505 



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 direc- 

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

 ratrices of the hyperboloid of one sheet, or the hyperbolic 

 paraboloid, 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 de- 

 duced 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 coordinates ^, rj,^. 

 Suppose p to be the radius of the sphere with respect to which 

 the surfaces A and B are reciprocal. Then if A be a cen- 

 tral surface expressed by the equation (11), and having 

 ^0) Vo> ^0 for the coordinates of its centre, the surface B will 

 be represented by the equation 



(P - Po) K' + (P' - Po) V' + (P" - Po) Z' ,,^. 



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

 equation of B will be 



(p -po)^^ + (y _;,„)^^ + (p-_^„)S^ 



=. 4p^ (g cos a + jj cos j3 -f ^ cos 7), 

 where a, j3, 7 are the angles which the axis of x makes with 



* This relation was first noticed by Mr, Salmon. 



