On the Surfaces of the Second Order. 277 



8. In this, as in the first class of surfaces, the right line 

 FA, joining a focus F with the foot of its corresponding direc- 

 trix, is perpendicular to the focal line ; and the focal and diri- 

 gent are reciprocal polars with respect to the section xy of the 

 surface. These properties are easily inferred from the preced- 

 ing results ; but, as they are general, it may be well to prove 

 them generally for both classes of surfaces. Supposing, there- 

 fore, the origin of co-ordinates to be anywhere in the plane of 

 xy, and writing the equation of the surface in the form 



(* - O 2 +{y- ytf + z* = L(x- xtf + M(y- ytf, (30) 

 which, when identified with (3), gives the relations 

 A = 1-L, B = 1-M, 



G = Lx z - x^ H = My z - y lt (31) 



K=Lxf + My?-x?-y?, 



we find, by differentiating the values of the constants G, H, 

 and K, 



Ldx-i, = dxi, Mdy z = dy^ 



Lx 2 dx z + My 2 dy-2, - Xi dx - y dy\ = 0. (32) 



Hence we obtain 



(# 2 - 0i) dx l + (y* - y^ ^i = 0; (33) 



an equation which expresses that the right line joining the 

 points F and A is perpendicular to the line which is the locus 

 of the point F. 



Again, the equation of the section xy of the surface being 



Ax 2 + f + 2Gx + 2Hy = K, (34) 



the equation of the right line which is, with respect to this 

 section, the polar of a point A whose co-ordinates are # 2 , y^ is 



(Ax, + G)x+ (By, + H}y = K- Gx 2 - Hy^ (35) 



but the relations (31) give 



Ax z + G = x z - x^ Ey-i. + H. = y^ - y^ 



(36) 

 K - 



