232 PROCEEDINGS OP THE AMERICAN ACADEMY 



]f we consider co fixed aud q variable, we shall find the equation of a 



plane containing all the points where tangents from co meet the surface. 



But this 



S{j(jpa) = 1 



is the same equation that we have just found for the polar plane of w. 

 And this is indeed what we should expect, for if two radii vectores 

 from CO become equal, their harmonic mean is equal to each of them, 

 and must reach the surface where they do. We see that the polar 

 plane of any point on a tangent plane must pass through the point 

 of contact, and that the polar plane of any point cuts the surface in 

 the locus of the points of contact of tangent lines drawn from that 

 point. 



Relations between Polar Planes and Conjugate Diameters. 



The function cpn is, as we have seen, not changed in direction 

 by varying the tensor of q. The polar jjlanes, then, of all points in 

 the same straight line from the origin are parallel, for they are all 

 represented by 



Sncpa = 1, 



where the tensor only of a varies. But the polar plane 



S(»qp« = 1, 



where a is a vector of the surface, becomes a tangent plane; and this 



is parallel to 



Sr»(jr« = 0, 



the diametral plane bisecting all chords parallel to «. Hence, we find 

 that the polar plane of any point is parallel to the diametral plane con- 

 jugate to the diameter passing through the point. From another prop- 

 erty of conjugate diameters, it is seen that the diameter through any 

 point bisects all chords that it meets parallel to the polar plane of that 

 point. Again, 



Saq}Q = 1 

 is the same as 



Sa(cpQ — « — 1) = ; 



and, in order that this should be equal to 



Saqpp = 0, 



« — 1 must vanish and a become infinite. Thus, in the same way that 

 a tangent plane is the polar plane of a point on the quadric, a diame- 

 tral plane is the polar plane of a point at infinity. 



