QUADRATIC VECTORS. 375 



2. The classification which I propose to make of various types of 

 quadratic vector depends upon the existence of directions called axes, 

 which are directions of the point-vector p such that we have simulta- 

 neously 



yZ - zY = 0, zX - xZ = 0, xY - yX = (3) 



Geometrically stated, an axis of F(p) is a direction of p such that F (p) 

 either vanishes in all its components or is parallel to p. Those axes, 

 if any, for which F(p) vanishes, may be called the zeros of F{p). 



It is well known that the number of sets of values of the ratios of 

 X, y, and z satisfying equations (3) is, when X, Y, and Z are ternary 

 forms of degree p, in general equal to p^ -[- p -{- 1} A quadratic 

 vector form has therefore, in general, seven axes. 



Of these seven axes, however, some may be in coincidence, giving 

 multiple axes. 



If, on the other hand, there are more than seven axes, there are 

 an infinite number. For let there be eight or more axes, and suppose 

 their number finite. Equations (3), if satisfied, will subsist after a 

 change of the coordinate system. We may therefore suppose no axis 

 to lie in the plane 3=0. The first two of equations (3) will have ten 

 or more solutions, viz. the two lines of intersection of the plane 2=0 

 with the quadric cone Z = 0, and the eight or more axes of the vector. 

 But two cubic equations with ten solutions have an infinite number. 

 The third equation is a consequence of the first two wherever z is 

 not zero. Hence the three equations have a common linear or quad- 

 ratic factor, and the number of axes cannot be finite. 



These facts suggest a division of quadratic vectors into three 

 types,— 



I. General type. There are seven, and only seven distinct axes. 



1 For two proofs from very different points of view, see Darboux, "Memoirs 

 sur les Equations Differentieles Algebriques," Bui. Sci. Math. 13, (1878), 

 p. 83, where the solutions of (3) give the singular points of a differential equa- 

 tion; and Clebsch, "Lecons sur la Geometrie," (Tr. Benoist), t. II p. 113 and 

 t. Ill p. 435, (Vorl. u. Geom. Bd. I s. 393, 1001), where these equations are 

 connected with a quadratic complex. 



Again, if we let a point transformation in homogeneous coordinates be 

 defined by the equations x = X, y = Y, z = Z, then (3) gives the fixed points, 

 together with the singular points. 



A short proof for the present case is as follows, — let the first two equations 

 of (3) define two cubic curves. They intersect, in general, in nine points, of 

 which two are the intersections of the line z = and the conic Z = 0. If z 

 and Z are not both zero the determinant xY — yX must vanish. Hence at 

 9-2 points equations (3) hold simultaneously. 



