where A'11, A'22,... A'12,... are certain coefficients depending on
the configuration. They have been called by Maxwell the _coefficients
of mobility_ of the system. When the form (19) is given, the values
of the velocities in terms of the momenta can be expressed in a
remarkable form due to Sir W.R. Hamilton. The formula (15) may be
written
p1q`1 + p2q`2 + ... = [Tau] + [Tau]', ... (20)
where [Tau] is supposed expressed as in (8), and [Tau]' as in (19).
Hence if, for the moment, we denote by [delta] a variation affecting
the velocities, and therefore the momenta, but not the configuration,
we have
p1[delta]q`1 + q`1[delta]p + p2[delta]q`2 + q`2[delta]p2 + ... = [delta][Tau] + [delta][Tau]'
dP[Tau] dP[Tau] dP[Tau]' dP[Tau]'
= ------- [delta]q`1 + ------- [delta]q`2 + ... + -------- [delta]p1 + -------- [delta]p2 + ... (21)
dPq`1 dPq`2 dPp1 dPp2