Mr. H. Hilton on van tier Waal** Equation, 117 



the area enclosed between it and the linos x=js 2 , .v = .r 1 , and 

 0=0 is 



It is shown in fig. 2. 



As in the case of the isothermals, so in the two families 

 treated of here, we have confined ourselves to a mathematical 

 discussion, and have left out physical considerations; as 

 before, the only part of the curves which has any physical 

 meaning is that for which x>\ 9 &>0 (and .*. y>— 27), and 

 the only part for which the physical interpretation of the 

 equation necessarily holds is that for which J?>|, 0>0 (and 



Instead of considering van der Waals' equation as repre- 

 sented by three families of curves, we may consider it 

 represented by the surface of the fourth degree, 



%^-(i/ + 80).r» + 9.c-3 = («) 



This is a ruled surface traced out by lines parallel to the 

 plane of 0, y. 



The line of striction is the intersection of (a) with the 

 cylinder 



O + 80-9) 3 +81(>-40 + 3) 2 =O. 



The plane x = meets the surface in no finite point; the 

 plane «£=g meets the surface in a line parallel to the axis of y< 

 Two sheets of the surface touch one another along the line at 

 infinity in <r=Q, which is parallel to y + 80 = O; the surface 

 also contains the line at infinity in the plane y = which is 

 parallel to # = 0j and the line at infinity in the plane #=J 

 which is parallel to y = Q. The normal at the point d J , y'. & is 



y-<( _Q-& x-x 1 



The tangent plane is therefore parallel to the axis of x when 



40V 8 =(3a'-l) a 



and 



.-. y.^ = ;W-2 and (y + 8^-9) 3 + 81(^-40' + 3) 2 =0, 



At the point (1, 1, 1) the principal radii of curvature are 



