298 On the Surfaces of the Second Order. 



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 adjacent 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 inter- 

 sected 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 inclined to the side of contact, the 

 arc intercepted between the points where the tangent circle of 

 the conic intersects the directive circles is bisected in the point of 

 contact ; therefore, either of the spherical triangles whose base is 

 the tangent arc so intercepted, and whose other two sides are the 

 directive circles, has a constant area ; because, if we suppose the 

 tangent arc to change its position through an indefinitely small 

 angle, and to be always terminated by the directive circles, the 

 two little triangles bounded by its two positions and by the two 

 indefinitely small directive arcs which lie between these posi- 

 tions, will have their nascent ratio one of equality, so that the 

 area of either of the spherical triangles mentioned above will 

 not be changed by the change in the position of its base. But 

 in each of these triangles the angle opposite the base is constant ; 

 therefore the sum of the angles at the base is constant. 



From this reasoning it appears that if a spherical triangle 

 have a given area, and two of its sides be fixed, the third side 

 will always touch a spherical conic having the fixed sides for its 

 directive arcs, and will be always bisected in the point of contact. 



7. The intersection of any given central surface of the 

 second order with a concentric sphere is a spherical conic, since 

 the cone which passes through the curve of intersection, and has 



