256 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
be a superficial distribution to represent the neutralized polarity at its surface. If 
denote the density of this distribution at any point; [/], [m], [n] the direction 
cosines, and [?'] the intensity of the magnetization of the solid close to it ; and (m, v 
the direction cosines of a normal to the surface, we shall have, as in the case of the 
uniformly magnetized solid previously considered, 
g>z= [i] cos 6= p/J . A+ [ini] . j«/+ p'wj .v ( 1 ). 
If, according to the usual definition of “ density,” h denote the density of the magnetic 
matter at P, in the continuous distribution through the interior, the expression found 
above for the quantity of matter in the element a, (3, y, leads to the formula 
, f d(if) , d(im) , d(m) } 
( 2 )- 
These two equations express respectively the superficial distribution, and the con- 
tinuous distribution through the solid, of the magnetic matter which entirely repre- 
sents the polarity of the given magnet. The fact that the quantity of northern matter 
is equal to the quantity of southern in the entire distribution, is readily verified by 
showing from these formulae, as may readily be done by integration, that the total 
quantity of matter is algebraically equal to nothing. 
43. If there be an abrupt change in the intensity or direction of the magnetization 
from one part of the magnetized substance to another, a slight modification in the 
formulae given above will be convenient. Thus we may take a case differing very 
little from a given case, but which instead of presenting finite differences in the in- 
tensity or direction of magnetization, on the two sides of any surface in the substance 
of the magnet, has merely very sudden continuous changes in the values of those 
elements : we may conceive the distribution to be made more and more nearly the 
same as the given distribution, with its abrupt transitions, and we may determine the 
limit towards which the value of the expression (2) approximates, and thus, although 
according to the ordinary rules of the differential calculus this formula fails in the 
limiting case, we may still derive the true result from it. It is very easily shown in 
this way, that, besides the continuous distribution given by the expression (2) applied 
to all points of the substance for which it does not fail, there will be a superficial dis- 
tribution of magnetic matter on any surface of discontinuity ; and that the density of 
this superficial distribution will be the difference between the products of the intensity 
of magnetization into the cosine of the inclination of its direction to the normal, on 
the two sides of the surface. 
44. This result, obtained by the interpretation of formula (2) in the extreme case, 
might have been obtained directly from the original investigation, by taking into ac- 
count the abrupt variation of the magnetization at the surface of discontinuity, as we 
did the abrupt termination of the magnetized substance at the boundary of the mag- 
net, and representing the un-neutralized polarity which results, by a superficial dis- 
tribution of magnetic matter. 
