Thirdly, the general determination of the true value of functions which present themselves under an indeterminate appearance for certain hypotheses made on the values of the corresponding variables; which is the least extensive and the least important of the three.
The first question is certainly the principal one in all points of view; it is also the most susceptible of receiving a new extension hereafter, especially by conceiving, in a broader manner than has yet been done, the employment of the differential calculus in the transformation of functions, on which subject Lagrange has left some valuable hints.
Having thus summarily, though perhaps too briefly, considered the chief points in the differential calculus, I now proceed to an equally rapid exposition of a systematic outline of the Integral Calculus, properly so called, that is, the abstract subject of integration.
THE INTEGRAL CALCULUS.
Its Fundamental Division. The fundamental division of the Integral Calculus is founded on the same principle as that of the Differential Calculus, in distinguishing the integration of explicit differential formulas, and the integration of implicit differentials or of differential equations. The separation of these two cases is even much more profound in relation to integration than to differentiation. In the differential calculus, in fact, this distinction rests, as we have seen, only on the extreme imperfection of ordinary analysis. But, on the other hand, it is easy to see that, even though all equations could be algebraically resolved, differential equations would none the less constitute a case of integration quite distinct from that presented by the explicit differential formulas; for, limiting ourselves, for the sake of simplicity, to the first order, and to a single function y of a single variable x, if we suppose any differential equation between x, y, and dy/dx, to be resolved with reference to dy/dx, the expression of the derived function being then generally found to contain the primitive function itself, which is the object of the inquiry, the question of integration will not have at all changed its nature, and the solution will not really have made any other progress than that of having brought the proposed differential equation to be of only the first degree relatively to the derived function, which is in itself of little importance. The differential would not then be determined in a manner much less implicit than before, as regards the integration, which would continue to present essentially the same characteristic difficulty. The algebraic resolution of equations could not make the case which we are considering come within the simple integration of explicit differentials, except in the special cases in which the proposed differential equation did not contain the primitive function itself, which would consequently permit us, by resolving it, to find dy/dx in terms of x only, and thus to reduce the question to the class of quadratures. Still greater difficulties would evidently be found in differential equations of higher orders, or containing simultaneously different functions of several independent variables.
The integration of differential equations is then necessarily more complicated than that of explicit differentials, by the elaboration of which last the integral calculus has been created, and upon which the others have been made to depend as far as it has been possible. All the various analytical methods which have been proposed for integrating differential equations, whether it be the separation of the variables, the method of multipliers, &c., have in fact for their object to reduce these integrations to those of differential formulas, the only one which, by its nature, can be undertaken directly. Unfortunately, imperfect as is still this necessary base of the whole integral calculus, the art of reducing to it the integration of differential equations is still less advanced.