OF AETS AND SCIENCES. 377 



the cyclic arcs of the conic. Since the cone is double, it will cut the 

 sphere in two closed curves ; and we therefore name the conic differently 

 according to the hemispliere considered. If the sphere be divided by 

 the principal plane of the cone, it gives a closed curve whose centre 

 will be the pole of the dividing circle, and whose principal diameters 

 will be the arcs of the greatest and least sections of the cone. The 

 cyclic arcs will intersect at the points where the arc of greatest section 

 meets the dividing circle, and will be symmetrical with reference to 

 the curve. This form of conic is a Spherical Ellipse. 



If the sphere be divided by the plane of least section of the cone, 

 the conic will consist of two branches. Its centre will be the pole of 

 the dividing circle, and its principal diameters will be the arcs made by 

 the plane of greatest section of the cone and the principal plane. The 

 cyclic arcs meet ordy once, and that at the centre. This curve is the 

 Spherical Hyperbola, and it will be found that its cyclic arcs have prop- 

 erties analogous to those of the asymptotes to the plane hyperbola. 



If, again, the sphere be bisected by a plane perpendicular to the two 

 already mentioned, there is still a third form of spherical conic, having 

 its centre at the pole of the bisecting circle. There is, properly speak- 

 ing, as might be expected from the method of projection used, no spher- 

 ical parabola. If a plane parabola be projected upon a sphere, points 

 at infinity are projected, and the spherical parabola is merely an 

 ellipse or an hyperbola. The conic of which the major axis is a quad- 

 rant has, however, the closest analogy to the Parabola. 



5. A spherical conic may also be defined as the locus of an equa- 

 tion of the second degree in spherical co-ordinates. The general equa- 

 tion is 



ax^ + 2hxy + hf + 2gx + 2/y + c = 0. 



This can be transformed to the centre as origin ; and, if we choose the 

 principal diameters as axes, it can be reduced to the form 



The equation for determining the centre is a cubic, and this shows that 

 a spherical conic has three centres. We are thus led in another way 

 to the results arrived at in Art. 4. 



This method of reducing the general equation is, however, on 

 account of the complex formulas for transformation of spherical co- 

 ordinates, long and tedious. It is better, therefore, to derive the equa- 

 tion referred to the centre from the central equation of plane conies. 



