304 On the Surfaces of the Second Order. 



of the central surface is a 'paraboloid having parabolas for its 

 focal curves. The limit of an ellipsoid, or of a hyperboloid of 

 two sheets, is an elliptic paraboloid, having one of its focals 

 modular and the other umbilicar, like each of the central sur- 

 faces from which it may be derived ; and the limit of a hyper- 

 boloid of one sheet is a hyperbolic paraboloid, having, like that 

 hyperboloid, both its focals modular. 



10. Let the plane touching at S the surface expressed by 

 equation (2) intersect the axis of x in the point X, and let the 

 normal applied at S intersect the planes yz, zz, xy, in the points 

 L, M, N respectively. Since the section made in the surface by 

 a plane passing through OX and the point S has one of its axes 

 in the direction of OX, it appears, by an elementary property of 

 conies, that the rectangle under OX and the co-ordinate x of 

 the point S is equal to the quantity P ; but that co-ordinate is 

 to LS as OS or a is to OX, and therefore the rectangle under a 

 and LS is equal to P. Similarly the rectangle under <r and MS 

 is equal to Q, and the rectangle under a and NS is equal to R. 

 Thus the parts of the normal intercepted between the point S 

 and each of the principal planes are to each other as the squares 

 of the semiaxes respectively perpendicular to these planes ; the 

 square of an imaginary semiaxis being regarded as negative, 

 and the corresponding intercept being measured from S in a 

 direction opposite to that which corresponds to a real semiaxis. 



The rectangle under a and the part of the normal inter- 

 cepted between two principal planes is equal to the difference 

 of the squares of the semiaxes which are perpendicular to these 

 planes. This rectangle is therefore constant, not only for a given 

 surface, but for all surfaces which are confocal with it. 



Hence the part of the normal intercepted between two prin- 

 cipal planes bears a given ratio to the part of it intercepted 

 between one of these and the third principal plane, whether the 

 normal be applied at any point of a given surface, or at any 

 point of a surface confocal with it. 



If therefore normals to a series of confocal surfaces be all 

 parallel to a given right line, they must all lie in the same plane 



