504 



Putting Pq and g'o alternately equal to zero in the equa- 

 tion (19), we get 



P P P q 9 2 



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 xz, the second on the focal which lies in the plane xij. 

 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 olher. And as cones 

 which have a common vertex, and circumscribe confocal 

 surfaces, are confocal, two such cones, when they intersect 

 each other, intersect at right angles. Therefore when a 

 right line touches two confocal surfaces, the tangent planes 

 passing through this right line are at right angles to each 

 other. 



§ 17. When two surfaces are reciprocal polars* with re- 

 spect 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 R' and R^to 

 be any corresponding points on these surfaces. Then the 

 plane which touches the surface A at the point R, intersects 

 the right line SR' perpendicularly in a point K, such that the 

 rectangle under SR' and SK is constant, being equal to the 



* Transactions of the Kojal Irish Academy, vol. xvii., p. 241 ; Exam. Papers, 

 An, 1841, p. cxxvi., quest. 4. 



