VI
173
ON HARDNESS
equally distributed over the surface of pressure. Besides Z,
the remaining two principal tensions at the origin can be
expressed in a finite form. It may be sufficient to state that
they are also pressures of the same order of magnitude as Z,,
and are of such intensity that, provided the material is at
all compressible, it will suffer compression in all three direc-
tions. When the curve of pressure is a circle, these forces
are to Z, in the ratio of (1+4℗)/2(1+20) : 1; for glass
about as 5/6:1.
5/61. The distribution of stress inside depends
not only on the form of the ellipse of pressure, but also
essentially on the elastic coefficient ; so that it may be
entirely different in the two bodies which are in contact.
When we compare the stresses in the same material for the
same form but different sizes of the ellipse of pressure and
different total pressures, we see that the stresses at points
similarly situated with regard to the surface of pressure are
proportional to each other. To get the pressures for one case
at given points we must multiply the pressures at similarly
situated points in the other case by the ratio of the total
pressures, and divide by the ratio of the compressed areas.
If we suppose two given bodies in contact and only the
pressure between them to vary, the deformation of the
material varies as the cube root of this total pressure.
It is desirable to obtain a clear view of the distribution
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FIG. 19.
of stress in the interior; but the formulæ are far too compli-
cated to allow of our doing this directly. But by considering
the stresses near the z-axis and near the surface we can form
a rough notion of this distribution. The result may be
expressed by the following description and the accompanying
diagram (Fig. 19), which represents a section through the axis of