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. What is necessary in order to deduce from this a mechanical explanation? It is necessary that _U_ can be regarded as the potential energy of a system and _T_ as the _vis viva_ of the same system. There is no difficulty as to _U_, but can _T_ be regarded as the _vis viva_ of a material system? It is easy to show that this is always possible, and even in an infinity of ways. I will confine myself to referring for more details to the preface of my work, 'Électricité et optique.' Thus if the principle of least action can not be satisfied, no mechanical explanation is possible; if it can be satisfied, there is not only one, but an infinity, whence it follows that as soon as there is one there is an infinity of others. One more observation. Among the quantities that experiment gives us directly, we shall regard some as functions of the coordinates of our hypothetical molecules; these are our parameters _q_. We shall look upon the others as dependent not only on the coordinates, but on the velocities, or, what comes to the same thing, on the derivatives of the parameters _q_, or as combinations of these parameters and their derivatives.