436 Prof. Hanumanta Rao, On a property of focal conks, etc. 



For this purpose eonsider a sub-group of the possible positions 

 of the moving plane, namely the planes all of which pass through 

 a fixed point. They are then tangent planes to an enveloping cone 

 of ^, and we have the result that through the curve of inter- 

 section of this enveloping cone and the quadric a there passes 

 another cone touched by the planes z, t. This is so in virtue of the 

 following simple result: 



Given two quadrics A, /x with two common conies z, t, through 

 the common curve of A and any enveloping cone of /x there passes 

 another cone touched by the planes of z, t. For if >S = and S = zt 

 be the two quadrics, any enveloping cone of the first is SS' = P^ 

 and meets the second in a curve lying on the cone P^ = S'zt, which 

 clearly has z, t for tangent planes. 



We have thus estabhshed the well-known result for the common 

 curve of two arbitrary quadrics, viz. that a varying tangent plane 

 to one of the cones through this curve makes with two fixed tangent 

 planes to a second such cone, angles with a constant sum. 



4. Quadrics having ring contact are just concentric spheres, and 

 two quadrics with two common conies are equivalent to two spheres. 

 But no such complete projective reduction can be effected in the 

 case of two arbitrary quadrics, and what we have done in Art. 3 is 

 to estabhsh a general result for the common curve of two arbitrary 

 quadrics by deducing it from the particular case where one quadric 

 degenerates into two planes. 



