232 ON 'THE ABSOLUTE MEASUREMENT OF HARDNESS. 
at the point of examination. Clearly, in such a case, hardness would 
increase with the curvature at the point. 
Evidence in favor of this surmise is already available, seeing that 
whenever the above experiments are carefully planned and executed, 
rupture always occurs in the plate, while the lens remains intact. 
Hence, in proportion as the lens is more convex it is also harder, and 
the value of the mean hardness of the system obtained from equation 
(4) must therefore increase with the lens curvature. This is what the 
experiments actually indicate. Pursuing this suggestion further, it 
follows that the equation expressing the hardness of the lens will be 
b 
Re 
where « is the hardness of a plane surface of the given material (a 
constant which might be called intrinsic hardness), and b the ecurva- 
ture constant, or, as it might be called, surface hardness. The close 
analogy between ) and the surface tension of liquids is obvious ata 
glance. 
As a second suggestion, I should like to propose a change of Hertz’s 
definition of hardness. Hertz’s characteristic contains three elements; 
it is (1) a pressure, (2) its direction, Z., is normal, and (3) it refers to 
the center of the impressed surface. 
Since the criterion in case of brittle bodies is the occurrence of rupt- 
ure, 7. €., a Separation of parts, the immediate cause can not be pressure 
but tension. Furthermore, the crack passes from the surface z = 0, not 
quite normally perhaps, but nearly so, into the interior; and hence it 
is not Z, but an oblique pressure, indeed alinost a lateral pressure, X,, 
which is pre-eminently active. Finally since the crack encircles the 
area of contact, the component XY, is here to be inserted. Unfortu- 
nately the complexity of the formule is such that a full solution of the 
problem can not be obtained for this case; they show however that 
AX, reaches itS maximum negative value on the outside of the surface 
of contact, and that the maximum is differently related to the lens 
curvature from the normal pressure. Obviously the latteris dependent 
on curvature in two dimensions, the other on the curvature in a single 
dimension. In short, even if the experiment leads to different values 
of (Z,)max according as different lens curvatures apply, it does not fol- 
low that these different values may not all correspond to one and the 
same (Xx)max- Perhaps these considerations may even be put more 
clearly by calling to mind that the maximum pressure on the surface 
produces no appreciable effect in this surface at all; its action, however, 
An allied analogy is given by the tensile strength of iron wire, which, according 
to Baumeister (Wied. Ann., vol. 18, 1883, p.578), is greater in proportion as the thick- 
ness of the wire is smaller. I have found that the law here is / = const. 7a, or 
identical with the above relations. Some exceptions may reasonably be taken to 
all of these points. 
a 
=. 
S: 
a ee a ae 
ate 
