where
| (r, s) = ( | ∂αr | − | ∂αs | )χ˙ + ( | ∂α′r | − | ∂α′s | )χ˙′ + .... |
| ∂qs | ∂qr | ∂qs | ∂qr |
(18)
These quantities (r, s) are subject to the relations
(r, s) = −(s, r), (r, r) = 0
(19)
The remaining dynamical equations, equal in number to the co-ordinates χ, χ′, χ″, ..., yield expressions for the forces which must be applied in order to maintain the velocities χ˙, χ˙′, χ˙″, ... constant; they need not be written down. If we follow the method by which the equation of energy was established in § 2, the equations (17) lead, on taking account of the relations (19), to
| d | (⅋ − T0) = Q1q˙1 + Q2q˙2 + ... + Qnq˙n, |
| dt |
(20)
or, in case the forces Qr depend only on the co-ordinates q1, q2, ... qn and are conservative,