Royal Irish Academy. 437 



5. If a variable spherical angle turn round a fixed point on the 

 surface of a sphere so as to intercept between its sides a constant 

 segment on a fixed arc, the arc joining the points in which its sides 

 meet two other fixed arcs will envelope a spherical conic touching 

 these two fixed arcs. 



6. If a constant spherical angle turn round a fixed point 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 spherical conic passuig 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 tw6 tangent arcs to a spherical conic intercept between them 

 a segment of a constant length on a fixed tangent 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 equa- 

 tion of a spherical conic, expressed in spherical coordinates : — 



1 1 . 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 given length ; if -^ — — 



' a ^ cos- a 



4- -^IBliL = 1, the moveable arc wiU envelope a spherical conic 



cos- Ip 

 whose principal diametral arcs are 2 a and 2 ft ; they will pass 

 through the fixed points, and the centre of the conic will be the pole 

 of the great circle passing through the two fixed points. 



12. The base of a spherical triangle being a quadrant, if its base 



anrfes, a, b, be such that ^"^ -f ---^' = 1, where a and /3 are 



tan- a tan- b 

 given arcs, the locus of the vertex will be a spherical conic, whose 

 principal diametral arcs arc 2 a and 2 ft; they will pass through the 

 extremities of the given quadrant, and the centre of the conic will be 

 the pole of the quadrant. 



Some of the preceding theorems lead to new and very general 

 properties of the conic sections : and one (No. G) gives rise to a new 

 and remarkably simple organic descrijjtion of thcni. It should be 

 observed that the arcs here spoken of are all arcs of great circles. 



