248 PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 
that if the magnetized substance be a regular crystal of magnetic iron ore, the mag 
netism is distributed through it according to some simple law ; but by taking an 
amorphous and heterogeneous fragment of ore presenting magnetic properties, by 
magnetizing in any way an irregular mass of steel, by connecting any number of 
morsels of magnetic matter so as to make up a complex magnet, or by bending a 
galvanic wire into any form, we may obtain magnets in which the magnetic property 
is distributed in any arbitrary manner, however irregular. Excluding for the present 
the last-mentioned case, let us endeavour to form a conception of the distribution of 
magnetism in actually magnetized matter, such as steel or loadstone, and to laydown 
the principles according to which it may in any instance be mathematically ex- 
pressed. 
18. In general we may consider a magnet as composed of matter which is mag- 
netized throughout, since, in general, it is found that any fragment cut out of a mag- 
netic mass is itself a magnet possessing properties entirely similar to those which 
have been described as possessed by any magnet whatever. It may be however that 
a small portion cut out of a certain position in a magnet, may present no magnetic 
phenomena ; and if we cut equal and similar portions from different positions, we 
may find them to possess magnetic properties differing to any extent both in intensity, 
and in the directions of their magnetic axes. 
19. If we find that equal and similar portions, cut in parallel directions, from any 
different positions in a given magnetic mass, possess equal and similar magnetic pro- 
perties, the mass is said to be uniformly magnetized. 
20. In general, however, the intensity of magnetization must be supposed to vary 
from one part to another, and the magnetic axes of the different parts to be not 
parallel to one another. Hence, to lay down determinately a specification of the dis- 
tribution of magnetism through a magnet of any kind, we must be able to express 
the intensity and the direction of magnetization at each point. Before attempting to 
define a standard for the numerical expression of intensity in magnetization, it will 
be convenient to examine the elementary laws upon which the phenomena of mag- 
netic force depend, since it is by these effects that the nature and energy of the mag- 
netism to which they are due must be estimated. 
Chapter II. On the Laws of Magnetic Force, and on the Distribution of 
Magnetism in Magnetized Matter. 
21. The object of the elementary magnetic researches of Coulomb was the deter- 
mination of the mutual action between two infinitely thin, uniformly and longitudi- 
nally magnetized bars. The magnets which he used were in strictness neither uni- 
formly nor longitudinally magnetized, such a state being unattainable by any actual 
process of magnetization ; but, as the bars were very thin cylindrical steel wires, and 
were symmetrically magnetized, the resultant actions were sensibly the same as if 
they were in reality infinitely thin, and longitudinally magnetized ; and from experi- 
