The way that the calculation goes in the dissertation is first to take the given expression for and (using the "simplify" function in Mathcad) add the four fractions together to form a single fraction
The procedure for turning this into a continued fraction is then to repeatedly invert it and divide the numerator by the denominator. The first steps will go like this:
(notice that 1.758 = 1/0.56882).
The next step will be to invert that last fraction so that the "remainder" becomes the denominator. This time, the numerator and denominator of the new fraction will both be cubic polynomials in s, so the quotient will be a constant (in fact 0.0023 according to the solution given in the dissertation), and the remainder will be of the form
Again, invert that fraction, divide top by bottom and get the remainder, and so on.