Page:Scientific Papers of Josiah Willard Gibbs.djvu/154

118 upwards (in the direction in which $$\zeta$$ is measured) the states represented will be stable in respect to adjacent states. This also appears directly from (162). But where the surface is concave upwards in either of its principal curvatures the states represented will be unstable in respect to adjacent states.

When the number of component substances is greater than unity, it is not possible to represent the fundamental equation by a single surface. We have therefore to consider how it may be represented by an infinite number of surfaces. A natural extension of either of the methods already described will give us a series of surfaces in which every one is the $$v\!-\!\eta\!-\!\epsilon$$ surface, or every one the $$t\!-\!p\!-\!\zeta$$ 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 components without change of temperature or pressure, we may more advantageously make one or both of the quantities $$t$$ or $$p$$ constant in each surface.

When there are three components, the position of a point in the $$X\!-\!Y$$ plane may indicate the composition of a body most simply, perhaps, as follows. The body is supposed to be composed of the quantities $$m_{1}, m_{2}, m_{3}$$ of the substances $$S_{1}, S_{2}, S_{3}$$, the value of $$m_{1}, m_{2}, m_{3}$$ being unity. Let $$P_{1}, P_{2}, P_{3}$$ be any three points in the plane, which are not in the same straight line. If we suppose masses equal to $$m_{1}, m_{2}, m_{3}$$ 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 $$m_{1}, m_{2}, m_{3}$$. Now if for every possible phase of the components, of a given temperature and pressure, we lay off from the point in the $$X-Y$$ plane which represents the composition of the phase a distance measured parallel to the axis of $$Z$$ and representing the value of $$\zeta$$ (when $$m_{1} + m_{2} + m_{3} =1$$), the points thus determined will form a surface, which may be designated us the $$m_{1}\!-\!m_{2}\!-\!m_{3}\!-\!\zeta$$ surface of the substances considered, or simply as their $$m\!-\!\zeta$$ 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 ourselves to the consideration of the properties of the $$m\!-\! \zeta$$ surface