On the Surfaces of the Second Order. 297 



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 applied 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 perpen- 

 dicular to the directive 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 ma^es 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 sphere ; and the 

 halves of the angles which they subtend at its centre are equal 

 to the angles in the greater segments of which they are the 

 chords, and consequently equal to the two adjacent acute angles 

 of the quadrilateral which has these chords for its diagonals. 

 Hence, as two opposite angles of the quadrilateral are together 

 equal to two right angles, it follows that the four angles of the 

 quadrilateral represent the four angles, the obtuse as well as.the 

 acute angles, which the tangent plane of the cone makes with the 

 planes of the directive sections ; the two angles of the quadrila- 

 teral which lie opposite to the same diagonal being equal to the 

 acute and obtuse angles made by the tangent plane with the 

 plane of the section of which that diagonal is the diameter. 



Thus any two adjacent angles of the quadrilateral may be 

 taken for the angles which the tangent plane of the cone makes 

 with the directive planes. If we take the two adjacent angles 

 which lie in the same triangle with the angle K contained by the 



