1888.] Bending and Vibration of thin Elastic Shells. 119 



showing that, except as to magnitude and sign, the curve of deforma- 

 tion is the same for all values of e l and z* 



If z = + !, the amplitudes are in the ratio 1 + 3z-f/l 2 ; and if, 

 further, z l = I, i.e., if the force be applied at one of the ends of the 

 cylinder, the amplitudes are as 2 : 1. The section where the 

 deformation (as represented by w) is zero, is given by Szz l + Z 2 = 0, 

 in which if z l = Z, z = JZ. 



When the condition as to the length of the cylinder is not imposed, 

 the ratio B/ : A/ is dependent upon s, and therefore the curves of 

 deformation vary with z t apart from mere magnitude and sign. If, 

 however, we limit ourselves to the more important term s = 2, we 

 have 



4m_A^_ f 4m Z 2 



z \ 



= 



so that w vanishes when 



This equation may be applied to find what is the length of the 

 cylinder when the deformation just vanishes at one end if the force 

 is applied at the other. If z l = z = I, 



L /J 3 ( m +*) 



a, " V \ 8m 



For many materials a [equation (11)] is about ^, or m = 2n. In 

 such cases the condition is 



It should not be overlooked that although w may vanish, u remains 

 finite. 



Reverting to (23), (24), (25) we see that, if the cylinder is open at 

 both ends, there are two types of deformation possible for each value 

 of s. If we suppose the cylinder to be closed at z by a flat disk 

 attached to it round the circumference, the inextensibility of the disk 

 imposes the conditions, w = 8r = 0, v = a 60 = 0, when z = O.f 

 Hence A 5 = 0, and the only deformation now possible is 



* That w is unaltered when z and z l are interchanged is an example of the general 

 law of reciprocity. 



f * being greater than 1. 



