210 Mr G. H. Bryan, On the Stability of Elastic Systems. [Feb. 27, 
forces, then if these forces be such as to bend or twist the wire in 
the position of equilibrium, the flexion and torsion thus caused 
will be finite the strains being of order e*. On the other hand the 
strains due to extension or compression of the middle line will 
be quantities of order e’. 
From the above results we can draw the following general 
conclusions as to the limit of the thickness of a thin wire or plate 
for which unstable equilibrium is possible under any system of 
forces, it being remembered that the body must not be strained 
beyond the elastic limits. 
1. Ifthe forces are such as to produce bending in the position 
of equilibrium this limiting thickness is a small quantity of the 
same order as the total increase in length in a bar of the same 
material and of length equal to the greatest linear dimension 
of the plate or wire, when the strain is the greatest it will bear. 
2. If the forces produce only extension or compression of the 
middle line or surface, unaccompanied by bending, the thickness 
may be much greater, and its limit is a small quantity of the same 
order as a mean proportional between the same length above 
mentioned and a length comparable with the greatest linear 
dimensions of the body. 
In no case can equilibrium be unstable for displacements other 
than those either composed wholly of pure bending or differing 
infinitely little from pure flexion and torsion. 
8. One consequence of this fact 1s worthy of notice. Gauss has 
proved that a thin shell in the form of a closed surface cannot be 
deformed by pure bending unaccompanied by extension or com- 
pression of the surface. Such a shell is, therefore, essentially 
stable, (although should there be a small crack or flaw in the 
material this would not be necessarily true). 
In particular a thin hollow elastic spherical shell surrounded 
by a medium subjected to uniform hydrostatic pressure, will 
always be perfectly stable. If the pressure be sufficiently in- 
creased we shall at last arrive at a pomt when the material of the 
shell will be ruptured, or a “‘doke” produced when the pressure is 
sufficiently great to cause “set”, but until this happens the shell 
will always continue to remain in the form of a sphere. 
* This is exemplified by the condition for instability of a wire when a couple is 
applied at both ends, investigated by Greenhill (loc. cit.). When such a wire is in 
unstable equilibrium the measure of torsion is finite, but the shear at any point 
is a small quantity proportional to its distance from the axis. If the wire is of 
circular section the integral twist must exceed 2m (3m —n)/m. 
