GIBBS' PAPERS I AND II 29 



(p", v") forms a curve which we call the critical locus. If the 

 temperature is high enough there will be no condensation. It 

 has been seen that f is constant for the rectilinear portion of the 

 isothermal including its extremities which lie upon the critical 

 locus. 



For any path connecting these two limiting points (p', v') and 

 {p", v") with p' = p" upon the isothermal t the total change of 

 f must be nil. Now 



6" - e' = fdt = fdQ - fpdv, 



n" - V = fdQ/t, 



p"v" - p'v' = fipdv + vdp). 



If the second equation be multiplied by —t' = —t" and the 

 three be added 



(c" - t"y)" + p"v") - W - t'v' + p'v') 



= fdQ - t' fdQ/t + fvdp = 0. 



Hence for any path joining the two points 



/ ^-^^ dQ -\- vdp = 0. 



In particular if the path be taken as a line v = v' rising 

 above the critical point to p = p" ', a line p = p'" to the value 

 V = v", and finally the Hne v = v" to p = p" (the three lines 

 forming three sides of a rectangle of which the straight por- 

 tion of the isothermal is the base), the value of fvdp is 

 {v" — v') {p' — p" ') and thus for this path 



/ 



^—-^dQ + {v"-v'){p' -p'") =0. 



6 



We have seen that pv = aMs a law satisfied within wide 

 limits. The law 



a at 



V = -,+ 



1.2 



V — b 



proposed by van der Waals, reduces essentially to pv = at when 

 V is large and is found to be an improvement on that equation 



