It matters little that _T_ and _U_ are now expressed by the aid of the parameters _q_ and their derivatives; it matters little that it is also by means of these parameters that we define the initial and final positions; the principle of least action remains always true.
Now here again, of all the paths that lead from one position to another, there is one for which the average action is a minimum, and there is only one. The principle of least action suffices, then, to determine the differential equations which define the variations of the parameters _q_.
The equations thus obtained are another form of the equations of Lagrange.
To form these equations we need to know neither the relations that connect the parameters _q_ with the coordinates of the hypothetical molecules, nor the masses of these molecules, nor the expression of _U_ as a function of the coordinates of these molecules.
All we need to know is the expression of _U_ as a function of the parameters, and that of _T_ as a function of the parameters _q_ and their derivatives, that is, the expressions of the kinetic and of the potential energy as functions of the experimental data.
Then we shall have one of two things: either for a suitable choice of the functions _T_ and _U_, the equations of Lagrange, constructed as we have just said, will be identical with the differential equations deduced from experiments; or else there will exist no functions _T_ and _U_, for which this agreement takes place. In the latter case it is clear that no mechanical explanation is possible.
The _necessary_ condition for a mechanical explanation to be possible is therefore that we can choose the functions _T_ and _U_ in such a way as to satisfy the principle of least action, which involves that of the conservation of energy.
This condition, moreover, is _sufficient_. Suppose, in fact, that we have found a function _U_ of the parameters _q_, which represents one of the parts of the energy; that another part of the energy, which we shall represent by _T_, is a function of the parameters _q_ and their derivatives, and that it is a homogeneous polynomial of the second degree with respect to these derivatives; and finally that the equations of Lagrange, formed by means of these two functions, _T_ and _U_, conform to the data of the experiment.