100 MR. R. V. SOUTHWELL ON THE GENERAL THEOKY OF ELASTIC STABILITY. 



of equilibrium. Hence every solution which we can obtain will add to the number 

 of these " test cases," which has not hitherto included solutions for any but plane 

 plates. 



A far more important advantage of the new method, from the practical point of 

 view, is the accuracy with which it follows the actual " stress history " in a body 

 which fails by instability under a gradually increasing stress. In cases where 

 instability precedes elastic break-down this difference of method is not important ; 

 but for the discussion of instability in overstrained material, where the stress-strain 

 relations are intimately dependent upon the previous stress history, its introduction 

 is absolutely necessary. 



The extension of EULEB'S theory to struts of practical dimensions and materials, 

 which forms the conclusion of this paper, suggests a large and new field for 

 investigation. The number of similar cases which can be treated, in the existing 

 state of our knowledge of plastic strain, is very small, and indications are given below 

 of the questions which still require an answer ; there is reason to believe that the 

 requisite experimental research would not present insuperable difficulties, and that 

 we may hope in the future to obtain an adequate theory of experimental results 

 which are at present very little understood. 



EQUATIONS OF NEUTRAL EQUILIBRIUM IN RECTANGULAR CO-ORDINATES. 



Method of Derivation. 



The question of stability arises in regard to any system in which there is a 

 possibility of slight displacement from the configuration of equilibrium. This possi- 

 bility may be afforded either by a more or less limited degree of mechanical freedom 

 in which case the problem is one of statical stability, and practically unaffected by 



Fig. 1. 



the tendency, which any actual body displays, to distort under the influence of 

 applied forces ; or it may be due, more or less entirely, to this tendency. In the 

 latter case the problem is one of elastic stability, and must be treated by distinct 

 methods. There is, however, no essential difference between the two types of 



