Page:Elementary Principles in Statistical Mechanics (1902).djvu/55

of a function of $a,\ldots h$, which we may denote by $a'$. We have then

$da'={\frac {1}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dr_{1}-{\frac {{\dot {r}}_{1}{}'}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}\,dt,$

(85)

which may be integrated by quadratures. In this case we may say that the principle of conservation of extension-in-phase has supplied the 'multiplier'

${\frac {1}{\frac {d(b,\ldots h)}{d(r_{2},\ldots r_{2n})}}}$

(86)

for the integration of the equation

$dr_{1}-{\dot {r}}_{1}\,dt=0.$

(87)

The system of arbitrary constants $a',b,\ldots h$ has evidently the same properties which were noticed in regard to the system $a,b',\ldots h$.