traced upon the Surface of the Sphere. 71 



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DEFINITIONS. 



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_ 

 THE circle may be considered either as a curve traced upon a sphere or 



upon a plane. When we consider it to be traced upon a plane, its centre 

 and radius are also in that plane, and may, for distinction, be called the 

 plane-centre and the plane-radius, respectively, of that circle. But when 

 it is considered as a curve traced upon the sphere, we may call the centre 

 and radius (what are usually called the pole and polar distance of the circle) 

 by the corresponding name of spherical centre and spherical radius of the 

 circle. 



In the investigations which follow, we rarely have occasion to speak of 

 the plane-centre and plane-radius, so that no confusion can arise from drop- 

 ping the adjective spherical, when speaking of the spherical centre and sphe- 

 rical radius. When, however, we apprehend that any misunderstanding 

 might arise from the omission, we shall be careful to supply the adjectives. 



When we consider any curve traced on the surface of the sphere, we con- 

 sider it, as in other parts of geometrical inquiry, to be the locus of a point, 

 which fulfils certain given conditions, and that the algebraical expression of 

 these conditions may be reduced to an equation between two variable arcs 

 of great circles, or between an arc and an angle. When between two arcs, 

 they are supposed at right angles, like the prime meridian and equator, in 

 describing the latitude and longitude of a place on the globe. The longi- 

 tude of the point we call the abscissa, and its latitude the ordinate. When 

 we refer to the polar angle made by the meridian of the point with the 

 prime meridian, and the polar distance of this point, we call it a polar equa- 

 tion, the polar distance being the radius-vector, and the polar angle is the 

 longitude of the point. 



These two methods can only just be said to differ, since in one, the polar 

 angle is measured by the abscissa of the other, and the radius- vector in one 

 is the complement of the ordinate of the other. In principle, then, they are 

 strictly the same, and in the details nearly so : though it will often happen 

 that one consideration is more directly applicable to particular problems than 



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