We may write— — as -• ( - — ). Now the dimensions of 1/x are 



COMPRESSIBILITY OF METALS. 221 



square centimeter, is seen to be 10~ 5 . The dimensions of this number 

 are those of the reciprocal of a pressure. 



The thermal expansion also decreases numerically with increasing 

 pressure in nearly all the cases above, but there are at least two sub- 

 stances in the list above which give a negative change of thermal 

 expansion with pressure that seems to be beyond the experimental 

 error. These are germanium and uranium. Unfortunately the ex- 

 pansion at atmospheric pressure of these substances does not seem 

 to have been determined, so that the proportional change could not be 

 computed, and blanks were accordingly left in the third column of 

 Table IV. The other blanks in this column are also due to missing 

 data, either my own, or that of expansion at atmospheric pressure. 

 Again the order of magnitude of the change of expansion with pressure 

 is 10- 5 . 



The physical significance of the fifth and sixth columns is easy to see. 



1 *x as 1 (I d * 



X 2 dp ' ' X \X d P 

 those of a pressure, and numerically 1/x is the pressure required to 



1 3x . 

 halve the volume. Hence — — is the proportional change of com- 



X 2 dp 



pressibility under a pressure that would halve the volume. Similarly 



llda. ., . . 



— is the proportional change of thermal expansion under a 



pressure which would halve the volume. Both of these quantities are 

 dimensionless, and the most immediately interesting thing about 

 them is their order of magnitude, which is that of a small number. 

 That this would be found to be the case was anticipated some years 

 ago by the theory of the solid state of Griineisen. 19 Griineisen's 

 theory went farther, and gave an exact expression for this number in 

 terms of the exponents in the assumed law of attraction and repulsion 

 between the atoms. This exact expression does not seem, however, 

 to fit at all well, and it is most doubtful whether the details of Griinei- 

 sen's theory can be maintained. The fact that these two ratios are 

 of the order of magnitude of a small number would probably be given 

 by a number of simple theories, simply because they are dimensionless. 

 The relative magnitude of the changes of compressibility and ex- 

 pansion with pressure is of some interest. The considerations which 

 I have already mentioned in the case of potassium would lead one to 

 expect that at very high pressures the thermal expansion becomes 

 vanishingly small. But if one plots the above values, ruling out those 

 metals which do not crystallize in the cubic system, it will be found 



