_Ft_ = (_N_)/(_C_) log (_Ct_ + _C_{1}_) + _C_{2}_;
or changing arbitrary constants, and remarking further that _Ft_ is 0
when _t_ = 0°,
_Ft_ = _A_ log (1 + (_t_)/(_B_)). (6)
The nature of the function _Ft_ would be thus determined, and we would
thus be able to estimate the motive power developed by any fall of heat.
But this latter conclusion is founded on the hypothesis of the constancy
of the specific heat of a gas which does not change in volume—an
hypothesis which has not yet been sufficiently verified by experiment.
Until there is fresh proof, our equation (6) can be admitted only
throughout a limited portion of the thermometric scale.
In equation (5), the first member represents, as we have remarked, the
specific heat of the air occupying the volume _v_. Experiment having
shown that this heat varies little in spite of the quite considerable
changes of volume, it is necessary that the coefficient _T′_ of log _v_
should be a very small quantity. If we consider it nothing, and, after
having multiplied by _dt_ the equation
_T′_ = 0,
we take the integral of it, we find
_T_ = _C_, constant quantity;
but
_T_ = _N_/_F′t_,
whence
_F′t_ = _N_/_T_ = _N_/_C_ = _A_;
whence we deduce finally, by a second integration,
_Ft_ = _At_ + _B_.