1887.] deformation of elastic plates. 151 
Thus the strains are independent of 2, y. Hence so also are 
the stresses, and the stress equations reduce to 
If we suppose that no tractions are applied to the bounding 
surfaces of the plate, these give us S, 7, A=0 throughout the 
element. 
These three equations enable us to find u,, v,, w,, and thus to 
express the strains in terms of o,, o,, o, «,, ,, K, and Z, 1e. we 
can express them in terms of the deformation of the middle-surface 
and the distance from it. 
The potential energy per unit volume due to the strain in the 
elementary prism can now be written down. If we integrate this 
alone the lme z, normal to the middle-surface, we obtain the 
potential energy per unit area of the piece of the middle-surface 
contained within the elementary prism. This is the same for all 
points of this piece of the middle-surface. If 6A be the area of 
this piece and W the potential energy per unit area for any point 
of 6A, then 2 W6A over the middle-surface is the potential energy 
due to the strain in the plate. This, in the limit, is | | WdadB, 
the integration extending over the middle-surface. 
We have next to shew how the theory can be made to give 
equations which shall be accurate to any desired order of approxi- 
mation. Returning to equations (6) we have in them to substitute 
the approximate values (9) found for w,v,w. Then apply the tests 
Ou a] 
that ay dndy dyou 
for u, v, w as functions of x, y. We shall then explain how to 
form the stress- -equations, and find approximate values for S, 7, R, 
and by utilizing the surface-values of these deduce expressions for 
u, v, w as functions of w, y, z. By repeating the process we can 
find expressions for the u, v, w which shall be true to any desired 
order of approximation. 
. and we shall on integrating deduce new values 
The work is interesting as the tests oe a .. shew that 
dandy ~ Oyo Oioke 
to the order of approximation to which we are working the 
measure of curvature, —(«,A,+«,)=0, but they introduce no 
other fresh condition. 
MOL, Vii, PIS Tl 11 
