354 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The sixteen principal directions are represented in the 64 by the 

 sixteen points of intersection of the four quadrics. 



The correspondence between t and t' will not be unique if t coincides 

 with one of the vertices of the self-conjugate five-point with respect to 

 any two of the quadrics. By choosing the defining hypersurfaces dif- 

 ferently the self- conjugate points can be chosen in cc^ different ways 

 These points are in fact the vertices of the cones belonging to the three- 

 parameter family of quadrics defined by the four quadrics above. Now, 

 if dui be taken as the homogeneous coordinates of the points of the /S4 

 above, it is seen at once that the coordinates of the vertices of these 

 cones are given by 



(17) 2(.^r^W = 0, ^=1,2,3,4,5. 



(Two parameters are independent.) We shall next show that these 

 directions are such that along them a hyperplane has two-point contact 

 with F5. 



The condition that the hyperplane (I, x) = 0h& tangent to V^ at x 

 is given by 



(18) te.)^o, (f.it)=o...(f.g=o. 



If it also passes through x + dx^ then 



(^, x-\-dx)=^ (^, x) + (^, dx) = (^, ^) + 2 U ^) d^h = 0. 



From (18) it is at once seen that this condition is verified. Now, in 

 order that the hyperplane be tangent at x + dx, we must have 



hence from (18) 



(^' 2 a^. '^"^ = 2 (f. i^) ^"^ = «- 



which is equation (17) again. Hence the directions defined by (17) 

 are those along which a hyperplane may have two points of contact 

 with Kj. These hyperplanes are called stationary hyperplanes. 



