1888. ] and Hquipotential Curves. 19g 
formation z=acnw+izbsnw. In this case the curve n = 0 is an 
ellipse having its semi-axes of lengths a and b; and we have 
@ = (—asnw+ibenw) dn w. 
Substituting the roots of f’(w) =0 for win the expression /(w) 
we obtain the six points 
+ Ja—B, £7 (ak'+b), +7 (ak'- Dd). 
The first pair of points contains the two foci of the ellipse, and 
although the other four points may be foci of the other members 
of the system, they are obviously not foci of the ellipse. 
(3) On the Stability of Elastic Systems. By G. H. Bryan, B.A., 
St Peter’s College. 
1. Kirchhoff was the first to shew* that if we are given the 
bodily forces acting on an elastic solid, and are also given either 
the surface tractions or surface displacements, there is one and only 
one state of strain in which the body can be in equilibrium, and 
that equilibrium is essentially stable for all displacements with 
the exception of rigid-body displacements. 
Euler found that a thin wire or shaft of length J and flexural 
rigidity HI becomes unstable if the thrust applied to its ends be 
greater than that given by the formula 
ise ae 
Bi TP 
Greenhill} has worked out the corresponding formulae when 
the wire transmits both end thrust and couple and is also sup- 
posed to be under the influence of “centrifugal force.” He has 
likewise determined} the greatest height of a thin vertical pole or 
tree consistent with stability, the diameter being a known function 
of the height. 
These appear to be the only instances in which the question of 
stability has been discussed in connection with the theory of 
Elasticity. It therefore appeared to me that it would be worth 
while to give a general investigation of the circumstances under 
which an elastic system can be in unstable equilibrium for other 
than rigid-body displacements of the various bodies forming the 
system. I have shewn that, in general, the only systems which 
* Vorlesungen iiber Math. Physik. 27, § 2. 
+ Proc. Institution of Mechanical Engineers, April, 1883, p. 182, 
£ Camb. Phil, Proceedings, Vol. 1v. p. 66, 
