Inductances and Coefficients of Rigidity. 
19 
similarly reckoned.” Moreover, it appears that the ratios of the in¬ 
ductances of wires of different sections (taking the circular wire as a 
standard), are as the ratios of the torsional rigidities of rods of various 
cross sections in the torsion problems as worked out by Saint Venant 
For example, the torsional rigidity of a circular cylinder being 
M 
j d6r 2 —mtr 
and that of an elliptical cylinder being as above, 
7ta 3 b 3 
—=-5 
a 2 + b 2 
the latter is essentially 
cib 
a' 2 + b 2 
times that of the circular cylinder. So if L=\u (ju being permeability) 
be the coefficient of self induction per unit of length of a round wire, 
then 
\n 
ab r 
Id + b 2 
is that of unit length of an elliptical one. So also Aklln is that of a 
square one; .3627/q that of one of triangular cross section (equilateral); 
%7tjna/b that for a flat strip or sheet whose thickness is a and breadth b, 
and so on. 
It will be noticed that these coefficients are the halves of those given 
by Saint Venant for the amounts by which the torsional rigidities of 
circular prisms of equal area are to be diminished when the cross sec¬ 
tions are as given above, and warping of the cross sections is a part of 
the strain. 
Now, as stated at the beginning of this paper, whatever may be the 
exact nature of electricity or magnetism, Clerk Maxwell has shown that 
the phenomena of both lead to the idea that electro-static lines of force 
and magnetic lines of force cause a stress in the medium they exist in, 
which is measured by 
along the lines; being of the nature of the tensile stresses transmitted 
along tie lines in elasticity, and are accompanied by corresponding pres¬ 
sures at right angles to them. The phenomena are essentially of the 
nature of elasticity and point to a medium capable of sustaining stresses 
The exact mechanism by which these stresses are produced is another 
question. It may be that mere attraction resisting displacement (as is 
sometimes assumed in theories of elasticity) is at the bottom of electro¬ 
static strains, but it seems pretty certain that ether rotations are at the 
bottom of magnetic strains, and Sir William Thomson has shown that a 
