Abstract. Plücker's and Klein's equations provide an upper bound on the number of real inflections on the coupler curve of a hinged planar four-bar mechanism. Generally, for any configuration of the four-bar, the coupler points whose trajectories exhibit inflections lie on a circle. The coupler plane is partitioned by the envelope of the inflection circles into connected regions within which every coupler point has the same number of inflections on its trajectory. This enables us to locate coupler curves exhibiting the maximum possible number of inflections. §1 Introduction Four-bar mechanisms provide a rich source of geometrical problems, some of considerable complexity. The problem of finding design parameters for which some coupler curve exhibits particular features dates back at least as far as James Watt. In his case the desire was to find a mechanism with approximate straight-line output and since that time many refinements and improvements have been found for this problem. To a considerable extent, this has depended on locating ordinary and higher inflections on coupler curves, though it must be said that this in itself is not sufficient for good straight-line approximation over a reasonable range of input angles-the approximation may be valid only very close to the inflection.One outstanding problem is to determine the maximum number of (real) inflection points that may be found on a single coupler curve. Here we seek to answer that question for hinged planar four-bar mechanisms (4R mechanisms), hereafter referred to simply is fourbar mechanisms. In theory this is a problem of algebraic geometry. That four-bar coupler curves are algebraic curves of degree six was discovered by Roberts [13]. Regarding them as complex projective curves and using standard results on their singularities, applications of 1991 Mathematics Subject Classification. Primary: 70B15, secondary: 14Q05, 53A17.

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