BOOK OF MATHEMATICAL PROBLEMS. 



1 1 1 G. The generators drawn through the point (x, y, z) of 

 the surface 



ayz + bzx + cxy + abc = 



will be at right angles, if 



1117. Normals are drawn to a conicoid at points lying along 

 a generator : prove that they will lie on a hyperbolic paraboloid 

 whose principal sections are equal parabolas. 



1118. The two conicoids 



2bcyz 2cazx 2abxy = t 4 , 

 2 b 2 c 2 _\ /ax by 



have their axes coincident in direction. 

 1119. The two conicoids 



ax* + ... +2a'yz+ ... = 1, 



have one, and in general only one, system of conjugate diameters 

 coincident in direction; but, if 



1 / 6V\ 1 /, ca 1 / a'b 



there will be an infinite number of such systems, the direction of 

 one of the diameters bein the same in all. 





1120. Prove that eight conicoids can in general be drawn, 

 containing a given conic and touching four given planes. 



1121. A, B is the shortest distance between two generators, 

 of the same system, of a conicoid; and any opposite generator 

 meets them in P, Q respectively : prove that the lengths x, y of 

 AP, BQ are connected by a constant relation of the form 



axy + bx + cy + d = 0. 



