States, and the Thermodynamic State- Equation. 155 



obtained complicated implicit relations connecting (v l5 p) 

 and (v 2 , p). A better method is due to Planck *, who 

 calculated the two-phase relations for C0 2 from Clausius' 

 state-equation (5) above. The three equations here under 

 consideration all give, by simple calculation, an implicit 

 relation of Vi and v 2 . Planck introduces an independent 

 variable yjr, by putting 



Vi — /3=q, cos 2 i/r/2, 



v 2 —/3=q. sin 2 ^/2. 



Substituting these values for i\, v 2 , an equation is obtained 

 which gives q explicitly in terms of yfr ; hence v 1? v 2 , and so 

 p, T are all obtained as functions of ty. The problem is thus 

 solved. The functions are, however, complicated, and are 

 not worked out beyond the equation giving q in terms of sjr. 



The method here adopted was suggested by that of Planck 

 and is essentially similar to it. Working with the algebraic 

 reduced equations, the independent variable, or parameter* 

 t is introduced, by writing : 



<h-l=q(l + *)/(2 + 0. 

 $,-l= ? /(2 + t). 



The rest of the calculation follows the same lines as in 

 Planck's method. The relations obtained are, however, 

 comparatively speaking, very simple, and have been worked 

 out completely for all three equations. In addition to p, v ly 

 t< 2 , T, the quantities 



M pvi M pv^ dlogp M r 



R ' T ' R * T ' <nogT' R ' T 



(where r is the Latent Heat), have all been calculated as 

 functions of t. It will be observed that each of those 

 quantities is of zero dimension and should therefore have the 

 same values at corresponding states in all substances. 

 Particularly simple expressions are obtained for Mr/RT and 

 dlp/dlT, which is of interest, since the former is the well- 

 known Trouton's constant, divided by R = 1*985, and the 

 latter is closely related to Crafts' correction constant for 

 boiling-point under different atmospheric pressures. 



The resulting functions are all tabulated on p. 156. For 

 purposes of abbreviation, the following symbols are used : — 



L = log c (l + 0- W^-L. W 2 = (1 + 0L-*. 



X = (2 + L -^- Y = t 2 -(l + t)L 2 . Z = t(2 + t)-2(l + t)L. 

 * Wied. Ann. xiii. p. 535 (1881). 



