VI
165
ON HARDNESS
have the same coordinates x, y, have approached each other
by a distance a-+ under the action of the pressure;
this approach must in the surface of pressure neutralise the
original distance A²+ By². Hence here we must have
51 — 5½ = a — Ax² - By, whilst elsewhere over the surfaces
51 - 52>a — A₂²- By. All these conditions can be satisfied
only by one single system of displacements; I shall give this
system, and prove that it satisfies all requirements.
-
-
-
=
As surface of pressure we take an ellipse, whose axes
coincide with those of the ellipses e constant, but whose
shape is more elongated than theirs. We reserve the deter-
mination of the lengths of its semi-axes a and b until later.
First we define a function P by the equation
P=
3p
16π
W
S
(
2/2 22
-
-
a² + λ b² + λ
dn
λ/√(a² +λ)(b² +λ)λ²´
² XX
where the lower limit of integration is the positive root of the
cubic equation
0=1-
2.2 y2
a² + u
22
b²+u 2
The quantity u is an elliptic coordinate of the point xyz; it
is constant over certain ellipsoids, which are confocal with
the ellipse of pressure, and vanishes at all points which are
infinitely close to the surface of pressure. The function P has
a simple meaning in the theory of potential. It is the poten-
tial of an infinitely flattened gravitating ellipsoid, which would
just fill the surface of pressure; in that theory it is proved
that P satisfies the differential equation
a²p a² ²P
V2P = + +
дл2 дуг 03
=
= 0.
Now from this P we deduce two functions II, one of which
refers to the one body, the second to the other, and we make
П1
II.
1
--
=
=
-
K,
1
-
1
∞
(P = 1 + 20, P.de),
-
1
- ½ (= − 1 + 20. [Pds)
K2
1+20₂