In the following discussion the unicursal quartic is regarded from two points of view. Chapter I treats of the curve in its correspondence to a conic section through a quadratic reciprocal transformation. This leads to an interesting classification of unicursal quartics and affords a convenient and ready method for determining the form of the curve. Incidentally, it brings to light a geometrical application of the well know "Group of Four." In Chapter II the curve is defined as the locus of intersection of corresponding rays of two projective pencils of the second order. This develops properties of the curve not readily obtained in the other treatment. The discussion shows that the two definitions are not independent, but that each is supplementary to the other.