def _gauss_first_equation(y, s, w):
    """Evaluates Gauss' first equation.

    Parameters
    ----------
    y: float
        The dependent variable.
    s: float
        First auxiliary variable.
    w: float
        Second auxiliary variable.

    Returns
    -------
    x: float
        The independent variable or free-parameter.

    Notes
    -----
    This is equation (5.6-13) from Bate's book [2].

    """
    x = w / y ** 2 - s
    return x