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extreme w-values, so 1 and 4, then the two-phase region 1 + 4 
will occupy all the available width in the spacial figure; this region 
forms a space which has the full width of the four-phase line as 
boundary. So with the same pressure and temperature no other 
stable two-phase equilibrium is possible there. The two other two- 
phase equilibria 1 +2 and 2-++4, which lie by the side of the 
three-phase line 1 +2 +4, and the equilibria 1 + 3 and 3+ 4, 
which lie by the side of the line 1 + 3 + 4, lie therefore always 
on the other side of the lines in question in the P,-7 projection. 
So in fig. 5 the situation of the region 
1 + 4 determines that of the two three- 
phase lines AO and BO, and at the same 
time that of the regions 1 +2, 2 +4, 
1 +3, and 3+ 4. So it now remains to 
decide what the situation is of the two 
remaining three-phase lines. It is now easy 
to see that the line OC lying on the right 
must represent the coexistence of 1 +2 +8 
Fig. 5. and the line OD that of 2 +3 +4. The 
line OC, namely, must bound on one side either the region 1 + 3 
or the region 3-+ 4. This can only take place by the three-phase 
line 1+2-+ 3, because in the other case besides 3 + 4, also the 
region 2+.3 would have to lie on the same side of the three- 
phase line, which can evidently not be the case. So now, the 
situation of the phases is quite determined. So it appears that one 
two-phase equilibrium occurs in the region AOS, two in the regions 
BOC and DOA, and three in COD. 
Now the angle AVS must contain the metastable prolongations 
of the two three-phase lines CO and DO. Suppose namely, that the pro- 
longation of CO should fall in DOA, then the region 1 + 2 should 
present an angle which is greater dan 180°; if the prolongation of 
DO lay in COB, then the region 3 + + would possess an angle 
greater than 180°. So it has been proved that only such a situation 
is possible that no prolongation falls in the angle COD. And this 
proves the stated rule. 
It will, moreover, be clear from the above proof, that the thesis 
might also be stated as follows : 
If the phases, arranged according to their z-values, are expressed 
by 1, 2, 3, and 4, the angle without metastable prolongations lies 
between the three-phase lines 1 + 2 +8 and 2 +3 +4. 
9. The application of this rule can naturally be twofold. At 
