38 REPRESENTATION BY SURFACES OF THE 



plane will change their positions, maintaining the aforesaid relations. 

 We may conceive of the motion of the tangent plane as produced 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 surface 

 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 

 surface 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 

 considerations one of the principal propositions in regard to such 



compounds. Let the tangent plane touch the pri- 

 mitive surface at the two points L and V (fig. 1), 

 which, to fix our ideas, we may suppose to repre- 

 sent liquid and vapor; let planes pass through 

 these points perpendicular to the axes of v and r\ 

 v respectively, 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 rj and v. Now the pressure and temperature represented by 



AC AB 



the tangent plane are evidently p^ and ^- respectively, and if we 



suppose the tangent plane in rolling upon the primitive surface to 

 turn about its instantaneous axis LV an infinitely small angle, so 



AA' AA' 



as to meet AB in A 7 , dp and dt will be equal to 



respectively. Therefore, 



dt~CV~v"-v" 



where i/ and rf denote the volume and entropy for the point L, 

 and v" and if those for the point V. If we substitute for rf rj 



T 



its equivalent - (r denoting the heat of vaporization), we have the 

 c 



equation in its usual form, -77 = ^* K- 



dt t(v v) 



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, 



/ 1 1\ 



instead of (a) we have p"-p'= T ( - + ~. ) , 



\r r J 



where r and / 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 T is what 

 is called the superficial tension. Equation (), 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 v=0 in the same line. 



