THERMODYNAMIC AL SYSTEM OF GIBBS 169 



(since de/drj, de/dv define the slope of the tangent plane at D). 

 Thus 



EE' = Ae - At' 

 d^e ■' dh O'e 



= ^^^^^ + d";s;^^^^ + ^^^^^' 



The expression on the right of this equation is positive when 

 dh d^e / dh \2 dh dh 



(the last condition is a consequence of the other two), so that 

 when these conditions are fulfilled E Hes above E'. Thus the 

 conditions which were obtained above signify that a phase is 

 stable with respect to continuous changes, when the rj-v-e 

 surface for adjacent phases Ues above the tangential plane at the 

 point representing the phase in question, except at the single 

 point of contact. 



It is often more convenient to use other sets of quantities as 

 the independent variables. Thus if we employ t, v, Wi, nh, 

 . . .rrin as independent variables, we have when dt = and 

 dm„ =. 0,* 



dp dp dp 



dp = -rdv+T-dm^... + 7-— dmn-i 



dv drrii ' ' ' dm„-i 



dni dyL\ dni 



dui =» -7- dv + ~ — dnii . . . + J drtin-i, 



dv drrii am„_i 



dun-i = ~3 — dv + — — ami . . . + :; drrin-i; 



dv dmi dm„-i 



whence, when dt = 0, dp = 0, dfxi = 0, . . . d^n-i = 0, 



Pn 



> (252) 



/ dHn-l \ 



Xdmn-i/t.v,^,, 



lin-2,mn t^n-\ 



(253) 



* In order that every variation considered shall represent a real 

 change of phase, it is necessary to make one of the quantities v, nii, m-i, 

 . . .ron constant. 



