204 Mr G. H. Bryan, On the Stability of Elastic Systems. [Feb. 27, 
the displacements and all the dimensions of the body are finite 
and comparable, it may readily be proved that the displacements 
must differ from rigid-body displacements by terms that are in- 
finitely small in comparison, (although this is not necessarily true 
for a body whose dimensions are not all finite and comparable). If | 
therefore we suppose the portions of du, dv, dw due to the rigid-body 
displacement alone to exist, the difference produced in the second 
and third terms of the inequality (4) will in general be infinitely 
small, while the first term will be zero instead of being positive, and 
the body must therefore a fortiorz be unstable for the rigid-body 
displacement alone. 
We may however have systems of finite bodies under tight- 
fitting constraints which allow of nearly, but not quite, rigid-body 
displacements or in which the potential energy of the forces differs 
considerably for small deviations from rigid-body displacements, in 
such cases unstable equilibrium may be broken by displacements 
accompanied by small variations of the strains, but these cases 
present no points of interest. A smooth sphere which, if slightly 
squeezed, can be pushed into the aperture in a solid anchor ring 
and fits it tightly is an illustration of such an unstable system. 
It is to be mentioned that in Kirchhoff’s investigation the 
variations of the surface tractions and bodily forces are supposed 
to be zero, while if the surface displacements are given their 
variations are zero. Hence the second variation of the potential 
energy is entirely due to strain. I have taken account of these 
variations of the impressed forces and discussed under what circum- 
stances alone their effect becomes important. 
Stability of Wires, Plates and Thin Shells. 
5. At the commencement of this paper I alluded to Euler’s 
and Greenhill’s determination of the criteria of instability of a thin 
wire. In like manner we may find the condition that a thin plate 
subject to given edge thrusts in its plane may be unstable. At 
present we shall content ourselves with the simple example of an 
infinitely long strip of breadth / and thickness 2h acted on by 
normal edge thrust in its plane and of magnitude P per unit of 
length of the edge. In this example, which closely resembles 
Euler’s problem of the wire with end thrust, the condition of 
instability is 
P>8nii ( i \e sles oth alta ca (11), 
m+n 
provided that, either the edges are fixed in position or the tangent 
planes along them are fixed in direction. This gives a very rough 
