230 ON THE ABSOLUTE MEASUREMENT OF HARDNESS. 
the less pronouncedly in harmony with the relations just adduced. 'This 
appears in the following summary : 
| Values. p= if 4 ale} Mean values. 
We ec or | i 2 
| J Theoretically con- ee Reese | bu) | iM kaa 
Glass! seas! | >\P/p2 )  RCERY (OM eM lotrel VOL GRT” PPR se cAkosoocsccc 
| | Rian | Dip 0.240 | 0,153 Os103. |<) 
| | )P/D32=| 54.4 | 55.4 53.5 | 54.4 £0,8 
| Glass I atl ROUTE EO ek ees Wes | 6.37 6.59 | 635 | 6.44 40:5 
|, stant. J pse— | o117 | O19 | 0)119\ |. lolis 20-001 
( . ee 1998 | 952° 66.5" = ee eee 
Quartz ....4| Dheoretically con |\pj,2— || 50,5. 7 |") ag |) Odo) ||iee see eee 
| stant. | D/p= } 0.188) ||) 0.08.) 170.070, 0/3 ite eee 
aes ana ee apeaneene eee 2—=| 64.3 | 65.8 65.2 65.1 +0.3 
| Quartz .--- | stant. Pip | 50.5 | 52.2 50. 5 5.11 +0. 04 
| | D3125— | 0. 0786 | 0. 0798 0.0775 | 0.0785 +0. 0004 
Here, as in the above case, the probable errors are between $ and 
1 per cent. 
On the basis of these results it follows, therefore, that if hardness be 
computed by the last of the equations (4), the Hertzian values, P, will 
vary with p. If, however, these data (P,;) are multiplied by yD, or, 
more conveniently, by 3 Vp, then the new values of hardness are con- 
stant qualities, irrespective of the curvature of the stylus used. In 
general, furthermore, the theoretical premises have been corroborated 
by experiment to a remarkably close degree of accordance; only in 
one point (and this happens to be the most important deduction) is 
there a wide divergence between predictions of the theory and the 
facts. Inasmuch as the disagreement evidences a well-defined law, it 
is worth while to examine the conditions under which the theory 
applies. 
(1) Hertz supposes the area of contact to be small relatively to the 
spherical surface. In the above experiments, however, it is quite 
doubtful whether this can at once be assumed in all cases. Indeed, the 
ratios of the limiting radius of the impressed area Rk and the lens 
radius preach values as high as 1 : 11, and they can not be at once dis- 
missed. Weare thus led to inquire in how far the theoretical state- 
ments, relatively to pressure direction and pressure components, curva- 
tures and area of the impressed surface, are affected by the large values 
R/p specified. I have done this and find, ina way which has already 
been suggested in the above text, that the theory still holds to a degree 
quite within the errors of experiment, at leastin the majority of obser- 
vations. The fixed values of q, moreover, is compatible with this result, 
for in the case of increasing loads q is pronouncedly constant when 
is smallest. 
(2) Again, the interesting fact that the locus of rupture surrounds 
the area of contact and is situated at a certain distance from it, may 
