1889.] Vibrations of an infinitely long Cylindrical Shell. 445 



. 2 -& 2 )U = ................... (13), 



=0 ......... (14), 



= ........ (15); 



and they may be satisfied in two ways. First let V = "W = ; then 

 U may be finite, provided 



s 2 -&2 = .................... (16). 



The corresponding type for U is 



U = cos 50 cos pt 



where F 5 = ...................... ( 18 )> 



pa? 



a being restored, as can be done at any moment by consideration of 

 dimensions. In this motion the material is sheared without extension, 

 every generating line of the cylinder moving along its own length. 

 The frequency depends upon the circumferential wave-length, and not 

 upon the curvature of the cylinder. 



The second kind of vibrations are those in which U = 0, so that 

 the motion is strictly in two dimensions. The elimination of the ratio 

 V/W from (14), (15) gives 



& 2 {& 2 -2(M + l)(l + s 2 )} = ............ (19), 



as the frequency equation. The first root is & 2 = 0, indicating in- 

 finitely slow motion. These are the flexural vibrations already 

 referred to, and the corresponding relation between V and W is by 

 (14) 



5 y+w = o .................... (20), 



giving by (4), (5), (6), 



The other root of (19) gives on restoration of a, 



&2 a 2 = -i!^(l+s 2 ) (21), 



m-f-n 



o 4imn 1-fs 2 /nr\ 



or & = - -V- ................. (22) ; 



m + n a z p 



while the relation between V and W is 



.................... (23). 



