130 M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 



ChavtehYII. A Digression on the Manner of expressing Arbitrary Func- 

 tions 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 formulfe 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 repre- 

 sent. 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 Avith 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 va- 

 riables. 



Chapter VIII. Continuation of the Digression on the Manner of re- 

 presenting Arbitrary Fimctions by Series of Periodical Quantities. — An 

 arbitrai-y function of two angles, one of which is comprised between zero 

 and 180°, and the other between zero and 360°, may always be repre- 

 sented 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 mecha- 

 nics 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 repro- 

 duce in this chapter, Avill 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 jireceding 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 inde- 

 finitely, which is a necessary consequence and sufficient to prevent the 

 series from becoming diverging, for which purpose its use is always al- 

 lowable. 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 va- 

 riables. This chapter terminates with the demonstration of another theo- 



