of States of Aggregation, 203 



may be taken to represent the state of the air at the given 

 moment (fig. 1, PI. VIII.). Suppose this construction repeated 

 for all values of p, v; then the geometrical locus of all the 

 points on which the extremity J can lie is a curved surface, 

 which for shortness may be called the " temperature-surface." 

 To every point of this surface corresponds a particular state of 

 the air, since a particular value of each of the variables p, v, T 

 is given when the position of the point is given. 



Through the point J draw a plane parallel to the vertical 

 plane OYZ; the temperature-surface is cut by this plane in 

 a straight line whose inclination e to the plane of pv is given 

 by the equation 



, dT v /Q , 



tan6= dp = W < 3 > 



in which v is to be considered constant. 



Again, draw a plane through J parallel to the vertical plane 

 OXZ ; its line of intersection with the surface is also straight, 

 its inclination being given by 



bmm= § = w < 4 ) 



wherein p must be taken constant. 



Thus the lines of constant volume and the lines of constant 

 pressure on this curved surface form two systems of straight 

 lines. The temperature-surface can therefore be conceived as 

 the geometrical locus of all the intersections of these two sys- 

 tems of lines. 



If, finally, we draw a horizontal plane through the point J, 

 the temperature-surface will be cut by it along a curved line 

 which represents a line of constant temperature (as it corre- 

 sponds to the equation T= const.), and may therefore be 

 termed an "isothermal" (fig. 2). The equation of these iso- 

 thermals may also be given in the form 



pv= const., (5) 



whence it appears that an isothermal lies in a horizontal plane 

 and is a rectangular hyperbola. 



If we consider the temperature-surface as the face of a 

 mountain, the isothermals will be represented by curved hori- 

 zontal paths along its slope, while the lines of constant volume 

 and the lines of constant pressure are straight paths leading 

 directly up the slope. Each given alteration in the state of 

 the air may then be looked upon as a movement over the moun- 

 tain by a given path whose successive points represent the 

 successive states through which the air passes. 



