240 BOOK OP MATHEMATICAL PROBLEMS. 



all intersect the same straight line, the normals at points on 

 the plane 



+i + *+i.o 



al bm en 



will also intersect that line. Prove also that the condition 

 for this is 



(roV - 1"') (6 2 - c 2 ) 2 + (nH* - m 2 ) (c 2 - a 2 ) 2 + (IV - n*) (a* - IJ = 0. 



If l = m=n=l t the straight line which the normals inter- 



sect is 



ax (b z - c 2 ) = by (c 2 - a 2 ) - cz (a 2 - 6 2 ). 



1151. The normals to the paraboloid 



f+--- 



b c 



at points on the plane px + qy + rz = 1 will all meet one straight 

 line if 



1152. PQ is a chord of a conicoid, normal at P ; any plane 

 conjugate to P<2 meets the conicoid in a curve A : prove that one 

 axis of the cone whose vertex is P, and base the curve A, is the 

 normal at P, and that the other axes are parallel to the axes of 

 any section parallel to the tangent plane at P. 



1153. Straight lines are drawn through the point (o? , y , z ), 

 such that their conjugates with respect to the paraboloid 



*+?-* 



a b 



are perpendicular to them respectively : prove that these straight 

 lines must lie on the cone 



and that their conjugates will envelope the parabola 



(-)'* (?)*+<-'-* 



