The reduction to normal form of a non-normal system of differential equations

This paper was edited by Sigismund Cohn, C. W. Borchardt and A. Clebsch from posthumous manuscripts of C. G. J. Jacobi. The solution of the following problem: "to transform a square table of m ² numbers by adding minimal numbers ℓ_i to each horizontal row, in such a way that it possess m transversal maxima", determines the order and the shortest normal form reduction of the system: the equations u _i = 0 must be respectively differentiated ℓ_i times. One also determines the number of differentiations of each equation of the given system needed to produce the differential equations necessary to reduce the proposed system to a single equation.