48 



2. The quantity f. Relations hetioeen f, r, and s. 



When neither a nor h is a function of T, it follows from (1) that 



and so also 



fdp\ R 1 / a\ 



T (dp\_ RT _ a 



p \dTJy p (v—b) ' • pv'' 



At the e.-i.ieal point (Q = (!;) = 



dp 

 = — , so that there 



Tu f dp \ RTjc vie 



Pk \.dTjk Pl{vIc — hk) v]c — hk' pkv\- 



After substitution of the value s from (5), j)/,. from (4), and vic = rbk, 

 we get : 



r 8 ;^ 27 1 



f=s- = *, or /-! = —- .... (6) 



. «' fK 



If in the second relation according to (5) we substitute rrl ^ 



for — , then 



*^ = --^ (7) 



s^ 64 A,^' 



As r no longer occurs in this, Van ukr Waat.s was justified in 



ƒ— 1 27 

 saying that this equation derived by him, viz. — —=i~, would be 



almost entirely exact, whereas the two others, viz, (5) and (6) with 

 r':=Vk-bg, would only be true by approximation. But as we already 

 observed above, when not r' but r = vk : l^k is taken, these latter 

 equations too will hold with the same degree of accuracy as (7). 



As, when association is disregarded, >j = 0,977 and P.^ = 0,980; 

 whereas, when it is taken into account, on an average )-^ = 1,007, 

 ;., = 1,0J3, the ratio X, -. X, will be =: 1 — 0,003 or =1 — 0,006 

 in (5) and (6), i.e. it may be put equal to unity. For X, -. X^"^ we 

 find 1,026 or 1,001, so that this ratio approaches unity still more 

 closely than P.j : \ on assumption of association, but with disregard 

 of it will not differ more from unity than 2 or 3 7o- Finally 1 : X^ 

 in the second equation (6) will remain either 2°/^ above, or about 

 17o below unity. 



The value 2,11 is found for r from (5) with .s- = 3,774, while 

 the second equation (6) with ƒ ^ 7 also yields the value 2,11. 

 Further 6- : (/—I) = 14,24 : 6 = 2,37, and also 64 : 27 = 2,37; so 



