(1)
whence
| ∂Τ | = κ, | ∂Τ | = κ′, | ∂Τ | = κ″, ..., |
| ∂χ˙ | ∂χ˙′ | ∂χ˙″ |
(2)
where κ, κ′, κ″, ... are the constant momenta corresponding to the cyclic co-ordinates χ, χ′, χ″, .... These equations are linear in χ˙, χ˙′, χ˙″, ...; solving them with respect to these quantities and substituting in the remaining Lagrangian equations, we obtain m differential equations to determine the remaining co-ordinates q1, q2, ... qm. The object of the present investigation is to ascertain the general form of the resulting equations. The retained co-ordinates q1, q2, ... qm may be called (for distinction) the palpable co-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence.
If, as in § 1 (25), we write
R = T − κχ˙ − κ′χ˙′ − κ″χ˙″ − ...,
(3)
and imagine R to be expressed by means of (2) as a quadratic function of q˙1, q˙2, ... q˙m, κ, κ′, κ″, ... with coefficients which are in general functions of the co-ordinates q1, q2, ... qm, then, performing the operation δ on both sides, we find
| ∂R | δq˙1 + ... + | ∂R | δκ + ... + | ∂R | δq1 + ... = | ∂T | δq˙1 + ... + | ∂T | δq1 + ... |
| ∂q˙1 | ∂κ | ∂q1 | ∂q˙1 | ∂q1 |