which, being decomposed into two, with respect to the two unknown functions _f_ and π, which are independent of each other, will completely express the analytical definition of the required curve. The only difficulty peculiar to this new analysis consists in the elimination of the characteristic δ, for which the calculus of variations furnishes invariable and complete rules, founded, in general, on the method of "integration by parts," from which Lagrange has thus derived immense advantage. The constant object of this first analytical elaboration (which this is not the place for treating in detail) is to arrive at real differential equations, which can always be done; and thereby the question comes under the ordinary transcendental analysis, which furnishes the solution, at least so far as to reduce it to pure algebra if the integration can be effected. The general object of the method of variations is to effect this transformation, for which Lagrange has established rules, which are simple, invariable, and certain of success. _Equations of Limits._ Among the greatest special advantages of the method of variations, compared with the previous isolated solutions of isoperimetrical problems, is the important consideration of what Lagrange calls _Equations of Limits_, which were entirely neglected before him, though without them the greater part of the particular solutions remained necessarily incomplete. When the limits of the proposed integrals are to be fixed, their variations being zero, there is no occasion for noticing them. But it is no longer so when these limits, instead of being rigorously invariable, are only subjected to certain conditions; as, for example, if the two points between which the required curve is to be traced are not fixed, and have only to remain upon given lines or surfaces. Then it is necessary to pay attention to the variation of their co-ordinates, and to establish between them the relations which correspond to the equations of these lines or of these surfaces.