| d²u | + | 1 | du | + ( 1 − | m² | ) u = 0, |
| dρ² | ρ | dρ | ρ² |
and may be expressed as the series
| ρm | { 1 − | ρ² | + | ρ4 | ... }; |
| 2m·m! | 2·2m + 2 | 2·4·2m + 2·2m + 4 |
the function of zero order is deduced by making m = 0, and is equivalent to the series 1 − ρ²/2² + ρ4/2²·4², &c. O. Schlömilch defines these functions as the coefficients of the power of t in the expansion of exp ½ρ(t − t−1). The symbol generally adopted to represent these functions is Jm(ρ) 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; this relation was discussed independently, in 1878, by Lord Rayleigh.
For the mathematical investigation see [Spherical Harmonics] and for tables see [Table, Mathematical].
See A. Gray and G.B. Matthews, Treatise on Bessel’s Functions (1895); Encyclopädie der math. Wissenschaften; F.W. Bessel, Untersuchung des Teils der planetarischen Störungen (1824).
BESSEMER, SIR HENRY (1813-1898), English engineer, was born on the 19th of January 1813, at Charlton, in Hertfordshire. Throughout his life he was a prolific inventor, but his name is chiefly known in connexion with the Bessemer process for the manufacture of steel, by which it has been rendered famous throughout the civilized world. Though this process is now largely supplemented, and even displaced, by various rivals, at the time it was brought out it was of enormous industrial importance, since it effected a great cheapening in the price of steel, and led to that material being widely substituted for others which were inferior in almost every respect but that of cost. Bessemer’s attention was drawn to the problem of steel manufacture in the course of an attempt to improve the construction of guns. Coming to the conclusion that if any advance was to be made in artillery better metal must be available, he established a small iron-works in St Pancras, and began a series of experiments. These he carried on for two years before he evolved the essential idea of his process, which is the decarbonization of cast iron by forcing a blast of air through the mass of metal when in the molten condition. The first public announcement of the process was made at the Cheltenham meeting of the British Association in 1856, and immediately attracted considerable notice. Many metallurgists were sceptical on theoretical grounds about his results, and only became convinced when they saw that his process was really able to convert melted cast iron into malleable iron in a perfectly fluid state. But though five firms applied without delay for licences to work under his patents, success did not at once attend his efforts; indeed, after several ironmasters had put the process to practical trial and failed to get good results, it was in danger of being thrust aside and entirely forgotten. Its author, however, instead of being discouraged by this lack of success, continued his experiments, and in two years was able to turn out a product, the quality of which was not inferior to that yielded by the older methods. But when he now tried to induce makers to take up his improved system, he met with general rebuffs, and finally was driven to undertake the exploitation of the process himself. To this end he erected steelworks in Sheffield, on ground purchased with the help of friends, and began to manufacture steel. At first the output was insignificant, but gradually the magnitude of the operations was enlarged until the competition became effective, and steel traders generally became aware that the firm of Henry Bessemer & Co. was underselling them to the extent of £20 a ton. This argument to the pocket quickly had its effect, and licences were applied for in such numbers that, in royalties for the use of his process, Bessemer received a sum in all considerably exceeding a million sterling.
Of course, patents of such obvious value did not escape criticism, and invalidity was freely urged against them on various grounds. But Bessemer was fortunate enough to maintain them intact without litigation, though he found it advisable to buy up the rights of one patentee, while in another case he was freed from anxiety by the patent being allowed to lapse in 1859 through non-payment of fees. At the outset he had found great difficulty in making steel by his process—in his first licences to the trade iron alone was mentioned. Experiments he made with South Wales iron were failures because the product was devoid of malleability; Mr Göransson, a Swedish ironmaster, using the purer charcoal pig iron of that country, was the first to make good steel by the process, and even he was successful only after many attempts. His results prompted Bessemer to try the purer iron obtained from Cumberland haematite, but even with this he did not meet with much success, until Robert Mushet showed that the addition of a certain quantity of spiegeleisen had the effect of removing the difficulties. Whether or not Mushet’s patents could have been sustained, the value of his procedure was shown by its general adoption in conjunction with the Bessemer method of conversion. At the same time it is only fair to say that whatever may have been the conveniences of Mushet’s plan, it was not absolutely essential; this Bessemer proved in 1865, by exhibiting a series of samples of steel made by his own process alone. The pecuniary rewards of Bessemer’s great invention came to him with comparative quickness; but it was not till 1879 that the Royal Society admitted him as a fellow and the government honoured him with a knighthood. Bessemer died at Denmark Hill, London, on the 15th of March 1898.