1888.] Mr G. H. Bryan, On the Stalility of Elastic Systems. 203 
can be comparable with I | | o'pdadydz. Writing these terms in 
the forms | | (8F8u + 8G-3v + 8H8w)dS and 
\ | (SP Se + 8QS/+ SR8q + 98a + 878b + 8U 8c) dadyde, 
respectively, we see, that if the variations of the strains are still 
supposed to be comparable with those of the surface displacements, 
the variations 6F, 6G, 6H of the surface tractions must be com- 
parable with the stress variations 6P,6Q,6R,65,67,6U, a condition 
which is otherwise almost evident. Now this will be the case if 
the surface tractions are due to the reactions of other elastic solids 
with which the body under consideration is in contact. The same 
is equally true if they are preduced by the pressure of liquids 
whose elasticity of volume is comparable with that of the solid, 
provided the variational displacement involves changes in the 
volume of the liquid. In such instances let us consider the 
potential energy of the whole system instead of that of a single 
body. Then the total potential energy due to work done by the 
actions and reactions of the various bodies of the system vanishes 
identically by Newton’s third law, and therefore also its successive 
variations vanish identically. On the other hand, the second 
variations of the potential energies of strain of all the bodies are 
essentially positive and therefore that of the whole system is 
essentially positive. Hence the whole system will in general be 
stable, and therefore any body of the system will necessarily a 
fortiori be in stable equilibrium for displacements of the kind 
here considered. In every other conceivable case, 6F, 6G, 6H will 
be linear functions of é6u, dv, dw and their differential coefficients, 
in which the coefficients are quantities of the same order of 
magnitude as the surface tractions F, G, H in the position of equi- 
librium, and are therefore small compared with the elastic con- 
stants, hence 6F, 6G, 6H are small compared with 6P, 6Q, dR, dS, 
67, 6U, and | [ &TdS is small compared with [ i | opdady dz. 
4. From these discussions it follows that the equilibrium of 
an elastic solid acted on by any system of bodily forces or surface 
tractions, which produce only small strains of the substance, is 
essentially stable for all displacements with the exception of those 
in which the variations produced in the strains are very small in 
comparison with the variations in the positions of the particles of 
the solid. 
Now if the strains were zero the displacement would be purely 
that of a rigid body. If they are infinitely small compared with 
VOL, VU. PII. 15 
