traced upon the Surface of the Sphere. 401 



Let the annexed figure referred to A and AB be that which is expressed 



by (5) ; A being the foot of the perpendicular from the summit B of the 

 cone. Put BA b : then if EB A <, we have 



AE = r = 6 tan< (5.) 



By eliminating r we have 



6* tan* (j> {tan* a cos 2 6 + % tan a tan /3 tan 7 sin 6 cos 6 + tan 2 j3 sin* 6} 

 + 26tan<{a / tanacos0 + 5 / tan/3sin(9}+d !l tan z J=0 (6.) 



If we multiply this by cos 2 <, we may write it under the form 



= A cos 2 (p + % sin <p cos <p' (E cos 6 + F sin 6) 



+ sin S! </>(Gcos 2 <9 + 2Hsin0cos0 + Ksin J 0) (7.) 



This, with the exception of the first side of the equation, is the same as 

 (xiv. 4.) ; and hence, as this difference is merely that of less generality, we 

 learn at once that our last equation (7) designates a conic section on the 

 surface of the sphere ; or, in other words, that the intersection of a cone of 

 the second degree with a concentric sphere gives a spherical conic section 

 such as was formerly defined (xxiv.), in analogy to a very common mode 

 of defining the plane sections of the cone. The analogy we see, then, is 

 perfect between the two. 



It may here, however, be suggested, that as the present equation (7) is 



