688 Mr. R. V. Southwell on the 



surfaces be subjected to pressures TL U II 2 . Then the condi- 

 tion of stability requires that a certain relation should exist 

 between these four quantities, which may be written 



F(a, h, n l5 n 2 )>o." 



Mr. Basset suggests two methods by which this problem 

 may be attacked : — " Ln the first place, let the flue be slightly 

 deformed, and let the period of the small oscillations be 

 found ; then the condition of stability requires, that the 

 periods should be real quantities. In the second place, let 

 the potential energy in the deformed state be found ; then 

 the condition of stability requires, that the potential energy 

 of the flue when in equilibrium should be a minimum. 

 Either of these methods will determine the mathematical 

 form of the function F." 



I propose to use a third method, practically equivalent to 

 both of the above, and to investigate the conditions which 

 must obtain if a circular tube is in neutral equilibrium under 

 the action of external pressure. This is of course the 

 limiting case between stable and unstable equilibrium, and 

 when we have found a value of the pressure for which the 

 equilibrium is neutral, it is easy, by changing the equation 

 into an inequality, to express the condition of stability. 

 When a configuration possesses neutral equilibrium, the 

 period of vibration, for small displacements from that con- 

 figuration, is infinite : that is to say, configurations of in- 

 finitesimal distortion exist which are also in equilibrium, and 

 the condition of neutral stability may be determined from 

 this consideration. 



The above remarks lead to no alteration in Mr. Basset's 

 methods, beyond the substitution of equations of equilibrium 

 for his equations (6) of small motion, which become 



tlT N 



0, 

 as o 



dN T 



-, 



as 



H-HM^H 



as 



(i) 



where p is the radius of the deformed middle surface. The 

 proposed alteration refers to the methods by which these 

 equations are to be treated. 



In equations (1), T, N and G are the values of the stress- 

 resultants and stress-couples in a position of slight distortion 



