661 



it is apparent from ^2) that we may also satisfy (6). Now follows 

 from (5) in connection with (6): 



X, [ii, A.r, + ^, Ay, + ...] + ;., [^t, A.r, + ^,Ay, -|- . . .] -f ; 



+ X„ [fi, A.r„ + n. Ay, + ...]:= \^'^ 



Now we have in (7) the equation sought for; for the ratios 



between A, . . . A„ we have yet to substitute their values from (2) 



and for the ratios between ft, , . . i^i,, still their values from (6). 



dP 

 In order to calculate—^ we add the n equations (3) after having 

 dl 



multiplied the 1^^ by X^, the[^2"<^ by /, etc. By using (2) and (4) 



we find : 



— JS" (A F) . A P+^ (;. /f ) . A r -f i ^ (;. d'Z) + i ^ (;i d'Z) + ..=0 (8) 



or: 



dP ^ 2{XH) 



dT :s{xV) 



Herein is : 



2aH) = X, H, -\-).,H, -{-... -\- x„ H„ 

 the increase of entropy, and 



2:(xV) = x,V,-^x,\\-^...-\- Xn Vn 

 the increase of volume, which occur at the phase-reaction 



X^F,+X,F,^...^Xr,Fn = 



The direction of the tangent to a turning-line Er is, therefore, 

 defined in eacli point by (9) consequently by the same conditions 

 as a system of n components in n -j- 1 phases. It appears from (9) 

 that this curve has a point of maximum or minimum-temperature 

 when the phase-reaction proceed swithoutchangeof volume [^(^ F'j^OJ; 

 it has a point of maximum-pressure, when ^ {X H) = 0, consequently 

 when no heat is taken up or given out with the phase-reaction. 



Now we shall examine whether a singular point may occur on 

 the turning-line; then AP and LT have to be of higher order. 

 For this purpose it is necessary that we are able to give the value 

 zero to LP and A 7' in (3) and (4) without all other increments 

 Lx^ l\y^ . . . becoming zero also. 



Consequently we must be able to solve from : 



x,d{x\ + yAy\ + ... = — A^j 



x,d{x\ + yAy\ ■+-... = — LK\ 



etc. and from : 



d{x), = d{x), = = d{a:)n = AiTx I 



d(y\ = d{y\ = .... = diy)n = LKy \ 



Proceedings Royal Acad. Amsterdam. Vol. XX. 



(10, 



• . (11) 



48 



