Elimination of n 2 from Eq. (10) and (11) gives the equation of the envelope: 

 y = 4\l 1 g 7 r2 o^fx^ + I {cc z + 2)x = 1.714 <X.\^T + | (cx z + 2)x ... (12) 



As can be seen from Eq. (12), since the ordinates increase with increase of & , 

 no intersection occurs, and we come to the conclusion: The minimum buckling 

 pressure always corresponds to a deformation without nodal points between the 

 points of support in the longitudinal plane, and, therefore, to the smallest possi- 

 ble value of cC . 



•Translators Note: Equation (12) derived to justify the conclusion in the preced- 

 ing paragraph is a good approximation to the envelope of Eq. (7) in the neighbor- 

 hood of the origin, that is, for small values of x and <*- . The exact equation of 

 the envelope has a very important property: It can be used in place of the original 

 equation (7) to determine y with sufficient accuracy for any given values of x 

 and °C. Graphically, this fact is evident from the proximity of the envelope and 

 those parts of the family of straight lines used to determine y represented by Eq. 

 (7) with n as a parameter. Analytically it is evident from the fact that the 

 equation of the envelope, obtained by eliminating n between Eq. (7) and (11), is 

 in effect Eq. (7) with that value of n substituted which for any given x and <%■ 

 makes y a minimum. 



Eq. (12) is an approximate equation of the envelope of Eq. (7), valid only 

 in the neighborhood of the origin, because of the nature of the assumptions which 

 led to Eq. (10). The equation of the envelope of Eq. (7) should be derived directly 

 from Eq. (7) itself. When so derived, its usefulness is no longer limited to the 

 neighborhood of the origin. This equation can be very accurately derived as follows: 



Differentiating Eq. (7) with respect to either n, (n 2 + ct z ), or p and 

 equating to zero we get 

 (n 2 + cc z f x - (n 2 + ot z f oc s x - 3 (n 2 + oc z ) <X* (1 - T z ) + ot 6 {\ - (T 2 ) = (11a) 



The solution of (11a) for n gives that value of n which will make Eq. (7) a 

 minimum for any given ot and x. A very approximate solution can be readily ob- 

 tained. Rewriting Eq. (11a) 

 (n 2 + oc z f [(n 2 + oc z ) x - * 2 xJ - 3**0 - CT Z ) [n 2 + oc z - #73] = 







(n* + oc^ 4 - 3*0 - <T 3 )(n a + 2* 2 /3) 



n2 + * 2= «^IEI^^ 1+ |*Vn 2 =* ^-<r 2 ) pr.,. .(11 



b) 



where k.= 3 + 2 #Vn 2 (11c) 



Substituting (11b) in (7) we obtain 



