322 J. TF. Gibhs 0)1 Graphical Methods in the 



The constant of integration l)ecomes 0, if we call the entropy for 

 the state of which the volume and energy are both unity. 



Any other equations Avhich subsist between v, p, t, e and // may be 

 derived from the three independent equations (a), (b) and (d). If we 

 eliminate £ from (b) and (d), we have 



1] •= a log V -{• c log t -\- c log c. (e) 



Eliminating » from (a) and (e), we have 



If z=z {a-\-c) log t — a log ]) -\- e log c -|- a log a. (f) 



Eliminating t from (a) and (e), we have 



7/ =z (« + c) log V -\- C log jt) -|- C log — . (g) 



If?; is constant, equation (e) becomes 



' 7]-=.c log t + Const., 



i. e., the isometrics in the entropy-temperature diagram are logarith- 

 mic curves identical with one another in form, — a change in the value 

 of V having only the effect of moving the curve parallel to the axis of 

 //. If p is constant, equation (f) becomes 



7] = (a-\-c) log t -f Const., 



so that the isopiestics in this diagram have similar properties. This 

 identity in form diminishes greatly the labor of drawing any consid- 

 erable number of these curves. For if a card or thin ])oard l)e cut in 

 the form of one of them, it may be used as a pattern or ruler to draw 

 all of the same system. 



The isodynamics are straight in this diagram (eq. b). 



To find the form of the isothermals and isentropics in the volume- 

 pressure diagram, we may make t and /; constant in equations (a) 

 and (g) respectively, which will then reduce to the well-known equa- 

 tions of these curves : — 



2W =z Const., 

 and i/ v"''^'^ = Const. 



The equation of the isodynamics is of course the same as that of the 

 isothermals. None of these systems of lines have that property of 

 identity of form, which makes the systems of isometrics and isopies- 

 tics so easy to draw in the entropy-temperature diagram. 



the last note on page 310. It will be observed, that there would be no real loss of 

 generality if we should choose, as the body to which the letters refer, such a quantity 

 of tlie gas that one of the constants a and c should be equal to unity. 



