238 BOOK OF MATHEMATICAL PROBLEMS. 



are drawn through the point (A", 0, Z) : prove that the angle 

 between them is 



1140. The perpendiculars let fall from the vertex of a hyper- 

 bolic paraboloid on the generators will lie on two cones of the 

 second degree, whose circular sections are parallel to the princi- 

 pal parabolic sections of the paraboloid. 



1141. If A, A' be one of the real axes of a hyperboloid of one 

 sheet, and P, P' the points where any generator meets the genera- 

 tors of the opposite system through A, A' respectively ; the rect- 

 angle APj A'P' will be constant. 



1142. In a hyperboloid of revolution of one sheet, the 

 shortest distance between two generators of the same system is 

 not greater than the diameter of the principal circular section. 



1143. The equation of the cone generated by straight lines, 

 drawn through the origin parallel to normals to the ellipsoid 



at points where it is met by the coufocal surface 



1144. The points on a conicoid, the normals at which inter- 

 sect the normal at a given point, all lie on a cone of the second 

 degree having its vertex at the given point. 



1 1 45. Straight lines are drawn in a given direction, and the 

 tangent planes drawn through each straight line to a given coni- 

 coid are at right angles to each other : prove that the locus of such 

 straight lines is a cylinder of revolution, or a plane. 



