are also identical with those given by Clausius for the corresponding quantities.
Equations (112) and (181) show that if or is known as function of and , , etc., we can obtain by differentiation or , and , etc. as functions of the same variables. We have in fact
|
(191)
|
|
(192)
|
The corresponding equation relating to kinetic energy,
|
(193)
|
which may be obtained in the same way, may be verified by the known relations (186), (187), and (188) between the variables. We have also
|
(194)
|
etc., so that the average values of the external forces may be derived alike from
or from
.
The average values of the squares or higher powers of the energies (total, potential, or kinetic) may easily be obtained by repeated differentiations of , , , or , , , with respect to . By equation (108) we have
|
(195)
|
and differentiating with respect to
,
|
(196)
|
whence, again by (108),
|
|