(41)
in the notation of Bessel’s functions, if z2 = 4kx. Since y must vanish at the upper end (x = l), the admissible values of σ are determined by
σ2 = gz2/4l, J0(z) = 0.
(42)
The function J0(z) has been tabulated; its lower roots are given by
z/π= .7655, 1.7571, 2.7546,...,
approximately, where the numbers tend to the form s − 1⁄4. The frequency of the gravest mode is to that of a uniform bar in the ratio .9815 That this ratio should be less than unity agrees with the theory of “constrained types” already given. In the higher normal modes there are nodes or points of rest (y = 0); thus in the second mode there is a node at a distance .190l from the lower end.
Authorities.—For indications as to the earlier history of the subject see W. W. R. Ball, Short Account of the History of Mathematics; M. Cantor, Geschichte der Mathematik (Leipzig, 1880 ... ); J. Cox, Mechanics (Cambridge, 1904); E. Mach, Die Mechanik in ihrer Entwickelung (4th ed., Leipzig, 1901; Eng. trans.). Of the classical treatises which have had a notable influence on the development of the subject, and which may still be consulted with advantage, we may note particularly, Sir I. Newton, Philosophiae naturalis Principia Mathematica (1st ed., London, 1687); J. L. Lagrange, Mécanique analytique (2nd ed., Paris, 1811-1815); P. S. Laplace, Mécanique céleste (Paris, 1799-1825); A. F. Möbius, Lehrbuch der Statik (Leipzig, 1837), and Mechanik des Himmels; L. Poinsot, Éléments de statique (Paris, 1804), and Théorie nouvelle de la rotation des corps (Paris, 1834).
Of the more recent general treatises we may mention Sir W. Thomson (Lord Kelvin) and P. G. Tait, Natural Philosophy (2nd ed., Cambridge, 1879-1883); E. J. Routh, Analytical Statics (2nd ed., Cambridge, 1896), Dynamics of a Particle (Cambridge, 1898), Rigid Dynamics (6th ed., Cambridge 1905); G. Minchin, Statics (4th ed., Oxford, 1888); A. E. H. Love, Theoretical Mechanics (2nd ed., Cambridge, 1909); A. G. Webster, Dynamics of Particles, &c. (1904); E. T. Whittaker, Analytical Dynamics (Cambridge, 1904); L. Arnal, Traitê de mécanique (1888-1898); P. Appell, Mécanique rationelle (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., 1896); G. Kirchhoff, Vorlesungen über Mechanik (Leipzig, 1896); H. Helmholtz, Vorlesungen über theoretische Physik, vol. i. (Leipzig, 1898); J. Somoff, Theoretische Mechanik (Leipzig, 1878-1879).
The literature of graphical statics and its technical applications is very extensive. We may mention K. Culmann, Graphische Statik (2nd ed., Zürich, 1895); A. Föppl, Technische Mechanik, vol. ii. (Leipzig, 1900); L. Henneberg, Statik des starren Systems (Darmstadt, 1886); M. Lévy, La statique graphique (2nd ed., Paris, 1886-1888); H. Müller-Breslau, Graphische Statik (3rd ed., Berlin, 1901). Sir R. S. Ball’s highly original investigations in kinematics and dynamics were published in collected form under the title Theory of Screws (Cambridge, 1900).