PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 257 
Chapter IV. Determination of the Mutual Actions between any Given Portions 
of Magnetized Matter. 
45. The synthetical part of the theory of magnetism has for its ultimate object the 
determination of the total action between two magnets, when the distribution of 
magnetism in each is given. The principles according to which the data of such a 
problem may be specified have been already laid down (§§ 28-31.), and we have seen 
that, with sufficient data in any case. Coulomb’s laws of magnetic force are sufficient 
to enable us to apply ordinary statical principles to the solution of the problem. 
Hence the elements of this part of the theory may be regarded as complete, and we 
may proceed to the mathematical treatment of the subject. 
46. The investigations of the preceding chapter, which show us how we may con- 
ventionally represent any given magnet, in its agency upon other bodies, by an ima- 
ginary magnetic matter distributed on its surface and through its interior ; enable us 
to reduce the problem of finding the action between any two magnets, to the known 
problem of determining the resultant of the attractions or repulsions exerted between 
the particles of two groups of matter, according to the law of force which is met with 
so universally in natural phenomena. The direct formulae applicable for this object 
are so readily obtained by means of the elementary principles of statics, and so well 
known, that it is unnecessary to cite them here, and we may regard equations (1) and 
(2) of the preceding chapter (§ 42.) as sufficient for indicating the manner in which 
the details of the problem may be worked out in any particular case. The expression 
for the “ potential,” and other formulse of importance in Laplace’s method of treat- 
ing this subject, are given below (§ 51.), as derived from the results expressed in 
equations (1) and (2). 
47 . The preceding solution of the problem, although extremely simple and often 
convenient, must be regarded as very artificial, since in it the resultant action is 
found by the composition of mutual actions between the particles of an imaginary 
magnetic matter, which are not the same as the real mutual actions between the 
different parts of the magnets themselves, although the resultant action between the 
entire groups of matter is necessarily the same as the real resultant action between 
the entire magnets. Hence it is very desirable to investigate another solution, of a 
less artificial form, in which the required resultant action may be obtained by com- 
pounding the real actions between the different parts into which we may conceive the 
magnets to be divided. The remainder of the chapter, after some preliminary explana- 
tions and definitions, will be devoted to this object. 
48. The ‘‘ resultant magnetic force at any point ” is an expression which will very 
frequently be employed in what follows, and it is therefore of importance that its 
signification should be clearly defined. For this purpose, let us consider separately 
the cases of an external point in the neighbourhood of a magnet, and a point in 
space which is actually occupied by magnetic matter. 
MDCCCLI. 2 L 
