and inquire, how and are to be taken as functions of , and in order that the value of this integral may be independent of the choice of the surface passing through the given closed twisted curve, i. e., of the choice of the function of the variables and . The integral has the form [Pg 41] and the vanishing of the first variation in the sense which the new formulation of the question demands, gives the equation i. e., we find for the functions and of the three variables , and the differential equation of the first order If we add to this differential equation the partial differential equation resulting from the equations the partial differential equation (I) for the function of the two variables and and the simultaneous system of the two partial differential equations of the first order (I*) for the two functions and of the three variables , , and stand toward one another in a relation exactly analogous to that in which the differential equations (1) and (1*) stood in the case of the simple integral. It follows from the fact that the integral is independent of the choice of the surface of integration that if we think of the right hand integral as taken over an integral surface of the partial differential equations [Pg 42] and with the help of this formula we arrive at once at the formula which plays the same rôle for the variation of double integrals as the previously given formula (4) for simple integrals. With the help of this formula we can now answer the question how far Jacobi's condition in conjunction with Weierstrass's condition 0"> is necessary and sufficient for the occurrence of a minimum. Connected with these developments is the modified form in which A. Kneser,[53] beginning from other points of view, has presented Weierstrass's theory. While Weierstrass employed to derive sufficient conditions for the extreme values integral curves of equation (1) which pass through a fixed point, Kneser on the other hand makes use of any simple family of such curves and constructs for every such family a solution, characteristic for that family, of that partial differential equation which is to be considered as a generalization of the Jacobi-Hamilton equation.