CHAPTER VI.

These general views which I have indicated give only an imperfect idea of the truly infinite extent and variety of the questions to which geometers have risen by means of this single consideration of series, so simple in appearance and so limited in its origin. It necessarily presents as many different cases as the algebraic resolution of equations, considered in its whole extent; and it is, by its nature, much more complicated, so much, indeed, that it always needs this last to conduct it to a complete solution. We may, therefore, anticipate what must still be its extreme imperfection, in spite of the successive labours of several geometers of the first order. We do not, indeed, possess as yet the complete and logical solution of any but the most simple questions of this nature.

Its identity with this Calculus. It is now easy to conceive the necessary and perfect identity, which has been already announced, between the calculus of finite differences and the theory of series considered in all its bearings. In fact, every differentiation after the manner of Taylor evidently amounts to finding the law of formation of a series with one or with several variable indices, from the expression of its general term; in the same way, every analogous integration may be regarded as having for its object the summation of a series, the general term of which would be expressed by the proposed difference. In this point of view, the various problems of the calculus of differences, direct or inverse, resolved by Taylor and his successors, have really a very great value, as treating of important questions relating to series. But it is very doubtful if the form and the notation introduced by Taylor really give any essential facility in the solution of questions of this kind. It would be, perhaps, more advantageous for most cases, and certainly more logical, to replace the differences by the terms themselves, certain combinations of which they represent. As the calculus of Taylor does not rest on a truly distinct fundamental idea, and has nothing peculiar to it but its system of signs, there could never really be any important advantage in considering it as detached from ordinary analysis, of which it is, in reality, only an immense branch. This consideration of differences, most generally useless, even if it does not cause complication, seems to me to retain the character of an epoch in which, analytical ideas not being sufficiently familiar to geometers, they were naturally led to prefer the special forms suitable for simple numerical comparisons.

PERIODIC OR DISCONTINUOUS FUNCTIONS.

However that may be, I must not finish this general appreciation of the calculus of finite differences without noticing a new conception to which it has given birth, and which has since acquired a great importance. It is the consideration of those periodic or discontinuous functions which preserve the same value for an infinite series of values of the corresponding variables, subjected to a certain law, and which must be necessarily added to the integrals of the equations of finite differences in order to render them sufficiently general, as simple arbitrary constants are added to all quadratures in order to complete their generality. This idea, primitively introduced by Euler, has since been the subject of extended investigation by M. Fourier, who has made new and important applications of it in his mathematical theory of heat.

APPLICATIONS OF THIS CALCULUS.

Series. Among the principal general applications which have been made of the calculus of finite differences, it would be proper to place in the first rank, as the most extended and the most important, the solution of questions relating to series; if, as has been shown, the general theory of series ought not to be considered as constituting, by its nature, the actual foundation of the calculus of Taylor.