eT A 4 eekly ary 
ON THE ABSOLUTE MEASUREMENT OF HARDNESS. 225 
Preliminary optical work showed that the departure from homogenity 
throughout the mass of glass examined was nearly inappreciable, and 
the same was true of the given faces of the quartz appliances. Had it 
not been the case, the tests themselves would, in the course of the work, 
have indicated deficient isotropy, seeing that both the position or the 
Shape of the lines of rupture depend on these conditions. Such results 
were indeed actually obtained in certain experiments which I made with 
this especial end in view, but with which I will not further detain the 
reader, for another question, and an important one, has since loomed 
into view and must now be answered. The theory sketched above 
makes mention only of isotropic media, and thus it is not warrantable 
to apply it to crystals. 
In a measure this is true, but it must be noticed that in the first of 
the equations (4) only the numerical coefficient is influenced by :eolo- 
tropy, and if equations (1) and (2) can be proved to hold for the erys- 
talline body empirically, then the last of the equations (4) can be wrong 
only as to its coefficient. Furthermore, the uncertainty can be tested 
by means of the equation (3), as compared with the second of the equa- 
tions (4), to a very small margin of uncertainty, by inserting known 
values of the elastic constant KE’. Aside from this an interpretation of 
the coefficient in question shows that it is necessarily inclosed within 
narrow limits. There is still another point of view. Hertz’s theory is 
true for elliptical contact surfaces quite as much as for spherical sur- 
faces, from the nature of the reasoning employed; and even in the 
more general case (ellipsoid) the numerical factor in questions turns 
out to be 3/2. If therefore the latter is independent of differences of 
direction considered geometrically, it will also be independent of the 
elastic assymmetry, a consideration, it is true, which applies primarily 
for lines lying in the plane surfurce of the plate, but does not apply to 
the normal dimensions or depths. 
Tam bound to acknowledge, therefore, that the data obtained with 
erystals are possibly not as accurate as the corresponding data for 
isotropic substances. This curious divergence in the behavior of erys- 
tals and glasses is borne out by the following qualitative result: 
Whereas impressed area and the line of rupture of an isotropic body is 
always a circular, only the impressed area retains this figure for quartz, 
while the lines of rupture is a figure midway between a circle and a 
hexagon. I will return to this matter elsewhere. The radii of curva- 
ture ofthe lenses used were widely varied, and the values 1,3, 4, 5, 10, 
12,15, and 30 ™™. enter the following experiments, the first and _ last, 
however, only in special work. If the radius is too small there is obvi- 
ous difficulty in measuring the area of contact. Large radii, on the 
other hand, are equally unsatisfactory. The initial point-contact is 
not always attainable in this ease, while the stress which must ulti- 
mately be brought to bear is a serious tax on the apparatus. Again, 
H. Mis. 334, pt. 1 15 
