12 
Davies.—Some Electric and Elastic Analogies . 
We have another interesting analogy in the case of the equations of 
free vibrations of an elastic medium once subjected to applied forces, 
surface tractions, or both, and then left to its own forces of recovery. 
We shall find that the final equations are similar to those of the dis¬ 
charge of a condenser, especially if a dissipatory term be introduced into 
the equations of elasticity. 
If u, v, and w, be displacements in an ordinary homogeneous elastic 
medium, at a point aj, y, z, of the same, then 
_ clu dv _div 
e ~~ dx ‘ — dy dz 
will be the so-called elongations about the point x, y, z; being rates of 
variation of absolute displacements in proceeding along the coordinate 
axes. 
a, b , c, are shears, if 
Aw dv , , dn dw . dv dn 
a =b^ + d5 ; + Tx' c= Ax + Hy' 
+ —,—=the cubical dilatation; 
/V'Y* /Vi/ /7 v 
not necessarily uniform in all directions. 
.-><£-©• *•->( 
dy 
are rotations. 
dv 
dx 
du" 
dy. 
P = (m + n) e -t- (m—n)(f+ g) 
Q — {m + n)f+{m—ri){g +e) 
R = (m + n~)g + {•m—ri){e +/) 
will be normal stresses across the three coordinate planes at the point 
x y z and 
S=na ; T—nb\ U—nc 
are the tangential stresses, reckoned per unit area, at the same place. 
The inversion of these formulae will give 
a = 
U 
e = — — d(Q + R)/q 
f = -S--6(R + P)/q 
g = ~-6(P + Q)/q 
Where q denotes Young’s modulus; n = what Sir W. Thomson calls dis_ 
torsional rigidity; 6— Poisson’s ratio (which Poisson considered always 
