ME. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 191 



instability, and a general discussion of the elastic type may be very conveniently 

 illustrated by reference to a mechanical example. 



In this connection we may consider the system illustrated by fig. 1, in which a 

 uniform heavy sphere rests in equilibrium within a hemispherical l>owl, under the 

 action of its own weight and of the pressure exerted by a pointed plunger, which is 

 free to move in a vertical line through the centre of the bowl. This system has been 

 chosen for the illustration which it affords of collapse under a definite " critical 

 loading." In this it bears an unusual resemblance to examples of elastic instability 

 the stability of most mechanical systems being dependent solely upon the relative 

 dimensions of their members. In the absence of friction, we find that the equilibrium 

 will Income unstable as the load on the plunger is increased through a critical value 

 given by Wr 



P ' = R^' 



W is the weight of the sphere, 



r is the radius of the sphere, 



R is the radius of the bowl. 



The above solution rests upon the assumption that the sphere, bowl and plunger 

 are absolutely smooth and rigid, and the possibility of slight displacement is afforded 

 by the freedom of the sphere to take up any position of contact with the bowl. To 

 discuss the equilibrium of the sphere in the position illustrated we must consider the 

 forces which act upon it in a position of slight displacement. These include two 

 systems, one tending to restore the initial conditions, the other tending to increase 

 the distortion, and stability depends upon the relative magnitude of the two effects. 

 We may investigate the problem by three methods, fundamentally equivalent, which 

 are described below : 



(1) TJie Energy Method. We may derive expressions for the potential energy of 



the system in a position of slight displacement from the equilibrium position. 

 The condition of stability requires that the expression for the potential energy 

 shall have a minimum value in the equilibrium position. 



(2) The Method of Vibrations. We assume that the slight displacement has been 



effected by any cause, and investigate the types of vibration possible to the 

 system when this cause is removed. The condition of stability requires that 

 all such types shall have real periods. 



(3) The Statical Method. We confine our attention to the special case in which 



the stability of the equilibrium position is neutral. In this case there must 

 exist some type of displacement for which the collapsing and restoring 

 effects, discussed above, are exactly balanced, so that it may be maintained 

 by the original system of applied forces. We have, therefore, to find 

 conditions for the equilibrium of a configuration of small displacement, under 

 the given system of applied forces. 



