_du_ = _F_(_t_ + _dt_) − _Ft_ = _F′tdt_.
This is evidently the quantity of motive power produced by the fall of
unity of heat from the temperature _t_ + _dt_ to the temperature _t_.
If the quantity of heat instead of being a unit had been _e_, its motive
power produced would have had for its value
_edu_ = _eF′tdt_. (4)
But _edu_ is the same thing as δ_r_; both are the power developed by the
fall of the quantity _e_ of heat from the temperature _t_ + _dt_ to the
temperature _t_; consequently,
_edu_ = δ_r_,
and from equations (3), (4),
_eF′tdt_ = _N_ log _vdt_;
or, dividing by _F′tdt_,
_e_ = (_N_)/(_F′t_) log _v_ = _T_ log _v_.
Calling _T_ the fraction (_N_)/(_F′t_) which is a function of _t_ only,
the equation
_e_ = _T_ log _v_
is the analytical expression of the law stated pp. 80, 81. It is common
to all gases, since the laws of which we have made use are common to
all.
If we call _s_ the quantity of heat necessary to change the air that we
have employed from the volume 1 and from the temperature zero to the
volume _v_ and to the temperature _t_, the difference between _s_ and
_e_ will be the quantity of heat required to bring the air at the volume
1 from zero to _t_. This quantity depends on _t_ alone; we will call it
_U_. It will be any function whatever of _t_. We shall have
_s_ = _e_ + _U_ = _T_ log _v_ + _U_.
If we differentiate this equation with relation to _t_ alone, and if we
represent it by _T′_ and _U′_, the differential coefficients of _T_ and
_U_, we shall get
(_ds_)/(_dt_) = _T′_ log _v_ + _U′_; (5)
_ds_/_dt_ is simply the specific heat of the gas under constant volume,
and our equation (1) is the analytical expression of the law stated on
page 86.
If we suppose the specific heat constant at all temperatures (hypothesis
discussed above, page 92), the quantity _ds_/_dt_ will be independent of
_t_; and in order to satisfy equation (5) for two particular values of
_v_, it will be necessary that _T′_ and _U′_ be independent of _t_; we
shall then have _T′_ = _C_, a constant quantity. Multiplying _T′_ and
_C_ by _dt_, and taking the integral of both, we find
_T_ = _Ct_ + _C_{1}_;
but as _T_ = _N_/_F′t_, we have
_F′t_ = (_N_)/(_T_) = (_N_)/(_Ct_ + _C_{1}_).