48G 



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 repre- 

 sent 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 quadrilateral 

 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 adja- 

 cent angles which lie in the same triangle with the angle k 

 contained by the two sides of the cone that help to form the 

 quadrilateral, the sum of these two angles will be equal to 

 two right angles diminished by k ; and if we take the two 

 remaining angles of the quadrilateral, their sum will be equal 

 to two right angles increased by k ; both which sums are 

 constant. But if we take either of the other pairs of adja- 

 cent angles, the difference of the pair will be constant, and 

 equal to k. 



The same conclusion may be deduced as a property of 

 the spherical conic. Let a great circle touching this curve 

 be intersected in two points, one on each side of the point of 

 contact, by the two directive circles, that is, by two great 

 circles whose planes are directive planes of the cone which 

 passes through the conic and has its vertex at the centre of 

 the sphere. Since the right lines in which the tangent plane 

 of a cone intersects the directive planes are equally incHned 

 to the side of contact, the arc intercepted between the points 

 where the tangent circle of the conic intersects the directive 



