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 



(* - *i) 2 + (y - y$ + z* = L(x- x,}* + M(y- y,}*, (30) 

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

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



G = Lx,- x^ H=My,- y l9 (31) 



K = LxJ + My? - x? - y? y 



we find, by differentiating the values of the constants 6r, H, 

 and K, 



Ldx, = d%i 9 Mdy, = dy^ 



Lx, dx, + My, dy, - #1 dxi - y\ dy = 0. (32) 



Hence we obtain 



(# a - #1) dsK 1 + (y* - yi) dy l = ; (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 + By* + 2Gx + %Hy = 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 o? 3 , y 3 , is 



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



but the relations (31) give 



Ax, + G = x, - #!, By, + H= y, - y l9 



(36) 

 K- Gx, - Hy, = ^ (x, - a?,) + y (y, - y^ ; 



