VI
175
ON HARDNESS
p infinite, if the pressure per unit length of the cylinder is
to be finite. We then have in the second of equations (I)
B = (P₁+P₂). Further, we may neglect u compared with.
a², take a outside the sign of integration, and put for the
indeterminate quantity p/aco/∞ an arbitrary finite constant,
say p'; then, as we shall see directly, p' is the pressure per
unit length of the cylinder. The integration of the equation
can now be performed, and gives
b= p'(9, +9)
1
π(P₂+P₂)
For the pressure Z, we find
Z₂
=
πb2
-
The
and it is easy to see that p' has the meaning stated.
distance of approach a, according to our general formula, be-
comes logarithmically infinite. This means that it depends
not merely on what happens at the place of contact, but also
on the shape of the body as a whole; and thus its determina-
tion no longer forms part of the problem we are dealing with.
I shall now describe some experiments that I have per-
formed with a view to comparing the formulæ obtained with
experience; partly that I may give a proof of the reliability
of the consequences deduced, and their applicability to actual
circumstances, and partly to serve as an example of their
application. The experiments were performed in such a way
that the bodies used were pressed together by a horizontal
one-armed lever. From its free end were suspended the
weights which determined the pressure, and to it the one body
was fastened close to the fulcrum. The other body, which
formed the basis of support, was covered by the thinnest
possible layer of lamp-black, which was intended to record the
form of the surface of pressure. If the experiment succeeded,
the lampblack was not rubbed away, but only squeezed flat;
in transmitted light the places of action of the pressure could
hardly be detected; but in reflected light they showed as
small brilliant circles or ellipses, which could be measured
fairly accurately by the microscope. The following numbers
are the means of from 5 to 8 measurements.