Looking for the order of a system of arbitrary ordinary differential equations

This paper was edited by S. Cohn and C. W. Borchardt from posthumous manuscripts of C. G. J. Jacobi. The various canonical forms that a given system ordinary differential equations may take are considered. Looking for the order of the system, without using a normal form, is reduced to a problem of inequalities: the affectation problem. A new type of formulas, the truncated determinants, is introduced. The non vanishing of this quantity means that the order will be equal to the value H, solution of this inequalities problem, which is obtained by an algorithm similar to Harold Kuhn's Hungarian method.