  On a W*-algebra M, for given two positive linear forms f,g and algebra
elements a,b a variational expression for the Bures-distance d_B(f^a,g^b)
between the inner derived positive linear forms f^a=f(a* . a) and g^b=g(b* . b)
is obtained. Along with the proof of the formula also some earlier result of
S.Gudder on non-commutative probability will be slightly extended. Also, the
given expression of the Bures-distance nicely relates to some system of
seminorms proposed by D.Buchholz and which occured along with the problem of
estimating the so-called `weak intertwiners' in algebraic quantum field theory.
In the last part some optimization problem will be considered.
