Page:Elementary Principles in Statistical Mechanics (1902).djvu/53

ing constant ($a$), is also a quadrature, since the equation to be integrated may be expressed in the form

$dt=F(r_{1})\,dr_{1}.$

Now, apart from any such considerations as have been adduced, if we limit ourselves to the changes which take place in time, we have identically

${\dot {r}}_{2}\,dr_{1}-{\dot {r}}_{1}\,dr_{2}=0,$

and ${\dot {r}}_{1}$ and ${\dot {r}}_{2}$ are given in terms of $r_{1},\ldots r_{2n}$ by the differential equations of motion. When we have obtained $2n-1$ integral equations, we may regard ${\dot {r}}_{2}$ and ${\dot {r}}_{1}$ as known functions of $r_{1}$ and $r_{2}$. The only remaining difficulty is in integrating this equation. If the case is so simple as to present no difficulty, or if we have the skill or the good fortune to perceive that the multiplier

${\frac {1}{\frac {d(c,\ldots h)}{d(r_{3},\ldots r_{2n})}}},$

(79)

or any other, will make the first member of the equation an exact differential, we have no need of the rather lengthy considerations which have been adduced. The utility of the principle of conservation of extension-in-phase is that it supplies a 'multiplier' which renders the equation integrable, and which it might be difficult or impossible to find otherwise.

It will be observed that the function represented by $b'$ is a particular case of that represented by $b$. The system of arbitrary constants $a,b',c\ldots h$ has certain properties notable for simplicity. If we write $b'$ for $b$ in (77), and compare the result with (78), we get

${\frac {d(r_{1},\ldots r_{2n})}{d(a,b',c\ldots h)}}=1.$

(80)

Therefore the multiple integral

$\int \ldots \int da\,db'\,dc\ldots dh$

(81)