Prof. Sylvester's Theory of Tatnisage. 85 the form considered in this memoir, in which (x = 0) is the determinate line passing through the cusp and the two points of osculation of the osculating conic. The general form of quartics, which are simply met by the .line at infinity in three coincident points and another point and which have a triple and a single asymptote (x 3 y-\-u s-\-u 2 = 0), is not here considered. [The assumed form KO S /3 = tt 2 , or, as this is afterwards written, 2Kx s y-ax l + 2bxy + cy i + 2ex + 2dy + \, is, I think, introduced without a proper explanation. Say, the form is xSj = z 2 (*j£aj, y, z) 2 , it ought to be shown how for a cuspidal quartic we arrive at this form; viz., taking the cusp to be at the point (x=0, z=0), z = 0 for the tangent at the cusp, and x-0 an arbitrary line through the cusp; then the line «=0 besides intersects the curve in a single point, and, if y = 0 is taken as the tangent at that point, the equation of the curve must, it can be seen, be of the form {x* + 0xh)y = z> (a, b;c,f, g, h%x, y, «)". The conic (a, b, c, / , g, h^x, y, z)* = 0 touches the quartic at each of the two intersections of the quartic with the arbitrary line # = 0 ; and we cannot, so long as the line remains arbitrary, find a conic which shall osculate the quartic at the two points in question; but, for the particular line x+±dz = 0, there exists such a conic, viz., writing x instead of x + \6z, the form is x*y = z 2 (a, &' , c',/', g, h'~Qx, y, z) 5 , and the new conic (a', ...Qx, y, a) 2 = 0 has the property in question. This is the adopted form, and it thus appears that in it the line x = 0 is a determinate line, viz., the line passing through the cusp and the two points of osculation of the osculating conic. It thus appears that in the assumed form the lines JC==O, y = 0, 8 = 0 are determinate lines.-A. C ] Note on an Excejitional Case in which the Fundamental Postulate of Professor Sylvester's Theory of Tamisage fails.