1911-12.] The Molecular Theory of Magnetism in Solids. 219 
5. These quantities represent also the similar components, at the origin, 
due to a similarly disposed magnet at (r, 0). 
Hence, by summation for all values of (r, 0) we obtain the value of the 
total components of force at the origin, due to any given distribution of 
co-directed magnets. 
6. If now we simplify and make definite the problem by taking the 
case of a homogeneously and rectangularly arranged infinite system of 
equal and co-directed magnets, one of which is situated at the origin, we 
can obtain the values of the parallel and transverse components of force at 
the poles of the magnet placed at the origin by means of the operator 
0 0 0 
°'i a dx^ r ^dy ""1-^02^’ w ^ ere a ’ P’ y are the direction cosines of the axis. If 
we denote the operator by A, and the term, in the parallel component of 
force, which involves the power of the ratio a/r, by L 2ja , the value of 
this term at a pole is 
L V = L 2^ + ^yA 2 L 2(M _ .i) + iA 4 L 2(jlJt _2)+ • • . + ^y, L 0- 
To evaluate this we have the conditions 
A r = a cos 0 , 
A cos $=■■ — sin 2 # , 
r 
A sin # = — sin # cos # , 
r 
r 2 = x 2 + y 2 + z 2 , 
a0S 6= a i±Py+y i. 
r 
By means of the known relations 
(2n+l) sin 2 # P'„(cos #) = n(n+ 1 )(P ?i _ 1 (cos #) - P n+1 (cos #) , 
('in + 1) cos # P w (cos #) = rcP n _i(cos #) + (n + l)P n+1 (cos #) , 
it is easily found that 
So 
__ A 2 ^L 
2 p\ 
2 (n-p) 
(2/*+l )! 
ip\ (2/a - 2p+ 1)! 
(V+l)! r 
^ip\(2p-ip + l)\’ 
But this sum is the sum of the even coefficients in the 
(l+x) 2fA+1 . Therefore 
BV = , 
“(4 
a A V 
2(/4+ 1)P2(|X+1) • 
expansion 
of 
( 1 ) 
