168
VI
ON HARDNESS
form 0.; the ordinary procedure for the evaluation of
such an indeterminate form gives
that is, since for u =
Z,
=
ap
= -
3p
ди
Əz
Əz 8πab uJu'
O we have
22
= u(1
әр
-2-
=
--
3p
a²
Əz 2πab
-
y
b2
-
x²
-
y²
b2
•
Here no quantity occurs which could be affected by an
index. Hence in the surface of pressure Z, is the same for
both bodies; pressure and counter-pressure are equal. Lastly,
the integral of Z, over the surface of pressure is 3p/4πab
times the volume of an ellipsoid whose semi-axes are 1, a, b;
i.e. it equals p, and therefore the total pressure has the
required value.
It remains to be shown that the fourth condition can be
satisfied by a suitable choice of the semi-axes a and b. For
this purpose we remark that
әп,
51=
Əz
z ӘР
+29, P = − K₁ dz
K,
1
=
-
+$P,
so that at the surface ₁₁P and ₂ = 9,P. Since inside
the surface of pressure the lower limit u of the integral is
constantly zero, inside that surface P has the form PL-
Mx²- Ny²; and therefore it is necessary so to determine a, b
and a that (+9₂)M = A, (9₁ +9₂)N = B, (9₁+9₂)L = a, so
as to satisfy the equation ₁₂ = a - Ax² — By², and this
determination is always possible. Written explicitly the
equations for a and b are
∞
du
-
-
A 16π
√(a²+u)³(b²+u) u d₁+d₂ 3p
2
du
B 16π
(a² + u)(b² + u)³ u d₁+D₂ 3p
2
(I)