CONDITIONS WHICH MUST BE FULFILLED FOR A TOTAL DIFFERENTIAL TO NOT CHANGE SIGN WHILE WE CHANGE THE VALUES ATTRIBUTED TO THE DIFFERENTIALS OF THE INDEPENDENT VARIABLES.

Abstract

After what we have seen in the preceding lectures, if we denote by u a function of the independent variables $x, y, z, \dots ,$ and if we disregard the values of these variables which render one of the functions $u, du, d^2u, \dots$ discontinuous, the function u can only become a maximum or a minimum in the case where one of the total differentials $d^2u,$ $d^4u,$ $d^6u,$ $\dots ,$ namely, the first of these that will not be constantly null, will maintain the same sign for all possible values of the arbitrary quantities $dx=h, dy=k, dz=l, \dots ,$ or at least for the values of these quantities which will not reduce it to zero. Add that, in the latter assumption, each of the systems of values of $h, k, l, \dots$ that work to make the total differential in question vanish, must change another total differential of even order into a quantity affected by the sign that maintains the first differential, as long as it does not vanish.