Page:Elementary Principles in Statistical Mechanics (1902).djvu/74

for the moment of these forces, we have for the period of their action by equation (3)

${\dot {p}}_{1}=-{\frac {d\epsilon _{p}}{dq_{1}}}-{\frac {d\epsilon _{q}}{dq_{1}}}+F_{1}=-{\frac {d\epsilon }{dq_{1}}}+F_{1}.$

The work done by the force $F_{1}$ may be evaluated as follows:

$\int F_{1}\,dq_{1}=\int {\dot {p}}_{1}\,dq_{1}+\int {\frac {d\epsilon }{dq_{1}}}\,dq_{1},$

where the last term may be cancelled because the configuration does not vary sensibly during the application of the forces. (It will be observed that the other terms contain factors which increase as the time of the action of the forces is diminished.) We have therefore,

$\int F_{1}\,dq_{1}=\int {\dot {p}}_{1}{\dot {q}}_{1}\,dt=\int {\dot {q}}_{1}\,dp_{1}={\frac {{\dot {q}}_{1}}{p_{1}}}\int p_{1}\,dp_{1}.$

(131)

For since the $p$'s are linear functions of the ${\dot {q}}$'s (with coefficients involving the $q$'s) the supposed constancy of the $q$'s and of the ratios of the ${\dot {q}}$'s will make the ratio ${\dot {q}}_{1}/p_{1}$ constant. The last integral is evidently to be taken between the limits zero and the value of $p_{1}$ in the phase originally considered, and the quantities before the integral sign may be taken as relating to that phase. We have therefore

$\int F_{1}\,dq_{1}={\tfrac {1}{2}}p_{1}{\dot {q}}_{1}={\tfrac {1}{2}}p_{1}{\frac {d\epsilon }{dp_{1}}}.$

(132)

That is: the several parts into which the kinetic energy is divided in equation (128) represent the amounts of energy communicated to the system by the several forces $F_{1},\ldots F_{n}$ under the conditions mentioned.

The following transformation will not only give the value of the average kinetic energy, but will also serve to separate the distribution of the ensemble in configuration from its distribution in velocity.

Since $2\epsilon _{p}$ is a homogeneous quadratic function of the $p$'s, which is incapable of a negative value, it can always be expressed (and in more than one way) as a sum of squares of