(2)
where X′, Y′, Z′ are the components of the impulse on m. Now let δx, δy, δz be any infinitesimal variations of x, y, z which are consistent with the connexions of the system, and let us form the equation
Σm(ẋδx + ẏδy + z˙δz) = Σ(X′δx + Y′δy + Z′δz),
(3)
where the sign Σ indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations δq1, δq2, ... of the generalized co-ordinates, we have
| ẋ = | ∂x | q˙1 + | ∂x | q˙2 + ..., &c., &c. |
| ∂q1 | ∂q2 |
(4)
| δx = | ∂x | δq1 + | ∂x | δq2 + ..., &c., &c. |
| ∂q1 | ∂q2 |
(5)
and therefore