446 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Volume on the Equilibrium Curves 535 



Approximate Compressibility 536 



Discussion of the Results 538 



For the Liquid 538 



The Nature of the Abnormalities of Water already Known . . 540 

 Compressibility and Dilatation of Water over the Range of this 



Paper 541 



Table of Volume for Equal Intervals of Pressure and Tem- 

 perature 539 



Change from an Abnormal to a Normal Liquid 541 



The IManner of Transition, Particularly below 0° . . . . 543 

 Comparison of Present Data with Values Extrapolated from 



Previous Formulas, Probable Behavior at High Pressures . . 548 



Difference of Specific Heats 550 



Change of Internal Energy on an Isothermal 551 



Adiabatic Compressibility and Temperature Effect of Com- 

 pression 552 



For the Change of State Liquid-Solid 553 



The Present Theories of Liquid-Solid; the Points at Issue . . 553 

 The General Shape of the Various Equilibrium Curves in their 



Bearing on this Question 554 



The New E\'idence given by the VI-L Curves and the Probable 



Course of the Curve at Higher Pressures 555 



Bearing on van Laar's Recent Theory 556 



For the Change Solid-Solid 557 



The Compressibility of "Water. 



A complete thermodynamic knowledge of any substance over any 

 range of temperature and pressure is afforded by a knowledge of the 

 characteristic equation of the substance over that range (that is, the 

 relation connecting volume with temperature and pressure), and a 

 knowledge of the specific heat along some curve of the pressure- 

 temperature plane not an isothermal. For water, the specific heat is 

 known in its dependence on temperature at atmospheric pressure, that 

 is, along a line not an isothermal, so that theoretically all that is 

 needed for a complete thermodynamic knowledge of water is the 

 knowledge of the characteristic equation. This knowledge is evidently 

 given by the change of volume with pressure along various isothermals, 

 together with the change of volume along some line not an isothermal. 

 Practically this means that the characteristic equation may be found 

 by finding the isothermal compressibility at several different tempera- 

 tures and combining with the known dilatation at atmospheric pressure. 



It does not follow, however, that all the thermodynamic data are 

 given with equal accuracy by the knowledge of these quantities. Thus 



