34 KEPRESENTATION BY SURFACES OF THE 



we will call the thermodynamic surface of the body for which it i 

 formed.* 



To fix our ideas, let the axes of v, rj, and e have the directions 

 usually given to the axes of X, Y, and Z (v increasing to the right, 

 tj forward, and e upward). Then the pressure and temperature of 

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

 tangents of the inclinations of the surface to the horizon at that 

 point, as measured in planes perpendicular to the axes of r\ and of v 

 respectively. (Eqs. 2 and 3.) It must be observed, however, that 

 in the first case the angle of inclination is measured upward from 

 the direction of decreasing v, and in the second, upward from the 

 direction of increasing tj. Hence, the tangent plane at any point 

 indicates the temperature and pressure of the state represented. It 

 will be convenient to speak of a plane as representing a certain 

 pressure and temperature, when the tangents of its inclinations to 

 the horizon, measured as above, are equal to that pressure and 

 temperature. 



Before proceeding farther, it may be worth while to distinguish 

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

 formed. The position of the plane v = Q in the surface is evidently 

 fixed, but the position of the planes ij = 0, e = is arbitrary, provided 

 the direction of the axes of r\ and e 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\ and e = for any 

 state of the body which we may choose, we may place the origin of 

 co-ordinates at any point in the plane v = 0. Again, it is evident 

 from the form of equation (1) 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 equation 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 would 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 con- 

 siderations enable us to foresee to a certain extent the nature of the 

 general properties of the surface which we are to investigate. They 



* Professor J. Thomson has proposed and used a surface in which the co-ordinates 

 are proportional to the volume, pressure, and temperature of the body. (Proc. Roy. 

 Soc., Nov. 16, 1871, vol. xx, p. 1 ; and Phil. Mag., vol. xliii, p. 227.) It is evident, 

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

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

 volume, entropy, and energy. For, while 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. 



