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systems, the way vvliicli we heave followed then [viz. with the aid 

 of the graphical representation of the i\-- and the ^-function] is not 

 appropriate however to be applied to systems with more components. 

 The following method is much simpler and leads to the result 

 desired for any system. 



We consider an invariant point with the phases F^, F„ . . . Fn+i 

 and two of the curves starting from this point, viz. (i^,) = F, -\- 

 + /; + ... F„+2 and {F,) = F, -\- F, . . . Fn+2. (see fig. 1). Between 

 the stable parts of these curves the region {F^F^) ^ F^ -\- F^ -\- . .. F„^2 

 is situated. When we consider stable conditions only, this region 

 terminates at the one side in cur\'e [F^), at the other side in curve 

 (F,). Now it is the quesliou in which of the two angles (F,) (F^) 

 the region (i'\i<\) is situated, viz. in the angle which is smaller or 



in the angle which is larger than 180°. 



The first case has been drawn in fig. 1 



in the latter case the region (F^F^) 



should extend itself over the metastable 



parts of the curve (F^) and {F.^). We 



call the angle of the region (F^F^) in 



the point o the region-angle of (i^,/*^,) ; 



we can prove now : "a region-angle is 



Fig- 1- always smaller than 180°." 



In order to prove this we imagine in fig. 1 the region {F^F^) in 



the angle {F,) o {F^), which is larger than 180\ The stable part 



of this region then extends itself on both sides of the metastable 



part of curve {F,) and also of (F,). This now is in contradiction 



with the property that the stable part of each region, which may 



arise from a curve, is situated only at one side of this curve. Hence 



it follows, therefore, tliat the region-angle must be smaller than 180°. 



Therefore, when we will draw in ng. 1 the region (F^F,), this 



must be situated in the angle {F^) (F^), which is smaller than 



180°. As in fig. 1 {F,) and {F^) are drawn on difi"erent sides of 



(i^i); the regions {F^F^) and (F^F^) fall outside one another; when 



we had taken (F,) and (F,) on the same side of (F,), the two 



regions should partly cover one another. 



Another property is the following: every region, which extends 

 itself over the metastable or stable part of a curve (F^) contains the 

 phase Fp, or in other words: each region which is intersected b}' 

 the stable or the metastable part of a curve (F^) contains the phase 

 Fp. In an invariant point the n-\-2 phases F, F, . . . F,-j-2 occur; 

 consequently arround this point | (« -\- 2) {n -}- 1) bivariant regions 

 extend themselves. In n-f-l of these regions the phase Fj is wanting, 



