\ , (17) 



Vapour-pressure and the Volumes of Vapour and Liquid. 387 

 and (14), we get 



27 [\-2 + (\ + 2>-^](l-2\(g-^-g-^) 



8 (l-6- A )(X-l + e- A ) 2 (l-e- A -Xe- A ) 2 



By equations (13), (14), (15), and (17) what was aimed at 

 is attained, namely to express the four quantities w } W, II, 

 and 9 by one and the same quantity X. 



§ 4. If we expand the expressions found in series which 

 proceed by powers of X, we encounter a peculiar behaviour : 

 in nearly all the factors which occur in the numerators and 

 denominators the terms which are independent of X and those 

 affected with low powers of X cancel one another, so that all 

 the numerators and denominators have pretty high powers of 

 X for factors, which, indeed, then cancel one another in the 

 fractions. The respective series presenting the factors are as 

 follows : — 



l- e -^=x( 1 -i\+ J_ \2__1_ x s + !_■ 



! T 3! 4! T 5! 





X 4 - 



X-2 + (X + 2>-* = X 3 ( 1 -^ X+ |j X 2 - A X^ + 

 l-2X*--*-=2X*( _!_ _ 4_ x+ 11 x2 _ |6 xi+ 57 ^_ 



(1 _^ )2 _^-^x^(^-| I x +3 ^x 2 -gx3 +i ^ ] x^ 



Applying these expressions to the equations (13) and (14), 

 and effecting in these the indicated multiplication and divi- 

 sion, we obtain 



W 



=7 ( 2+x+ ^. + _1_V + 2 -^* + j-^X«+...) (19) 



From this we see, what can also be otherwise demonstrated 

 to be necessary, that the terms with even powers of X are equal 

 in the two expressions, and the terms with odd powers are 

 equal and with opposite signs. Hence we can introduce two 

 new quantities, M and N, containing only even powers of X, 



