CYLINDRICAL AND SPHERICAL THIN" ELASTIC SHELLS. 
437 
the variational equation will give the correct values of the tensions T 3 , which are 
unknown, and will also reproduce the values of the other stresses which have been 
obtained directly. This is the first test. The second test is furnished by the 
consideration that, if we substitute the values of the sectional stresses which we have 
obtained from the variational equation, in the first three of our original equations of 
motion in terms of these stresses, we ought to reproduce the equations of motion in 
terms of the displacements, Avhich have been obtained from the variational equation. 
This is found to be the case both when the shell is cylindrical and when it is spherical; 
and I therefore think that the fundamental hypothesis is sufficiently established. 
Having obtained the values of the sectional stresses, the boundary conditions can be 
deduced by means of Stokes’ theorem, which enables us to prove that it is possible 
to apply a certain distribution of stress to the edge of a thin shell, without producing 
any alteration in the potential energy due to strain. 
The fundamental hypothesis that R', S', T' may be treated as zero is not true when 
the surfaces of the shell are subjected to external pressures or tangential stresses; for 
if the convex and concave surfaces of the shell were subjected to pressures 11 ^, Ho, the 
value of R' as we pass through the substance of the shell from its exterior to its 
interior surface, must vary from — IT^ to — IIj, and consequently (exce])ting in very 
special cases) R will contain a term independent of the thickness. Hence the theory 
developed in the present paper is not applicable to problems relating to the collapse 
of boiler flues, or to the communication of the vibrations of a vibrating body to the 
atmosphere. In order to obtain a theory V 7 hich would enable such cpiestions to be 
mathematically investigated, it would be necessary to find the values of the additional 
terms in the variational equation of motion, which depend upon the external 
pressures; and this is a problem which awaits solution. 
It will be convenient briefly to state the notation employed. 
In the case of a cylindrical shell, OA is measured along a generating line, and OB 
along a circular section. In the case of a spherical shell, OA is measured along a 
meridian, and OB along a parallel of latitude. 
The three extensional strains along OA, OB, 00 are denoted by ctj, o-g, 0 - 3 ; and the 
three shearing strains about those lines l)y 7(7p5 ^ 2 ? ^^ 3 * We shall also use the letters 
[ji, p ; p, p to denote the first and second differential coefficients of ct ;^, 0 - 3 , 
with respect to r, when r = a. We shall also write 
E = (w — ri)/(m + n), 
^ = CT]^ -fi E ((T| + cr,), 
%C = X -f- E (k “h p)} 
K = cr^ + 0-3, 
B = 0-3 + E (o-i + 0-3), 
d? = p- + E (X + p.), 
dF — p + E(X + p). 
