404 J. W. Gibhs on a Representation by Surfaces, etc. 



the body and medium) ; if a line l)e 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 can be given to or taken 

 from another body at a constant tempei'ature equal to the tangent of 

 the inclination of the line to the horizon. (It represents the great- 

 est amount which can be given to the body of constant temperature, 

 if this temperature is greater 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 case, it represents the greatest amount which can be withdrawn from 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, tlie point of contact between the 

 plane and the sui-face 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 dis- 

 sipated energy will represent in respect to volume, entropy, and 

 energy all the states into which the l)ody can be brought by reversi- 

 ble processes, without producing permanent changes in external 

 bodies (except in the medium), and the solid figure included between 

 this plane figure and the surface of dissipated energy will represent 

 all the states into which the body can be brought by any kind of pro- 

 cesses, without producing permanent changes in external bodies 

 (except in the medium).* 



* The body under discussion has been supposed throughout tliis paper to be homo- 

 geneous in substance. But if we imagine any material system whatever, and suppose 

 the position of a point to be determined for every possible state of the system, by 

 making the co-ordinates of the point equal to the total volume, entropy, and energy of 

 tlie system, the points thus determined will evidently form a solid figure bounded in 

 certain directions by the surface representing the states of dissipated energy. In 

 these states, the temperature is necessarily uniform throughout the system ; the pres- 

 sure may vary (e. g., in the case of a very large mass like a planet), but it will always 

 be possible to maintain the equilibrium of the system (in a state of dissipated energy) 

 by a uniform normal pressure applied to its surface. This pressure and the uniform 

 temperature of the system will be represented by the inclination of the surface of dis- 

 sipated energy according to the rule on page 38S. And in regard to such problems as 

 have been discussed in the last five pages of this paper, this surface will possess, rela- 

 tively to the system which it represents, properties entirely similar to those of the sur- 

 face of dissipated energy of a homogeneous body. 



