Page:Scientific Memoirs, Vol. 1 (1837).djvu/387

1000° to 2000° higher than that of the boilers, there is an enormous loss of vis viva in the passage of the heat from the furnace into the boiler. It is therefore only from the employment of caloric at high temperatures, and from the discovery of agents proper to realize its motive force, that important improvements may be expected in the art of utilizing the mechanical power of heat.

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NOTE.

The integral of the general equation

${\frac {d\,Q}{d\,v}}\centerdot {\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}{\frac {d\,T}{d\,v}}=C,$

is, as we have seen,

$Q=F(T)-C\phi (p,z)$

(1)

$F(T)$ is an arbitrary function of the temperature $T$, varying from one body to another; $C$ is a function of the temperature which is the same for all the substances of nature, and $\phi (p,v)$ is a particular function of $p$, and of $v$ satisfying the equation

${\frac {d\,T}{d\,v}}\centerdot {\frac {d\,\phi }{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi }{d\,v}}=1$

(2)

This function $\phi$ may be determined in the following manner. Let

$\phi =\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}+\phi ',$

be substituted in the equation (2), it will be

${\frac {d\,T}{d\,v}}{\frac {d\,\phi '}{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi '}{d\,v}}={\frac {d\,T}{d\,p}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}};$

this equation is satisfied by putting

$\phi '=\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}+\phi '';$

$\phi ''$ satisfying the equation

${\frac {d\,T}{d\,v}}{\frac {d\,\phi ''}{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi ''}{d\,v}}={\frac {d\,T}{d\,p}}{\frac {d}{d\,v}}\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}},$

we shall have equally

$\phi ''=\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}+\phi '''.$