- ON THE ABSOLUTE MEASUREMENT OF HARDNESS. 215 
in any manner whatever from the former. The variations possible in 
such a case are twofold: (1) The spheres may have different radii, and. 
(2) the bodies may have different elastic constants than those which 
obtained in the first experiment. The theory of the experiment shows, 
with regard to the first of these points, that (other things being eaual) 
the radius of the impressed area is proportional to the cube root of the 
radius of the sphere, or that the area of the surface of contact varies 
as the two-thirds power of the radius. For the case of equal total 
pressures at the center of the latter, the pressure per unit of area, and 
hence also the maximum pressure in the impressed surface, must be 
proportional to the cube root of the curvatures of the sphere. To the 
extent, therefore, in which all reference is made to the stated central or 
maximum pressure (per unit of area), the data for limiting values of 
elastic resistance must be independent of the curvature of the impress- 
ing sphere. Hence the limiting value of total pressure is proportional 
to the square of the limiting or final radius of the area of contact; or, 
if the radius of the latter is expressed in terms of the total pressure 
and the radius of the sphere by aid of the above relations, then the 
value of total pressure, just sufficient to produce set, must increase with 
the square of the radius of the sphere. In regard to the second of the 
above queries, no special inention is expedient here. I will only remark 
that under conditions which are otherwise identical, the area of con- 
tact is expressible in terms of values of the elastic constants of the two 
contiguous bodies. To avoid this complication, I will at the outset 
confine myself to the state of things observed when both bodies are 
identical as to material. For this case the relations to be formulated 
admit of simple expressions. 
It may be worth while, by way of recapitulation, to express the laws — 
just enunciated symbolically. Let p be the radius of curvature of the 
sphere in millimeters, p the total pressure applied at its center, P its 
superior limit, 7. ¢., the value of p at the time of occurrence of the per- 
manent set. Let p,; be the pressure per unit of area at the center of 
the impressed surface, ¢. ¢., the maximuin of pressure in kilograms per 
square millimeter, P; the superior limit of p,, 7. e., the absolute hard- 
ness of the body. Let d be the diameter of the area of contact (this 
quality is immediately given by observation, and is in so far preferable 
to the radius embodied in the above text), D the superior limit of d, 
both in millimeters. Let H be the true hardness, which, as will be 
shown in the sequel, differs slightly from the theoretic value P,, q an 
abbreviation of the quotient p/d*, Q its limiting value. Let / be the 
area of contact, Fits limiting value, both in square millimeters. Fin- 
ally, let # be the modulus of elasticity of the material in kilograms per 
square millimeter, 4 Poisson’s coefficient, 7. e., the ratio of radial con- 
traction to longitudinal extension, H’ an abbreviation of the quotient 
H/ (1-)°). Brackets may serviceably be used to show that the quan- 
