The ordinary differential equation of the second order (1) and the partial differential equation (1*) stand in the closest relation to each other. This relation becomes immediately clear to us by the following simple transformation
We derive from this, namely, the following facts: If we construct any simple family of integral curves of the ordinary differential equation (1) of the second order and then form an ordinary differential equation of the first order
which also admits these integral curves as solutions, then the function
is always an integral of the partial differential equation (1*) of the first order; and conversely, if
denotes any solution of the partial differential equation (1*) of the first order, all the non-singular integrals of the ordinary differential equation (2) of the first order are at the same time integrals of the differential equation (1) of the second order, or in short if