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 of November 1472, at Ravenna. Bessarion was one of the most learned scholars of his time. Besides his translations of Aristotle’s Metaphysics and Xenophon’s Memorabilia, his most important work is a treatise directed against George of Trebizond, a violent Aristotelian, entitled In Calumniatorem Platonis. Bessarion, though a Platonist, is not so thoroughgoing in his admiration as Gemistus Pletho, and rather strives after a reconciliation of the two philosophies. His work, by opening up the relations of Platonism to the main questions of religion, contributed greatly to the extension of speculative thought in the department of theology. His library, which contained a very extensive collection of Greek MSS., was presented by him to the senate of Venice, and formed the nucleus of the famous library of St Mark.

 BESSBOROUGH, EARLS OF. The Ponsonby family, who have contributed many conspicuous men to Irish and English public life, trace their descent to Sir John Ponsonby (d. 1678), of Cumberland, a Commonwealth soldier who obtained land grants in Ireland. His son William (1657–1724) was created Baron Bessborough (1721) and Viscount Duncannon (1723), and the latter’s son Brabazon was raised to the earldom of Bessborough in 1739. He was the father not only of the 2nd earl (1704–1793), but of (q.v.), speaker of the Irish House of Commons. The 2nd earl was a well-known Whig politician, who held various offices of state; and his son the 3rd earl (1758–1844) was father of the 4th earl (1781–1847), first commissioner of works in 1831–1834, lord privy seal from 1835 to 1839 and lord-lieutenant of Ireland in 1846. He was succeeded by his three sons, the 5th earl (d. 1880), 6th earl (1815–1895), a famous cricketer and chairman of the Bessborough commission (1881) to inquire into the Irish land system, and 7th earl (d. 1906), and the last named by his son the 8th earl.  BESSÈGES, a town of south-eastern France, in the department of Gard, on the Cèze, 20 m. north of Alais by rail. Pop. (1906) 7662. The town is important for its coal-mines, blast-furnaces and iron-works.  BESSEL, FRIEDRICH WILHELM (1784–1846), German astronomer, was born at Minden on the 22nd of July 1784. Placed at the age of fifteen in a counting-house at Bremen, he was impelled by his desire to obtain a situation as supercargo on a foreign voyage to study navigation, mathematics and finally astronomy. In 1804 he calculated the orbit of Halley’s comet from observations made in 1607 by Thomas Harriot, and communicated his results to H. W. M. Olbers, who procured their publication (Monatliche Correspondenz, x. 425), and recommended the young aspirant in 1805 for the post of assistant in J. H. Schröter’s observatory at Lilienthal. A masterly investigation of the comet of 1807 (Königsberg, 1810) enhanced his reputation, and the king of Prussia summoned him, in 1810, to superintend the erection of a new observatory at Königsberg, of which he acted as director from its completion in 1813 until his death. In this capacity he inaugurated the modern era of practical astronomy. For the purpose of improving knowledge of star-places he reduced James Bradley’s Greenwich observations, and derived from them an invaluable catalogue of 3222 stars, published in the volume rightly named Fundamenta Astronomiae (1818). In Tabulae Regiomontanae (1830), he definitively established the uniform system of reduction still in use. During the years 1821–1833, he observed all stars to the ninth magnitude in zones extending from −15° to +45° dec., and thus raised the number of those accurately determined to about 50,000. He corrected the length of the seconds’ pendulum in 1826, in a discussion re-published by H. Bruns in 1889; measured an arc of the meridian in East Prussia in 1831–1832; and deduced for the earth in 1841 an ellipticity of. His ascertainment in 1838 (Astr. Nach., Nos. 365-366) of a parallax of 0″·31 for 61 Cygni was the first authentic result of the kind published. He announced in 1844 the binary character of Sirius and Procyon from their disturbed proper motions; and was preparing to attack the problem solved later by the discovery of Neptune, when fatal illness intervened. He died at Königsberg on the 17th of March 1846. Modern astronomy of precision is essentially Bessel’s creation. Apart from the large scope of his activity, he introduced such important novelties as the effective use of the heliometer, the correction for personal equation (in 1823), and the systematic investigation of instrumental errors. He issued 21 volumes of Astronomische Beobachtungen auf der Sternwarte zu Königsberg (1815–1844), and a list of his writings drawn up by A. L. Busch appeared in vol. 24 of the same series. Especial attention should be directed to his Astronomische Untersuchungen (2 vols. 1841–1842), Populäre Vorlesungen (1848), edited by H. C. Schumacher, and to the important collection entitled Abhandlungen (4 vols. 1875–1882), issued by R. Engelmann at Leipzig. His minor treatises numbered over 350. In pure mathematics he enlarged the resources of analysis by the invention of Bessel’s Functions. He made some preliminary use of these expressions in 1817, in a paper on Kepler’s Problem (Transactions Berlin Academy, 1816–1817, p. 49), and fully developed them seven years later, for the purposes of a research into planetary perturbations (Ibid. 1824, pp. 1-52).

 BESSEL FUNCTION, a certain mathematical relation between two variables. The Bessel function of order m satisfies the differential equation $d&#8202;^{2}u⁄d^{2}$＋$1⁄1$ $du⁄d$＋1 − $m^{2}⁄^{2}$ u＝0 and may be expressed as the series $^{m}⁄2^{m}.m!$ 1−$^{2}⁄2.2m+2$＋$^{4}⁄2.4.2m+2.2m+4$ ; the function of zero order is deduced by making m＝0, and is equivalent to the series 1−$^{2}⁄2^{2}$＋$^{4}⁄2^{2}.4^{2}$, &c. O. Schlömilch defines these functions as the coefficients of the power of t in the expansion of exp (t−t&#8202;−1). The symbol generally adopted to represent these functions is Jm where m denotes the order of the function. These functions are named after Friedrich Wilhelm Bessel, who in 1817 introduced them in an investigation on Kepler’s Problem. He discussed their properties and constructed tables for their evaluation. Although Bessel was the first to systematically treat of these functions, it is to be noted that in 1732 Daniel Bernoulli obtained the function of zero order as a solution to the problem of the oscillations of a chain suspended at one end. This problem has been more fully discussed by Sir A. G. Greenhill. In 1764 Leonhard Euler employed the functions of both zero and integral orders in an analysis into the vibrations of a stretched membrane; an investigation which has been considerably developed by Lord Rayleigh, who has also shown (1878) that Bessel’s functions are particular cases of Laplace’s functions. There is hardly a branch of mathematical physics which is independent of these functions. Of the many applications we may notice:—Joseph Fourier’s (1824) investigation of the motion of heat in a solid cylinder, a problem which, with the related one of the flow of electricity, has been developed by W. E. Weber, G. F. Riemann and S. D. Poisson; the flow of electromagnetic waves along wires (Sir J. J. Thomson, H. Hertz, O. Heaviside); the diffraction of light (E. Lömmel, Lord Rayleigh, Georg Wilhelm Struve); the theory of elasticity (A. E. Love, H. Lamb, C. Chree, Lord Rayleigh); and to hydrodynamics (Lord Kelvin, Sir G. Stokes).

The remarkable connexion between Bessel’s functions and spherical harmonics was established in 1868 by F. G. Mehler, who proved that a simple relation existed between the function of zero order and the zonal harmonic of order n. Heinrich Eduard Heine has shown that the functions of higher orders may be considered as limiting values of the associated functions;