384 PROCEEDINGS OF THE AMERICAN ACADEMY 



18. The two tangents drawn from any point to a spherical conic 



p make equal angles with the arcs 



joining that point to the foci. 

 The poles of pt and pt' lie on 

 the supplementary conic ; the 

 poles of PF and pf' lie on the 

 7~/ supplementary cyclic arcs. More- 

 over, all these poles lie on the 

 circle of which p is the pole. 

 Hence, by Art. 1 1 (j3), the poles 

 of FT and PF are the same dis- 

 tance apart as those of pt' and 

 pf', and therefore the angles tpf and t'pf' are equal. 



When the point p is on the conic, this theorem becomes the follow- 

 ing, which was one of the first discovered properties of spherical 

 conies : — The two arcs from the foci to any point of the conic make 

 equal angles wiih tlie tangent at that point. It follows immediately 

 also that, if from any point of one of two confocal conies tangent arcs 

 be drawn to the other, these tangents make equal angles with the tan- 

 gent to the first conic at the given point. 



19. The theorem of Art. 18 also proves that, if two confocal 

 conies intersect, they intersect at right angles. 



Since two confocal conies imply two supplementary concyclic conies, 

 we see that, if a common tangent arc be drawn to two concyclic conies, 

 the part intercepted between the two points of contact will be a quad- 

 rant. This is evident from the first part of this article, for this arc 

 measures the right angle made by the two confocal conies. 



20. AVe now come to an important theorem : — The projection of a 

 spherical conic on a tangent plane to the sphere at one of the foci is 

 a plane conic having the point of contact for a focus. Let p -[- (/ = 

 2a; then if ^ = the angle which q makes with the axis, 



cos (/ = cos 2« cos (I -|- sin 2« sin q ; 



but we also have by spherical trigonometry 



cos (>' = cos Q cos 2c -}~ sin q sin 2c cos Q, 



cos 1c — cos la 



tan p =:: 



in 2u — sin 2c cos 0' 



which is the polar equation of a spherical conic referred to a focus as 

 pole. Since tan q is the projection of q on the plane, and d remains 

 the same in the plane, — being the angle of the tangents to the sphere 



