PROF. W. THOMSON ON THE MATHEMATICAL THEORY OP MAGNETISM. 259 
matter, explained above (§§ 32-44), enable us to give the following simple statement 
of the definition, including both the cases which we have been considering. 
The resultant magnetic force at any point, whether in the neighbourhood of a mag- 
net or in its interior, is the force that a unit of northern magnetic matter would expe- 
rience if it were placed at that point, and if all the magnetized substance were re- 
placed by the corresponding distribution of imaginary magnetic matter. 
50. The determination of the resultant force at any point is, as we shall see, much 
facilitated by means of a method first introduced by Laplace in the mathematical 
treatment of the theory of attraction, and developed to a very remarkable extent by 
Green in his “ Essay on the Application of Mathematical Analysis to the Theories of 
Electricity and Magnetism” (Nottingham, 1828), and in his other writings on the 
same and on allied subjects in the Cambridge Philosophical Transactions, and in the 
Transactions of the Royal Society of Edinburgh. Laplace’s fundamental theorem 
is so well known that it is unnecessary to demonstrate it here ; but for the sake of 
reference, the following enunciation of it is given. The term “ potential,” defined in 
connection with it, was first introduced by Green in his Essay (1828). It was at a 
later date introduced independently by Gauss, and is now in very general use. 
Theorem (Laplace). — The resultant force produced by a body, or a group of at- 
tracting or repelling particles, upon a unit particle placed at any point P, is such that 
the difference between the values of a certain function, at any two points p and p' in- 
finitely near P, divided by the distance pp', is equal to its component in the direction 
of the line joining p and p'. 
D^nition (Green). — This function, which, for a given mass, has a determinate 
value at any point, P, of space, is called the potential of the mass, at the point P. 
It follows from the general demonstration, that, when the law of force is that of the 
inverse square of the distance, the potential is found by dividing the quantity of 
matter in any infinitely small part of the mass, by its distance from P, and adding 
all the quotients so obtained. 
51. The same demonstration is applicable to prove, in virtue of Coulomb’s funda- 
mental laws of magnetic force, the same theorem with reference to any kind of 
magnet that can be conceived to be composed of uniformly magnetized bars, either 
finite or infinitely small, put together in any way, that is, of any magnet other than 
an electro-magnet ; and the investigation, in the preceding chapter, of the resulting 
distribution of magnetic matter that may be imagined as representing in the simplest 
possible way the polarity of such a magnet, enables us to determine at once, from 
equations (1) and (2) of § 42, its potential at any point. Thus if V denote the poten- 
tial at a point P, whose coordinates are |, rj, ^ and if c?S denote an element of the 
surface of the magnet, situated at a point whose coordinates are [.r], \_y], [;z], we 
have, by the proposition enunciated at the end of § 49, — 
d{U) d{im) d{in) 
2 L 2 
