Page:Scientific Papers of Josiah Willard Gibbs.djvu/74

plane will change their positions, maintaining the aforesiad relations. We may conceive of the motion of the tangent plane as produced by rolling upon the primitive surface, while tangent to it in two points, and as it is also tangent to the derived surface in the lines joining these points, it is evident that the latter is a developable surface and forms a part of the envelop of the successive positions of the rolling plane. We shall see hereafter that the form of the primitive surface is such that the double tangent plane does not cut it, so that this rolling is physically possible.

From these relations may be deduced by simple geometrical considerations one of the principal propositions in regard to such compounds. Let the tangent plane touch the primitive surface at the two points $L$ and $V$ (fig. 1), which, to fix our ideas, we may suppose to represent liquid and vapor; let planes pass through these points perpendicular to the axes of $v$ and $\eta$ respectively, intersecting in the line $AB$, which will be parallel to the axis of $\epsilon$. Let the tangent plane cut this line at $A$, and let $LB$ and $VC$ be drawn at right angles to $AB$ and parallel to the axes of $\eta$ and $v$. Now the pressure and temperature represented by the tangent are evidently ${\frac {AC}{CV}}$ and ${\frac {AB}{BL}}$ respectively, and if we suppose the tangent plane in rolling upon the primitive surface to turn about its instantaneous axis $LV$ an infinitely small angle, so as to meet $AB$ in $A'$, $dp$ and $dt$ will be equal to ${\frac {AA'}{CV}}$ and ${\frac {AA'}{BL}}$ respectively. Therefore,

${\frac {dp}{dt}}={\frac {BL}{CV}}={\frac {\eta ^{\prime \prime }-\eta ^{\prime }}{v''-v'}},$

where $v'$ and $\eta ^{\prime }$ denote the volume and entropy for the point $L$, and $v''$ and $\eta ^{\prime \prime }$ those for point $V$. If we substitute for $\eta ^{\prime \prime }-\eta ^{\prime }$ its equivalent ${\frac {r}{t}}$ ($r$ denoting the heat of vaporization), we have the equation in its usual form, ${\frac {dp}{dt}}={\frac {r}{t(v''-v')}}\cdot$

the work done when the body passes from one state to the other. The equation may also be derived at once from the general equation (1) by integration. It is well known that when states being both fluid meet in a curved surface,

instead of (α) we have $p''-p'=T\left({\frac {1}{r}}+{\frac {1}{r'}}\right),$

where $r$ and $r'$ are the radii of the principal curvatures of the surface of contact at any point (positive, if the concavity is toward the mass to which $p''$ refers), and $T$ is what is called the superficial tension. Equation (β), however, holds good of such cases, and it might easily be proved that the same istrue of equation (γ). In other words, the tangent planes for the points in the thermodynamic surface representing the two states cut the plane $v=0$ in the same line.