208 Mr G. H. Bryan, On the Stability of Elastic Systems. [Feb. 27, 
of any such small element will differ from a rigid-body displace- 
ment by components which are small quantities of order e in com- 
parison. Ifthe wire be taken sufficiently thin these terms may be 
made as small as we please in proportion, and a displacement, 
differing infinitely little from one of pure bending or twist, will 
therefore be exactly that kind of displacement for which alone, an 
element, all of whose dimensions are comparable, may be in un- 
stable equilibrium. 
The same thing is true for a thin plate or shell. We suppose 
such a plate divided into elementary parallelopipeds whose length 
and breadth are both comparable with the thickness (2h) and the 
relative displacements in such an element are referred to certain 
rectangular axes which move with the element*. The bending 
being determined by the quantities «,, ,, «,, if these be supposed 
of order of magnitude represented by @, the components of 
translation of the centre of any element as well as the rotations of 
the axes of x, y, z in that element will be quantities of order @. 
On the contrary if the displacement be one in which the middle 
surface is either unextended or the’ stretching is infinitely small in 
comparison with the bending, it appears just as before that the 
relative displacements of a point in the element referred to these 
moving axes are of order h’@, the strains being of order h@. If the 
plate be infinitely thin, such a displacement, therefore, differs 
infinitely little from a rigid-body displacement, so far as any 
element, all of whose dimensions are comparable, is concerned. 
7. It only remains for us to determine what is the order of 
magnitude of the small strains produced in such a thin elastic 
solid when the forces acting on it are so great that it is possible 
that equilibrium may be unstable. Let us take a thin plate both 
bent and stretched in the state of equilibrium so that o,, o,, a 
are the principal extensions along two perpendicular lines in the 
middle surface, and the shear of the angle between these lines 
respectively. The total potential energy due to strain is 
4nh W, + 4nh’ W,, 
where} 
W, = = Nn [tc tn a4) at 2 qd = 7) Goa? =F o,0,)} ds aoeoos (22). 
W,= = ffi {(«, —A,)" + 2 (1 —p) («7 —A,«,)} dS......-2- (23). 
Let the whole potential energy of the impressed forces whether 
* See Mr Love’s ‘‘Note on Kirchhoff’s theory of the deformation of elastic 
plates,” Camb. Phil. Proceedings. 
Kirchhoff, loc. cit. p. 454. 
