588 Mr. H. Hilton on van der Waals 9 Equation. 



of the point on (S) corresponding to this value of is 



hence the tangent to (S) at point (J ; 0) makes an angle 



< tan -1 



27 



Jl)*j 



= 



with the axis of x ; i. £., it touches the axis of x at this point. 



The curve (S) is shown in fig. 1. 



The relation between the reduced temperature and volume 

 at the boiling-point is found by eliminating y between (a) 

 and (3) ; it is : — 



3 j(166> + 27> 2 — 36a; + 9 



Of Q , 



6\6x 



V 



(27-32fl)c? 3 + 9^ 2 -3^-l 

 3.?;-l 



| | - Vq&0 + 27x 2 - 12# + 1 

 + (3*-!)^/^ ^ 



x ^ f(27-16^ 2 -3 + 3(3a;-l) ^ 27 -^^ 



1~~ (3.i?-l)(16^f 2 -18cB + G) ) 



between x = ^ and x = l ; and the same with the sign of 

 the radical changed between x=l and x = ^o . 

 When y is so large that the equation 



3y% s -(y + 89)x 2 + 9x-3 = 



has only one real root, this root is (cf. Works on Theory of 

 Equations) 



X r?/ + 80 3 /i ( ^~ 



2y\/n$y + 54(^-202/^- 80*) +(y + 80) 3 } + \J\ { 18y(y-4»J 



+ ^(2/ + 8^) 3 -2y V7292/ + 54(^-20^-8^) + 



b+m>}] 



If we assume that this expression, which is the real root of 

 x when y is large, by the principle of continuity represents 

 the smallest root of the equation when y becomes so small that 

 the equation has three real roots, we obtain the relation be- 



