On the Surfaces of the Second Order. 279 



for the fixed point, the ratio of FF' to A A' is the same as the 

 ratio of FN to AP, or of F'N to AT ; and consequently the 

 distances FF' and AA' are similarly divided in the points N, 



and P. 



9. In the equation (30), considered as equivalent to the 

 equation (1), the constants L and M are both positive; but the 

 properties which have been deduced from the former equation 

 are independent of this circumstance, and equally subsist when 

 one of these constants is supposed to be negative (for they can- 

 not both be negative). This leads us to inquire what surfaces 

 the equation (30) is capable of representing when the constants 

 L and M have different signs; as also, for a given surface, what 

 lines are traced in the plane of xy by points F and A, of which 

 #!, yu and # 2 , 11* are the respective co-ordinates. After the ex- 

 amples already given, this question is easily discussed, and the 

 result is, that the only surfaces which can be so represented are 

 the ellipsoid, the hyperboloid of two sheets, the cone, and the 

 elliptic paraboloid that is to say, the umbilicar surfaces to- 

 gether with the cone ; and that, for an umbilicar surface, the 

 locus of F is the umbilicar 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 



f(x - a*) + g(y - y*), /(* -**)-g(y- y a ), 



in which /and g are constant; and these factors are evidently 

 proportional to the distances of a point whose co-ordinates are 

 a?,'y, s, from two planes whose equations are 



f(x - x z ] +g(y - y) = 0, f(x - a,) -g(y - y 8 ) = 0, 



which planes always pass through a directrix, and are inclined 

 at equal and constant angles to the axis of x or of y. There- 

 fore, if F be the focus which belongs to this directrix, the square 



