206 Mr G. H. Bryan, On the Stability of Elastic Systems. [Feb. 27, 
Now 3ra,?/r°a,? will have a minimum value = 1 when all the 
constants a,a@,... vanish, and a, is not =0. Hence the plane 
form will be stable or unstable, according as P is less or greater than 
WT 
Sh? ais) ae 
Snh & seal 
Tf equilibrium in the plane form is critical so that P is equal to 
the above value, the pressure per unit of surface over the edge is 
p where 
J m \1r 
= ma Lipp fhe eS 
P= 95 4nh ie @ -| JE crtteetieeeeeeenees (16), 
and if o is the measure of the compression of the surface 
Sih ( a Vers 
m+n 
mh? 
o= 3 E bee c eer ce esc es cesses ceces (17). 
In the wire problem of Euler 
3m — n\ 1 
aye ss 
p=uk.( = \5 a ge cep ecilata shane (18). 
2 
Gis 2 DUAR. ee (19), 
where & is the radius of gyration of the cross section of the wire 
about the line about which bending takes place. 
The values of ¢ will therefore be equal in the respective cases if 
By taking h and & sufficiently small compared with / these values 
of « may be made as small as we please, and therefore the lamina, 
as well as the wire, will, if thin enough, be unstable for thrusts far 
less than those required to produce “set.” 
6. We shall now explain the possible instability of such thin 
solids from general observations. 
We may point out that the surface integrals both in dW, ®W 
increase indefinitely in importance compared with the volume 
integrals, if one or two of the dimensions of the body become 
indefinitely diminished, and this is in accordance with the known 
fact that the surface tractions on an infinitely thin wire or shell 
must be infinitely small. Hence, although the ratios of 6, 6G, dH 
to the displacement variations, when (4) is satisfied, diminish 
indefinitely, yet if we suppose these ratios to be always comparable 
