454 Mr. A. W. Rucker on the [June 18, 



and volume defined by the point in the plane of pv from which it is 

 drawn in only one or in several states. 



This surface will be represented by the equation 



and curves may be drawn on it showing the relations between the 

 pressure, volume, and temperature when the state of the water is altered 

 in any determinate manner, the projections on the plane of pv of those 

 represented by the equation to the surface, combined with 



forming the boundaries of the region from all points in which ordinates 

 can be drawn parallel to the axis of t which intersect the surface in two 

 or more points. The ordinary adiabatics drawn on the plane of pv are 

 the orthogonal projections of curves on the surface, each of which is 

 denned by the condition that the water in passing through the series of 

 states indicated by its successive points neither gains nor loses heat, and 

 which, to avoid confusion, will be called complete adiabatics. 



Let now the line LM in the plane of pv (fig. 1) be the line p 1 atmo- 

 sphere. Draw an ordinate from L meeting the surface in A and B ; 

 then, if different complete adiabatics pass through A and B, their pro- 

 jections on the plane of pv will intersect; and the only hypothesis on 

 which we can avoid the assumption of the intersection of adiabatics is 

 that the complete adiabatics are the intersections of the characteristic 

 surf ace f(\pvi]=0 with cylindrical surfaces, the director curves of which 

 are the plane adiabatics, and the generating lines parallel to the axis 

 of t. In this case the same complete adiabatic would pass through 

 every such pair of points as A and B, which is evidently impossible, as 

 in performing the cycle AQBPA the water would absorb heat along 

 AQB without at any time emitting it, and yet would neither increase its 

 internal energy nor perform any external work, since the cycle projects 

 into a straight line and a discontinuous curve meeting it in only one point. 

 As, therefore, a complete adiabatic cannot pass through A and B, and as 

 a similar train of reasoning would hold for the third point in which AB 

 meets the surface, three adiabatics as well as three isothermal s pass 

 through the point on the plane of pv, which is the common projection of 

 these points. 



As this conclusion disproves M. Verdet's theorem, we may proceed to 

 consider a few simple propositions based on the hypothesis of the possi- 

 bility of the intersection of adiabatics ; and in so doing it will be advisable 

 to use a new term to distinguish between two classes of points, of inter- 

 section of the projections on the plane of pv of curves on the surface ; 

 and reserving the usual expressions (intersect, cut, meet, &c.) for the pro- 

 jections of points of intersection on the surface, we shall say that two curves 

 cross one another when they meet in a point which does not correspond 



