and the forms of Arr, Ars are as given by § 1 (7). It is to be remembered that the coefficients α0, α1, α2, ... A11, A22, ... A12 ... will in general involve t explicitly as well as implicitly through the co-ordinates q1, q2,.... Again, we find
Σm (ẋδx + ẏδy + z˙δz) = (α1 + A11q˙1 + A12q˙2 + ...) δq1 + (α2 + A21q˙1 + A22q˙2 + ...) ∂q2 + ...
| = | ∂Τ | δq1 + | ∂Τ | δq2 + ... = p1δq1 + p2δq2 + ..., |
| ∂q˙1 | ∂q˙2 |
(6)
where pr is defined as in § 1 (13). The derivation of Lagrange’s equations then follows exactly as before. It is to be noted that the equation § 2 (15) does not as a rule now hold. The proof involved the assumption that Τ is a homogeneous quadratic function of the velocities q˙1, q˙2....
It has been pointed out by R.B. Hayward that Vieille’s case can be brought under Lagrange’s by introducing a new co-ordinate (χ) in place of t, so far as it appears explicitly in the relations (1). We have then
2Τ = α0χ˙² + 2(α1q˙1 + α2q˙2 + ...) χ˙ + A11q˙1² + A22q˙2² + ... + 2A12q˙1q˙2 + ....
(7)
The equations of motion will be as in § 2 (10), with the additional equation
| d | ∂Τ | − | ∂Τ | = X, | |
| dt | ∂χ˙ | ∂χ |