296 On the Surfaces of the Second Order. 



property which it is easy to prove without the aid of the reci- 

 procal cone. 



The two directive sections drawn through any point S of 

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

 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 that point are tangents to each cone as well 

 as to the given surface; therefore the side of contact of each 

 cone bisects one of the angles 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 that 

 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 the angles 

 (properly reckoned) which any tangent plane of a cone makes 

 with its two directive planes is constant. This property may 

 be otherwise proved as follows : 



Through a point assumed anywhere in the side of contact 



* This property, and that to which it is reciprocal, as well as some other pro- 

 perties of the cone, were, together with the idea of reciprocal cones and of spherical 

 conies, suggested hy my earliest researches connected with the mechanical theory 

 of rotation and the laws of douhle refraction. I was not then aware that the focal 

 lines of the cone had been previously discovered, nor that the spherical conic had 

 been introduced into geometry. Indeed all the properties of the cone which are 

 given in this Paper were first presented to me in my own investigations. Its double 

 modular property, related to the vertex as focus, was one of the propositions in the 

 theory of the rotation of a solid body, and was used in finding the position of the axis 

 of rotation within the body at a given time. But the modular property common to 

 all the surfaces of the second order was not discovered until some years later. 



