FROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 275 
obtained in their simplest forms by eliminating the arbitrary function cp by differentia- 
tion ; and are of course 
^ ^—0 
d2 dy—^ 
dx dz 
— — — =0 
dy dx 
(III.) 
Cor . — It follows from the first part of the preceding investigation that equations 
{h.) express that the distribution, if not lamellar, is complex-lamellar. By elimi- 
nating the arbitrary function from those equations, (which merely express that 
ocdx-^-f^d^-j-ydz is integrable by a factor,) we obtain the well-known equation 

as the simplest expression of the condition that a, j3,y must satisfy, in order that the 
distribution which they represent maybe complex -lamellar ; and we also conclude 
that if this equation be satisfied the distribution must be complex-lamellar, unless 
each term of the first number vanishes by equations (III.) being satisfied, in which 
case the distribution is, as we have seen, lamellar. 
76. The resultant force at any point external to a lamellarly-magnetized magnet 
will, according to § 73 (Cors. 2 and 4 .), depend solely upon the edges of the shells 
into which it may be divided by surfaces perpendicular to the lines of magnetization 
(or the bands into which those surfaces cut the bounding surface), and not at all on 
the forms of these shells, within the bounding surface, nor upon any closed shells of 
which part of the magnet may consist ; and the resultant force at any internal point 
may (§ 73. Cors. 2, 4 , and 7,) be obtained by compounding a force depending solely 
on those edges, with a force in the direction contrary to that of the magnetization of 
the substance at the point, and equal to the product of i-r into the intensity of the 
magnetization. For either an external or an internal point, the resultant force may be 
expressed by means of a potential, according to ^ 49 ; and the value of this potential 
may be obtained by means of the theorems of § 73, in the following manner. 
Let us suppose all the open shells, that is to say all the shells cut by the bounding 
surface of the given magnet, to be removed, and an imaginary series of shells 
having the same edges, and the same magnetic strengths, and coinciding with the 
bounding surface, substituted for them ; and, for the sake of definiteness, let us 
suppose each of these shells to have its north polar side outwards, and to occupy a 
part of the surface for which the value of p is greater than at its edge. The whole 
surface will thus be occupied by a series of superimposed magnetic shells, constituting 
a complex magnetic shell which will produce a potential at any external point the 
same as that due to the whole of the given magnet ; and it will produce a potential 
at any internal point, which, together with the potential due to the closed shells which 
2 N 2 
