116 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



Comparing the two methods, we observe that in one 



v = x, rj = y, e = z t (187) 



dz dz . dz dz /IQQ\ 



P- ~& t= ^' * = *-*-&*- dy y ' (188) 



and in the other 



t = x,p = y, !* = =*, (189) 



dz dz dz dz 



n=-j->v='j-> = z ^- x j~y' 



dx dy dx dy y 



Now -=- and -r- are evidently determined by the inclination of the 



y dz dz 



tangent plane, and z-j-x-j-y is the segment which it cuts off 



on the axis of Z. The two methods, therefore, have this reciprocal 

 relation, that the quantities represented in one by the position of 

 a point in a surface are represented in the other by the position 

 of a tangent plane. 



The surfaces defined by equations (187) and (189) may be dis- 

 tinguished as the v-fj-e surface, and the t-p- surface, of the substance 

 to which they relate. 



In the t-p- surface a line in which one part of the surface cuts 

 another represents a series of pairs of coexistent states. A point 

 through which pass three different parts of the surface represents a 

 triad of coexistent states. Through such a point will evidently pass 

 the three lines formed by the intersection of these sheets taken two 

 by two. The perpendicular projection of these lines upon the p-t 

 plane will give the curves which have recently been discussed by 

 Professor J. Thomson.* These curves divide the space about the 

 projection of the triple point into six parts which may be dis- 

 tinguished as follows : Let f (v} , (L \ (s) denote the three ordinates 

 determined for the same values of p and t by the three sheets passing 

 through the triple point, then in one of the six spaces 



?"<?<?, (191) 



in the next space, separated from the former by the line for which 



ML) _ S) 



, f<n < W < *>, (192) 



in the third space, separated from the last by the line for which 



in the fourth f (5) < f (L > < f< r >, (194) 



in the fifth < n < <n (195) 



in the sixth fw < f (r > < ?*>. (196) 



* See the Reports of the British Association for 1871 and 1872 ; and Philosophical 

 Magazine t vol. xlvii. (1874), p. 447. 



