452 



it is easy to see that the right line AF is a normal to the 

 focal curve ; for the quantities a?2 — Xi and ?/2 — ^i ^^^ P'^O" 

 portional to the cosines of the angles which that right line 

 makes with the axes of a; and // 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 nor- 

 mal 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 intei'sect 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 join- 

 ing 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 



^j! + S!l = i, ^ + yl=i, (10) 



Pi Ql P2 Q2 



so that the quantities p, q, r may represent the squares of the 

 semiaxes of the surface, and Pi, Qi, P2, Q2 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 



K K 



P=Z -, Q = -, R = K, 



A B 



p, = p(l— a), Qi = q(1— b), (U) 



P Q 



Q2 = 



whence it follows that 



PlP2 = P-, QiQ2 = q'^, (12) 



