QUADRATIC VECTORS. 419 



jzy - i {xy + bz-), (160) 



as the value of (159). The cones (3) become 



(a) zizy + bz') = 0, ) 



(6) xyz = 0, > (161) 



(c) yix^— xy — bz^) = 0. ) 



The tangent plane to the quadric cone whose equation is xy -{- bz^ = 

 is, at the element i, the coordinate plane ?/ = 0. The cones (a) and 

 (b) both have i for a double element. Thus the cone (c), which at 

 the element i consists of the plane sheet y = 0, has three coincident 

 elements in common with the other cones (b) and (c). That is, i is 

 a triple axis of (160). 



At the element j, the quadric cones 



xy -\- bz"^ = and x^ — xy — bz^ = 



have the common tangent plane x = 0. The cones (a) and (b) have 

 j for a double element. Thus the cone (c), which at the element 

 j consists of the quadric sheet x^— xy — bz^ = 0, touches one sheet of 

 each of the other cones, and cuts one sheet, giving triple intersec- 

 tion, but the three consecutive elements are not coplanar. By 

 change of coordinate planes, we can, if we wish, obtain non-de- 

 generate cubics having the same order of contact at these elements. 

 The remaining axis is the intersection of the plane z = with the 

 quadric cone x^— xy — bz^ = 0, i. e. we have x = y. 



As an example of greater generality, we may take a and b both 

 arbitrary, with the axis ^^ not coplanar with either of the triple axes 

 182 or iSo. Suppose i = ^2, and j = /Sa as before, and put k = /St. Let 

 the tangent planes to the cones (3) meet in the axis jSy, that is, k = 

 /Sa = ^i. We may take /3i any vector not coplanar with /34 and jSs, 

 (for we assume now b different from zero), and may put j3i = i. We 

 may take /Se any vector not coplanar with ^2 and 183, (for we assume a 

 not zero), and may put /Se = 3. All the three-row determinants 

 become +1, —1, or 0. The vector (159) becomes, aside from a 

 constant factor, 



j{a - b)yz ^ k{xy - az^) (162) 



The cones (3) become 



y[{2a - b)z^ - xy] = 0, 



x{xy - az") = 0, \ (163) 



xyz = 0. 



