493 



points, the focal curves approach indefinitely to parabolas, 

 as do also all sections of the surface which pass through the 

 primary axis, while the surface itself approaches indefinitely 

 to a paraboloid ; so that the limit 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 ellip- 

 tic paraboloid, having one of its focals modular and the 

 other umbilicar, like each of the central surfaces from which 

 it may be derived; and the limit of a hyperboloid of one 

 sheet is a hyperbolic paraboloid, having, like that hyberbo- 

 loid, both its focals modular. 



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

 by equation (2), intersect the axis of a; in the point X, and let 

 the normal applied at S intersect the planes y?i, xz, 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 coordinate x of the point S is equal to the quantity 

 p ; but that coordinate is to LS as OS or a is- to OX, and 

 therefore the rectangle under o- and LS is equal to p. Simi- 

 larly the rectangle under a 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 correspondingintercept being measured 

 from S in a direction opposite to that which corresponds to a 

 real semiaxis. 



The rectangle under <t and the part of the normal inter- 

 cepted between two principalplanes,is equal to the difference 

 of the squares of the semiaxes which are perpendicular to 

 these planes. This rectangle is therefore constant, not only 



