Some new criteria for the absolute stability of the feedback system of Fig. 1 with a monotone (odd monotone) nonlinearity are derived in the familiar multiplier form. These criteria generalise the results of the author (Ref.1). The main theorem of the paper states that the feedback system with an odd monotonic ??(.) is absolutely stable, if a multiplier function of the form Z(j??) = 1 + ??j?? + Y1(j??) + Y2(j??) (where ?? is a positive constant, and the inverse Fourier transforms y1(t), y2(t) of Y1(j??), Y2(j??) are defined on [o, ??) and (-??, o] respectively), can be found such that (i) Re Z(j??) G(j??) ?? 0 for all real ??, (ii) y1(t), y2(t) satisfy a certain time domain constraint, and (iii) k(t) lies, depending upon the multiplier chosen and the nonlinear function, within a band formed by two monotonic functions. When the nonlinearity is monotonic, it is required, in addition, that y1(t) and y2(t) be nonpositive.