485 



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 con- 

 tact, let two directive planes be drawn. As the circles in 

 which the cone is cut by these planes have a common chord, 

 they are circles of the same sphere; and a tangent plane ap- 

 plied to this sphere, at the aforesaid point, coincides with 

 the tangent plane of the cone, because each tangent plane 

 contains the tangents drawn to the two circles at that point. 

 The common chord of the circles is bisected at right angles 

 by the principal plane which is perpendicular to the direct- 

 ive axis, and therefore that principal plane contains the 

 centres of the two circles and the centre of the sphere. Now 

 the acute angle made by a tangent plane of a sphere with 

 the plane of any small circle passing through the point of 

 contact, is evidently half the angle subtended at the centre 

 of the sphere by a diameter of that circle ; therefore the 

 acute angles, which the common tangent plane of the cone 

 and of the sphere above-mentioned makes with the planes of 

 the directive sections, are the halves of the angles subtended 

 at the centre of the sphere by the diameters of the sections. 

 But the diameters which lie in the principal plane already 

 spoken of, and are terminated by two sides of the cone, are 

 chords of the great circle in which that plane intersects the 



spherical conies, suggested by my earliest researches connected with the mechani- 

 cal theory of rotation and the laws of double 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 investiga- 

 tions. 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. 



