18 
Davies.—Some Electric and Elastic Analogies. 
The last term is the time integral of the continuous couple by which an 
angular velocity r would be communicated to the liquid in a closed infin¬ 
itely light box of the same internal shape as the given prism to which 
torsion is imparted, and of length unity, liquid density n, as was shown by 
Stokes in 1843. (Math, and Phys. Papers, vol. I., p. 17.) The effective 
moment of inertia* of the liquid equals the correction to the torsional 
elasticity of the prism calculated by Coulomb’s law as if the cross sec¬ 
tion were circular, a and p would in this case represent the compon¬ 
ents parallel to x and y of the velocities of the liquid relatively to the 
box. If in any case this couple be divided by r we have the so-called 
torsional rigidity of the prism. For a prism of eliptical section 
n £f 
where J and L are moments of inertia around x and y respectively. 
For a triangular prism (equilateral) 
J+L) ] 1— 
(a 2 -V) 
(a 2 + b 2 j 
-nr 
7ta 3 5 3 
a- + b' 2 
N=r- 
na 4 
~l$Vf 
and so on. 
Now Oliver Heaviside has shown that, calling H 1 and H 2 , the two 
components of the magnetic force in a cylinder in which an electric cur¬ 
rent is flowing, and which is surrounded by a return conductor in the 
form of a closely fitting sheath (like the Deptford mains, for example) 
then 
H 1 =-27t r r o 
dDj m 
dx ’ 
H 2 =27txr c 
dn 
dy 
Likewise also, in analogy with the expression above given for the 
energy of a twisted prism, viz.: 
^(u 2 + /4 2 ) 
where a and denote the x and y components of the twisting strain, we 
have, calling T the magnetic energy per unit of length of a rod carrying 
a current of density P Q and which is enclosed by a return conducting 
sheath, 
= Energy of magnetic field per unit length of rod carrying current. 
“ The lines of tangential stress in the torsion problem and the lines of 
magnetic force in the electrical problem are identical,, and the energy is 
* By the effective moment of inertia is meant “ the moment of inertia 
of a rigid solid which may be fixed within the box, if the liquid be re¬ 
moved, to make its motions the same as they are with the liquid in it.”— 
Thompson & Tait, Nat. Phil., part II, p. 242. 
