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The decomposition into its linear factors of a decomposable quadric function cannot be effected in a symmetrical manner otherwise than by formulæ containing supernumerary arbitrary quantities; thus, for a binary quadric (which of course is always decomposable) we have (a, b, c)(x, y)2 = 1/(a, b, c)(x1, y1)2 Prod. {(a, b, c)(x, y)(x1, y1) ± √ac-b2(xy1-x1y)}; or the expression for a linear factor is 1/√(a, b, c)(x1, y1)2{(a, b, a, b, c)(x, y)(x1, y1) ± √ac-b2(xy1-x1y)}, which involves the arbitrary quantities (x1, y1). And this appears to be the reason why, in the analytical theory of the conic, the questions which involve the decomposition of a decomposable ternary quadric have been little or scarcely at all considered: thus, for instance, the expressions for the coordinates of the points of intersection of a conic by a line (or say the line-equations of the two ineunts), and the equations for the tangents (separate each from the other) drawn from a given point not on the conic, do not appear to have been obtained. These questions depend on the decomposition of a decomposable ternary quadric, which decomposition itself depends on that for the simplest case, when the quadric is a perfect square. Or we may say that in the first instance they depend on the transformation of a given quadric function U = (*)(x, y, z)2 into the form W2+ V, where W is a linear function, given save as to a constant factor (that is, W = 0 is the equation of a given line), and V is a decomposable quadric function, which is ultimately decomposed into its linear factors, = QR, so that we have U = W2 + QR. The formula for this purpose, which is exhibited in the eight different forms I, II, III, IV, I(bis), Il(bis), Ill(bis), IV(bis), is the analytical basis of the whole theory; and the greater part of the memoir relates to the establishment of these forms. The solution of the geometrical questions above referred to is (as shown in the memoir) involved in and given immediately by these forms. It is also shown that the formulæ are greatly simplified in the case e. g. of tangents drawn to a conic from a point in a conic having double contact with the first-mentioned conic, and that in this case they lead to the linear Automorphic Transformation of the ternary quadric. The memoir concludes with some formulæ relating to the case of two conics, which however is treated of in only a cursory manner.

XXIX. On the analytical theory of the conic