202 Mr G. HH. Bryan, On the Stability of Elastic Systems. [Feb. 27, 
Thus the paraboloidal jelly discussed by Greenhill* may some- 
times be unstable under its own weight, even if its transverse 
section is so large that his mode of treatment (in which the for- 
mulae for the bending moment of a thin wire whose middle line 
is the axis of the paraboloid are employed) gives only very roughly 
approximate results. 
But these cases are of little interest. In the solids with which 
we are dealing, such finite strains cannot exist without “set” 
being caused, and therefore the second variation of the potential 
energy of the forces must be small compared with that due to 
strain. 
If the body is self-attracting the same thing is equally true. 
For the potential energy of self-attraction is 
ff aor 
Wa biede R ; 
where 
R=(@a@t+u-—wv—-wyt(ytou-y—-wvypt+(z+w-27-wy, 
and the integral extends to every pair of elements of the body. 
The first and second variations of this are 
Ill] letedyazotavayar 
(om _ 8) = (=) +(O0= Bs, (z) Gn sa) - (a) | 
d 
an 
| I | i | pdadydzp dx'dy'dz' 
, Rv) Qi 
{(au— au) x + (80 bv) = + (bu — Bu) =| a 
respectively, 
By the variational equation of equilibrium pp’ or p* must be 
comparable with the stresses in the position of equilibrium and 
therefore in general small compared with the elastic constants. 
Hence, just as in the previous case, the second variation of the 
potential energy of self-attraction cannot in general be made com- 
parable with that due to strain, except by taking the displacement 
variations to be such that the strain variations are small in com- 
parison. 
Next let us examine whether the surface integral | | &TdS, 
which is the second variation of the energy of the surface tractions 
* Camb, Phil, Proc, Vol. tv, loc. cit, 
