PROF. W. THOMSON ON THE MATHEMATICAL THEORY OP MAGNETISM. 253 
the values of [jij at the points A and B, the investigation of § 36, with the elementary 
principles and notation of the differential calculus, leads at once to the determination 
of the ultimate distribution of magnetic matter by which such a bar-magnet may be 
represented. Thus if AP be denoted by 5 ; will be a function of s, which may be sup- 
posed to be known, and its differential coefficient will express the continuous distri- 
bution of magnetic matter which replaces the group of material points at Kj, Kg, &c. ; 
so that the entire distribution of polarity in the bar and at its ends will be as follows 
in any infinitely small length, <7, of the bar, a quantity of matter equal to 
and, besides, terminal accumulations, of quantities 
— [}//] at A 
and (jW/) at B. 
It follows that if through any part of the length of a bar, the strength of the mag- 
netism is constant, there will be no magnetic matter to be distributed through this 
portion of the magnet ; but if the strength of the magnetism varies, then, according 
as it diminishes or increases from the north to the south pole of any small portion, 
there will be a distribution of northern or southern magnetic matter to represent the 
polarity which results from this variation. 
Corresponding inferences may be made conversely, with reference to the distribu- 
tion of magnetism, when the distribution of the imaginary magnetic matter is known. 
Thus Coulomb found that his long thin cylindrical bar-magnets acted upon one 
another as if each had a symmetrical distribution of the two kinds of magnetic matter, 
northern within a limited space from one end, and southern within a limited space 
from the other, the intermediate space (constituting generally the greater part of the 
bar) being unoccupied ; from which we infer that no variation in the magnetism was 
sensible through the middle part of the bar, but that, through a limited space on 
each side, the intensity of the magnetization must have decreased gradually towards 
the ends*. 
39. The distribution of magnetic matter which represents the polarity of a uni- 
formly magnetized body of any form, may be immediately determined if we imagine 
* This circumstance was alluded to above, in § 21. Interesting views on the subject of the distribution 
of magnetism in bar-magnets are obtained by taking arbitrary examples to illustrate the investigation of the 
text. Thus we may either consider a uniform bar variably magnetized, or a thin bar of varying thickness, cut 
from a uniformly magnetized substance ; and according to the arbitrary data assumed, various remarkable 
results may be obtained. We shall see afterwards that any such data, however arbitrary, may be actually pro- 
duced in electro-magnets, and we have therefore the means of illustrating the subject experimentally, in as 
complete a manner as can be conceived, although from the practical non-rigidity of the magnetism of magnetized 
substances, ordinary steel or loadstone magnets would not afford such satisfactory illustrations of arbitrary 
cases as might be desired. The distribution of longitudinal magnetism in steel needles actually magnetized in 
different ways, and especially “ magnetized to saturation,” has been the object of interesting experimental and 
theoretical investigations by Coulomb, Biot, Green and Riess. 
