Chapter VII. A Digression on the Manner of expressing Arbitrary Functions by Series of Periodical Quantities.—Lagrange was the first to give a series of quantities proper to represent the values of an arbitrary function, continuous or discontinuous, in a determined interval of the values of the variable. This formula supposes that the function vanishes at the two extremes of this interval; it proceeds according to the sines of the multiples of the variable; many others exist of the same nature which proceed according to the sines or cosines of these multiples, even or uneven, and which differ from one another in conditions relative to each extreme. A complete theory of formulæ of this kind will be found in this chapter, which I have abstracted from my old memoirs, and in which I have considered the periodical series which they contain as limits of other converging series, the sums of which are integrals, themselves having for limits the arbitrary functions which it is our object to represent. Supposing in one or other of these expressions in series, the interval of the values of the variable for which it takes place to be infinite, there results from it the formula with a double integral, which belongs to Fourier; it is extended without difficulty, as well as each of those which only subsists for a limited interval, to two or a greater number of variables.
Chapter VIII. Continuation of the Digression on the Manner of representing Arbitrary Functions by Series of Periodical Quantities.—An arbitrary function of two angles, one of which is comprised between zero and 180°, and the other between zero and 360°, may always be represented between those limits by a series of certain periodical quantities, which have not received particular denominations, although they have special and very remarkable properties. It is to that expression in series that we have recourse in a great number of questions of celestial mechanics and of physics, relative to spheroids; it had however been disputed whether they agreed with any function whatever; but the demonstration of this important formula, which I had already given and now reproduce in this chapter, will leave no doubt of its nature and generality. This demonstration is founded on a theorem, which is deduced from considerations similar to those of the preceding chapter. We examine what the series becomes at the limits of the values of the two angles; we then demonstrate the properties of the functions of which its terms are formed; then it is shown that they always end by decreasing indefinitely, which is a necessary consequence and sufficient to prevent the series from becoming diverging, for which purpose its use is always allowable. Finally, it is proved, that for the same function there is never more than one development of that kind; which does not happen in the developments in series of sines and cosines of the multiples of the variables. This chapter terminates with the demonstration of another theo-
