1862.] 107 



or the expression for a linear factor is 



f) - 



which involves the arbitrary quantities (#',*/'). 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 in- 

 stance, the expressions for the coordinates of the points of inter- 

 section 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. All these questions depend on the 

 decomposition of a decomposable ternary quadric, which decompo- 

 sition 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=(#^5, y, z)* into the form W 2 -f V, where W is a linear function 

 given, save as to 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=W 2 + 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. 



It will be sufficient for the present abstract to quote one only of 

 these forms, viz., 



(a, . .fa, y, z) 2 = Quotient by (a, . . fa', y', z') 2 of , 



+ Quotient by (A, . . fans' ny\ . ,) 2 of Product 



y> 



L J *?,y>,J 



I, m, n 



where the notation (which is of course explained in the Memoir) 

 will, I think, be understood without difficulty, and I do not stop to 

 explain it here. 



The solution of the geometrical questions above referred to is, as 



