J. ^V. (xibhs — Equilibrium of Heterogeneous ISuhsfances. 17;5 



of a constant quantity of a substance are represented by rectangular 

 co-ordinates. This method corresponds to the first kind of limda- 

 raental equation described on pages 140-144. Any other kind of 

 fundamental equation for a substance of invariable composition will 

 suggest an analogous geometrical method. Thus, if we make m con- 

 stant, the variables in any one of the sets (99)-(103) are reduced to 

 three, which may be represented by rectangular co-ordinates. This 

 will, however, afford but four different methods, for, as has already 

 (page 150) been observed, the two last sets are essentially equivalent 

 when n ■= \. 



The method described in the preceding volume has certain advan- 

 tages, especially for the purposes of theoretical discussion, but it may 

 often be more advantageous to select a method in which the proper- 

 ties represented by tM'o of the co-ordinates shall be such as best serve 

 to identify and describe the different states of the substance. This 

 condition is satisfied by temperature and pressiire as well, perhaps, as 

 by any other properties. We may represent these by two of the 

 co-ordinates and the potential by the third. (See page 143.) It 

 will not be overlooked that there is the closest analogy between these 

 three quantities in respect to their parts in the general theory of 

 equilibrium. ( A similar analogy exists between volume, entropy, and 

 energy.) If we give m the constant value unity, the third co-ordinate 

 will also represent C, which then becomes equal to /<. 



Comparing the two methods, we observe that in one 



v = x, i] — y, € = z, (187) 



dz dz ^ dz dz ,^^^ 



and in the other 



t-z.x^ p=-y, i.i^'C,z=.z, (189) 



dz dz dz dz , ^ 



uX clx 



Now ^— and ^— are evidently determined by the inclination of the 

 dx dy 



(Txi (XX 



tangent plane, and z — -^ x — -^y is the segment which it cuts ofi" 



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. 



