  A theorem of Kirby states that two framed links in the 3-sphere produce
orientation-preserving homeomorphic results of surgery if they are related by a
sequence of stabilization and handle-slide moves.
  The purpose of the present paper is twofold: First, we give a sufficient
condition for a sequence of handle-slides on framed links to be able to be
replaced with a sequences of algebraically canceling pairs of handle-slides.
Then, using the first result, we obtain a refinement of Kirby's calculus for
integral homology spheres which involves only +/-1-framed links with zero
linking numbers.
