1888.] Sending and Vibration of thin Elastic Shells. 123 



difference between p s ' and p s becomes very small. Approximately in 

 this case 



'/ -11 



Ps IPs- +- 



or if we take m = 2n, 5 = 2, 



In my former paper I gave the types of vibration for a circular 

 cone, of which the cylinder may be regarded as a particular case. 

 In terms of columnar coordinates (z, r, 0) we have 



= (A 5 -fBsZ" 1 ) cos 50 .................... 



= 5 tan 7 (A s z+~B s ) sin 50 ................ (54), 



= tan2 7 [ s -iB 5 -s (A 5 z + B 5 )] sin s0 ........ (55), 



<y being the semi- vertical angle of the cone. For the calcula- 

 tion of the energy of bending it would be simpler to use polar co- 

 ordinates (r, 0, 0), r being measured from the vertex instead of from 

 the axis. 



If the cone be complete up to the vertex, we must suppose, in 

 (53) &c., B s = 0. And if we proceed to calculate the potential 

 energy, we shall find it infinite, at least when the thickness is uni- 

 form. For since A s is of no dimensions in length, the square of the 

 change of curvature must be proportional to A s z z~%. When this is 

 multiplied by zdz, and integrated, a logarithm is introduced, which 

 assumes an infinite value when 2 = 0. The complete cone must 

 therefore be regarded as infinitely stiff, just as the cylinder would be 

 if one rim were held fast. 



If two similar cones (bounded by circular rims) are attached so 

 that the common rim is a plane of symmetry, the bending may be 

 such that the common rim remains plane. If the distance of this plane 

 from the vertex be z l9 the condition to be satisfied in (53) &c., is 

 that 8z = where z = . Hence 



s S .............. (56), 



= s tan 7 A s < z ^ ^ > sin s0 .......... (57), 



= s tan 2 7 A s {z l z} sin 50 .............. (58). 



