General Equations of Elasticity 
13 
to be i; Wertheim but which may apparently have any value from £ 
for incompressible bodies, to infinity as in the case of elastic gums. It is 
questionable, however, if the latter can properly be ranked as elastic 
solids.)* 
In the general case of elastic distortion involving shear, dilatation, and 
variation of rotation we have either 
§ 
m + nZ-u + p( u — X ) =0 
8 A 
m—^ + n^v + p( v — Y) —0 
+ p{w — Z) =0 
oz 
wheiein the rotations (or their variations) do not explicitly appear; or 
the following due to Lame wherein they do appear, viz.: 
(m+ n) 
^L_ 2n ( s A?_ s h\ 
dx \SiJ Sz / 
+ p{u — X)—0 
with a similar equation for y and z each. 
If all applied forces and surface tractions are zero then these equa¬ 
tions become those of an elastic system in motion under its own elastic 
recovery, or intermolecular forces, whatever these may be, and the most 
general motion possible consists of a series of superposed small har¬ 
monic vibrations of the points of the body about their positions of 
equilibrium, translations and rotations of the body as a whole being ex¬ 
cluded. These vibrations may give rise to sounds and are ultimately 
dissipated as heat. The equations then simply lose the terms contain¬ 
ing X, Y, Z, and their most general possible solution is of the form 
u=u 1 r 1 +u 2 r 2 +u s r s + .... u-t\- 1 - . 
v=v 1 r 1 ' + v 2 t 2 ' + v 3 r 3 ' . vfl 4- 
w=w 1 r 1 +w 2 r 2 +w 6 r i 
The u i v i w i are functions of position {x y z ) only, and the r. are 
functions of the time only. Any set, such as u=u i r i v—v^^ w=w, i r i B 
*As before stated, the relations between these constants are 
3kn 
=m=lc+^n. 6- 
3k — 2n 
q > 2(3k + n) 
generally. This makes 6—\ (Poisson) if n—%k\ k is the pressural rigidity 
or the bulk modulus of Sir W. Thomson’s writings; n is his rigidity; 
q is Young’s modulus. 
