ON THE ABSOLUTE MEASUREMENT OF HARDNESS. 213 
classified with the allied properties encountered in case of tension, tlex- 
ure, ete. It is of course necessary to go into further detail, in par- 
ticular to determine how pressure is distributed and varies within the 
surface of contact, for upon these conditions the effects of stress and 
the resistance of the material will depend. The solution of this problem 
has enabled Hertz* to propound a fundamental principle. In his 
attempt to verify his theory experimentally Hertz was however much 
less successful, and as a consequence soon abandoned the work. The 
only data which he adduced refer to glass, and his results for hard- 
ness were: 
Kg. /mm?. 
Pressure of a hard steel lens against plate glasse........--!-..:.....--..---2-- 185 
DY SE ola egty tog ale ae of | ee a ee ae eee. - SNe ane: fe 150 
PSLOSSUNGOD, UV Onumlns OLASS MOCKS ce io fc Soiete os aio HS eis oe ean eee as 190 
Thus the data obtained are not satisfactorily constant. Moreover, 
my results show that not more than the third or fourth part of the dis- 
crepancies observed are referable to the material. Differences, there- 
fore, necessarily remain. It would be inexpedient to attempt to account 
for them here, chiefly because the number of experiments made is much 
too small relatively to the conditions (form, material, stress, impact, 
ete.), under which the results were obtained. Nor has Hertz given a 
sufficiently detailed statement of the dimensions of the bodies exam- 
ined. 
II. THEORY. 
The pressureless contact between a sphere and a plane is a point. 
If pressure be applied at the center of the sphere, normally, both sur- 
faces will change form near the point in question, until the strain has 
reached a given value. In other words, the sphere will be flattened 
and the plane curved, and the original point is now replaced by a sur- 
face of contact. ! shall call this the impressed surface or area (Druch- 
fliiche). Itis neither plane nor of the curvature of the sphere; but the 
radius will obviously lie somewhere between these limiting values, and 
will depend (cet. par.) on the elastic properties of the two contiguous 
bodies. Furthermore, under the conditions stated, the impressed area 
is clearly cireumscribed by a circle. 
If pressure acting normally through the center of the sphere is 
increased the impressed surface will also increase in size, and the pres- 
sure is now brought to bear ona larger surface. But the strain to which 
the materialis put will depend on the stress per unit of impressed surface, 
and we are thus led to inquire as to the law compatible with which the 
pressure per unit of area increases with the total pressure, for ob- 
viously both magnitudes must increase simultaneously. It is also 
easily seen that the relation between total pressure and pressure per 
unit of the impressed surface is closely allied with the relation of total 
“Hertz: Crelle’s Journal, 1882, vol. xcut, p. 156. 
