BOOK OK MATHEMATICAL PROBLEMS. 



der of revolution whose radius is J(a 9 -p 1 )', a being the semi 

 major axis, and p the perpendicular from the centre on the tan- 

 gent plane normal to the given direction. 



1 1 83. If with a given point as vertex, a cone of revolution 

 be described whose base is a plane section of a given conicoid ; 

 this bast- will touch a fixed cone whose vertex lies on one of the 

 axes of the enveloping cone drawn from the given poiut. 



1184. The straight line joining the points of contact of a 

 common tangent plane to the two conicoids, 



bubtends a right angle at the centre. 



1185. Through a given point can in general be drawn two 

 straight lines, either of which is a focal line of any cone, en- 

 veloping a given conicoid, and having its vertex on the straight 

 line. If two enveloping cones be drawn with their vertices one 

 on each of these straight lines, a prolate conicoid of revolution 

 can be inscribed in them, having its focus at the given point. 



1186. If any point be taken on the umbilical focal conic 

 of a conicoid, there exist two fixed points Z, such that if any 

 plane A be drawn through L and a be its pole, Oa is at right 

 angles to the plane through and the line of intersection of A 

 with the polar of L, 



1187. With a given point as vertex there can in general be 

 drawn one tetrahedron self-conjugate to a given conicoid, and 

 such that the edges meeting in the point are two and two at 

 right angles; but if the given point lie on a focal curve, an 

 infinite number. 



