188 MR R V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 



are concerned with the integration of certain differential equations, fundamentally 

 the same for all problems, and the satisfaction of certain boundary conditions ; and 

 by a theorem due to KIKCHHOFF* we are entitled to assume that any solution which we 

 may discover is unique. In these problems we are confronted with the possibility of 

 two or more configurations of equilibrium, and we have to determine the conditions 

 which must be satisfied in order that the equilibrium of any given configuration may 

 I stable. 



The development of both branches has proceeded upon similar lines. That is to 

 say, the earliest discussions were concerned with the solution of isolated examples 

 rather than with the formulation of general ideas. In the case of elastic stability, a 

 comprehensive theory was not propounded until the problem of the straight strut had 

 been investigated by EuLER.t that of the circular ring under radial pressure by 

 M. LEVY} and G. H. HALPHEN, and A. G. GREENHILL had discussed the stability of 

 a straight rod in equilibrium under its own weight, || under twisting couples, and when 

 rotating.^ 



In a paper which has become the foundation of the theory in its existing form,** 

 G. H. BRYAN has brought these isolated problems for the first time within the range 

 of a single generalization. Examining the conditions under which KIRCHHOFF'S 

 theorem of determinacy may fail, he was led to the conclusion that instability is only 

 possible in the case of such bodies as thin rods, plates, or shells, and in these only 

 when types of distortion can occur which do not involve extension of the central line 

 or middle surface, so that it is legitimate to discuss any problem in elastic stability 

 by methods which have been devised for the approximate treatment of such 

 bodies. He showed, moreover, that the stability of the equilibrium of any given 

 configuration depends upon the condition that the potential energy shall be a 

 minimum in that configuration. 



A closer examination of BRYAN'S theory suggests that some of the conclusions 

 which have len drawn from it are scarcely warranted. The contention that no 

 closed shell can fail by instability, because any distortion would involve extension of 

 the middle surface, will be discussed later.ft For our present purpose it is sufficient 

 to remark that the whole theory is based upon the assumption that the strains 

 occurring previously to collapse must be kept to the extremely narrow limits within 

 which, in the case of ordinary materials, HOOKE'S Law is satisfied. This assumption, 

 of course, expresses a restriction necessarily imposed upon the range of practical 



A. E. H. LOVE, 'Mathematical Theory of Elasticity ' (second edition), 118. 

 t ' Hist. Acad. Berlin,' XIII. (1757), p. 252. 

 } 'LiouviLLE's Journal,' X. (1884), p. 5. 

 'Comptes Rendus,' XCVIII. (1884), p. 422. 

 II 'Proc. Camb. Phil. Soc.,' IV. (1881), p. 65. 

 H 'Proc. Inst. Mech. Eng.,' 1883, p. 182. 

 ** 'Proc. Camb. Phil. Soc.,' VI. (1888), p. 199. 

 tt Cf. pp. 222, 236. 



