MR. R. V. SOUTH WKI.L ON THK GENERAL THEORY OF ELASTIC STABILITY. 189 



problems which can he treated by the ordinary theory of elasticity ; but it is not 

 legitimate to conclude that instability is only possible, even if its conditions were only 

 calculable, in the case of materials which obey HOOKE'S Law, and there is no warrant 

 for tin- rmployment of "crushing formulae" in the design of short struts and thick 

 boiler flues.* 



A more serious weakness in the existing theory of elastic stability, when regarded 

 from tin- mathematical standpoint, is the fact that the methods which it employs are 

 admittedly only approximate. The higher the elastic limitt of the material under 

 consideration, the less adequate are these methods to deal with the whole range of 

 problems which should come within its scope. In fact, we are faced with the 

 anomaly that, while in its ordinary applications the theory of elasticity is not 

 concerned with the conception of an elastic limit, in questions of stability the 

 existence of finite limits is an essential condition for the adequacy of its results. In 

 an ideal material, possessing perfect elasticity combined with unlimited strength, 

 types of instability could occur with which existing methods would be quite 

 insufficient to deal. 



The theory of elastic stability is thus in much the same position as that of the 

 ordinary theory of elasticity before the discovery of the general equations, and oue 

 aim of the present paper is to remedy its defects by the investigation of general 

 equations, which may be termed " Equations of Neutral Equilibrium," and which 

 express the condition that a given configuration may be one of limiting equilibrium. 

 These equations are universally applicable only to ideal material of indefinite strength, 

 and the possibility of elastic break-down must receive separate investigation ; but 

 they are also applicable, even with materials of finite strength, to any problem which 

 comes within the restrictions imposed by BRYAN'S discussion, and therefore enable us 

 to test the accuracy of his treatment of problems, such as that of the boiler Hue, for 

 which the ordinary Theory of Thin Shells has been thought insufficiently rigorous.^ 



In every problem of this paper it is found that the Theory of Thin Shells gives a 

 solution which is correct as a first approximation, and the practical advantage* of the 

 new method of investigation are, therefore, not immediately apparent. But it must 

 be remembered that the approximate theory of thin plates and shells has not as yet 

 been rigorously established, and that much work has recently lx;en undertaken with 

 the object of testing it by comparison with accurate solutions of isolated problems. 

 Now in finding conditions for the neutrality of the equilibrium of any given 

 configuration we are at the same time obtaining the solution of a statical problem ; 

 for a configuration of slight distortion from the equilibrium position will also be one 



* W. C. UNWIN, ' Elements of Machine Design ' (1909), Part I., p. Ill ; S. E. SI/XJUM, "The Collapse 

 i >f Tubes under External Pressure," ' Engineering,' January 8, 1909. 



t By "elastic limit" is intended, here and throughout this paper, the limit of linear elasticity. 



J I'f. pp. 210, 224. 



LOVE, op. tit., Introduction, p. 29, and Chapter XXII. 



