sleep. From old observations he calculated the orbit of Halley's comet. Bessel introduced himself to Olbers, and submitted to him the calculation, which Olbers immediately sent for publication. Encouraged by Olbers, Bessel turned his back to the prospect of affluence, chose poverty and the stars, and became assistant in J. H. Schröter's observatory at Lilienthal. Four years later he was chosen to superintend the construction of the new observatory at Königsberg.[92] In the absence of an adequate mathematical teaching force, Bessel was obliged to lecture on mathematics to prepare students for astronomy. He was relieved of this work in 1825 by the arrival of Jacobi. We shall not recount the labours by which Bessel earned the title of founder of modern practical astronomy and geodesy. As an observer he towered far above Gauss, but as a mathematician he reverently bowed before the genius of his great contemporary. Of Bessel's papers, the one of greatest mathematical interest is an "Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne ensteht" (1824), in which he introduces a class of transcendental functions, , 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 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
