332 



one side is a binary qiiantic in .v, on the other side it represents a 

 curve in the plane of the conic K. An arbitrary quadric, for instance, 

 can always be thrown into the form h^x^^^ -\- h^ip^ -]- h^\^^, and 

 therefore it represents a right line meeting the conic Km two 

 points, at which the parameter x is equal to either of the roots of 

 the quadric h, -^^ -}- h^tp^ -\- h^^^. In particular, since the quadrics 

 *,,§2'^8' ^^'® perfect squares, the right lines ^j, |,, §3, are tangents 

 of K with the points of contact, say, ^1,^45,^3. In this way each 

 element of the involution J corresponds with a conic and the in- 

 volution J itself defines a linear system S of conies, such that the 

 system is determined by three of its elements. Evidently the system 

 S thus constructed must contain : the conic K, the three pairs of 

 right lines Ijip,, ^^V's» -sV's and lastly the double line T. Since S 

 contains a double line, it is not a w^holly general system. Its Jacobian 

 breaks up into the right line T and into a conic H. The Jacobian 

 passes through every point of contact of two conies belonging to S, 

 hence the conic H passes through the points A^,A2,As, and meets 

 K in a fourth point A^. The tangents to K at the points A^, A „A^, 

 i.e. the right lines ^,, §,,§,, and also the tangent i;^ at the point A^ 

 intersect i^ again respectively in the points i^^, i?,, ^j, S,. The latter 

 points, lying on the Jacobian, are the centres of degenerate conies 

 êi^i,êjV's,^3»ï'3 and of a fourth degenerate conic S,tf%, and thus we 

 have proved that the involution J besides the three elements i^xp^, 

 §^■^^,^^1^^^ each having a double root, necessarily must contain a 

 fourth element ij^tf^ that has the same peculiarity. 



In a certain sense the tangent 5^ considered as a binary quadric 

 is directly connected with the reducible integral. Let us seek for 

 the points in which the conic K is touched by any conic of the 

 system S. If R, and B^ are two arbitrary elements of S, the 

 equation of the system is 



In order to find the values of the parameter .r at the points of 

 contact, we must again conceive R, and R, as binary quartics and 

 the required parametric values are the roots of the sextic 



{R^R,' -R,'R,) = 0. 



Now, as was said before, the points of contact in question are 

 the points in which K meets the Jacobian. Hence the roots of the 

 sextic are the parametric values of .x at the points A^,A^,As,A^ 

 and at the points of intersection of K with the right line T. There- 

 fore the sextic is the product of the five quantics ^'^§^> ^^§2, ^^£3, ^^i4 



