314 On the Surfaces of the Second Order. 



a plane parallel to the plane ?, since the quantities p 1 -p and 

 p' ' -p" are proportional to the squares of the semiaxes of the 

 section which are parallel to ? and respectively. The focal 

 lines are therefore the generatrices of the hyperbolic paraboloid 

 at the point S. 



Putting p and q alternately equal to zero in the equation 

 (19), we get 



? + &- : i, M+-0, (21) 



p p' p" q q' q" 



the equations of two cones which have a common vertex at S, 

 the first of them standing on the focal which lies in the plane 

 ass, the second on the focal which lies in the plane xy. The 

 mean axis of each of these cones is the normal at S to the 

 hyperbolic paraboloid; the internal axis of either cone is the 

 normal to the elliptic paraboloid which has the base of that 

 cone for its modular focal. 



As the cones which have a common vertex, and stand on 

 the focals of any surface of the second order, are confocal, they 

 intersect at right angles. Therefore when two planes passing 

 through a bifocal right line touch the focals, these planes are 

 at right angles to each other. And as cones which have a 

 common vertex, and circumscribe confocal surfaces, are confo- 

 cal, two such cones, when they intersect each other, intersect at 

 right angles. Therefore when a right line touches two confo- 

 cal surfaces, the tangent planes passing through this right line 

 are at right angles to each other. 



17. When two surfaces are reciprocal polars* with respect 

 to any sphere, and one of them is of the second order, the other 

 is also of the second order. Let the surface B be reciprocal to 

 the surface A before mentioned, with, respect to a sphere of 

 which the centre is S ; and suppose B' and E to be any cor- 

 responding points on these surfaces. Then the plane which 



* Transactions of the Eoyal Irish Academy, VOL. xvn. p. 241 ; "Examination 

 Papers," An. 1841, p. cxxvi., question 4. 



