indeterminate. Their objection appears, however, to rest upon a
misapprehension as to the meaning of the sum of the series; if x1 be
the point of discontinuity, it is possible to make x approach x1,
and n become indefinitely great, so that the sum of the series
takes any assigned value in a certain interval, whereas we ought
to make x = x1 first and afterwards n = ∞, and no other way of
going to the double limit is really admissible. Other papers by
Dircksen (Crelle, vol. iv.) and Bessel (Astronomische Nachrichten, vol.
xvi.), on similar lines to those by Dirichlet, are of inferior importance.
Many of the investigations subsequent to Dirichlet’s have the object
of freeing a function from some of the restrictions which were imposed
upon it in Dirichlet’s proof, but no complete set of necessary and
sufficient conditions as to the nature of the function has been obtained.
Lipschitz (“De explicatione per series trigonometricas,”
Crelle’s Journal, vol. lxiii., 1864) showed that, under a certain condition,
a function which has an infinite number of maxima and
minima in the neighbourhood of a point is still expansible; his
condition is that at the point of discontinuity β, |ƒ(β + δ) − f(β)| < Bδα
as δ converges to zero, B being a constant, and α a positive exponent.
A somewhat wider condition is
δ = 0
for which Lipschitz’s results would hold. This last condition is adopted by Dini in his treatise (Sopra la serie di Fourier, &c., Pisa, 1880).
The modern period in the theory was inaugurated by the publication by Riemann in 1867 of his very important memoir, written in 1854, Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe. The first part of his memoir contains a historical account of the work of previous investigators; in the second part there is a discussion of the foundations of the Integral Calculus, and the third part is mainly devoted to a discussion of what can be inferred as to the nature of a function respecting the changes in its value for a continuous change in the variable, if the function is capable of representation by a trigonometrical series. Dirichlet and probably Riemann thought that all continuous functions were everywhere representable by the series; this view was refuted by Du Bois-Reymond (Abh. der Bayer. Akad. vol. xii. 2). It was shown by Riemann that the convergence or non-convergence of the series at a particular point x depends only upon the nature of the function in an arbitrarily small neighbourhood of the point x. The first to call attention to the importance of the theory of uniform convergence of series in connexion with Fourier’s Series was Stokes, in his memoir “On the Critical Values of the Sums of Periodic Series” (Camb. Phil. Trans., 1847; Collected Papers, vol. i.). As the method of determining the coefficients in a trigonometrical series is invalid unless the series converges in general uniformly, the question arose whether series with coefficients other than those of Fourier exist which represent arbitrary functions. Heine showed (Crelle’s Journal, vol. lxxi., 1870, and in his treatise Kugelfunctionen, vol. i.) that Fourier’s Series is in general uniformly convergent, and that if there is a uniformly convergent series which represents a function, it is the only one of the kind. G. Cantor then showed (Crelle’s Journal, vols. lxxii. lxxiii.) that even if uniform convergence be not demanded, there can be but one convergent expansion for a function, and that it is that of Fourier. In the Math. Ann. vol. v., Cantor extended his investigation to functions having an infinite number of discontinuities. Important contributions to the theory of the series have been published by Du Bois-Reymond (Abh. der Bayer. Akademie, vol. xii., 1875, two memoirs, also in Crelle’s Journal, vols. lxxiv. lxxvi. lxxix.), by Kronecker (Berliner Berichte, 1885), by O. Hölder (Berliner Berichte, 1885), by Jordan (Comptes rendus, 1881, vol. xcii.), by Ascoli (Math. Annal., 1873, and Annali di matematica, vol. vi.), and by Genocchi (Atti della R. Acc. di Torino, vol. x., 1875). Hamilton’s memoir on “Fluctuating Functions” (Trans. R.I.A., vol. xix., 1842) may also be studied with profit in this connexion. A memoir by Brodén (Math. Annalen, vol. lii.) contains a good investigation of some of the most recent results on the subject. The scope of Fourier’s Series has been extended by Lebesgue, who introduced a conception of integration wider than that due to Riemann. Lebesgue’s work on Fourier’s Series will be found in his treatise, Leçons sur les séries trigonométriques (1906); also in a memoir, “Sur les séries trigonométriques,” Annales sc. de l’école normale supérieure, series ii. vol. xx. (1903), and in a paper “Sur la convergence des séries de Fourier,” Math. Annalen, vol. lxiv. (1905).
