1. _Use of the Differential Calculus as preparatory to that of the Integral._ In forming the differential equations of any phenomenon whatever, it is very seldom that we limit ourselves to introduce differentially only those magnitudes whose relations are sought. To impose that condition would be to uselessly diminish the resources presented by the transcendental analysis for the expression of the mathematical laws of phenomena. Most frequently we introduce into the primitive equations, through their differentials, other magnitudes whose relations are already known or supposed to be so, and without the consideration of which it would be frequently impossible to establish equations. Thus, for example, in the general problem of the rectification of curves, the differential equation, _ds_² = _dy_² + _dx_², or _ds_² = _dx_² + _dy_² + _dz_², is not only established between the desired function s and the independent variable _x_, to which it is referred, but, at the same time, there have been introduced, as indispensable intermediaries, the differentials of one or two other functions, _y_ and _z_, which are among the data of the problem; it would not have been possible to form directly the equation between _ds_ and _dx_, which would, besides, be peculiar to each curve considered. It is the same for most questions. Now in these cases it is evident that the differential equation is not immediately suitable for integration. It is previously necessary that the differentials of the functions supposed to be known, which have been employed as intermediaries, should be entirely eliminated, in order that equations may be obtained between the differentials of the functions which alone are sought and those of the really independent variables, after which the question depends on only the integral calculus. Now this preparatory elimination of certain differentials, in order to reduce the infinitesimals to the smallest number possible, belongs simply to the differential calculus; for it must evidently be done by determining, by means of the equations between the functions supposed to be known, taken as intermediaries, the relations of their differentials, which is merely a question of differentiation. Thus, for example, in the case of rectifications, it will be first necessary to calculate _dy_, or _dy_ and _dz_, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only _ds_ and _dx_; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus.