For real symmetric or Hermitian matrices with tridiagonal form, the secular equation may be written as a continued fraction equation f(λ)=0. f(λ) is a member of a recursively defined sequence R(n)(λ) of n continued fractions if the secular equation is of the nth order. The basis for a new method of computing the eigenvalues of such tridiagonal matrices is given. The method requires the determination of an integervalues function Pn(γ) for a succession of values of γ, where Pn(γ) is a function only of n and the signs of the n terms in R(n)(γ).