CYLIJ^DRICAL AXD SPHERICAL THIN ELASTIC SHELLS. 
435 
If the edges AD, BD were of finite length, there would also be a couple whose axis 
is parallel to the normal, but since this couple is proportional to the cube of the edge, 
it vanishes in comparison with the other stresses when the rectangle OADB is 
indefinitely diminished. 
We shall denote the components of the bodily forces per unit of mass in the 
directions OA, OB, OC by X, Y, Z ; but for reasons wdiich will be more fully explained 
hereafter, we shall suppose that these forces aidse solely from external causes, such as 
gravity and the like. All forces arising from pressures or tangential stresses applied 
to the surface of the shell will be expressly excluded. 
The first step is to write down the equations of motion of an element of the shell 
in terms of the sectional stresses,* which can be done by the usual methods ; we shall 
thus obtain six equations, three of which are formed by resolving the forces parallel 
to OA, OB, OC, and three more by taking moments about these lines. 
These equations will not, however, enable us to solve any statical or dynamical 
problems ; in order to do this we require the equations of motion in terms of the dis¬ 
placement of a point on the middle surface and their space variations with respect to 
the coordinates of that point. 
3. The values at P' of all the quantities with which we are concerned are functions 
of the position of P', and are, therefore, functions of (r, z, (f)) or (r, 9, cf)), according 
as the shell is cylindrical or spherical. If, therefore, be the value of any such 
quantity at P', and the value of the same quantity at the point P, which is the 
projection of P' on the middle surface, it follows that 
CEl'= F (r) = F (a + C) 
by Taylor’s theorem, where the brackets are employed, as wall be done throughout 
this paper, to denote the values of the differential coefficients at the middle surface 
where r = a. 
Objections have been raised by Saint-Venant and endorsed by Mr. Love, to the 
supposition that the first few terms of the expansion by Taylor's theorem of the 
quantities involved may be taken as a sufficient approximation. If, however, this 
objection were valid, it w'ould appear to me to upset the greater part of most physical 
investigations ; inasmuch as it is always assumed as a general principle, that when a 
quantity is known to be a function of the position of a point P, its value at a neigh¬ 
bouring point P' may be obtained by Taylor’s theorem, unless some physical 
discontinuity exists in passing from P to P'. If, therefore, we put B, we may 
write 
E' = A + A/'+iA,/f2-f .(2) 
where the A’s are functions of the position of P and also of the thickness of the shell. 
* See Besant “ On the EquilibHum of a Bent Lamina,” ‘ Quart. Journ. Math.,’ vol. 4, p. 12. 
3 K 2 
