519 
When the two indifferent phases have opposite sign, then the 
singular equilibrium J/ is not transformable. 
Let the composition of the invariant equilibrium be: 
A, Hi + see + Joly Fo +- Arti Ht =f see + An+e Prize 
in which A,, A, ete. indicate the quantities of the phases, they are 
consequently positive. When in this equilibrium reaction (1) takes 
place, then the composition becomes: 
(A Kai) Fy +... + (Ap—Ka,) Fy + (Api — Kaa) Bit... 9) 
in which K is arbitrary. In order that (9) may represent the singular 
equilibrium M, the coefficients of the singular pbases /, and 44 
must be zero and the coefficients of the other phases must be positive. 
The latter condition can always be satistied; for this it is only 
necessary to take large enough the quantities A,, A, ete. of those 
phases. The first condition gives: 
Ai Fs Ay 
Cpl kn Ap 4 
As A, and 4,4, are both positive, a, and a,41 must have, therefore, 
the same sign. Moreover it appears that the invariant equilibrium 
may be converted into the equilibrium J/ only then when tbe 
quantities A, and 4, of the indifferent phases are in the ratio of 
ap to a. 
A, — Kap =0 and A,41 — Kap —= 0 consequently 
Now we shall put the question, which P,7-diagramtypes answer 
those particular cases. We consider an invariant point with the 
phases F,....#,42, in which HW, and F4 are the indifferent phases 
and the other ones the singular phases. 
Then we have the singular equilibria: 
M= f+... + Mit Boe +…+ Pipe 
(Ep) == F, + ze a ee “bE es + Fore ed ane a Pie =e = Fo 
BH) =d Bat + Bed + Fie == M + Ho 
When M is no singular equilibrium, no phase-reaction may occur, 
therefore, between its phases, then it is the bivariant equilibrium 
WF) which is represented ina P,7-diagram by a region between 
the curves f, and F141. 
When JM is indeed a singular equilibrium, so that between its 
phases a phase-reaction can occur, then it is represented in the 
P,T-diagram by a curve (M). 
Now we may show that the 3 singular curves (J/) (/,) and (47,44) 
coincide, This coincidence may take place in two different ways. 
