_,._ MK. IL V. SOrTIIWKLI. OX THE GENERAL THEORY OF ELASTIC STABILITY. 

 so Unit tlu- \aln. - <>f <;. for which collapse by instability may be expected to occur, 



G l-2oc< cosech 



~ 2m (m+l ) V ~ 2m"- \-2ett cosech 2at 



and 



G _ 1 + 2ai cosech 2ett 



~ 2m(m + l)C ~ 2m 3 '- 1 + 2at cosech 2at ' 

 respectively, the total thrust, per unit length of edge, being 



JJ = -2*G ........... (33) 



The first approximations to a solution, in terms of t, are 



G =_$.--.aV' . (34) 



m 1 



and 



=-E ......... (35) 



m 



respectively. Since the complete wave-length of the corrugations into which the 

 plate distorts is 



A = ^, (36) 



we see that (34) is equivalent to (17), and that the latter formula is therefore 

 supported by our investigation as a first approximation. The second solution (35) is 

 without practical interest, owing to the magnitude of the thrust required to produce 

 collapse. It refers to a type of distortion, theoretically possible for _an ideal material 

 without limits of elasticity, which is approximately realized in actual specimens of 

 ductile material, when tested to failure under compressive stress. Since Q = R = 0, 

 we see from (23) that in this type the middle surface remains plane. In the first 

 tyjxj of failure, where P = S = 0, we find that U. = when y = 0, so that the middle 

 surface undergoes no change of extension in the distortion given by ', v', w'.* 



EQUATIONS OF NEUTRAL EQUILIBRIUM IN CYLINDRICAL CO-ORDINATES. 



Derivation, of the Equations. 



The equations (15) of neutral equilibrium are expressed in a form which is 

 unsuitable for the investigation of problems concerned with the stability of thin 

 tubes, and we have next to obtain the corresponding equations in cylindrical 



* Besides the harmonic solutions to (20) we may have 



*' = gx, v = hy, ?/' = fc; 

 but g, h, and k vanish in virtue of the boundary conditions. 



