The differential Calculus.

either calculus, which are quite a different study from that of the abstract principles of differentiation and integration. The consequence of the common practice of confounding these principles with their application, especially in geometry, is that it becomes difficult to conceive of either analysis or geometry. It is in the department of Concrete Mathematics that the applications should be studied.

The first division of the differential calculus is grounded on the condition whether the functions to be differentiated are explicit or implicit; the one giving rise

Two portions. to the differentiation of formulas, and the other to the differentiation of equations. This classification is rendered necessary by the imperfection of ordinary analysis; for if we knew how to resolve all equations algebraically, it would be possible to render every implicit function explicit; and, by differentiating it only in that state, the second part of the differential calculus would be immediately included in the first, without giving rise to any new difficulty. But the algebraic resolution of equations is, as we know, still scarcely past its infancy, and unknown for the greater number of cases; and we have to differentiate a function without knowing it, though it is determinate. Thus we have two classes of questions, the differentiation of implicit functions being a distinct case from that of explicit functions, and much more complicated. We have to begin by the differentiation of formulas, and we may then refer to this first case the differentiation of equations, by certain analytical considerations which we are not concerned with here. There is another view in which the two general cases of differentiation are distinct. The relation obtained between the differentials is always more indirect, in comparison with that of the finite quantities, in the differentiation of implicit, than in that of explicit functions. We shall meet with this consideration in the case of the integral calculus, where it acquires a preponderant importance. Subdivisions. Each of these parts of the differential calculus is again divided: and this subdivision exhibits two very distinct theories, according as we have to differentiate functions of a single variable, or

SUBDIVISIONS OF THE DIFFERENTIAL CALCULUS. 79

functions of several independent variables,-the second branch being of far greater complexity than the first, in the case of explicit functions, and much more in that of implicit. One more distinction remains, to complete this brief sketch of the parts of the differential calculus. The case in which it is required to differentiate at once different implicit functions combined in certain primitive equations must be distinguished from that in which all these functions are separate. The same imperfection of ordinary analysis which prevents our converting every implicit function into an equivalent explicit one, renders us unable to separate the functions which enter simultaneously into any system of equations; and the functions are evidently still more implicit in the case of combined than of separate functions: and in differentiating, we are not only unable to resolve the primitive equations, but even to effect the proper elimination among them.

Reduction to the elements.

We have now seen the different parts of this calculus in their natural connection and rational distribution. The whole calculus is finally found to rest upon the differentiation of explicit functions with a single variable, the only one which is ever executed directly. Now, it is easy to understand that this first theory, this necessary basis of the whole system, simply consists of the differentiation of the elementary functions, ten in number, which compose all our analytical combinations; for the differentiation of compound functions is evidently deduced, immediately and necessarily, from that of their constituent simple functions. We find, then, the whole system of differentiation reduced to the knowledge of the ten fundamental differentials, and to that of the two general principles, by one of which the differentiation of implicit functions is deduced from that of explicit, and by the other, the differentiation of functions of several variables is reduced to that of functions of a single variable. Such is the simplicity and perfection of the system of the differential calculus.

The transformation of derived Functions for new variables is a theory which must be just mentioned, to avoid the omission of an indispensable complement of the system of diffe

Transformation of derived functions for new variables.

rentiation. It is as finished and perfect as the other parts of this calculus; and its great importance is in its increasing our resources by permitting us to choose, to facilitate the formation of differential equations, that system of independent variables which may appear to be most advantageous, though it may afterwards be relinquished, as an intermediate step, by which, through this theory, we may pass to the final system, which sometimes could not have been considered directly.

