398 PROCEEDINGS OF THE AMERICAN ACADEMY § lO 







The left hand term in the parenthesis is essentially posi- 

 tive; the right hand term essentially negative, since />► 

 /'> o and 4 >> r. When / becomes very great, the left 

 hand term disappears in comparison with the other; as 

 / — r approaches zero the right hand term becomes infi- 

 nite, so that at both extremes the curvature is concave 



T 



upwards. Between the two, it is evident that, if -— - is 



sufficiently small, the right hand term will become less 

 than the left hand, and the curvature will become convex 

 upward. There will therefore in general be two middle 

 points where the curve is perfectly straight. 



From I. we see that when /= /', ^= 00, that is, the 

 curve becomes parallel to the axis of pressures with an 

 infinite downward slope; when / increases indefinitely, 

 the curve approaches the axis of volumes as an asymp- 

 tote, and the slope becomes positive before it vanishes 

 owing to the disappearance of the second term in parenthe- 

 sis in comparison with the first. Between these values, 



T 



if is sufficiently small, the slope evidently becomes 



negative somewhere, and in changing to negative and then 

 to positive again, must twice pass through the value zero. 

 Hence there will be in general two points, D and jF, 

 where the curve is parallel to the axis of volumes. When, 

 however, T is increased, it is at last impossible for the 

 slope to become negative; there must be some value for T 

 when the slope reaches the value zero without crossing it; 

 for we may put ^ and its derivative simultaneously equal 

 to zero, and the isothermal in question fulfils all the con- 

 ditions which should hold at the critical temperature. 



It is easy to show that all the curves in the neighbor- 

 hood of the critical temperature have the general charac- 



