252 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
magnetic matter, which represents in the simplest possible manner the polarity of 
any given magnet, is of much interest, and even importance, in the theory of mag- 
netism, and we may therefore make this an object of investigation, before going 
farther. 
36 . Let it be required to find the distribution of imaginary magnetic matter to repre- 
sent the polarity of any number of uniformly magnetized needles. Si Nj, S^Nj, ... S„N„ 
of strengths ... respectively, when they are placed together, end to end (not 
necessarily in the same straight line). 
If A denote the position occupied by Sj when the bars are in their places; if N, 
and Sg are placed in contact at Ki ; Ng and S3, at Ka ; and so on until we have the 
last magnet, with its end S„, in contact with N„_i, at K„_i, and its other end, N„, free, 
at a point B ; we shall have to imagine 
l/jy units of southern magnetic matter to be placed at A ; 
(jb^ units of northern, and units of southern matter at K, ; 
(Jb.2 units of northern, and of southern matter at Ka ; 
units of northern, and of southern matter at K„. i ; 
and lastly, units of northern matter at B. 
Hence the final distribution of magnetic matter is as follows : — 
— i«/i at A 
(^2 1^1 
(^2 ^3 L-a 
1 [^n ^^n— 
and [bb^ B. 
37. The complex magnet AKiK2...K„_iB consists of a number of parts, each of 
which is uniformly and longitudinally magnetized, and it will act in the same way as 
a simple bar of the same length, similarly magnetized ; and hence the magnetic matter 
which represents a bar-magnet AB of this kind is concentrated in a series of points, 
at the ends of the whole bar, and at all the places where there is a variation in the 
strength* of its magnetization. 
38 . If the length of each part through which the strength of the magnetism is 
constant, be diminished without limit, and if the entire number of the parts be in- 
creased indefinitely, a straight or curved infinitely thin bar may be conceived to be 
produced, which shall possess a distribution of longitudinal magnetism varying con- 
tinuously from one end to the other according to any arbitrary law. If the strength 
of the magnetism at any point P of this bar be denoted by (Jb, and if [jW-] and (fjb) denote 
* This expression is equivalent to the product of the intensity of magnetization into the section of the har ; and 
by retaining it we are enabled to include cases in which the bar is not of uniform section. 
