130 M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 
CuaprterVII. 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 formule of this kind will be found in 
this chapter, which I have abstracted from my old memoirs, and in which 
T 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 with a double integral, which belongs to 
Fourier; it is extended without difficulty, as wellas each of those which 
only subsists for a limited interval, to two or a greater number of va- 
riables. 
Cuaprer VIII. Continuation of the Digression on the Manner of re- 
presenting 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 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, will leave no doubt of its nature and genérality. 
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 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 cosinés of the multiples of the va- 
riables. This chapter terminates with the demonstration of another theo- — 
