1904 - 5 .] Magnetic Quality in Molecular Assemblages. 1035 
desirable to obtain the expression in a form in which the direction 
cosines of the magnetisation are separate from those giving 
positions of centres. In the summations of last section, we must 
carefully remember that each sum is not merely a sum of distinct 
numerical values. Account must also be taken of the number of 
times that a given numerical value appears. In this way, from 
the condition 2p = A. 2 + /x 2 4 - v 2 , we get 32(A 2 ) = 2p2-N, where N is 
the number of times that a given A. 2 occurs with the given p. 
Remembering again that sums of quantities which involve odd 
powers of A, ,u, or v vanish, we find 
(aA + fig + yv) 4 — 2(a 4 A 4 + /3 4 /x 4 4- y 4 c 4 ) + 62(a 2 /3 2 A 2 //, 2 + (3 2 y 2 g 2 v 2 4- y 2 a 2 i/ 2 A 2 ) 
- (a 4 + /3 4 + y 4 )2-A 4 + 6(a 2 ) S 2 4 - fly 4- y 2 a 2 )2-Ay 
= (a 4 4- p 4 4- y 4 )2-A 4 4- (a 2 /3 2 + /5 2 y 2 4- y 2 a 2 )[(2p) 2 S-N - 32-A 4 ] , 
since 62(A> 2 ) = 32.A 2 (/a 2 4- v 2 ) = 32-A 2 (2 \p - A 2 ). 
But 1 = a 4 4- P 4 + y 4 4- 2 (a 2 j8 2 4- /3 2 y 2 4- y 2 a 2 ), and so we get 
2(aA + pg + yv) 4 = J(a 4 4- /3 4 4- y 4 )(52-A 4 - 2-(2^) 2 ) - 1(32-A 4 - 2-(2 p) 2 ) . 
Therefore, writing 2-(2 \p) 2 in the form (2p) 2 5-N, the parallel 
component takes the form 
! [^{ (a 4 + / 34 + / )(52.^-2.N) 
_\y 
WY 
( 32 ' 
A. 4 
m 
-S-N)] 
442-N 
(2 P) 
- 5/2 
where the 2 p outside the square bracket comes under the signs of 
summation also. The summations are easily carried out to a 
sufficient approximation, and will be given subsequently (§ 22). 
The expression reduces to 
A — B(a 4 + p 4 + y 4 ) , 
where A and B are positive, and A>B. Since the k in Wallerant’s 
formula for the induction (§ 5) is a negative quantity, it appears that 
the form here deduced for the internal component of force parallel 
to the induction is identical with that which Wallerantgave for the 
component of induction parallel to the external force. The a used 
here corresponds to his cos a, etc. This result justifies to some 
extent the remarks made at the end of § 5. Further justification 
will appear when we consider the transverse components of force 
and induction (§§ 14, 16). 
