. 



traced upon the Surface of the Sphere. 315 



If we write it under the temporary form of 



sin 2 2i = A cos* + 2B cos (p sin (f) + C sin* <p, 



we get 



s jn (h - -4- , / c A sin* 2i B* \/B 2 2(C A) sin* 2s + sin 4 2i , 5 , 



~ V ~ (C A) 2 + B 2 



This value of sin <p is fourfold, composed of two equal pairs of values, 

 the individuals of each pair being equal, and affected with opposite signs. 

 This indicates pairs of points on the radius-vector, the individuals of each 

 pair being at the distance of TT from each other, that is, diametrically oppo- 

 site ; thus forming two equal and opposite branches, situated one on the 

 convex and the other on the concave hemisphere. This is clear, if we re- 

 flect that sin = sin (sin TT + <p) ; and that hence whatever curve sin (p 

 traces out, the equation of sin (p will trace out a similar one upon the 

 opposite hemisphere. 



In the process of forming equation (2.) we lost all traces of the distinc- 

 tion of cases of the problem, whether 2i be equal to the sum or to the 

 difference of the focal distances of the points of the curve whose general 

 equation is (3.) : but we find it re-appearing in the radical of equation (4.), 

 for the + refers to the locus where the sum, and where the difference, is 

 given. When the sum is taken, the locus is composed of two isolated but 

 equal and similar branches, situated point for point diametrically opposite 

 to one another, as in Fig. 20. When the difference is taken, the locus is 

 composed of four isolated branches, ranged in two pairs, the corresponding 

 points in either pair being diametrically opposite, as in the former case. 

 The discussion of the general equation of the spherical conic sections is 

 full of interesting results, especially for the analogies which they bear to 

 lines of the second degree, and the oftentimes curious modifications which 

 that analogy undergoes. This we cannot, however, for want of room, enter 

 upon in the present paper*. 



The general equation (3.) is not, however, necessary when our object is 

 simply to investigate the character of this pair of curves isolated from all 

 others which may be traced on the same sphere. For this purpose, we may 

 suppose the equator to pass through the foci, and the prime meridian to bi- 



* See also Note (C) at the end. 



