1887.] deformation of elastic plates. 155 
On integrating this we obtain two equations for the «,,X,, «,, 
g,, o, aS functions of the surface-values of A, and these equations 
involve one unknown constant, thus they are equivalent to one 
relation between these quantities. Determining the constant from 
one surface-equation and substituting in the equation 
R=(m—n) (e+ f+g) + 2ng, 
Ow, 
0z 
2 
We find that # is a quantity of the order (x, =i) ee and 
X,7,¢ where e is of the order of the thickness, and the ieee 
along the normal to the middle-surface of the bodily force is a 
quantity of the same order. The way in which the bodily force 
occurs in the equations of equilibrium is that the line integrals 
of the X, Y, Z and of the Xz, Yz enter, the form of the equation 
just written down shews that the effect produced by the surface- 
forces R is the same as if their resultant were applied directly to 
the middle-surface. This is a result otherwise obtained by M. 
Boussinesq. 
For an isotropic plate we should still find ee ; Me = 0, so that 
this order of approximation will not enable us to include the effect 
of the forces S, 7. 
The method we have adopted is capable of being extended so 
as to give any desired order of approximation, but it is much 
simpler to form the equation of virtual work directly from the 
strains as given by equations (10) as was done by Kirchhoff. I 
have given this second approximation partly for the sake of the 
manner in which it appears that the measure of curvature is very 
small, and partly because it serves to take away the reproach of 
M. Boussinesq, that the stresses S 7, R cannot be found by Kirch- 
hoff’s method. The above work would be different for an exolo- 
tropic plate, and would, generally speaking, yield the values of S, 7’ 
as well as that of R. It will be readily seen that it is quite 
unnecessary to know the values of S, 7, & for the purpose of 
finding the form assumed by the plate under given forces, though 
it may be interesting if we wish to discover whether they can 
ever be important enough to produce a sensible tendency to 
rupture. The result is that they are always small quantities of 
the order of the square of the thickness. 
