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(Read before the American Mathematical Society, December 27, 1915.) IN this paper we shall give a new method of deriving the equation of a rational plane curve from its parametric equations. Let the parametric equations of the rational plane curve R of order n be (1) Xi = dit + Cxbit + C2Cit ~ H (i = 0, 1, 2). In particular if n = 2, equations (1) become (2) Xi = afi + 2bit + a (i = 0, 1, 2), which are the parametric equations of a conic or R. The coordinates of the points of the R whose parameters are h and t2 are aA 2 + 26^i + c* and a42 2 + 2bit2 + Ci, where i = 0, 1, 2. Hence the equation of the line on t± and t2, written out in full, is \a0ti 2 + 2b0ti + Co aiti 2 + 2biti + cx a2h 2 + 2b2t\ + c2\ (3) \a0t2 2 + 2b0t2 + Co ait2 2 + 2bit2 + C\ a2t2 2 + 2b2t2 + c2 = 0. I XQ XI X2 I By subtracting the second row from the first and removing the factor h — t2, equation (3) reduces to \ao(fi + t2) + 2b0 ai(h + t2) + 2&i a2(h + t2) + 2b21 (4) \aot2 2 + 2b0t2 + Co a\t2 2 + 2b\t2 + Ci a2fe 2 + 2b2t2 + c2| = 0. I #o #1 #2 The expanded form of (4) is (5) 2 | ate \*ht2 + | aac |(<i + fe) + 2 | to | = 0.

A new method of finding the equation of a rational plane curve from its parametric equations