The Rayleigh-Ritz method yields a decreasing sequence
that approximates the desired eigenvalue
of Mathieu function or of spheroidal wave function. The method produces also a sequence of elementary functions
that approximates the desired eigenfunction
of the same functions. This paper provides answers to some questions about the rate of convergence
, to the error bounds for the maximum norm
and to other convergence properties of the method. Errors are investigated by interesting numerical experimentations.