CONTACT OF ELASTIC SOLIDS
153
where u, the inferior limit of integration, is the positive root
of the cubic equation
مرد
a² + u
+
b² + u
+
22
-
u
=
1.
Inside the surface of pressure, which is bounded by the given
ellipse, we have u = 0, PL-Ma²- Ny²; where L, M, N
denote certain positive definite integrals. The condition (2 d)
is satisfied by choosing a and b so that
(9₁+9₂)M = A, (9₁+9₂)N = B,
which is always possible. The unknown a which occurs in
the condition is then determined by the equation
(9₁+9₂)L = a.
It follows directly from the equation
น
=
3p
2παι
1
-
x² y²
a b2
that the first of the conditions (2 e) is satisfied.
To show that the second also is satisfied is to prove that
when z=0 and x²/a²+y²/l²>1, (Î₂+9₂)P>a− Ax² — By².
For this purpose we observe that here
P=L-Mx² - Ny²
3p
16π
π
1
2:2
y²
dr
a² + λ¯¯ b² +λ/ √(a² +λ)(b² +λ)λ´
2
and hence P>L- Ma² - Ny², for the numerator of the ex-
pression under the sign of integration is negative throughout
the region considered. Multiplying by 9, +9, we get the
inequality which was to be proved. Finally, a simple integra-
tion shows that the last condition (2 f) also is satisfied;
therefore we have in the assumed expression for P and the
corresponding system, n, a solution which satisfies all the
conditions.