288 Mr G. H. Bryan, Application of the Energy Test [Oct. 29, 
through insufficient strength of their substance. As I pointed 
out in a former number of the Proceedings*, thinness and 
flexibility are necessary qualifications for this kind of collapse. 
I shall here discuss (1) Euler's wire under end thrust, a simple 
illustration of the method, (2) the two dimensional collapse of 
an infinitely long thin tube under external pressure. The latter 
was worked out before I was aware that Prof. Greenhill was 
extending these investigations; the result differs from one given 
by Prof. Unwin+, owing to the fact that in the latter certain 
assumptions are made which cannot be regarded as approximate. 
(1) Letus first consider a wire of length J under end thrust 7, 
showing that the results agree with Eulev’s. 
Suppose that in the state of equilibrium @ is the co-ordinate 
of any point on the rod measured from one end, and let the rod 
be displaced in any plane so that the point receives a lateral 
displacement z Then (as in Lord Rayleigh’s Sound, Vol. 1., p. 
136) the potential energy of the whole rod due to longitudinal 
compression, will be diminished by the quantity 
1 rds mide? 
r{ (3) dx or aT | (=) da, 
while the potential energy due to bending will be increased by 
l 1 20) 2 
| Lee me per | (53) a 
0p o \dax 
If the rod is in stable equilibrium the potential energy must 
be increased by the displacement, therefore for all displacements 
1 d2\? trde\? 
1 eee perl i ae 
ser] (33) a at] (Z) dso oa (1). 
If the ends are fixed in position, we may, by Fourier’s theorem, 
take 
z= a, sin nve/l, 
and if they are fixed in direction, we must take 
z= da, cos nra/l. 
Hither form substituted in (1) leads to the condition 
4 EI za’ n'a [Ut — Ta’ n'a’ /P > 0, 
for all values of the co-ordinates a,. If we suppose all of these 
co-ordinates, except one (a,), vanish, we find 
T < EIn'a’ |’. 
* “On the Stability of Elastic Systems.” Camb. Phil. Soc. Vol. v1. Pt. rv. 
(1888), p. 199. 
+ Min. Proc. Inst. C. E. 1875. 
