Page:Scientific Papers of Josiah Willard Gibbs.djvu/89

initial state a straight line be drawn in the plane perpendicular to the axis of $v$, so that the tangent of the angle which it makes with the direction of the axis of $\eta$ shall be equal to the given temperature $t'$, it may easily be shown that the vertical projections of the two segments of this line made by the point of the initial state and the surface of dissipated energy represent the two quantities required.^[1]

These problems may be modified so as to make them approach more nearly the economical problems which actually present themselves, if we suppose the body to be surrounded by a medium of constant pressure and temperature, and let the body and the medium together take the place of the body in the preceding problems. The results would be as follows:

If we suppose a plane representing the constant pressure and temperature of the medium to be tangent to the surface of dissipated energy of the body, the distance of the point representing the initial state of the body from this plane measured parallel to the axis of $\epsilon$ will represent the available energy of the body and medium, the distance of the point to the plane measured parallel to the axis of $\eta$ will represent the capacity for entropy of the body and medium, the distance of the point to the plane measured parallel to the axis of $v$ will represent the magnitude of the greatest vacuum which can be produced in the body or medium (all the power used being derived from the body and medium); if a line be drawn through the point in a plane perpendicular to the axis of $v$, the vertical projection of the segment of this line made by the point and the tangent plane will represent the greatest amount of heat which chan be given to or taken from another body at a constant temperature equal to the tangent of the inclination of the line to the horizon. (It represents the greatest amount which can be given to the body of constant temperature, if this temperature is greter than that of the medium; in the reverse case, it represents the greatest amount which can be withdrawn from that body.) In all these cases, the point of contact between the plane and the surface of dissipated energy represents the final state of the given body.

If a plane representing the pressure and temperature of the medium be drawn through the point representing any given initial state of the body, the part of this plane which falls within the surface of dissipated energy will represent in respect to volume, entropy, and energy all the states which the body can be brought by reversible processes, without producing permanent changes in external bodies (except in the medium), and the solid figure included between

1. Thus, in figure 3, if the straight line $MAN$ be drawn so that $\tan {NAC}=t'$, $MR$ will be the greatest amount of heat which can be given to the body of constant temperature and $NS$ will be the greatest amount which can be taken from it.