| κ = | ∂Τ | , κ′ = | ∂Τ | , κ″ = | ∂Τ | , ... |
| ∂χ˙ | ∂χ˙′ | ∂χ˙″ |
(24)
These equations, when written out in full, determine χ˙, χ˙′, χ˙″, ... as linear functions of q˙1, q˙2, ... q˙m, κ, κ′, κ″,... We now consider the function
R = Τ − κχ˙ − κ′χ˙′ − κ″χ˙″ − ... ,
(25)
supposed expressed, by means of the above relations in terms of q˙1, q˙2, ... q˙m, κ, κ′, κ″, ... Performing the operation δ on both sides of (25), we have
| ∂R | δq˙1 + ... + | ∂R | δκ + ... = | ∂Τ | δq˙1 + ... + | ∂Τ | δχ˙ + ... − κ∂χ˙ − χ˙δκ − ... , |
| ∂q˙1 | ∂κ | ∂q˙1 | ∂χ˙ |
(26)
where, for brevity, only one term of each type has been exhibited. Omitting the terms which cancel in virtue of (24), we have
| ∂R | δq˙1 + ... + | ∂R | δκ + ... = | ∂Τ | δq˙1 + ... − χ˙δκ − ... |
| ∂q˙1 | ∂κ | ∂q˙1 |