1888.] Mr G. H. Bryan, On the Stability of Elastic Systems. 209 
bodily forces, surface tractions, edge tractions or couples round the 
edge be denoted by V,. So that the total potential energy of the 
system is 
W =4nh WE enhPW, Voce kc ences. (24), 
Let the system receive a small variational displacement of 
order 60. The conditions of equilibrium (4) and instability (5) 
become 
OW = 4nhdW,+4nV?dW,+6V,=0  ......... (25). 
OW = 4nh& W, + 4nVPOW, + 0°V, <0 «0. (26), 
of which the former must hold for all variations and the latter 
for some variations. 
Now as we have already shewn 6’V will in general be a 
quantity of the same order of magnitude as 6V 60. Moreover 
o&W,, o W, are of order 60. Suppose that equilibrium is unstable 
for a displacement of pure bending, then we must put 6c,, d0,, da 
all =0 and therefore &W,=0. In order that the inequality (26) 
may be satisfied 6’V, must be a quantity of order nh’d0°. Hence 
dV is of order nh’ 60. 
Again dW, involves products of «,, ,, «, into their variations 
and is of order «,60, and therefore if the plate is bent in the 
position of equilibrium it follows from (25) that «,, «,, , are all 
finite. But the strains when this is the case are small quantities 
of order h. 
To find the strains due to extension of the middle surface of 
the plate in the position of equilibrium we must use the equation 
of virtual velocities (25) for variations of order 6@ in which 6c,, éa,, 
da are not zero. We at once find that o,,o,, a, the strains due to 
stretching, are smail quantities of order h’. 
In the example we have worked out, the plate was not bent in 
the position of equilibrium and the result there found confirms our 
present general result. 
In the case of the bent wire exactly the same conclusions hold 
good. The potential energies due to stretching and bending for 
an isotropic wire are 
1 [Baotds and 4 | Ha (a Lie ken? ay 7 he ) dS, 
where # = Young’s modulus, «=area of cross section k,, k, being 
its principal radii of gyration, k, a constant of the same magnitude 
and «, % are the curvatures about the principal axes of the 
section and 7 the twist. Now a is of order e’, k,, k,, k, of order e, 
e being as before comparable with the thickness of the wire. 
It readily follows that if such a wire be unstable under any given 
