Störungen, welcher aus der Bewegung der Sonne ensteht" (1824), in which he introduces a class of transcendental functions, J n ( x ) , much used in applied mathematics, and known as "Bessel's functions." He gave their
principal properties, and constructed tables for their evaluation. Recently it has been observed that Bessel's functions appear much earlier in mathematical literature.98 Such functions of the zero order occur in papers of Daniel Bernoulli (1732) and Euler on vibration of heavy strings suspended
from one end. All of Bessel's functions of the first kind and of integral orders occur in a paper by Euler (1764) on the vibration of a stretched elastic membrane. In 1878 Lord Rayleigh proved that Bessel's functions are merely particular
cases of Laplace's functions. J. W. L. Glaisher illustrates
by Bessel's functions his assertion that mathematical branches growing out of physical inquiries as a rule "lack the easy flow
or homogeneity of form which is characteristic of a mathematical theory properly so called." These functions have been studied by C. Th. Anger of Danzig, O. Schlömilch of Dresden,
R. Lipschitz of Bonn (born 1832), Carl Neumann of Leipzig
(born 1832), Eugen Lommel of Leipzig, I. Todhunter of St. John's
College, Cambridge.
Prominent among the successors of Laplace are the following:
Siméon Denis Poisson (1781–1840), who wrote in 1808
a classic Mémoire sur les inégalités séculaires des moyens mouvements des planètes. Giovanni Antonio Amadeo Plana (1781–1864)
of Turin, a nephew of Lagrange, who published in 1811 a italianMemoria sulla teoria dell' attrazione degli sferoidi ellitici, and contributed to the theory of the moon. Peter Andreas Hansen
(1795–1874) of Gotha, at one time a clockmaker in Tondern, then Schumacher's assistant at Altona, and finally director of the observatory at Gotha, wrote on various astronomical subjects, but mainly on the lunar theory, which he elaborated in his work Fundamenta nova investigationes orbitæ veræ quam Luna perlustrat (1838), and in subsequent investigations embracing extensive lunar tables. George Biddel Airy (1801–1892),
royal astronomer at Greenwich, published in 1826 his Mathematical Tracts on the Lunar and Planetary Theories. These researches have since been greatly extended by him. August Ferdinand Möbius (1790–1868) of Leipzig wrote, in 1842,
Elemente der Mechanik des Himmels. Urbain Jean Joseph Le Verrier (1811–1877) of Paris wrote the Recherches Astronomiques,