200 Mr G. H. Bryan, On the Stability of Elastic Systems. [Feb. 27, 
are not essentially stable, except for displacements differmg in- 
finitely little from those of a system of rigid bodies are such thin 
wires, plates, or shells as are capable of “being deformed by pure 
bending or twisting. 
It is to be remembered that the strains which most substances 
are capable of undergoing without losing their elastic properties 
must be confined within certain extremely small limits, it is only 
such imperfectly elastic substances as jelly or indiarubber, which 
can be subjected to finite strain without breaking. We shall 
therefore usually suppose the limits of the elasticity to be small, 
the elastic constants being large quantities so that the limits of 
the stresses are finite. 
2. Let an elastic solid be in equilibrium in a state of strain 
under any system of external forces and constraints, the displace- 
ments of any point being wu, v, w and the strains being e, f, g, a, b, ¢ 
as usual. Let ¢@ be the elastic potential or potential energy of 
strain per unit volume, so that for an isotropic body 
b(¢,f,9,4,b,c)=4(m+n)(e+f+g)y 
+ $n (a’+ 0'+ c'— 4fg —4ge — 4ef)......... Cy: 
Let the potential of the bodily forces at the point (v+u, y+v, 2+w) 
be V and let TdS be the potential energy of the surface tractions 
on the surface element dS. The whole potential energy of the 
system in the position of equilibrium will therefore be 
W= |[[edeayde + |[[pVaedyde + | Tas. ey 
if we suppose the bodily forces to be due to external causes, not 
to self-gravitation of the solid. 
Equilibrium will be stable if W be a true minimum, unstable 
if W be a maximum or minimax. Let the displacements of any 
point receive small variations du, dv, ’w. The condition of equi- 
librium gives the ordinary variational equation 
sw =|[[opdedyde + [I] OT ined +f STdS =0...(8). 
Proceeding to the second variations we see that the condition that 
equilibrium may be unstable requires that for some variations 
SW = [[[esacdyae + [[[pstV away ae + [[eras< 0...(49. 
* Employing the elastic constants of Thomson and Tait, 
