PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 261 
bar of finite length. If m denote the strength of the bar, and if N and S be its north 
and south poles respectively, its potential at any point, P, will be, according to 
34 and 50, 
m m 
Let A denote the distance of the point of bisection of the bar from P, and 0 the angle 
between this line and the direction of the bar measured from its centre towards its 
north pole. Then, if a be the length of the bar, the expression for the potential becomes 
r ^ 1 1 
— aAcosS + ia^)^ (A^+ aA cos 9 + ' 
By expanding this in ascending powers of a, and neglecting all the terms after the 
first, we find for the potential of an infinitely small bar magnet, 
ma cos 9 
If now we suppose any number of such bar-magnets to be put together so as to 
constitute a mass magnetized in parallel lines, infinitely small in all its dimensions, 
COS 0 
the values of d and A, and consequently the value of will be infinitely nearly 
the same for all of them, and the product of this into the sum of the values of ma 
for all the bar-magnets will express the potential of the entire mass. Hence, if the 
total magnetic moment be denoted by [/j, the potential will be equal to 
j«.cos9 
Now if we conceive the bars to have been arranged so as to constitute a uniformly 
magnetized mass, occupying a volume <p, we should have (§ 30.) for the intensity of 
magnetization, * Hence if <p denote the volume of an infinitely small element of 
uniformly magnetized matter, and i the intensity of its magnetization, the potential 
which it produces at any point P, at a finite distance from it, will be 
i<p . cos 9 
where A denotes the distance of P from any point, E, within the element, and 6 the 
angle between E P and a line drawn through E, in the direction of magnetization of 
the element, towards the side of it which has northern polarity. 
55. Let us now suppose the element E to be a part of a magnet of finite dimen- 
sions, of which it is required to determine the total potential at an external point, P. 
Let I, n, ^ be the coordinates of P, referred to a system of rectangular axes, and let 
X, 3/, ^ be those of E. We shall have 
