is an integral equation of the first order of the differential equation (1) of the second order,
represents an integral of the partial differential equation (1*) and conversely; the integral carves of the ordinary differential equation of the second order are therefore, at the same time, the characteristics of the partial differential equation (1*) of the first order.
In the present case we may find the same result by means of a simple calculation; for this gives us the differential equations (1) and (1*) in question in the form
where the lower indices indicate the partial derivatives with respect to
. The correctness of the affirmed relation is clear from this.
The close relation derived before and just proved between the ordinary differential equation (1) of the second order and the partial differential equation (1*) of the first order, is, as it seems to me, of fundamental significance for the calculus of variations. For, from the fact that the integral