220 Proceedings of the Royal Society of Edinburgh. [Sess. 
Similarly, using T 2m to represent the term in F sin ( 0 + 0) which involves 
the power 2 p of the ratio a/v, and employing the other known relations 
sin 2 0. P" n (cos 0) = 2 cos 0Pk(cos 0) - n(n + l)P„(cos 0 ) , 
(2n+ 1) cos 0P' n (cos 0) = ??P' n+1 (cos 6) + (n + l)P' w _ 1 (cos 0 ) , 
(2 n + l)Pn(cos 6) = P' n+1 (cos 6) - P' (n - 1) (cos 6) , 
we get 
• (2) 
Finally, therefore, 
F COB (* + *>52 •§2( 4 ^)'‘ 2 (/ i + 1 ) P 2(M+l), • 
7 O''/ 
• (3) 
F sin (£ + £) = 2 • ^S( 4 ^) Msinep Vfi>> • 
• (4) 
where the first 2 in each expression indicates summation for all the con- 
stituent magnets in the infinite system, omitting the one at the origin. If, 
in any crystalline system of molecular magnets, the arrangement is not 
singly homogeneous, so that products and odd powers of co-ordinates do 
not vanish on summation, these expressions require modification, for odd 
powers of A would appear in the developments of I/ 2m and T' 2m . 
7. The most powerful term in (3) and (4) respectively is the first. 
Each successive term is of order two higher in the ratio a/r than the 
preceding. Therefore the effect of the first term is felt at much greater 
distances than that of any other, and is more strongly felt at any one 
distance. Consequently that term, in each series, is of greatest importance 
in the determination of the so-called ferromagnetic quality of a system of 
molecular magnets. 
In the expression (3), for the component of the internal field in the 
direction of magnetisation, the first term is 
(5) 
If p, pp, qp represent respectively the distances between successive planes 
normal to the three principal rectangular directions in the space-lattice 
determined by the magnet-centres, we have 
where X, p, v are integers to which all values, positive and negative, from 
1 to oo are to be given, and we may presume q <f 1. Also, if a, f3, y be 
