458 
MR. A, B. BASSET ON THE EXTENSION AND ELEXHRE OP 
whicli appear in the solution of (59) can be eliminated, and the resulting determinantal 
equation, combined with (60), will give the frequency.* 
If a complete cylinder of finite length were vibrating in this manner, it would be 
necessary to satisfy the conditions at the circular ends, and this would require that 
= 0 , G 3 = 0 at the ends for all values of ^; and from the first and fourth of ( 43 ) 
we see that this requires that p, = 0 , or 
whence 
d^w 
V) = 0, 
tv = A cos B sin (j) 
for all values of (f>. Since it is impossible to satisfy this condition for the kind of motion 
considered, it follows that when the cylinder is of finite length it would be necessary 
to apply at every point of the ch’cular boundary a tension and a couple Gg of the 
requisite amount. 
This is the question upon which Lord Rayleigh and Mr. Love are at issue; and 
the preceding investigation shows that Mr. Love is right in supposing that it is 
impossible to satisfy the boundary conditions along the curved edges of a cylindrical 
shell when these edges are free, although he does not appear to have noticed that it 
is possible to satisfy these conditions when the free edges are generating lines. In 
order to obtain a complete mathematical solution of this question, it would be 
necessary to work out the problem of the free vibrations of a complete cylindrical 
shell of given length 2 ?, which is deformed in such a manner that dvldtj) tv = 0, 
where v and w are functions of ^ alone, and is then let go, without assuming that 
the middle surface remains unextended during the subsequent motion. 
Owing unfortunately to the extremely complicated nature of the general 
equations, a rigorous solution of this problem would be exceedingly difficult. We 
shall, however, be able to throw some light upon this question, by solving and 
discussing the following much simpler statical problem. 
14. Let us consider a heavy cylindrical shell, whose cross section is a semicircle, 
and which is suspended by means of vertical bands attached to its straight edges, so 
that its axis is horizontal; and let us investigate the state of strain produced by its 
own weight. 
In order to simplify the problem as much as possible, we shall suppose that the 
displacement of every point of the middle surface lies in a plane perpendicular to the 
axis, and we shall afterwards investigate the stresses which must be applied to the 
circular edges, in order to maintain this state of things. 
* [This problem is of a similar character to that of a bar, whose natural form is eii’cular, and which 
has been discussed by Lamb. ‘London Math. Soc. Proc.,’ vol. 19, p. 365.—June, 1890.] 
