SOLID GEOMETRY. 241 



11-11. If a straight line be perpendicular to its conjugate with 

 respect to any conicoid, it will be perpendicular to its conjugate 

 with respect to any conicoid confocal with the former. 



1155. Any generator of the surface y s +s? - x? = m will be 

 perpendicular to its conjugate with respect to the surface 



<u + ... + 2a'yz+ ... = 1, 

 if &c-a /1 = ca-&' t = a&-c /> ; and aa'=b'c. 



115G. In the two conicoids 



eight generators of the first will be perpendicular respectively 

 to their conjugates with respect to the second. 



1 157. is a fixed point, P a point such that the polar planes 

 of 0, P with respect to a given conicoid arc perpendicular to each 

 "tli-r ; prove that the locus of P is the plane bisecting chords 

 ular to the polar plane of 0. 



1 1 58. A hyperbolic paraboloid, whose principal sections are 



equal, i- li:iwn through two given non-intersecting straight lines; 

 prove that the locus of its vertex is a straight 1 



1 159. If two conicoids have two common generators of tin- 

 same system, they will also have two common generators of the 

 opposite system. 



1 I ''.'. If two given straight lines be generators of the same 

 system of a conic. given point, with 



respect to the conicoid, will puns i fixed p< 



1101. If two conicoida touch each other in thn- 

 will touch ea ; or will have 



common gcnciat>n. 



