Page:Scientific Papers of Josiah Willard Gibbs.djvu/220

and by (345) and (346)

${\frac {\epsilon }{m}}={\frac {L_{1}(c_{1}t+E_{1})+L_{2}(c_{2}t+E_{2})t^{\frac {c_{2}-c_{1}}{a}}e^{\frac {E_{1}-E_{2}}{at}}}{L_{1}+L_{2}t^{\frac {c_{2}-c_{1}}{a}}e^{\frac {E_{1}-E_{2}}{at}}}}\cdot$

(351)

By this equation we may calculate directly the amount of heat required to raise a given quantity of the gas from one given temperature to another at constant volume. The equation shows that the amount of heat will be independent of the volume of the gas. The heat necessary to produce a given change of temperature in the gas at constant pressure, may be found by taking the difference of the values of $\chi$, as defined by equation (89), for the initial and final states of the gas. From (89), (350), and (351) we obtain

${\frac {\chi }{m}}={\frac {L_{1}(c_{1}t+at+E_{1})+L_{2}(c_{2}t+at+E_{2})t^{\frac {c_{2}-c_{1}}{a}}e^{\frac {E_{1}-E_{2}}{at}}}{L_{1}+L_{2}t^{\frac {c_{2}-c_{1}}{a}}e^{\frac {E_{1}-E_{2}}{at}}}}\cdot$

(352)

By differentiation of the two last equations we may obtain directly the specific heats of the gas at constant volume and at constant pressure.

The fundamental equation of an ideal ternary gas-mixture with a single relation of convertibility between its components is

$p=a_{1}e^{\frac {H_{1}-c_{1}-a_{1}}{a_{1}}}t^{\frac {c_{1}+a_{1}}{a_{1}}}e^{\frac {\mu _{1}-E_{1}}{a_{1}t}}+a_{2}e^{\frac {H_{2}-c_{2}-a_{2}}{a_{2}}}t^{\frac {c_{2}+a_{2}}{a_{2}}}e^{\frac {\mu _{2}-E_{2}}{a_{2}t}}+a_{3}e^{\frac {H_{3}-c_{3}-a_{3}}{a_{3}}}t^{\frac {c_{3}+a_{3}}{a_{3}}}e^{\frac {\lambda _{1}\mu _{1}+\lambda _{2}\mu _{2}-E_{3}}{a_{3}t}},$

(353)

where $\lambda _{1}$ and $\lambda _{2}$ have the same meaning as on page 168.

^[1]The Conditions of Internal and External Equilibrium for Solids in contact with Fluids with regard to all possible States of Strain of the Solids.

In treating of the physical properties of a solid, it is necessary to consider its state of strain. A body is said to be strained when the relative position of its parts is altered, and by its state of strain is meant its state in respect to the relative position of its parts. We have hitherto considered the equilibrium of solids only in the case in which their state of strain is determined by pressures having the same values in all directions about any point. Let us now consider the subject without this limitation.

If $x',y',z'$ are the rectangular co-ordinates of a point of a solid body in any completely determined state of strain, which we shall call

1. [This paper was originally printed in two parts, divided at this point. For dates see heading, p. 55.]