164
VI
ON HARDNESS
=
consequently depend only to the very smallest extent on the
forces applied to other parts of the bodies. Hence it is suffi-
cient to know the form of the surfaces infinitely near the point
of contact. To a first approximation, if we consider each
body separately, we may even suppose their surfaces to coin-
cide with the common tangent plane z = 0, and the common
normal to coincide with the axis of z; to a second approxima-
tion, when we wish to consider the space between the bodies,
it is sufficient to retain only the quadratic terms in xy in the
development of the equations of the surfaces. The distance
between opposite points of the two surfaces then becomes a
homogeneous quadratic function of the x and y belonging to
the two points; and we can turn our axes of x and y so that
from this function the term in xy disappears. After com-
pleting this operation let the distance between the surfaces
be given by the equation e Ax²+ By². A and B must of
necessity have the same sign, since e cannot vanish; when we
construct the curves for which e has the same value, we obtain
a system of similar ellipses, whose centre is the origin. Our
problem now is to assign such a form to the surface of pressure
and such a system of displacements and stresses to its neigh-
bourhood, that (1) these displacements and stresses may satisfy
the differential equations of equilibrium of elastic bodies, and
the stresses may vanish at a great distance from the surface of
pressure; that (2) the tangential components of stress may
vanish all over both surfaces; that (3) at the surface the
normal pressure also may vanish outside the surface of pressure,
but inside it pressure and counterpressure may be equal;
the integral of this pressure, taken over the whole surface of
pressure, must be equal to the total pressure p fixed before-
hand; that, lastly (4) the distance between the surfaces, which
is altered by the displacements, may vanish in the surface of
pressure, and be greater than zero outside it. To express the
last condition more exactly, let 1, 71, 1 be the displacements
parallel to the axes of x, y, z in the first body, 2, 72, 2 those
in the second. In each let them be estimated relatively to
the undeformed parts of the bodies, which are at a distance
from the surface of pressure; and let a denote the distance
by which these parts are caused by the pressure to approach
each other. Then any two points of the two bodies, which