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

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

(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'

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

(86)

for the integration of the equation

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

(87)

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