56 



on the surface of a sphere, the arcs joining the points in 

 which its sides meet a fixed arc with two other fixed points 

 will intersect in a point, the locus of which will be a sphe- 

 rical conic passing through these two last mentioned fixed 

 points. 



If two tangents to a parabola intersect at a constant 

 angle, the radii vectores drawn from the focus to the two 

 points of contact will also contain between them a constant 

 angle. But, as is well known, in any conic section, the 

 point of concourse of the tangents at the extremities of two 

 focal radii vectores, which contain between them a constant 

 angle, will generate a conic section. Hence we deduce the 

 following very general properties of spherical conies. 



7. If two tangent arcs to a spherical conic intercept be- 

 tween them a segment of a constant length on a fixed tan- 

 gent arc to the curve, their point of concourse will generate 

 a second spherical conic. 



8. If a constant spherical angle turn round a fixed point 

 on a spherical conic, the arc joining the points, in which 

 its sides meet the curve, will envelope a second spherical 

 conic. 



9. In theorem 7, if the segment intercepted on the fixed 

 tangent arc be a quadrant, the point of concourse of the 

 tangent arcs will move along an arc of a great circle, 



10. In theorem 8, if the constant angle be right, the arc 

 which it subtends in the spherical conic will pass through a 

 fixed point. 



The two following theorems may be obtained by the aid 

 of the equation of a spherical conic, expressed in spherical 

 coordinates: 



11. From two fixed points on the surface of a sphere, 



the distance between which is 90°, let arcs p, p', be drawn 



perpendicular to a moveable arc, and let a, |3, be arcs of a 



, . . „ sin ^p , sin ^p' ^ , 



given length; if- — -f--\ ^=: 1, the moveable arc will 



cos ^a cos ^p 



