Spheroidal wave-functions

The solution of problems relating to vibrations, in connection with spheroids— or, in two dimensions, elliptic cylinders—has hitherto only been attempted in one manner. If the vibrations have a time factor for their fundamental vector, of the form ℯ ipt , the equation of wave motion becomes, if ø is the fundamental vector, (∇ 2 + k 2 ) ø = 0 where the wave-length is 2π/ k ,and if C is the velocity of propagation of the wave in the external region, k = p/c . When oblate spheroidal co-ordinates are used, defined in terms of Cartesians by x= a √ { (1 + μ 2 ) (1 + ζ 2 ) } cos ω, y = a √ {1 - μ 2 ) (1 + ζ 2 ) } sin ω, z = a μζ, this can be transformed, after the usual manner, to ∂/∂μ (1 - μ 2 ) ∂ø/∂μ + ∂/∂ζ (1+ζ 2 ) ∂ø/∂ζ + k 2 a 2 (μ 2 + ζ 2 ) ø = 0, (1) when there is symmetry round the axis of z .