260 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
where A and [A] are respectively the distances of the points oc^y^z and [x^y,z\ from 
the point P. and are given by the equations 
The double and triple integrals in the first and second terms of this expression are to 
be taken respectively over the whole surface bounding the magnet, and throughout 
the entire magnetized substance. Since, as is easily shown, the value of that portion 
of the triple integral in the second member which corresponds to an infinitely small 
portion of the solid containing (|, ^), when this point is internal, is infinitely small, 
it follows that the magnetic force at any internal point, as defined in § 48, is derivable 
from a potential expressed by equation (3). 
52. The expressions for the resultant force at any point, and its direction, may be 
immediately obtained when the potential function has been determined, by the rules 
of the differential calculus. Thus, if V has been determined in terms of the rectan- 
gular coordinates, I, n, of the point P, the three components, X, Y, Z, of the resultant 
force on this point will be given, in virtue of Laplace’s fundamental theorem enun- 
ciated in § 50, by the formulae. 
( 4 ), 
where the negative signs are introduced, because the potential is estimated in such a 
way that it diminishes in the direction along which a north pole is urged. If we 
take the expression (3) for V, and actually differentiate with reference to n, ^ 
under the integral signs, we obtain expressions for X, Y, and Z which agree with 
the expressions that might have been obtained directly, by means of the first prin- 
ciples of statics (see § 46), and thus the theorem is verified. Such a verification, 
extended so as to be applicable to a body acting according to any law of force, consti- 
tutes virtually the ordinary demonstration of the theorem. 
53. The formulae of the preceding paragraphs are applicable for the deterrnination 
of the potential, and the resultant force at any point, whether within the magnetized 
substance or not, according to the general definition of § 49. The case of a point in 
the magnetized substance, according to the conventional second definition of § 48, 
cannot present itself in problems with reference to the mutual action between two 
actual magnets. This case being therefore excluded, we may proceed to the investi- 
gations indicated in § 47. 
54. In the method which is now to be followed, the magnetized substances con- 
sidered must be conceived to be divided into an infinite number of infinitely small 
parts, and the actual magnetism of each part will be taken into account, whether in 
determining the potential of the magnet at a given external point, or in investigating 
the mutual action between two magnets. In the first place, let us determine the 
potential due to an infinitely small element of a magnetized substance, and for this 
purpose we may commence by considering an infinitely thin, uniformly magnetized 
