VI
179
ON HARDNESS
as to the force necessary to drive in another point. (2) Since
finite and permanent changes of form are employed, elastic
after-effects, which have nothing to do with hardness, enter
into the results of measurement to a degree quite beyond
estimation. This is shown only too plainly by the introduc-
tion of the time into the definition of Crace-Calvert and
Johnson, and it is therefore doubtful whether the hardness of
bodies thus measured is always in the order of the ordinary
scale. (3) We cannot maintain that hardness thus measured
is a property of the bodies in their original state (although
without doubt it is dependent upon that state). For in the
position in the experiment the point already rests upon per-
manently stretched or compressed layers of the body.
I shall now try to substitute for these another definition,
against which the same objections cannot be urged. In the
first place I look upon the strength of a material as determined,
not by forces producing certain permanent deformations, but
by the greatest forces which can act without producing de-
viations from perfect elasticity, to a certain predetermined
accuracy of measurement. Since the substance after the action
and removal of such forces returns to its original state, the
strength thus defined is a quantity really relating to the original
substance, which we cannot say is true for any other definition.
The most general problem of the strength of isotropic bodies
would clearly consist in answering the question-Within what
limits may the principal stresses X, Y, Z, in any element lie
so that the limit of elasticity may not be exceeded?
If we
represent X, Y, Z, as rectangular rectilinear coordinates of a
point, then in this system there will be for every material a
certain surface, closed or in part extending to infinity, round
the origin, which represents the limit of elasticity; those values
of X, Y, Z, which correspond to internal points can be borne,
the others not so. In the first place it is clear that if we
knew this surface or the corresponding function (X, Y,
Z₁) = 0 for the given material, we could answer all the
questions to the solution of which hardness is to lead us. For
suppose a point of given form and given material pressed
against a second body. According to what precedes we know
all the stresses occurring in the body; we need therefore only
see whether amongst them there is one corresponding to a