266 On the Surfaces of the Second Order. 



curve ; for the quantities # 2 - x^ and y z - y^. are proportional to 

 the cosines of the angles which that right line makes with the 

 axes of x and y respectively, while the values just given for 

 these quantities are, in virtue of the equation (6), proportional 

 to the cosines of the angles which the normal to the focal curve 

 at the point F makes with the same axes. 



It may also be shown, that if the directrix prolonged through 

 A intersects a directive plane in a certain point, and if a right 

 line drawn through F, parallel to the directrix, intersect the 

 same plane in another point, the right line joining those points 

 will be a normal to the curve described in that plane by the first 

 point. 



3. To find in what way the focal and dirigent curves are 

 connected with the surface, let the equations (5), (6), (7) (when 

 K does not vanish), be put under the forms 



+ = 1, (9) 



so that the quantities P, Q, R may represent the squares of the 

 semiaxes of the surface, and Pi, Qi> P 2 , Qz the squares of the 

 semiaxes of the curves, these quantities being positive or ne- 

 gative^ according as the corresponding semiaxes are real or 

 imaginary. Then we have 



(11) 



P- P Q Q 



~' ~ 



whence it follows that 



PiP.-P 8 , Q:Q 2 =Q 2 , (12) 



and also that 



Pi = P - -R, Q^Q-M. (13) 



