180
VI
ON HARDNESS
point outside the surface (X, Y, Z) = 0, to be enabled to
tell whether a permanent deformation will ensue and, if so, in
which of the two bodies. But so far there has not even been
an attempt made to determine that surface. We only know
isolated points of it: thus the points of section by the positive
axes correspond to resistance to compression; those by the
negative axes to tenacity; other points to resistance to torsion.
In general we may say that to each point of the surface of
strength corresponds a particular kind of strength of material.
As long as the whole of the surface is not known to us, we
shall let a definite discoverable point of the surface correspond
to hardness, and be satisfied with finding out its position.
This object we attain by the following definition,-Hardness
is the strength of a body relative to the kind of deformation
which corresponds to contact with a circular surface of pressure.
And we get an absolute measure of the hardness if we decide
that-The hardness of a body is to be measured by the normal
pressure per unit area which must act at the centre of a circular
surface of pressure in order that in some point of the body the
stress may just reach the limit consistent with perfect elasticity.
To justify this definition we must show (1) that the neglected
circumstances are without effect; (2) that the order into
which it brings bodies according to hardness coincides with
the common scale of hardness. To prove the first point,
suppose a body of material A in contact with one of material
B, and a second body made of A in contact with one made of
C. The form of the surfaces may be arbitrary near the point
of contact, but we assume that the surface of pressure is.
circular, and that B and C are harder or as hard as A. Then
we may simultaneously allow the total pressures at both con-
tacts to increase from zero, so that the normal pressure at the
centre of the circle of pressure may be the same in both cases.
We know that then the same system of stresses occurs in both
cases, therefore the elastic limit will first be exceeded at the
same time and at points similarly situated with respect to
the surface of pressure. We should from both cases get the
same value for the hardness, and this hardness would cor-
respond to the same point of the surface of strength. It is
obvious that the elements in which the elastic limit is first
exceeded may have very different positions relatively to the