31. In a medical implant assembly having first and second bone anchors cooperating with a longitudinal connecting member, the improvement wherein the longitudinal connecting member comprises: a) an anchor member portion having a first width and also having an integral inner core extension having a second width that is smaller than the first width and extending from the anchor member portion along a central axis thereof; the anchor member portion passing through and being secured to the first bone anchor; the extension slidably passing through the second bone anchor; b) an elastic and pre-compressed outer spacer, the longitudinal connecting member being slidingly received through the spacer along the axis, the spacer being positioned between the bone anchors, the spacer having at least a first curved end surface; c) at least one inelastic sleeve, the core extension being slidingly received within the sleeve along the axis, the sleeve being in engagement with the second bone anchor, the sleeve having at least a second curved end surface; and d) at least one elastic and pre-compressed bumper disposed near an end of the core extension and not between the first and second bone anchors, the elastic bumper being more elastic than the spacer, allowing for the sleeve to move back and forth along the core a greater distance toward the bumper than toward the spacer.
31. In a medical implant assembly having first and second bone anchors cooperating with a longitudinal connecting member, the improvement wherein the longitudinal connecting member comprises: a) an anchor member portion having a first width and also having an integral inner core extension having a second width that is smaller than the first width and extending from the anchor member portion along a central axis thereof; the anchor member portion passing through and being secured to the first bone anchor; the extension slidably passing through the second bone anchor; b) an elastic and pre-compressed outer spacer, the longitudinal connecting member being slidingly received through the spacer along the axis, the spacer being positioned between the bone anchors, the spacer having at least a first curved end surface; c) at least one inelastic sleeve, the core extension being slidingly received within the sleeve along the axis, the sleeve being in engagement with the second bone anchor, the sleeve having at least a second curved end surface; and d) at least one elastic and pre-compressed bumper disposed near an end of the core extension and not between the first and second bone anchors, the elastic bumper being more elastic than the spacer, allowing for the sleeve to move back and forth along the core a greater distance toward the bumper than toward the spacer. 38. The connecting member according to claim 31 , wherein the sleeve curved end surface is cup shaped.