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absolutely convergent approximately assume asymptotic expansion Bessel Functions Bessel's Equation calculated Chapter coefficients conductor considered constant contour convergent coordinates corresponding cosh curve cylindrical coordinates deduced definite integrals denote differential equation disk distance electric electrode equations of motion expression finite follows formula give given Green's function Hence imaginary axis infinite number infinity J. J. Thomson Jn(x Laplace's equation Lommel Lord Kelvin Lord Rayleigh magnetic force maxima maxima and minima membrane minima negative obtain orifice oscillations parallel plane positive integer positive zeros potential problem propagation Prove quantities radius real and positive real axis real zeros result roots satisfy the equation Similarly sinh solution of Bessel's substituted suppose surface Taylor's Theorem temperature tends to zero Theorem theory values vanish velocity velocity of propagation verified vibration wave wire