150
CONTACT OF ELASTIC SOLIDS
dition that inside the surface of pressure Z, is everywhere
positive, and the condition that outside the surface of pressure
51-5a — Ax² - By2, otherwise the one body would overflow
into the other.
-
(f) Lastly the integral (Zds, taken over the part of the
surface which is bounded by the curve of pressure, must be
equal to the given total pressure, which we shall call p.
The particular form of the surface of the two bodies only
occurs in the boundary condition (2 d), apart from which
each of the bodies acts as if it were an infinitely extended
body occupying all space on one side of the plane z = 0, and
as if only normal pressures acted on this plane. We there-
fore consider more closely the equilibrium of such a body.
Let P be a function which inside the body satisfies the
equation ²P = 0; in particular, we shall regard P as the
potential of a distribution of electricity on the finite part of
the plane z = 0.
Further let
II
-
1
14+20) | Pde - J}.
zP
+
KK(1+20)
where i is an infinitely great quantity, and J is a constant so
chosen as to make II finite. For this purpose J must be
equal to the natural logarithm of i multiplied by the total
charge of free electricity corresponding to the potential P.
From the definition of II it follows that
V²II
= -
2 ap
Kaz
Introducing the contraction 9=
=
әп
дх
"
•
2(1+0)
K(1+20)
we put
әп
ап
η
= dy'
5 =
+ 29P,
Əz
ap
2
ap
=
σ = √²II+ 29
=
Əz K(1+20) Əz
This system of displacements is easily seen to satisfy the