Page:Scientific Papers of Josiah Willard Gibbs.djvu/109

obtain conditions in regard to $T,P,M_{1},M_{2},...M_{n}$, some of which will be inconsistent with others or with equations (43). These conditions we will represent by

$A\geqq 0,\,\,B\geqq 0,\,\,etc.$

(46)

$A,B$, etc. being linear functions of $T,P,M_{1},M_{2},...M_{n}$. Then it will be possible to deduce from these conditions a single condition of the form

$\alpha A+\beta B\geqq 0,$

(47)

$\alpha ,\beta$, etc. being positive constants, which cannot hold true consistently with equations (43). But it is evident from the form of (47) that, like any of the conditions (46), it could have been obtained directly from (45) by applying this formula to a certain change in the system (perhaps not restricted by the equations of condition (39), (40), (41)). Now as (47) cannot hold true consistently with eqs. (43), it is evident, in the first place, that it cannot contain $T$ or $P$, therefore in the change in the system just mentioned (for which (45) reduces to (47))

$\sum \delta \eta +\sum D\eta =0,$ and $\sum \delta v+\sum Dv=0,$

so that the equations of condition (39) and (40) are satisfied. Again, for the same reason, the homogeneous function of the first degree of $M_{1},M_{2},...M_{n}$ in (47) must be one of which the value is fixed by eqs. (43). But the value thus fixed can only be zero, as is evident from the form of these equations. Therefore

$(\sum \delta m_{1}+\sum Dm_{1})M_{1}+(\sum \delta m_{2}+\sum Dm_{2})M_{2}...+(\sum \delta m_{n}+\sum Dm_{n})M_{n}=0$

(48)

for any values of $M_{1},M_{2},...M_{n}$ which satisfy eqs. (43), and therefore

$(\sum \delta m_{1}+\sum Dm_{1}){\mathfrak {S}}_{1}+(\sum \delta m_{2}+\sum Dm_{2}){\mathfrak {S}}_{2}...+(\sum \delta m_{n}+\sum Dm_{n}){\mathfrak {S}}_{n}=0$

(49)

for any numerical values of ${\mathfrak {S}}_{1},{\mathfrak {S}}_{2},...{\mathfrak {S}}_{n}$ which satisfy eqs. (38). This equation (49) will therefore hold true, if for $r$ of the letters ${\mathfrak {S}}_{1},{\mathfrak {S}}_{2},...{\mathfrak {S}}_{n}$ we substitute their values in terms of the others taken from eqs. (38), and therefore it will hold true when we use ${\mathfrak {S}}_{1},{\mathfrak {S}}_{2},...{\mathfrak {S}}_{n}$, as before, to denote the units of the various components. Thus understood, the equation expresses that the values of the quantities in the parentheses are such as are consistent with the equations of condition (41). The change in the system, therefore, which we are considering, is not one which violates any of the equations of condition, and as (45) does not hold true for this change, and for all values of $T,P,M_{1},M_{2},...M_{n}$ which are consistent with eqs. (43), the state of the system cannot be one of equilibrium. Therefore it is necessary, and it is evidently sufficient for equilibrium, that it shall be possible to assign to $T,P,M_{1},M_{2},...M_{n}$ such values,