PvE ON THE ABSOLUTE MEASUREMENT OF HARDNESS. 
pressure to the increase of the impressed surface or area of contact. 
Now Hertz’s theory shows the radius of the latter to increase propor- 
tionally to the cube root of the total pressure applied, and hence the 
impressed area will increase as the two-thirds power of total pressure. 
To this degree, therefore, the effect of total pressure is abortive; and in 
view of the enlargement of the impressed area stress per unit of area 
increases only as the cube root of the total stress. Furthermore, the 
manner in which pressure is distributed throughout the surface of con- 
tact is fully given by the theory. It is found that at any given time 
pressure decreases gradually from the center of the area towards its 
boundary where stress is necessarily zero, in accordance with the 
expression 
Vi=7 
where x is the fraction of the total radius of the impressed area by 
which any of its points is symmetrically located relatively to the cen- 
ter. The reference roughly made above to pressure per unit area is, 
therefore, of the nature of a mean value; and the maximum pressure at 
the center of area is related to the mean value here in question in the 
ratio of 3 to 2. Now if the total pressure at the center of the sphere is 
gradually increased, the maximum pressure per unit of area at the cen- 
ter of the impressed surface will also continually increase; and at a 
certain value one of the two bodies, or both (supposing them to be made 
of the same material), will necessarily reach the limits of elasticity. 
Hyvidence as to whether this has occurred or not is not far to seek; ina 
plastic body the strain will be permanent, There will, in other words, 
be an evidence of “set,” for the parts affected fail to return to their 
original positions when the stress is relieved. Furthermore, in a brittle 
body, set will be actually accompanied by rupture at the parts too 
highlystrained. Wemay therefore in all instances conclude ay follows: 
The least value of the (central) pressure per unit of area necessary to pro- 
duce permanent set (or rupture) at the center of the impressed surface is 
Hertz’s datum for the hardness of the body under examination. In 
addition to the normal pressures every point of the area of contact is 
also actuated by lateral pressures, and it is quite feasible to obtain 
Some general notion of their value. At the center of contact they are 
positive, 7. e., the body is uniformly compressed, whence it follows that 
in our method of testing a crack is not to be looked for here. The case 
is pronouncedly different near the boundary of the area, where the lat- 
eral stresses are all negative and of the nature of tensions; and since 
the loci of like stresses are circles concentric with the center of area, 
we may look for a circular line of rupture. 
Thus far our considerations were only extended to a system of two 
given bodies in contact. The question arises how the condition will 
change if the original system is replaced by a second system differing 
