444 
MR. A. B. BASSET ON THE EXTENSION AND ELEXURE OF 
third and fourth consist of the products of the extensions and the quantities which 
principally depend upon the bending. 
8 . Having obtained the value of the potential energy, we must in the next place 
form the variational equation of motion. This equation may symbolically be written 
SW + = SU + 8E.. . (25), 
where SfT is the term which depends upon the time vPvriations of the displacements, 
SU is the work done by the bodily forces, and represents the work done upon the 
edges of the portion of the shell considered, in producing the displacements, Sti, Sv, Sw, 
by the forces arising from the action of contiguous portions of the shell. It, therefore, 
follows that S% is a line integral taken round the edge of the portion of the shell 
which is being considered; and as one of our objects is to calculate the values of the 
sectional stresses in terms of the displacements by means of (25), it will be convenient 
to apply the variational equation to a curvilinear rectangle bounded by four lines of 
curvature. 
We must now calculate SC. We have 
Now 
ch 
sc = p j'jj' {vf Su + v' Sv + Sir') (1 + h'/a) dh'd^. 
d'f^ ' dr'^ 
+ 
2/d / • • dhit, . du 
3a. 
u — -b “w Sw 
dr di 
= 2hu Su + I i 
dz dz 
iu 
dgK 
dz 
+ 
3a 
• • dSiu , dw » 
U “p oil 
dz dz 
by (8) and (9). Treating the other terms in a similar way, we shall find that the 
value of Sc is 
SC = 2ph j J (u Su + y Sr + iv Siv) cZS 
^ ^ ^ • (g^ ^ S^) _{_ Sa -1- - ^ Sy - (X + p) Sw 1 dS 
2 ^ 
3a 
f r • • dSio d'W cs . V /dSw 
dz 
d,z a d(f) ' dz a d(j) 
+ E(w8K + K8w)|.dS . . (26). 
