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 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; [Pg 39] 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 is independent of the path of integration it follows that if we think of the left hand integral as taken along any path and the right hand integral along an integral curve of the differential equation With the help of equation (3) we arrive at Weierstrass's formula where designates Weierstrass's expression, depending upon , Since, therefore, the solution depends only on finding an integral which is single valued and continuous in a certain neighborhood of the integral curve , which we are considering, the developments just indicated lead immediately—without the introduction of the second variation, but only[Pg 40] by the application of the polar process to the differential equation (1)—to the expression of Jacobi's condition and to the answer to the question: How far this condition of Jacobi's in conjunction with Weierstrass's condition 0"> is necessary and sufficient for the occurrence of a minimum. The developments indicated may be transferred without necessitating further calculation to the case of two or more required functions, and also to the case of a double or a multiple integral. So, for example, in the case of a double integral to be extended over a given region , the vanishing of the first variation (to be understood in the usual sense) gives the well-known differential equation of the second order for the required function of and . On the other hand we consider the integral