Authorities.—The foregoing historical account has been mainly drawn from A. Sachse’s work, “Versuch einer Geschichte der Darstellung willkürlicher Functionen einer Variabeln durch trigonometrische Reihen,” published in Schlömilch’s Zeitschrift für Mathematik, Supp., vol. xxv. 1880, and from a paper by G. A. Gibson “On the History of the Fourier Series” (Proc. Ed. Math. Soc. vol. xi.). Reiff’s Geschichte der unendlichen Reihen may also be consulted, and also the first part of Riemann’s memoir referred to above. Besides Dini’s treatise already referred to, there is a lucid treatment of the subject from an elementary point of view in C. Neumann’s treatise, Über die nach Kreis-, Kugel- und Cylinder-Functionen fortschreitenden Entwickelungen. Jordan’s discussion of the subject in his Cours d’analyse is worthy of attention: an account of functions with limited variation is given in vol. i.; see also a paper by Study in the Math. Annalen, vol. xlvii. On the second mean-value theorem papers by Bonnet (Brux. Mémoires, vol. xxiii., 1849, Lionville’s Journal, vol. xiv., 1849), by Du Bois-Reymond (Crelle’s Journal, vol. lxxix., 1875), by Hankel (Zeitschrift für Math. und Physik, vol. xiv., 1869), by Meyer (Math. Ann., vol. vi., 1872) and by Hölder (Göttinger Anzeigen, 1894) may be consulted; the most general form of the theorem has been given by Hobson (Proc. London Math. Soc., Series II. vol. vii., 1909). On the theory of uniform convergence of series, a memoir by W. F. Osgood (Amer. Journal of Math. xix.) may be with advantage consulted. On the theory of series in general, in relation to the functions which they can represent, a memoir by Baire (Annali di matematica, Series III. vol. iii.) is of great importance. Bromwich’s Theory of Infinite Series (1908) contains much information on the general theory of series. Bôcher’s “Introduction to the Theory of Fourier’s Series,” Annals of Math., Series II. vol. vii., 1906, will be found useful. See also Carslaw’s Introduction to the Theory of Fourier’s Series and Integrals, and the Mathematical Theory of the Conduction of Heat (1906). A full account of the theory will be found in Hobson’s treatise On the Theory of Functions of a Real Variable and on the Theory of Fourier’s Series (1907). (E. W. H.)
FOURMIES, a town of northern France, in the department
of Nord, on an affluent of the Sambre, 39 m. S.E. of Valenciennes
by rail. Pop. (1906) 13,308. It is one of the chief centres in
France for wool combing and spinning, and produces a great
variety of cloths. The glass-works of Fourmies date from
1599, and were the first established in the north of France. Iron
is worked in the vicinity, and there are important forges and
foundries. Enamel-ware is also manufactured. In 1891 labour
troubles brought about military intervention and consequent
bloodshed. A board of trade arbitration and a school of commerce
and industry are among the public institutions.
FOURMONT, ÉTIENNE (1683–1745), French orientalist, was
born at Herbelai, near Saint Denis, on the 23rd of June 1683.
He studied at the Collège Mazarin, Paris, and afterwards in the
Collège Montaigu, where his attention was attracted to Oriental
languages. Shortly after leaving the college he published a
Traduction du commentaire du Rabbin Abraham Aben Esra sur
l’ecclésiaste. In 1711 Louis XIV. appointed Fourmont to
assist a young Chinese, Hoan-ji, in compiling a Chinese grammar.
Hoan-ji died in 1716, and it was not until 1737 that Fourmont
published Meditationes Sinicae and in 1742 Grammatica Sinica.
He also wrote Réflexions critiques sur les histoires des anciens
peuples (1735), and several dissertations printed in the Mémoires
of the Academy of Inscriptions. He became professor of Arabic
in the Collège de France in 1715. In 1713 he was elected a
member of the Academy of Inscriptions, in 1738 a member
of the Royal Society of London, and in 1742 a member of that
of Berlin. He died at Paris on the 19th of December 1745.
His brother, Michel Fourmont (1690–1746), was also a member of the Academy of Inscriptions, and professor of the Syriac language in the Royal College, and was sent by the government to copy inscriptions in Greece.
An account of Étienne Fourmont’s life and a catalogue of his works will be found in the second edition (1747) of his Réflexions critiques.
FOURNET, JOSEPH JEAN BAPTISTE XAVIER (1801–1869),
French geologist and metallurgist, was born at Strassburg on
the 15th of May 1801. He was educated at the École des Mines
at Paris, and after considerable experience as a mining engineer
he was in 1834 appointed professor of geology at Lyons. He was
a man of wide knowledge and extensive research, and wrote
memoirs on chemical and mineralogical subjects, on eruptive
rocks, on the structure of the Jura, the metamorphism of the
Western Alps, on the formation of oolitic limestones, on kaolinization
and on metalliferous veins. On metallurgical subjects
also he was an acknowledged authority; and he published
observations on the order of sulphurability of metals (loi de
Fournet). He died at Lyons on the 8th of January 1869. His
chief publications were: Études sur les dépôts métallifères (Paris,
1834); Histoire de la dolomie (Lyons, 1847); De l’extension
des terrains houillers (1855); Géologie lyonnaise (Lyons, 1861).
FOURNIER, PIERRE SIMON (1712–1768), French engraver
and typefounder, was born at Paris on the 15th of September
1712. He was the son of a printer, and was brought up to his
father’s business. After studying drawing under the painter
