1387 
Now we can show: 
1. “the concentration-regions of the equilibria, which are represented 
by a point of a P,7-diagram, never coincide completely or partially.” 
2. “the sum of the concentration-regions of all equilibria, which 
are represented by a point of the ?,7-diagram, is equal to the con- 
centration-region of the invariant equilibrium.” 
We shall call these properties the rules of the concentration-regions. 
Before proving those rules, we shall first elucidate them by 
some examples. 
In the P,7-diagram of a binary system | fig. 2 (I)) we find between 
the curves (1) and (4) the bivariant equilibria 12, 23, and 34; the 
corresponding concentration-regions F\/,, FF, and F,/', do not 
cover one another, and together they are equal to the invariant con- 
eentration-region #,/. The same appears when we consider a point 
between two other curves. 
When we take a point on curve (2), then this represents, as no 
region extends itself over this curve, the monovariant equilibrium 
(F,) = F, + F, + F, only; the concentration-region of this equili- 
brium is again the invariant region / #,. When we take a point 
on curve (3) then this represents the equilibrium (/,)=/,+F,+ £;; 
as the region of this equilibrium is again the line 4’, /’,, consequently 
no region is allowed to extend over curve (3). 
Over curve (1) indeed a region has to extend, this is apparent 
from the following. A point of this curve represents the equilibrium 
(F,) = F, + F, + F,, the concentration-region of which is indicated 
by the line F, F,. Consequently over curve (1) another region has 
to extend itself, the concentration-region of which is represented by 
the line #’, F,, therefore, the region 12. 
In the same way it is apparent that over curve (4) still the region 
34 has to extend. 
Let us consider now a ternary system, of which the concentration- 
and the P,7-diagram are represented by the figs. 1 (II) and 2 (II). 
The concentration-region of the invariant equilibrium is here the 
quintangle of fig. 1 (11). A point between the curves \L) and (2) 
represents the three bivariant equilibria 134, 245, and 345. It appears 
from fig. 1 (II) that those equilibria cover one another neither com- 
pletely nor partly, and that they form together the quintangle. The 
same is true for the points between the other curves. 
It follows also at once that over each of the curves a region has 
to extend itself. A point of curve (1) represents viz. the monovariant 
equilibrium (1) = 2 +3 +4 +5, the region of which is represented 
in fig. 1 (11) by quadrangle 2534. As the sum of the regions of 
