GIBBS' PAPERS I AND II 43 



One may plot in the same diagram the isothermals from 



b^ _ Rht/a _ 1 

 a ~ V ~ (7 + 1)2' 



and the locus of the limit of stability from 



¥p 2V 1 



a (7 + 1)3 (7 + 1)2 



The table is good for any substance satisfying van der Waals' 

 equation. 



Lecture XXVI. li \}/ = e —tr], d\p = —'i]dt —pdv, and 



_ _ (^\ - ("^ ?L 



\dv / 1 \v'^ V — h 



may be integrated to find 



), 



^ = -^ - ntAog(,v -h)+^ (t), (8) 



V 



v = - (^)^ = R log (v-h)- $' (t), (9) 



e = _ ^ + $(^) -t^'{t), (10) 



V 



^•^ (I). = -'*"«• (!') 



If the volume is very great the specific heat for constant volume 

 is ordinarily constant, say c. Then —^'{t) = c log t + const., 

 and the constant may be taken as zero without loss of gen- 

 erality. Hence 



*(0 = d - d log t, (12) 



and for a substance satisfying van der Waals' equation we have 



\p = -- - Rt log {v - h) + d - d log t, (13) 



V 



7] = R\og(v - h) -{- c log t, (14) 



e = - - + d, (15) 



V 



