QUADRATIC VECTORS. 431 



y{xy + 2' - yz) = 0, a-t/2 = 0, x{xy + 2^) = 0. 



Any linear function of the left members gives a uniquely determined 

 tangent plane ?/ = at the element i, provided that the coefficient of 

 the first is not zero. That j, k, and j+k are axes verifies by direct 

 substitution, as also the quadruple character of the axis i. 



As a second example, let i be a quadruple, and j a triple axis. Let 

 the tangent plane at t be ?/ = and at j be z = 0. Let g = I and 

 (/i = 0. The vector 



j{xy + 2^) + iyz 



satisfies these conditions, the cones (3) becoming 



z{xy + 2^) = 0, yz^ = 0, and xhj + xz^ — yH = 0. 



That there are no axes except i and j is obvious by inspection, and the 

 quadruple character of i verifies easily. This is a special case of (186). 



The constant ^i is closely related to the aberrancy ^^ of any cubic 

 curve obtained from the cones (3) by plane sections through jSi, and 

 vanishes if the aberrancy of such a cubic vanishes at /3i. 



Conditions that a quadratic vector may have an axis of order 

 higher than four may be obtained by similar methods, but such 

 conditions are not needed for the complete determination of the axes 

 of a quadratic vector. For suppose a vector to have an axis of the 

 fifth order. It is evident that there can be, at most, two other axes 

 if the vector is irreducible. An examination of these is sufficient 

 proof of the quintuple character of the multiple axis. Or again, 

 suppose a vector to have only two axes. If these be tested, by the 

 rules above given, one will be foimd to be of the fourth order, at least. 

 It will then be of the fifth, or of the sixth, order, according as the 

 remaining axis is of order two or one. 



If a quadratic vector has only one axis, that must be of the seventh 

 order. As a simple example, 



%{xy + 2^ + 1/2) + J7/2 



has i for its only axis, the cones (3) becoming 



yH = 0, zixy + 1/2 + 22) = 0, i/(i/2 + 22) = 0. 



The tangent plane at i is not determinate, all combinations of these 

 equations having a double element at i. This is, therefore, not a 

 special case of (186). 



18 Salmon, Higher Plane Curves, 3rd. Ed. Art. 407. 



