CYLINDRICAL AND SPHERICAL THIN ELASTIC SHELLS. 
453 
of finite dimensions is under the influence of forces and couples applied to its edges, 
these equations would give the values of such forces or couples, and that the 
conditions to be satisfied at a free edge would require that eacli of the above five 
stresses should vanish at a free edge. Kirchhoff'"" has, however, shown that this is 
not the case, but that the boundary conditions are ordy four in number; and the 
reason of this is, that it is possible to apply a certain distribution of stress to the edge 
of a shell, without producing any alteration in the potential energy. 
By Stokes’ theorem. 
Sic d- H' ) (Z 2 ; =: 0 ; 
the integration extending round any curvilinear rectangle bounded by four lines of 
curvature OA, AD, DB, BA. If, therefore, we apply to the side AD the stresses 
Md^HVa, No'=-^, = 
' "" a ci xp 
to the side DB the stresses 
clR' 
and to the sides BO, OA, corresponding and opposite stresses respectively, the 
preceding integral becomes 
N3' B w + 
Hfi idSw 
a \ dcf) 
which shows that the work done by these stresses is zero. Such a system of stresses 
may, therefore, be applied or removed without interfering with the equilibrium or 
motion of the shell. 
Let us now suppose that the rectangle OADB, instead of being under the action of 
the remainder of the shell, is isolated, and that its state of strain is maintained by 
means of constraining stresses applied to its edges; then it follows that if, instead of 
the torsional couples Hj, Hj, due to the action of contiguous portions of the shell, we 
apply torsional couples ^1^, where 
SJ. = H, + H-.(53). 
= .(54), 
* ‘ Crelle,’ vol. 40, p. 51, 1850, and Collected Works, p. 237. 
