Page:Scientific Papers of Josiah Willard Gibbs.djvu/142

remaining case, in which the phase can be varied without altering the value of (133) can hardly be expected to occur. The phase concerned would in such a case have coexistent adjacent phases. It will be sufficient to discuss the condition of stability (in respect to continuous changes) without coexistent adjacent phases.

This condition, which for brevity's sake we will call the condition of stability, may be written in the form

$\epsilon ''-t'\eta ''+p'v''-\mu _{1}'m''_{1}...-\mu _{n}'m''_{n}>0,$

(142)

in which the quantities relating to the phase of which the stability is in question are distinguished by single accents, and those relating to the other phase by double accents. This condition is by (93) equivalent to

$\epsilon ''-t'\eta ''+p'v''-\mu _{1}'m''_{1}...-\mu _{n}'m''_{n}-\epsilon '-t'\eta '+p'v'-\mu _{1}'m'_{1}...-\mu _{n}'m'_{n}>0,$

(143)

and to

$-t'\eta ''+p'v''-\mu _{1}'m''_{1}...-\mu _{n}'m''_{n}+t''\eta ''+p''v''-\mu _{1}''m''_{1}...-\mu _{n}''m''_{n}>0.$

(144)

The condition (143) may be expressed more briefly in the form

$\Delta \epsilon >t\Delta \eta -p\Delta v+\mu _{1}\Delta m_{1}...+\mu _{n}\Delta m_{n},$

(145)

if we use the character $\Delta$ to signify that the condition, although relating to infinitesimal differences, is not to be interpreted in accordance with the usual convention in respect to differential equations with neglect of infinitesimals of higher orders than the first, but is to be interpreted strictly, like an equation between finite differences. In fact, when a condition like (145) (interpreted strictly) is satisfied for infinitesimal differences, it must be possible to assign limits within which it shall hold true of finite differences. But it is to be remembered that the condition is not to be applied to any arbitrary values of $\Delta \eta ,\Delta v,\Delta m_{1},...\Delta m_{n}$, but only to such as are determined by a change of phase. (If only the quantity of the body which determines the value of the variables should vary and not its phase, the value of the first member of (145) would evidently be zero.) We may free ourselves from this limitation by making v constant, which will cause the term $-p\Delta v$ to disappear. If we then divide by the constant $v$, the condition will become

$\Delta {\frac {\epsilon }{v}}>t\Delta {\frac {\eta }{v}}+\mu _{1}\Delta {\frac {m_{1}}{v}}...+\mu _{n}\Delta {\frac {m_{n}}{v}},$

(146)

in which form it will not be necessary to regard $v$ as constant. As we may obtain from (86)

$d{\frac {\epsilon }{v}}>=td{\frac {\eta }{v}}+\mu _{1}d{\frac {m_{1}}{v}}...+\mu _{n}d{\frac {m_{n}}{v}},$

(147)