On the Surfaces of the Second Order. 315 



touches the surface A at the point E intersects the right line 

 SB,' perpendicularly in a point K, such that the rectangle under 

 SR' and SK is constant, being equal to the square of the radius 

 of the sphere. Now if the point K approach indefinitely to S, 

 the distance SR' will increase without limit, the surface B being 

 of course a hyperboloid ; and if through S any plane be drawn 

 touching the surface A, a right line perpendicular to this plane 

 will evidently be parallel to a side of the asymptotic cone of the 

 hyperboloid. The asymptotic cone of B is therefore reciprocal 

 to the cone which, having its vertex at S, circumscribes the 

 surface A. Hence, as the directive planes of a hyperboloid 

 are the same as those of its asymptotic cone, it follows that the 

 directive planes of the surface B are perpendicular to the gene- 

 ratrices of the hyperboloid of one sheet, or the hyperbolic para- 

 boloid, which passes through S, and is confocal with the surface 

 A. And this relation between two reciprocal surfaces ought to 

 be general, whatever be the position of the point S with respect 

 to them ; * for though it has been deduced by the aid of the 

 circumscribing cone aforesaid, it does not, in its enunciation, 

 imply the existence of such a cone. This conclusion may be 

 verified by investigating the equation of the surface B in terms 

 of the co-ordinates , j, . Suppose p to be the radius of the 

 sphere with respect to which the surfaces A and B are recipro- 

 cal. Then if A be a central surface expressed by the equation 

 (11), and having , r/o, o for the co-ordinates of its centre, the 

 surface B will be represented by the equation 



(P - Po) V + (P'- Po) 2 + (P"- Po) F 



(22) 

 -VtW+^ + W)-!* 1 ; 



but if A be a paraboloid expressed by the equation (18), the 

 equation of B will be 



(p-p } + (p'-p^r? + (p"- Po }? 



(23) 



= /o4 2 ( cos a + r) cos /3 + cos 7), 



* This relation was first noticed by Mr. Salmon. 



