APPENDIX D. 



NOTE ON THE CORRECTION FOR THE COMPRESSIBILITY OF THE 



PIEZOMETER. 



The usual correction neglects the fact that when the compressibility of the liquid 

 is different from that of the walls, the liquid under pressure does not occupy the same 

 part of the vessel as before pressure. 



Let V be the volume of the part of the vessel occupied by liquid ; a that of the 

 tube between the two positions of the index, both measured at 1 atmosphere ; e, e, the 

 average absolute compressibility of liquid and vessel per ton for the first p additional 

 tons. Equate to one another the volume of the liquid, and the volume of the part of 

 the vessel into which it is forced, both at additional pressure p. We have thus — 



Y{\-ep) = (Y-a)(\-< P ) 

 whence 



"•(>-t) + ,w 



As ^ is usually small, this equation is treated as equivalent to 



a 



j,Y 



i.e., the absolute compressibility of the liquid is equal to its apparent compressibility, 

 added to the absolute compressibility of the envelop. 



One curious consequence of the exact equation is that, if the compressibilities were 

 both constant, or were known to change in a given ratio by pressure, it would be 

 possible (theoretically at least) to measure absolute compressibilities by piezometer 

 experiments alone, without employing a substance whose absolute compressibdity is 

 determined by an independent process. For the additional term in the exact equation 

 makes the coefficients of e and e numerically different ; whereas in the approximate 

 equation they are equal, but with opposite signs, and therefore can give e — e only. 



In my experiments described above, a/Y rarely exceeds 0'02, so that this correction 

 amounts to (0-02 x 26 in 500, or) 5 units in the fourth significant place ; and thus 

 just escapes having to be taken account of. When 4 places are sought at lower 

 pressures than 3 tons, or 3 places at pressures of 4 tons and upwards, it must be taken 

 account of. 



