ON THE ABSOLUTE MEASUREMENT OF HARDNESS. 231 
be looked into. But ifin the above formule the impressed surface is 
replaced by the area within the circle of rupture, the empiric law stated 
above is not changed. The source of discrepancy is not, therefore, to 
be found here. 
(3) Nor does the assumption that the impressed surfaces may be rel- 
atively too large help us out of the dilemma. For in such a case the 
differences between theory and fact would vanish in proportion as R/p 
is smaller. The results do not show this. In case of glass III, for 
instance, the ratios P/D? are still enormously different, for p=4 and 
p=i2 P/ DP=83.9 and 56.4, respectively), whereas the quantity R/p has 
already decreased to 51; and 5',, respectively. 
To decrease R/p even beyond this, a new lens was made of the same 
glass with a radius as large as p=30 millimeters. In this case R/p= 
gy and P/ DP? ought therefore now either to coincide with the corre- 
sponding quantity for p=12, or at least to differ inappreciably from it. 
The data found for P/D®, however (59.6 and 56.4), are very far from 
being constant, while P/ D2/3 shows the same fixed values as above. 
(4) I may instance, in passing, that in the case of different substances 
the quotients P/D? are independent of p. Thus, for the substances 
tested the values given dimensions of lens are: Glass I, 100; glass IT, 
105; glass III, 115; quartz, 135. Hence it is possible to obtain a rela- 
tive scale of hardness which is not affected by the discrepancies here 
discussed, and therefore some certain progress has been reached, from 
a practical point of view at least. 
(5) Summarizing the above, | am bound to confess that the cause of 
the discrepancy between theory and experiment has thus far eluded 
me. <A gap must therefore be left in the theoretical side of the inquiry, 
with reference to which I would like to hazard the following sugges- 
tions: Compatibly with the relations which I have found experi- 
mentally, the last of the equations (4) leads to very different values of 
P, when different test lenses are employed. If therefore P, be termed 
the hardness of the material, the formula has no concrete meaning, 
Hence, either the stated definition of hardness must be rejected or one 
of the conditions, subject to which the equation was deduced, is not 
applicable. Saliently among these is the assumption that plate and 
lens are of the same material, and are therefore necessarily identical as 
to hardness. If this is not the case, then the hardness of one of the 
parts of the system is to be expressed in terms of the other (the equa- 
tions for this computation are of an involved character), and with the 
aid of the observed data; or equation (4) can only yield a rough value for 
the mean hardness of the system of plate and lens at best. The point 
which I am approaching is this: Even if the lens and plate be cut from 
the same homogeneous solid it does not follow that they are necessa- 
rily equally hard, for hardness may reasonably be conceived to vary 
both with the substance and with the superficial curvature of the parts 
