484 



a reciprocal property. Thus, we have seen that the tangent 

 plane of a cone makes equal angles with two planes passing 

 through the side of contact and through each of the focal 

 lines; therefore, drawing right lines perpendicular to the 

 planes, and planes perpendicular to the right lines here men- 

 tioned, we have, in the reciprocal cone, a side making equal 

 angles with the right lines in which the directive planes of 

 this cone are intersected by a plane touching it along that 

 side. It is therefore a property of the cone, that the inter- 

 sections of a tangent plane with the two directive planes 

 make equal angles with the side of contact; a property which 

 it is easy to prove without the aid of the reciprocal cone. 



The two directive sections drawn through any point S of 

 a given surface of the second order may, when they are cir- 

 cles, be made the directive sections of a cone, and this may 

 obviously be done in two ways. Each of the two cones so 

 determined will be touched by the plane which touches the 

 given surface at the point S, because the right lines which 

 are tangents to the two circular sections at tiiat point, are 

 tangents to each cone as well as to the given surface; there- 

 fore the side of contact of each cone bisects one of the an- 

 gles made by these two tangents; and hence the two sides of 

 contact are the principal directions in the tangent plane at 

 the point S, that is, they are the directions of the greatest 

 and least curvature of the given surface at tliat point ; for 

 these directions are parallel to the axes of a section made in 

 the surface by a plane parallel to the tangent plane, and the 

 axes of any section bisect the angles contained by the right 

 lines in which the plane of section cuts the two directive 

 planes. 



§ 6. It has been shown that the sum of the angles which 

 any side of a cone makes with its focal lines is constant. 

 Hence we obtain the reciprocal property, that* the sum of 



• This propertj', and that to which it is reciprocal, as well as some other 

 properties of the coue, were, together with the idea of reciprocal cones and of 



