500 



describes a cone condirective with the surface to which it 

 belongs ; the two cones so described are reciprocal, and the 

 focal lines of the cone which belongs to one surface are 

 perpendicular to the directive planes of the other surface. 



When two confocal paraboloids intersect, if normals be 

 applied to them at any point S of their intersection, and a 

 bifocal chord of each surface be drawn parallel to the nor- 

 mal of the other, the two chords so drawn will be equal and 

 of a constant length, wherever the point S is taken in the 

 line of intersection of the surfaces ; that constant length 

 being equal to the difference between the parameters of 

 either pair of coincident principal sections. 



§ 15. The point S being the common intersection of a 

 given system of confocal surfaces, of which the equations 

 are 



t+l + ^ = ,, ^ + t + l.-^, 



r « B P' « K ^1^^ 



o;^ y^ _i_ ^^ _ I 



suppose that another surface A confocal with these, and ex- 

 pressed by the equation 



|! + | + ^=i, (11) 



ro ^O "0 



is circumscribed by a cone having its vertex at S. If the 

 normals applied at S to the given surfaces, taken in the order 

 of the equations (10), be the axes of new rectangular coor- 

 dinates ^, r\, Z,, the equation of the cone, referred to these 

 coordinates, will be* 



* The equation (12) was obtained in the year 1832, and was given at my 

 lectures in Hilary Term, 1836. The most remarkable properties of cones 

 circumscribing confocal surfaces, are immediate consequences of this equation. 

 That such cones, when they have a common vertex, are confocal, their focal lines 

 being the generatrices of the hyperboloid of one sheet passing through the ver- 



