264 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
59. If m, n denote the direction cosines of a line drawn along the bar, from its 
middle towards its north pole, and if a be the length of the bar, we shall have 
ri — yl=am, X^—Xl=-an. 
Hence, if the bar be infinitely short, and if x, y, z denote the coordinates of its 
middle point, we have 
and 
dX 
, , dX 
dx 
dY 
, . AY 
, i'i 
dx 
dZ 
7 . AZ 
, <iZ 
dx ■ 
“^+ 4 - 
Multiplying each member of these equations by /3, we obtain the expressions for the 
components of the force in this case ; and the expressions for the components of the 
couples are found in their simpler forms, by substituting for | &c. their values 
given above ; and, on account of the infinitely small factor which each term contains, 
taking 2X, 2Y, and 2Z, in place of X+X', Y+Y', and Z+Z'. 
60. Let us now suppose an infinite number of such infinitely small bar-magnets 
to be put together so as to constitute a mass, infinitely small in all its dimensions, 
uniformly magnetized in the direction (/, m, n) to such an intensity that its magnetic 
moment is (/j. We infer, from the preceding investigation, that the total action on 
this body, when placed at the point x, y, z, will be composed of a force whose com- 
ponents are 
acting at the centre of gravity of the solid supposed homogeneous ; and a couple of 
which the components are 
jW/(Zm— Yw), 
— Zl), 
(^{Yl-Xm). 
61. The preceding investigation enables us, by means of the integral calculus, to 
determine the total mutual action between any two given magnets. For, if we take 
X, Y, Z to denote the components of the resultant force due to one of the magnets, 
at any point {x,y, z) of the other, and if i denote the intensity and (/, m, n) the direc- 
tion of magnetization of the substance of the second magnet at this point, we may 
take fA=i. dxdydz in the expressions which were obtained, and they will then express 
