176 J. W. Gibhs — Equilibrium of Heterogeneous Substances. 



which every one is the v-7]-e surface, or every one the t-p-l surface for 

 a body of constant composition, the proportion of the components 

 varying as we pass from one surface to another. But for a simultaneous 

 view of the properties which are exhibited by compounds of two or 

 three comj^onents without change of temperature or pressure, we may 

 more advantageously make one or both of the quantities t or p con- 

 stant in each surface. 



Surfaces and Curves in tchich the Composition of the Body repre- 

 sented is Variable and its Temperature and Pressure are Constant. 



When there are three components, the position of a point in the 

 J^I^plane may indicate the composition of a body most simply, per- 

 haps, as follows. The body is supposed to be composed of the quan- 

 tities ?7«j, //ig, i^a '^^ tlie substances ^S*,, /S'g, S^^ the value of m^ -(" 

 mg + mg being unity. Let Pj, P^, P3 be any three points in the 

 plane, which are not in the same straight line. If we suppose masses 

 equal to m^, mg, m^ to be placed at these three points, the center of 

 gravity of these masses will determine a point which will indicate 

 the value of these quantities. If the triangle is equiangular and has 

 the height unity, the distances of the point from the three sides will 

 be equal numerically to «?j, m,, m^. Now if for every possible 

 phase of the components, of a given temperature and pressure, we 

 lav off from the point in the X- Y plane which represents the compo- 

 sition of the phase a distance measured parallel to the axis of Z and 

 representing the value of C (when ni^-\-n)2-\-'mQ=.\), the points 

 thus determined will form a surface, which may be designated as the 

 mj-mg-^Vg-C surface of the substances considered, or simply as their 

 m-t, surface, for the given temperature and pressure. In like manner, 

 when there are but two component substances, we may obtain a 

 curve, which we will suppose in the X-Z plane. The coordinate y 

 may then represent temperature or pressure. But we will limit our- 

 selves to the consideration of the properties of the m-X, surface for 

 w =r 3, or the m-l curve for n =z 2, regarded as a surface, or curve, 

 which varies with the temperature and pressure. 



As by (96) and (92) 



and (for constant temperature and pressure) 



d'Q = f.1^ dm J -f- yWg ^^'^2 + /^3 dm^, 

 if we imagine a tangent plane for the point to which these letters 

 relate, and denote by l' the ordinate for any point in the plane, 



