stances. In most cases, the annexed table will be sufficient. The values given 

 here differ as much as 20^ or more from the Lorenz values and in the direction of 

 a better agreement with observation. The essential characteristic of the buckling 

 phenomena, the unlimited increase of the number of lobes as the length of the tube 

 decreases, has been previously emphasized In the investigations mentioned above. 



The extension of the investigation to the 

 region above the elastic limit, and the possibility 

 of a theory for the corrugated tube, will be dis- 

 cussed In a section at the close of this article. 

 1. The Elastic Equations. 

 Let a point of the cylindrical surface (See 

 Elg. 2) have the coordinates x, measured in the di- 

 rection of the axis, and (p , measured on the cir- 

 cumference of the cross section. Let the elastic 

 displacement of this point be u in the direction of the axis, v in the direction 

 of the tangent to the circle, and w in the direction of the radius, measured in- 

 ward. We then express the strains d, , and £i for the u- and v- directions and the 

 angular change r in the u-v plane as follows: 



-—(1) 



Flgo 2. Stresses and 



displacements in a 



volume elements 



^■=t-"=»=T(f-'-)' f-t*&-^ 



(See Love, The Mathematical Theory of Elasticity, p. 543) where as above, a de- 

 notes the radius of the cylinder. 



Let 2h designate the shell thickness, E the elastic modulus, and (7= O.J, 

 Polsson's ratio. With the simplifying substitution 



2 Eh 



•(2) 



1 -a^ 



we can easily find the values for the various longitudinal and transverse forces, 

 namely, the normal forces T^ and Tg and the shear force S, Fig. 2, as follows: 

 (Translator's note: These forces are really stresses multiplied by the thick- 

 ness). 



Ti - c(e, H-o-cfa), Tg = o(d;tf a<f, ), s = |.(i-ff);^ -(3) 



Since the shear modulus Is 



the solution of the first two e- 



^{1 +0-)' 

 quations of (3) for <Sj, and <5 2 yields the well known simple relation 



1 Ti -O-Tg 



2h 



1 Tg -O-T^ 

 E 2E 



(Translator's note: In the text the shear modulus is given ^( 1 + 0" ) ). 



