Tliermodynamic Pfoperties represented by Surfaces. 383 



To fix our ideas, let the axes of u, //, and e have the directions usu- 

 ally given to the axes of X, Y, and Z (y; increasing to the right, i] 

 forward, and s upward). Then the pressure and temperature of the 

 state represented by any point of the surface are equal to the tan- 

 gents of the inclinations of the surface to the horizon at that point, as 

 measured in planes perpendicular to the axes of ?/ and of v respect- 

 ively. (Eqs. 2 and 3). It must be observed, however, that in the 

 first case the angle of inclination is measured upward from the direc- 

 tion of decreaning y, and in the second, upward from the direction of 

 increasing if. Hence, the tangent plane at any point indicates the 

 temperature and ])ressure of the state represented. It will be conve- 

 nient to speak of a plane as representing a certain pressure and tem- 

 perature, when the tangents of its inclinations to the horizon, meas- 

 ured as above, are equal to that pi'essure and tem})ei'ature. 



Before proceeding farther, it may be worth while to distinguish 

 between what is essential and what is arbitrary in a surface thus 

 formed. The ])osition of the plane y=0 in the surface is evidently 

 fixed, but the position of tlie planes ?/=0, f:=0 is arl)itrary, provided 

 the direction of the axes of // and s be not altered. This results from 

 the nature of the definitions of entropy and energy, which involve 

 each an arbitrary constant. As we may make ?/r=0 and 6z::0 for any 

 state of the body which we may choose, we may place the oi-igin of 

 co-ordinates at any point in the plane v=zO. Again, it is evident from 

 the form of equation (I) that whatever changes we may make in the 

 units in which volume, entropy, and energy are measured, it will 

 always be possible to make such changes in the units of temperature 

 and pressure, that the ecpiation will hold true in its present form 

 without the introduction of constants. It is easy to see how a change 

 of the units of volume, entropy, and energy woidd affect the surface. 

 The projections parallel to any one of the axes of distances between 

 points of the surface would be changed in the ratio inverse to that in 

 which the corresponding unit had been changed. These considera- 

 tions enable us to foresee to a certain extent the nature of the o-ene- 

 ral properties of the surface which we are to investigate. They must 

 be such, namely, as shall not l)e aftected by any of the changes men- 

 tioned above. For example, we may find properties which concern 



however, that the relation between the vohime, pressure, and temperature affords a 

 less complete knowledge of the properties of the body than tlie relation between the 

 volume, entropy, and energy. For, wliile the former relation is entirely determined by 

 the latter, and can be derived from it by differentiation, the latter relation is by no 

 means determined by the former. 



