As most of the standard treatises on dynamics contain satisfactory proofs of Lagrange's equations, I do not see that any useful purpose is served by proposing an additional one. The important point is this:—That amongst the numerous forms in which the kinetic energy of a dynamical system can be expressed, there is only one form which can be employed in using Lagrange's equations, and that is the Lagrangian form in which T is expressed as an n-ary quadric of velocities which are the time variations of the coordinates.