OF ARTS AND SCIENCES. 233 



From llie relation between polar planes and conjugate diatneters, 

 it is evident that there are thi'ee rectangular directions for which the 

 polar plane of a point is perpendicular to the vector from the origin to 

 that point. It is, perhaps, needless to add that in cases where two 

 of the roots Cj, c.^, c^ are equal, — that is, where the directions for which 

 cpQ is parallel to (», degenerate into one vector and any two in a plane 

 perpendicular to it, — the directions for which central radii and polar 

 planes are perpendicular degenerate in the same way, and we have 

 surfaces of revolution. 



In the central equation of the cone, we have already noticed 'that 

 the constant term vanishes. If, now, we take the general central 

 equation of the cone in the form 



Spqp? = 0, 



a tangent plane to the surface at any point a is represented by 



So(jpa = ; 

 but this is satisfied by 



Q = 0. 



Every tangent plane to a cone, then, passes through the centre, or 

 vertex. The equation of the polar plane of any point is 



^Qcpa = 0, 



and this also passes through the centre. It is parallel to the diametral 

 plane conjugate to «, and since both of these planes pass through the 

 centre they must coincide. Indeed, both are represented by the same 

 equation 



S(»g)« = 0. 



We see, moreover, that this is the polar plane of all points on the line 



Q = xa, 

 because all such planes are parallel and all pass through the origin. 



Cyclic Normals. 



I wish to say only a word about these remarkable vectors, in order 

 to show the connection between the equation of the central quadrics 

 and the self-conjugate part of that of the paraboloids. If we take 

 the cyclic transformation of the equation 



