THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 143 
For a Z force of 47u(a+1) units at (a’, y’, 2’) we have therefore 
y=0 
eal a dr 
OS ter oa ee ees) 
ge 
fre, Die 
It is easy to verify that these values of 1, 6, p substituted in (4) do actually reproduce 
equations (6). 
Similarly for an X force of 47~(a+1) units at (x’, y’, 2’) we find 
d= —_ —~ 9 
“de de® 2 de de (°) 
YY ded r> 
PF aa de 
Here “ =F ~ denotes a potential function having “|”; * for z-derivative, and is defined 
by Bio Piostions 
ad 7a , ] , h ’ 
— _ Co 2 — Z 
7S (2-2) log (r+z-z)-r when z>2 ; ; (10) 
= —(z-z) log (r—z+2)-r when z<z’ 
It may be observed that the necessity for dealing separately with the two regions 
z>z and z<7~ in these cases is not inconsistent with the theorem of (4), which refers 
only to a displacement free from singularity in the space considered. 
3. Solution of the problem of normal traction. 
Coming now to the problems relating to a solid bounded by the two parallel planes 
z=h and z= —h, we begin with the simplest of these, and seek a solution of the 
equations of equilibrium giving 
X=Y=Z=0 throughout the body ; 
the normal stress 
n=fla, M) Of Bla, 
on 2=—-h 
the tangential stresses zx, 2,=0 on both faces z=-+h. The arbitrary function f(a, y), 
which we shall suppose to vanish at all points without a given finite area A, is expressed 
in a form amenable to analytical treatment in the familiar theorem 
Lee f(x’, y da’ a, ,= afta, y) . ; 5 Ol) 
eyP+(y-y 24 2} 
the integral being taken over the area A. 
(If we imagine the plane z=0 to be covered with attracting matter of surface density 
f(x, y), then the theorem expresses the well-known relation between the density at 
