495 



planes, it will be found that the points of contact all lie in a 

 bifocal right line, and that the normals at these points lie in 

 a plane parallel to the axis of the surfaces ; so that the part 

 of any normal which is intercepted by the two principal planes 

 is constant. This theorem may be proved from the two follow- 

 ing properties of the paraboloid: — 1. A normal being ap- 

 plied to the surface at the point S, the segments of the 

 normal, measured from S to the points where it intersects 

 the planes of the two principal sections, are to each other in- 

 versely as the parameters of these sections. 2. Supposing 

 the axis of j: to be that of the surface, the difference between 

 the coordinates x of the point S and of the point where the 

 normal meets the plane of one of the principal sections, is 

 equal to the semiparameter of the other principal section. 



§ 11. Let a tangent plane, applied at any point S ofa 

 surface of the second order, intersect the plane of one of its 

 focals in the right line 0, and let P be the foot of the per- 

 pendicular dropped from S upon the latter plane. The pole 

 of the right line 9, with respect to the principal section 

 lying in this plane, is the point P. Let N be its pole with 

 respect to the focal. Then if T be any point of the right 

 line 9, the polar of this point with respect to the section 

 will pass through P, and its polar with respect to the focal 

 will pass through N ; and if the former polar intersect the 

 dirigent curve in A, A', and the latter intersect the focal in 

 F, F', the points F, F' will correspond respectively to the 

 points A, A', and the distances A A' and FF' will be similarly 

 divided by the points P and N (See Part L § 8). But since 

 the point S is in the plane of the two directrices which pass 

 through A and A', the lengths AP and A'P, which are the 

 perpendicular distances of S from the directrices, are pro- 

 portional to the lengths FS and F'S. Therefore FN is to 

 F'N as FS is to F'S and the right line NS bisects one of the 

 augles niude by the right lines FS and F'S. And as this holds 

 wherever the point T is taken on the right Une 9, that is, 



