254 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
it divided into infinitely thin bars, in the directions of its lines of magnetization ; for 
each of these bars will be uniformly and longitudinally magnetized, and therefore 
there will be no distribution of matter except at their ends. Now the bars are all 
terminated on each side by the surface of the body, and consequently the whole 
magnetic effect is represented by a certain superficial distribution of northern and 
southern magnetic matter. It only remains to determine the actual form of this 
distribution ; but, for the sake of simplicity in expression, it will be convenient to 
state previously the following definition, borrowed from Coulomb’s writings on elec- 
tricity. 
40. If any kind of matter be distributed over a surface, the su'perjicial density at 
any point is the quotient obtained by dividing the quantity of matter on an infinitely 
small element of the surface in the neighbourhood of that point, by the area of the 
element. 
41. To determine the superficial density at any point in the case at present under 
consideration, let a be the area of the perpendicular section of an infinitely thin uni- 
form bar, of the solid, with one end at that point. Then, if i be the intensity of 
magnetization of the solid, iu will be, as may be readily shown, the ‘‘strength” of the 
bar-magnet. Hence at the two ends of the bar we must suppose to be placed quan- 
tities of northern and southern imaginary magnetic matter each equal to iu. In the 
distribution over the surface of the given magnet, these quantities of matter must be 
imagined to be spread over the oblique ends of the bar. Now if 6 denote the incli- 
nation of the bar to a normal to the surface through one end, the area of that end 
will be and therefore in that part of the surface we have a quantity of matter 
equal to ia spread over an area Hence the superficial density is 
i cos L 
This expression gives the superficial density at any point, P, of the surface, and its 
algebraic sign indicates the kind of matter, provided the angle denoted by 6 be 
taken between the external part of the normal, and a line drawn from P in the same 
direction as that of the motion of a point carried from the south pole, to the north 
pole, of a portion close to P, of the infinitely thin bar-magnet which we have been 
considering. 
42. Let it be required, in the last place, to determine the entire distribution of 
magnetic matter necessary to represent the polarity of any given magnet. 
We may conceive the whole magnetized mass to be divided into infinitely small 
parallelepipeds by planes parallel to three planes of rectangular coordinates. Let 
a, (3, y denote the three edges of. one of these parallelepipeds having its centre at a 
point P (x, 3 /, z). Let i denote the given intensity, and I, m, n the given direction 
cosines of the magnetization at P. It will follow from the preceding investigation 
that the polarity of this infinitely small, uniformly magnetized parallelepiped, may be 
represented by imaginary magnetic matter distributed over its six faces in such a 
