466 



equally subsist when one of these constants is supposed 

 to be negative (for they cannot both be negative). This 

 leads us to inquire what surfaces the equation (30) is 

 capable of I'epresenting when the constants l and m have 

 different signs ; as also, for a given surface, what lines are 

 traced in the plane oi xy by points F and A, of which xi, 

 yi and x^, yi are the respective coordinates. After the ex- 

 amples already given, this question is easily discussed, and 

 the result is, that the only surfaces which can be so repre- 

 sented are the ellipsoid, the hyperboloid of two sheets, the 

 cone, and the elliptic paraboloid — that is to say, the um- 

 bilicar surfaces together with the cone ; and that, for an 

 umbilicar surface, the locus of F is the urabilicar focal, and 

 therefore the locus of A is the corresponding dirigent ; while 

 for the cone the points F and A are unique, coinciding with 

 each other and with the vertex of the cone. A geometrical 

 interpretation of this case is readily found ; for as l and m 

 have different signs, the right-hand member of the equation 

 (30), if M be the negative quantity, is the product of two 

 factors of the form 



Six - x2) + g{y - y^), fix - x^ -giy-y^), 



in which /and ^ are constant; and these factors are evi- 

 dently proportional to the distances of a point whose coordi- 

 nates are x, y, %, from two planes whose equations are 

 f{x -x.2)+g{y- 2/2) =0, fix- X2) - giy - 7/,) = 0, 

 which planes always pass through a directrix, and are in- 

 clined at equal and constant angles to the axis of a: or of y. 

 Therefore, if F be the focus which belongs to this directrix, 

 the square of the distance of F from any point of this surface 

 is in a constant ratio to the rectangle under the distances of 

 the latter point from the two planes. And these planes are di- 

 rective planes ; because, if a section parallel to one of them 

 be made in the surface, the distance of any point of the sec- 

 tion from the other plane will be proportional to the square 



