quaternary systems four different positions [figs. 1, 3, 5 and 7 (III)]. 
As to each of these different positions a definite P, 7-diagram 
belongs, we found for binary systems one |fig. 2 (I)], for ternary 
systems three [figs. 2, 4 and 6 (ID) and for quaternary systems four 
(figs. 2, 4, 6, and 8 (III) different types of P,7-diagrams. 
As we can no more represent the concentration-diagram for sys- 
tems with more than four components (unless in a space with more 
than three dimensions) we can no more apply in the same way the 
method which we have followed till now. 
Yet we may deduce, as we shall see further, for each arbitrary 
system the different types of P,T-diagrams. Before discussing this 
question we shall first indicate in what way we can deduce in 
each definite case the corresponding P, 7-diagram. 
We consider a system of 7 components, in the invariant point of 
which the n+ 2 phases F,.../,42 occur. The 7 +2 monovariant 
curves (/,)...(/,42) start from this point. When the compositions 
of the 7+ 2 phases are known, then, as we have seen in commu- 
nication I, the reactions, which oecur in each of the monovariant 
equilibria (/)...(/,42) are completely detined. 
We write for the reaction between the phases of the equilibrium (F’): 
a,F, -+-a,P,+...+-:Gi4eMye=0. . . . (4 
for the reaction of the equilibrium (/,): 
Oy Pigg te sa One nts 0 4 ns ara (5) 
for the reaction of the equilibrium (4): 
eF, Heb, Hel, +... teypehi42=0. . . (6) 
etc. As in all n+ 2 monovariant equilibria occur, consequently 
we have also n + 2 equations of reaction. 
These „+ 2 equations, however, are not independent of one 
another, but they are mutually dependent, viz. in this way that two 
of them are sufficient to define the others. When we know for 
instance the equations of reaction for the equilibria (#,) and (F,), 
then we find them for (/’,), by eliminating /', from both the first. 
Therefore, when we eliminate /’, from (4) and (5), then we find (6). 
In order to find the equation for the equilibrium (#,), we eliminate 
F, from (4) and (5); ete. 
Consequently we find: when we know the equations of reaction 
for 2 of the 7 + 2 equilibria (7)... Ue), then those for the n 
other equilibria are also known. 
We may express this also in the following way: we can represent 
