464 



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



and K, 



Lf?A'2= dxif udy^i = dyi, 



i,X2 dx2 + M?/2 dy2 — Xi dxi — y, dy^ = 0. 



(32) 



Hence we obtain 



{x2 - a-i) dxi + (?/2 — yi) dyi=0; (33) 



an equation which expresses that the right hne 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^ 4- By"^ + 2gx + 2Hy =. k, (34) 



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



section, the polar of a point A whose coordinates are X2i 



y2, is 



(ax2 + g)x + (uy-i ■\- K)y = K - GX2 — nyii (35) 



but the relations (31) give 



Aa?2 + G = 372 — a;i , ^y2 + ^ = y2- yi, 



K — GX2-ny2 = Xt{x2—Xi)+yi{y2-yi); 



and hence the equation (35) becomes 



{X2 - a?i) {x — xi) + (?/2 - yi) (y — Vi) = 0, (37) 



which, as is evident from (33), is the equation of a tangent 

 applied to the focal at the point F corresponding to A. 

 This shows that the focal and dirigent are reciprocal polars 

 with respect to the section xy, and that in this relation, as 

 well as in the other, the points F and A are corresponding 

 points. 



Supposing F' and A' to be two other corresponding 

 points on the focal and dirigent, if tangents applied to the 

 focal at F and F' intersect each other in T, the point T will 

 be the pole of the right line AA' with respect to the section 

 xy, as well as the pole of the right line FF' with respect to 



