limited to the differential -1, and others similar with respect to the other variables, must be satisfied. The values of the differential coefficients may be readily found in terms of the differential coefficients of U, by means of the relations found before; if we substitute them in the equation above, we shall find

and we shall have similar equations for each of the other variables; all these joined to the equations which express that U is the differential of a function U, must be satisfied, in order that Ŭ should be a second differential of a function U. Similar considerations will prove that U, in order to be a third differential of a function U must satisfy, besides the preceding, the equation

We have supposed, in order to be more general, none of the first differentials dx, dy, &c. to be constant; if it were not the case, then the equations relative to the variables, the differentials of which are supposed to be constant, should be suppressed. If we suppose dr, for instance, to be constant, it is obvious that all the differential coefficients taken with respect to x, would vanish.

TABLE

OF THE

PRINCIPAL MATTERS

IN THE

TREATISE OF DIFFERENTIAL AND INTEGRAL CALCULUS

ALPHABETICALLY ARRANGED.

N. B. The Numbers in Parentheses refer to the Articles, and the others to the Pages.