JJ9 



In order to define the direction of tliese curves in tlie /*, 7-diagram, 

 we may use the following thesis ') : the systems which are formed 

 on addition of heat at an isovolumetrical reaction exist at higher 

 — those which are formed on withdrawal of heat exist at lower 

 temperatures. The systems which are formed on decrease of volume 

 at an isentropical reaction exist under higher — those which are 

 formed on increase of volume exist under lower pressures. 



Let us considei' now tlie equilibrium (F^) := F.^ -\- F,-^ . . . Fn-\-2, 

 which is represented in tig. 1 by curve {F^) at a temperature 7^ 

 and under a |n'essure P„, which are i-epresented t)y the point (/. 

 On addition of heat under a constant pressure oi' on change of 

 volume at a constant temperature a reaction, which is completely 

 defined, occurs between these n-\^i phases. Let us write this reaction: 



y. f. + !U /'; + ...//„ + 2 i^\, +2 = (5) 



The n relations between the n. -\- 1 reaction-coefticients are fixed 

 then by the n equations (2) in which, however, we must omit all 

 terms which refer to the phase F^, [consequently y^, (,/,',),, (,/■,"), etc.]. 



Now we let reaction (5) occur until one of Ihe phases of the 

 equilibrium {F,) disappears; then an eqiiilil)riiiin of n phases 

 arises, which is consequently bivariant. In all // -|- 1 bivariant 

 equilibria can arise from the equilibrium (F^). As in each of these 

 equilibria two of the phases of the invariant point are wanting, we 

 represent a bivariant equilibrium by putting between parentheses 

 the failing phases. (F^F,) represents consequently the equililirium 

 F, -\~ F, -{■... F„-^-2. B'rom the equilibrium ( F^). therefore, the 

 bivariant equilibria {F^F^), {F^F,) . . . {F,F„^2) may arise in the 

 manner, which is treated above. 



In a bivariant equilibrium P and 7' can be considered as inde- 

 pendent variables; each bivariant equilibrium can, therefore, be 

 represented in the 7^, 7'-diagram by the points of the plane of this 

 diagram, consequently by a region. 



C'Onsequenlly n -{- 1 bivariant regions, which may arise from the 

 equilibrium (7*\), go through each monovariant curve (T*',). Each of 

 these regions is divided into two parts b}' the curve (7*^,) ; the one 

 part represents stable conditions, the other metastable conditions. 

 When we limit ourselves to the stable parts of these regions, we 

 may say : in a system of n components n -{- 1 bivariant regions 

 start from each monovariant curve. 



1) F. A. H. ScHREiNEMAKERS. Heterog. Gleichgewichte von H. \V. Baichuis 

 RoozEBOOM. HI': vvc find liercin the proofs for lürnary systems on p. 12:20—221 

 aud 298 — 301. These, however, arc also true tor systems of n components. 



