1911-12.] The Molecular Theory of Magnetism in Solids. 235 
limiting these quantities, it is not possible thereby to determine the sign 
of 231 A'g — 315 C 3 + 3 OR/ 3 . Its sign can only be found by summation with 
respect to the given values of X, /jl, v, which depend upon the special space- 
lattice involved. Its form is identical with that of the full-line curve 
in fig. 5. 
A point of special interest lies in the fact, deducible from (20), that y T' 2 
and y T' 4 have the same geometrical form, and in the fact that y L' 2 and y L' 4 
also have the same geometrical form. 
16. Equation (23) becomes applicable to the hexagonal system by the 
substitution r p = J 3. Taking the case y = 0, we get 
y L' 4 = 6 { 23 1 [a«p + /36Q + 1 5a 2 /3 2 (a 2 S + />' 2 T)] 
- 315[a 4 X + /3 4 Y + 6a 2 /3 2 X’] + 105(« 2 W + /? 2 W') - 5N } (26) 
if we write P = 2 . Q = 2 . 27/x 6 / _18/2 , etc. In the general case (23) 
contains twenty coefficients, between which ten independent relations, each 
involving four coefficients, are obtainable by evaluation of the quantities 
(A 2 + 3 m 2 + g V)A 4 , (X 2 + 3/x 2 + g V)g M 4 , (X 2 + 3 M 2 + g V)g V,( X 2 + 3/x 2 + g V) 3A V, 
(X 2 + 3 m 2 + gV)3gVV, (X 2 + 3 M 2 + g V)g VA 2 , (X 2 + 3 M 2 + g V)X 2 , 
(X 2 + 3/ul 2 + q 2 r 2 )3 / a 2 , (X 2 + 3 /a 2 + g V)g V, and (X 2 + 3/>t 2 -l-g 2 j/ 2 ), each divided by 
the proper power of (X 2 + 3^ 2 -l-gV). So far, therefore, as these relations 
go, the ten coefficients P, Q, etc., are independent. But the conditions of 
symmetry of the hexagonal arrangement in the plane y = 0 show that y L , 4 
has equal values with a = 1, and a = 1/2, and that it has also equal values 
with a = 0 and a= ^/3/2. From these relations we find 
2|(P_q + S-T)-3(X-Y) + (W-W') = 0, 
16 7 . (27) 
11(P + Q-5(S + T))-10(X + Y-6X') = 0. I 
whence we may express P and Q, say, in terms of the remaining eight 
quantities 
Equation (26) shows that the parallel component of the internal field 
in the plane y = 0 is symmetrical in the sextants 0° to 60°, etc. ; and has 
maxima or minima in the directions 0°, 60°, etc., and minima or maxima 
in the directions 30°, 90°, etc. 
In this case the transverse component is 
Y T' 4 = [33{|8‘Q - a 4 P + 5(a 4 S - /3 4 T) + 2a 2 /3 2 (T - S)} 
- 30{/3 2 Y - a 2 x+ 3(a 2 - /3 2 )X'} + 5(W'- W>] (28) 
This vanishes at a = 0, /3 = 0; and, with the conditions (27), it vanishes also 
