THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE 141 
In terms of the displacements, the equations of equilibrium are therefore 
py2u + (+p) aE, Gan 0) 
CRE, 
9 dA 
py?v + Cas + Y = 04 (3) 
py?w + (+m Ly =O) 
dz 
When the bodily force is null, or X = Y=Z=0, the following forms are easily shown to 
satisfy equations (8), 
s € 
(i) w= a 
- oth 
da 
dé 
dk 
dé 
ae 
dz 
L+3y db,» Cd 
A+ pe dx dz di 
A+ 3u db, 5, Pd 
A+p dy dz dy 
_ At+3pu dd ioe dd 
Xr + yu dz lz 
(ii) uw = 
where ¥, 0, ¢ are potential functions, so that 
vy=0, y0=0, y%=0. 
These solutions have been used by BovssinEsq in his treatment of the problem of a 
solid bounded by a single plane z=0. They are equally effective when the boundary 
consists of two parallel z-planes. Thus, as will explicitly appear in the sequel, and as 
might be proved at once, any solution of (3), with X=Y=Z=0, in the space between 
the planes z= + h, can be expressed in the form 
Se ee ash o, ate 
i tN ads 
Paes, ott a fake) at al? 2. Wy x: : ; ; (4) 
dix dy dy dz dy 
dé dd. 5, Uh 
de dz nn dz 
(Wb) == 
Here, and throughout the paper, the symbol a is used to denote the fraction (A + 3u)/ 
(A+). 
With these values of wu, v, w the stresses across a z-plane, VizZ., zz, zy, zz, are given by 
ze 4s dy Pi a6 dh ay ad 
2p dydz° dudz d«adz dx dz 
A. dy PO, Ph | 5, do (5) 
2a dadz dydz dydz dy d# 
Re a dO dh On ah 
= _— ad he 
2p dz dt Ga, 
