of the ThtrtHodtjnaiidc Properties of Substances. 387 



and pressure of the compound to vary, the two points of tlie primi- 

 tive surface, the line in the derived surface uniting them, and tlie tan- 

 gent plane will change their positions, maintaining the aforesaid rela- 

 tions. We may conceive of the motion of the tangent plane as pro- 

 duced by rolling upon the primitive surface, while tangent to it in 

 two points, and as it is also tangent to the derived surface in the lines 

 joining these points, it is evident that the latter is a developable and 

 forms a part of the envelop of the successive positions of the rolling 

 plane. We shall see hereafter that the form of the primitive sur- 

 face is such that the double tangent plane does not cut it, so that this 

 rolling is physically possible. 



From these relations may be deduced by simple geometrical consid- 

 erations, one of the principal propositions in regard to such com- 

 pounds. Let the tangent plane touch the primitive surface at the 

 two points L and V (fig. 1), which, to fix our ideas, we may suppose 

 to represent liquid and vapor; let planes pass through these points 

 perpendicular to the axes of v and // respect- 

 ively, intersecting in the line AB, which will be 

 parallel to the axis of e. Let the tangent plane 

 cut this line at A, and let LB and VC be drawn 

 at right angles to AB and parallel to the axes of 

 7; and V. NoAV the pressure and temperature 

 represented by the tangent plane are evidently 



AC , AB 



^r— and -j^Y respectively, and it we suppose the 



never seen the phenomenon of the coexistence of tliese two states, or of any other two 

 states of this substance. 



Equation (y) may be put in a form in wliich its vahdity is at once manifest for two 

 states which can pass either into tlie other at a constant pressure and temperature. 

 If we put p' and t' for the equivalent ;/' and t", the equation may be written 



e"— e'= f {Tj"— v')~p' {v"— v'). 

 Here the left hand member of the equation represents the difference of energy in the 

 two states, and the two terms on the right represent severally the heat received and 

 the work done when the body passes from one state to the other. The equation may 

 also be derived at once from the general equation (1) by integration. 



It is well known that when the two states being both fluid meet in a curved surface, 



instead of (a) we have p"— n'= T I 1 ), 



\ r r'J 



where r and r' are the radii of the principal curvatures of the surface of contact at any 



point (positive, if the concavity is toward the mass to which p" refers), and 7" is what 



is called the stiperficial tension. Equation (/3), however, holds good for such cases, and 



it might easily be proved that the same is true of equation (7). In other words, the 



tangent planes for the points in the thermodynamic surface representing the two states 



cut the plane t;=0 in the same line. 



Trans. Connecticut Academy, Yol. II. 33 Dec, 1873. 



