VI
171
ON HARDNESS
modulus, and can therefore be found by means of Legendre's
tables without further quadratures. But the calculations are
wearisome, and I have therefore calculated the table given
below, in which are found the values of μ and for ten
values of the argument ; presumably interpolation between
these values will always yield a sufficiently near approxima-
tion. We may sum up our results thus: The form of the
ellipse of pressure is conditioned solely by the form of the
ellipses e constant. With a given shape its linear di-
mensions vary as the cube root of the pressure, inversely as
the cube root of the arithmetical mean of the curvatures, and
also directly as the cube root of the mean value of the elastic
coefficients 9; that is, very nearly as the cube root of the
mean value of the reciprocals of the moduli of elasticity.
is to be noted that the area of the ellipse of pressure in-
creases, other things being equal, the more elongated its form.
If we imagine that of two bodies touching each other one be
rotated about the common normal while the total pressure is
kept the same, then the area of the surface of pressure will be
a maximum and the mean pressure per unit area a minimum
in that position in which the ratio of the axes of the ellipse
of pressure differs most from 1.
It
Our next inquiry concerns the indentations experienced
by the bodies and the distance by which they approach each
other in consequence of the pressure; the latter we called a
and found its value to be (9₁+92)L Transforming the
integral L a little, we get
3p 91 + Ꮽ
a=
2 -
8π a
dz
√(1+k² z ²)(1 + x²)
The distances by which the origin approaches the distant
1
T 90
80 70 60
50
40
30
20
10
0
μ
1.000 1.128 1.284 1.486 1-754 2.136 2.731 3.778 6.612
∞
V
1.000 0.893 0.802 0.717 0.641 0.567 0.493 0.408 0.319
0