258 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
(1.) The resultant force at a point in space, void of magnetized matter, is the force 
that the north pole of a unit-bar (or a positive unit of imaginary magnetic matter), if 
placed at this point, would experience. 
(2.) The resultant force at a point situated in space occupied by magnetized matter, 
is an expression the signification of v/hich is somewhat arbitrary. If we conceive the 
magnetic substance to be removed from an infinitely small space round the point, the 
preceding definition would be applicable ; since, if we imagine a very small bar-mag- 
net to be placed in a definite position in this space, the force upon either end would 
be determinate. The circumstances of this case are made clear by considering the 
distribution of imaginary magnetic matter required to represent the given magnet, 
without the small portion we have conceived to be removed from its interior ; which 
will differ from the distribution that represents the entire given magnet, in wanting 
the small portion of the continuous interior distribution corresponding to the removed 
portion, and in having instead a superficial distribution on the small internal surface 
bounding the hollow space. If we consider the portion removed to be infinitely 
small, the want of the small portion of the solid magnetic matter will produce no 
finite effect upon any point ; but the superficial distribution at the boundary of the 
hollow space will produce a finite force upon any magnetic point within it. Hence 
the resultant force upon the given point round which the space was conceived to he 
hollowed, may be regarded as compounded of two forces, one due to the polarity of 
the complete magnet, and the other to the superficial polarity left free by the removal 
of the magnetized substance*. The former component is the force meant by the 
expression ‘‘ the resultant force at a point within a magnetic substance,” when em- 
ployed in the present paper-j". 
49. The conventional language and ideas with reference to the imaginary magnetic 
* If the portion removed be spherical and infinitely small, it may be proved that the force at any point within 
it, resulting from the free polarity of the solid at the surface bounding the hollow space, is in the direction of 
the lines of magnetization of the substance round it, and is equal to — . This theorem (due to Poisson) will 
3 
be demonstrated at the commencement of the Theory of Magnetic Induction, because we shall have to consider 
the “ magnetizing force ” upon any small portion of an inductively magnetized substance as the actual 
resultant force that would exist within the hollow space that would be left if the portion considered were re- 
moved, and the magnetism of the remainder constrained to remain unaltered. 
j* If we imagine a magnet to be divided into two parts by any plane passing through the line of magnetiza- 
tion at any internal point, P, and if we imagine the two parts to be separated by an infinitely small interval 
and a unit north pole to be placed between them at P, the force which this pole would experience is “ the re- 
sultant force at a point, P, of the magnetic substance.” This is the most direct definition of the expression that 
could have been given, and it agrees with the definition I have actually adopted ; but I have preferred the ex- 
planation and statement in the text, as being practically more simple, and more directly connected with the 
various investigations in which the expression will be employed. 
[Note added June 15, 1850. — Some subsequent investigations on the comparison of common magnets and 
electro-magnets have altered my opinion, that the definition in the text is to be preferred ; and I now believe 
the definition in the note to present the subject in the simplest possible manner, and in that which, for the 
applications to be made in the continuation of this Essay, is most convenient on the whole.] 
