spacing of two successive nodal points of the lobe line appearing in a longitudi- 

 nal section of the tube. If, as is the case in the fire tubes of a boiler, the 

 tube ends can be regarded as fixed radially, then 1 can be taken equal to the total 

 length between the ends of the tube. If the end supports are yielding, a somewhat 

 higher value of 1 must be used. If the tube is divided into several belts by 

 sufficiently rigid frames, the effective length 1 is an average value of the in- 

 dividual frame spacings. It is possible to obtain more exact data for determining 

 the average value from the known calculated results for the buckling of rods with 

 several bays. However, since the frames in submarine hulls are usually equally 

 spaced, no difficulty is presented. 



It is a question whether in many cases a higher number of waves between 

 two successive frames, that is to say dividing 1 into halves, thirds, etc. , may 

 not result in a smaller value of the buckling pressure. This assumption seems not 

 improbable when we consider that the smallest possible number of lobes, n = 2, in 

 a circumferential belt between two frames, does not always result in the smallest 

 critical pressure. In reality Eq. (7) indicates, in contrast to (C) and (D) of Z, 

 that for fixed n and x the ordinate y does not increase unconditionally with cC . 

 For sufficiently small values of x, y decreases; for example, if e>c is made three 

 times as large while n remains constant. However, it does not follow from this 

 that the foregoing assumption is true since it is not yet certain whether a lower 

 buckling pressure occurs by changing n while oC is increased. The question to de- 

 cide is whether the polygons that are constructed for a single OL intersect in the 

 region of very small x or whether the polygons always lie above each other and 

 meet only in the origin. The answer is apparent if we consider that with small x 

 on the one hand the number of lobes increases without limit, while on the other 

 hand the vertexes of the polygons increase without limit. From the first condi- 

 tion it follows that Eq. (7), by expanding in powers of 1/n 8 and neglecting the 

 higher powers, can be simplified to: 



J - < 1 - S'>* 4 d-*fc*) *n* (l.i*£l), (10, 



From the second condition it follows that we may with sufficient accuracy, in the 

 neighborhood of the origin, consider the polygon as the envelope of the family of 

 straight lines determined by Eq. (10) for successive values of n. This envelope is 

 obtained through Eq. (10) in conjunction with the following equation obtained by 

 the differentiation of Eq. (10) with respect to n 8 : 



= _?0-f)* 4 



- 2(5*? - 2) 

 1 3n a 



+ x (11) 



