-directed magnets into coincidence with the ternary axis. 
Similarly, we find that the transverse force always tends to turn the 
system from coincidence with a quaternary axis. It acts more towards, 
or from, a binary axis according as the displacement of the direction of 
magnetisation from the binary axis is, on the whole, towards a quaternary, 
or a ternary, axis. 
From these conditions we see that, when G>D, the ternary axes are the 
only directions of stable equilibrium for the co-directed system of magnets 
when the external field is removed. On the other hand, if D>G, the 
quaternary axes are directions of stable equilibrium. 
14. To determine the values of L' 2 , and the components of T' 2 , in the 
hexagonal system, we have to put p — J 3 , q = q, in the preceding section 
if we choose the axes as in § 11 . Equation (18) is then subject to the 
conditions 
D = 35]T 
A 4 
(A 2 + 3/x 2 + £ 2 i/ 2 ) 9/2 
E = 35 £- 
v 
(A 2 + 3/x 2 -f- q 2 v 2 ) 9/2 ’ 
F = 35 2 
g 4 v 4 
(A 2 + 3//, 2 4- $ 2 v 2 ) 9/2 ; 
9AV Tr _ - ( r, V 3gVX 2 
(A 2 + 3ju. 2 + gV 2 ) 9/2 ’ 4- • (A 2 + 3/t 2 + qV) m ’ • (A 2 + 3/x 2 + qW)™ ’ 
A 2 30 2 • (A 2 + 3 j a 2 + gV 2 ) W! ’ B * 
3 fx 
(A 2 + 3/x 2 + g 2 v 2 ) 
,^ 2 „ 2 \ 7 / 2 ’ C 2 30 2 */\2 
q 2 v 2 
(A 2 +3^ 2 + 2 2 r 2 ) 7/2 ’ 
T_ V 3 
^-‘(A 2 + 3 /x 2 + 2 Vf' 
Relations amongst these quantities are at once given by the conditions 
of molecular symmetry in the directions a = l, /3 = y = 0, and a = 1/2, 
/3= JS/2, y = 0 , and those of symmetry in the directions a = y = 0 , /3=1, 
and a= JS/2, /3= 1/2, y = 0 . Expressing these conditions in (18) we find, 
under limitation of the problem to the plane y = 0, 
D- 
E- 
* , T D 9E 
An 4" J == 4- 4- 
2 16 + 16 + 
■p i 9D E 
-D.^ + J = (- — -f- 
2 16 16 
3G 
8 
3G 
8 
3 A_^ + J 
Hence D — E = A 2 — B 2 , and G — E = D — G, and therefore, in the hexagonal 
system, 
L 2 = 
2Ma 2 
~~P*~ 
(D-A 2 + J), . 
. ( 21 ) 
that is, the parallel component of this order , is absolutely constant in the 
principal plane. Its sign depends on the relative values of D, J, and A 2 . 
