45G 
MR. A. B. BA.SSET OR THE EXTENSION AND FLEXURE OF 
the product Acr“. At the same time, inasmuch as the production of change of 
curvature involves some extension or contraction of all but the central layers, and 
consequently of those portions of the shell which are near its external surface, it does 
not seem unreasonable to suppose that in the neighbourhood of a free edge, an exten¬ 
sion or contraction of the middle surface may take place, which is comparable with the 
change of curvature. 
In the fourth place, Mr. Love appears to have argued as if the equations of 
motion of a shell, whose middle surface undergoes no extension or contraction 
throughout the motion, might be obtained from his general equations (30), (31), (32), 
by putting cr^ = 0-3 = r? = 0 ; but it has already been pointed out, that the correct 
equations for this kind of motion must be obtained by taking the variation subject 
to the conditions of inextensibility, and introducing indeterminate multipliers. It 
will be shown in the next section, that in the case of the flexural vibrations of an 
indefinitely long complete cylindrical shell considered by Hoppe and Lord Ray¬ 
leigh,'" the differential equation for the tangential displacement v is of the sixth 
degree, and that when the cross section of the shell consists of a circular arc, this 
equation contains sufficient constants to enable all the conditions of the problem to be 
satisfied. 
13. The first problem which we shall consider will be that of the flexural vibrations 
of an indefinitely long cylinder, in which the disjolacement of every element lies in a 
plane perpendicular to the axis of the cylinder, and which has been discussed by 
Hoppe and Lord Rayleigh. 
In this problem the middle surface is supposed to undergo no extension or contrac¬ 
tion throughout the motion, and the solution is most easily obtained by means of the 
general equations (11). In these equations we must omit all the terms on the right 
hand sides which involve li^, for they would, if retained, give rise to a term involving 
/d in the period equation, which must be rejected, since we do not carry the approxi¬ 
mation further than /d in determining the period. 
We evidently have! = Ng = Ho = 0 ; also none of the quantities are functions 
of 2 :. The equations of motion are thus 
clT, 
d(j) 
dNi 
d(f) 
fZCh 
d(j) 
N| = 2phav, 
—- T 3 = 2p]iaw, 
-p a = 0, 
* ‘ Theoiy of Soiiiicl,’ vol. 1, p. 324 ; ‘ Roj. Soc. Proc.,’ vol. 45, p. 120. Equation (51). 
t We sliall pi'esently see that these conditions imply a constraint at infinity. 
