SPHERES AND CYLINDERS 127 



Now let u denote the distance between the circle and the elh'pse 



measured radially. Then 



r = u a, 



or, if u is assumed to be positive when it lies outside the circle and 



negative when it lies inside, 



r = u 4- a. 



Differentiating both sides of this equation with respect to a, 



di- _ du d*r _ d*u 



da da da* da* 



Also, if dl is the length of an infinitesimal arc of the circle, ada = dl. 

 Substituting these values in equation (68), it becomes 



(69) E 



which is the required differential equation of the elastic curve in the 

 curvilinear coordinates I and u. 



109. Crushing strength of hollow circular cylinder. As a continu- 



i of the preceding article, let it be required 

 to find the external pressure which is just suffi- 

 cient to cause the cylinder to retain its flattened 

 form, or, in other words, the critical external pres- 

 sure just preceding collapse. 



In Fig. 95 let OA and OB be axes of symmetry; 

 then it is sufficient to consider merely the quadrant 

 AOB. Let c denote the length of the chord AC, Flo 95 



and let w be the unit external pressure. Then for 

 a section of the cylinder of unit length the external pressure P on 



the curved strip A C is 



P = wc. 



N >w let 3/ denote the bending moment at the point A. The tangen- 

 tial force at this point is equal to the resultant pressure on OA, or wb. 

 Consequently the bending moment M at the point C is 



M= M n 4- wb - AD wc-- = M n + wlb' AD 



