PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 255 
manner that the density will be uniform over each face, and that the quantities of 
matter on the six faces will be as follows : — 
—il . jSy, and il .j3y; on the two faces parallel to YOZ ; 
— im . ya, and irn . ya ; on the two faces parallel to ZOX ; 
— in . a|3, and in . aj3 ; on the two faces parallel to XOY. 
Now if we consider adjacent parallelepipeds of equal dimensions, touching the six 
faces of the one we have been considering, we should find from each of them a second 
distribution of magnetic matter, to be placed upon that one of those six faces whicii 
it touches. Thus if we consider the first face |8y, or that of which the distance from 
YOZ is X— ^ a; we shall have a seconddistribution upon it derivedfrom a parallelepiped 
the coordinates of the centre of which are x—a, y, z ; and the quantity of matter in 
this second distribution will be 
This, added to that which was found above, gives 
diil) f \ n ^ d{iT) 
for the total amount of matter upon this face, 
distribution on the other face, /3y, is equal to 
Again, the quantity in the second 
and therefore the total amount of matter on this face will be 
By determining in a similar way the final quantities of matter on the other faces of 
the parallelepiped, we find that the total amount of matter to be distributed over its 
surface is 
\d{iT) 
, d[im) 
, d{in)'\ 
1 dx 
•“ dy 
1 dz ] 
|aj3y. 
Now as the parallelepipeds into which we imagine the whole mass divided are infinitely 
small, we may substitute a continuous distribution of matter through them, in place 
of the superficial distributions on their faces which have been determined ; and in 
making this substitution, the quantity of matter which we must suppose to be spread 
through the interior of anyone of them must be half the total quantity on its surface, 
since each of its faces is common to it and another parallelepiped. Hence the 
quantity of matter to be distributed through the parallelepiped a|3y is equal to 
^d{il) d{im) d{in) 
dy ' dz 
Besides this continuous distribution through the interior of the magnet, there must 
