δA = ∫t′t δΤdt + ∫t′t Σm (ẋδẋ + ẏδẏ + z˙δz˙) dt + 2Τ′δt′ − 2Τδt

= ∫t′t δΤdt + [ Σm (ẋδx + ẏδy + z˙δz)]t′t

− ∫t′t Σm (ẍδx + ÿδy + z¨δz) dt + 2Τ′δt′ − 2Τδt.

(4)

Now, by d’Alembert’s principle,

Σm (ẍδx + ÿδy + z¨δz) = −δV,

(5)

and by hypothesis we have

δ(Τ + V) = 0.

(6)