462 
MR. A. B. BASSET ON THE EXTENSION AND ELEXURE OF 
see that is negative, and, therefore, the strain tends to increase the curvature of 
the circular sections. Now when a cylindrical shell is hent about a generating line in 
such a manner that its curvature is increased, all lines parallel to the axis which lie 
on the convex side of the middle surface will be contracted, whilst all such lines 
which lie on the concave side will be extended, and this contraction and extension 
will give rise to a couple about the circular sections which tends to produce anticlastic 
curvature of the generating lines. In order to prevent this taking place it is necessary 
to apply at every point of the circular edges a couple Gg tending to produce synclastic 
curvature, and a tension Tj parallel to the axis, whose values are given by (68) and 
(69). If this couple and tension were removed, the middle surface would bend about 
its circular sections, and anticlastic curvature of the generating lines would be pro¬ 
duced, and this would necessarily involve extension or contraction parallel to the axis, 
so that the problem could no longer be treated as one of two dimensions. 
It must, however, be within the experience of everyone that when a thin cylindrical 
shell of finite length, whose cross section is the arc of a circle, is bent about its 
generating lines, the shell does not assume a saddle-back form, and consequently the 
anticlastic curvature of the generating lines must he so small as to be inappreciable. 
This circumstance furnishes an additional argument in favour of the supposition that 
the extension of the middle surface is only sensible in the neighbourhood of the free 
edges. 
We therefore conclude that if the circular edges were free, some extension or con¬ 
traction of the middle surface must necessarily take place, but that this extension or 
contraction is small compared with the change of curvature along a circular section, 
except just in the neighbourhood of the edges. From these considerations the infer¬ 
ence is, that if by means of proper constraints applied to the circular edges, a 
cylindrical shell were enabled to execute the non-extensional vibrations discussed in 
§13, the vibrations would cease to be non-extensional if the constraints were 
removed ; but that the amplitudes of those portions of the displacements upon which 
the extension depends, would be very small compared with the amplitudes of those 
portions upon which the change of curvature along a circular section depends, except 
just in the neighbourhood of the edges. Moreover, the theory of plane plates shows, 
that the frequency of the extensional vibrations is expressible* by means of a series of 
even powers of li, commencing ivith a term independent of h; whilst the frequency of 
the flexural vibrations is expressible by means of a similar series commencing loitli h^. 
It therefore follows, that the pitch of the notes arising from the former class of vibra¬ 
tions, is high compared with the pitch of those arising from the latter class. And 
although, except under special circumstances, it is not possible in the case of curved 
shells whose edges are free, for these two classes of vibrations to coexist independently, 
as in the case of a plane plate; yet recent investigations show, that the pitch of 
* Lord Rayleigh, ‘ London Math. Soc. Proc.,’ vol. 20, p. 225. See especially equations (38), (45), 
and (53). 