Analytical

Though we cannot here consider the conapplications. crete applications of this calculus, we must glance at those which are analytical, because they are of the same nature with the theory, and should be looked at in connection with it. These questions are reducible to three essential ones. First, the development into series of functions of one or more variables; or, more generally, the transformation of functions, which constitutes the most beautiful and the most important application of the differential calculus to general analysis, and which comprises, besides the fundamental series discovered by Taylor, the remarkable series discovered by Maclaurin, John Bernouilli, Lagrange and others. Secondly, the general theory of maxima and minima values for any functions whatever of one or more variables: one of the most interesting problems that analysis can present, however elementary it has become. The third is the least important of the three :-it is the 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. In every view, the first question is the most eminent; it is also the most susceptible of future extension, especially by conceiving, in a larger manner than hitherto, of the employment of the differential calculus for the transformation of functions, about which Lagrange left some valuable suggestions which have been neither generalized nor followed up.

It is with regret that I confine myself to the generalities which are the proper subjects of this work; so extensive and so interesting are the developments which might otherwise be offered. Insufficient and summary as are the views of the Differential Calculus just offered, we must be no

THE INTEGRAL CALCULUS.

81

less rapid in our survey of the Integral Calculus, properly so called; that is, the abstract subject of integration.

The Integral Calculus.

The Integral
Calculus.

of implicit

Its divisions.

The division of the Integral Calculus, like that of the Differential, proceeds on the principle of distinguishing the integration of explicit differential formulas from the integration differentials, or of differential equations. The separation of these two cases is even more radical in the case of integration than in the other. In the differential calculus this distinction rests, as we have seen, only on the extreme imperfection of ordinary analysis. But, on the other hand, it is clear that even if all equations could be algebraically resolved, differential equations would nevertheless constitute a case of integration altogether distinct from that presented by explicit differential formulas. Their integration is necessarily more complicated than that of explicit differentials, by the elaboration of which the integral calculus was originated, and on which the others have been made to depend, as far as possible. All the various analytical processes hitherto proposed for the integration of differential equations, whether by the separation of variables, or the method of multipliers, or other means, have been designed to reduce these integrations to those of differential formulas, the only object which can be directly undertaken. Unhappily, imperfect as is this necessary basis of the whole integral calculus, the art of reducing to it the integration of differential equations is even much less advanced.

Subdivisions.

One variable, or several.

As in the case of the differential calculus, and for analogous reasons, each of these two branches of the integral calculus is divided again, according as we consider functions with a single variable or functions with several independent variables. This distinction is, like the preceding, even more important for integration than for differentiation. This is especially remarkable with respect to differential equations. In fact, those which relate to several independent variables may evidently present this

characteristic and higher difficulty- that the function sought may be differentially defined by a simple relation between its various special derivatives with regard to the different variables taken separately. Thence results the most difficult, and also the most extended branch of the integral calculus, which is commonly called the Integral Calculus of partial differences, created by D'Alembert, in which, as Lagrange truly perceived, geometers should have recognized a new calculus, the philosophical character of which has not yet been precisely decided. This higher branch of transcendental analysis is still entirely in its infancy. In the very simplest case, we cannot completely reduce the integration to that of the ordinary differential equations.

Orders of

A new distinction, highly important here, differentiation. though not in the differential calculus, where it is a mistake to insist upon it, is drawn from the higher or lower order of the differentials. We may regard this distinction as a subdivision in the integration of explicit or implicit differentials. With regard to explicit differentials, whether of one variable or of several, the necessity of distinguishing their different orders is occasioned merely by the extreme imperfection of the integral calculus; and, with reference to implicit differentials, the distinction of orders is more important still. In the first case, we know so little of integration of even the first order of differential formulas, that differential formulas of a high order produce new difficulties in arriving at the primitive function which is our object. And in the second case, there is the additional difficulty that the higher order of the differential equations necessarily gives rise to questions of a new kind. The higher the order of differential equations, the more implicit are the cases which they present; and they can be made to depend on each other only by special methods, the investigation of which, in consequence, forms a new class of questions, with regard to the simplest cases of which we as yet know next to nothing.

The necessary basis of all other integrations is, as we see from the foregoing considerations, that of explicit differential formulas of the first order and of a single variable and we cannot succeed in effecting other integrations but

;