454 MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
the state of strain will remain unchanged, provided we apply in addition the stresses 
and 
+ B'/a 
1 dW 
= N 3 + 
a d<p 
. AT , dW 
(55) 
(56), 
whence, eliminating H' between (53), (55), and (54), (56) respectively, we obtain 
. . . (57), 
= MgCt — H, 
- - = No - - 
a d(j} “ n d(j) 
and 
Pi + 
dz 
Ni + 
dR^ 
dz 
(58). 
In these equations the Homan letters denote the stresses due to the action of 
contiguous portions of the shell, whose values are given (43) and (44), whilst 
the Old English letters denote the values of the actual stresses applied to the 
boundary. If, therefore, the shell consists of a portion of a cylinder which is bounded 
by four lines of curvature and whose edges are free, the boundary conditions along 
the circular edges are obtained by equating the right hand sides of the first and 
fourth of (43), and the right hand sides of (57) to zero, the first two of which express 
the condition that the tension perpendicular to, and the flexural couple about, a Ime 
element of the circular edge must vanish when the edge is free ; and the boundary 
conditions along the straight edge are similarly obtained by equating the right hand 
sides of the first, second, and fourth of (44), and the right hand side of (58) to zero, 
the first three of which express the conditions that the tangential shearing stress, the 
tension and the flexural couple must vanish when the free edge is a generating line. 
We may also, if we do not wish to introduce the time and the bodily forces into these 
equations, substitute for u — X, v — Y, lu — Z their approximate values from (32). 
12 . We have now obtained all the materials we require, for a perfectly accurate 
approximate solution of any problem relating to the vibrations of a thin cylindrical 
shell as far as the terms involving the cube of the thickness, but before proceeding 
to discuss any problems, it will be necessary to make some remarks respecting 
Mr. Love’s paper. The first line of my expression for the potential energy which is 
given in (24), and which involves h and not agrees with the expression obtained by 
Mr. Love and other writers; also the approximate equations of motion (32) agree, as 
has been already pointed out, with the corresponding equations obtained by him, and 
by means of which he has discussed the extensioual vibrations of a cylinder. It will 
