MR. K. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 243 



We are considering an infinitesimal distortion in the third position, so that if we 

 represent the increase of strain in fig. 12 by \z, the increase of longitudinal com- 

 pressive stress to the right of F may be taken as E'Xz, and the decrease of this stress 

 to the left of F as EXz. Hence we obtain, for the section under consideration, the 

 diagram of longitudinal stress which is shown in the lower part of fig. 12. The 

 uniform stress of the second configuration is shown by the horizontal line In, and it is 

 a condition for neutral stability in the second configuration that no increase of thrust 

 shall be required to maintain the distortion. If the cross-section of the strut is 

 rectangular, of dimensions a x 2t, it follows that the triangles Imk and qmn must be 

 equal in area, or 



19? -E (111) 



PF~E'' 



This relation fixes the position of F on the cross-section, and in terms of the 

 dimensions shown in fig. 12 we may write for the moment of resistance about G, 



M =r + *E'a\2(z -6)^2+1 Ea\z(z-b)dz 



But if y is the deflection of the strut at the point G, in the infinitesimal 

 distortion, we have, as in the ordinary theory of bending, 



where x denotes the distance of the section from one end ; and for equilibrium in the 

 third configuration M must be equal to the bending moment due to thrust, or 2atpy : 

 hence, 



Equation (112) shows the modification which must be introduced, to take account 

 of elastic break-down, into EULER'S equation 



(113) 



and it is easy to see that if I is the length, calculated from (113), of a strut which 

 can just support the stress p, and I' the length as calculated from (112), then 



But, from (111), 



t + b _ /E 

 t^b~ V W 



2 I 2 



