  Suppose that f has continuous derivatives thru order r+1 for x>0, and let
P_{c} denote the Taylor polynomial to f of order r at x=c,c>0. In a previous
paper of the author, it was shown that if r is an odd whole number and the
(r+1)st derivative of f is nonzero on [a,b], then there is a unique
x_{0},a<x_{0}<b, such that P_{a}(x_{0})=P_{b}(x_{0}). This defines a mean,
depending on f and r, given by m(a,b)=x_{0}. In this paper we discuss the real
parts of the pairs of complex conjugate non-real roots of P_{b}-P_{a}. We prove
some results for r in general, but our most significant results are for the
case r=3. We prove in that case that if f(z)=z^{p}, where p is an integer, p
not equal to 0,1,2, or 3, then P_{b}-P_{a} has non-real roots with real part
strictly between a and b for any 0<a<b. This defines a countable family of
means. We construct a cubic polynomial, g, whose real root gives the real part
of the pair of complex conjugate non-real roots of P_{b}-P_{a}. Instead of
working directly with a formula for the roots of a cubic, we use the
Intermediate Value Theorem to show that g has a root in (a,b).
