traced upon the Surface of the Sphere. 



303 



When the longitude is greater than IT, the polar distance will be rec- 

 koned on a dotted meridian PR'P'M, which comes up (on account of its 

 also being greater than TT) on the convex side in P'M //7 ; and it therefore lies 

 in EP'O so long as 



And it arrives again at O, when < and 6 have attained each the value of 

 three quadrants. 



During the interval between the values of -=- and 2?r taken by <p and 



6, the point will be in the octant POQ, and at the termination of that pe- 

 riod, it will have returned to P, the origin. It is also evident, that, by 

 continuing the process, the same succession of changes will take place, and 

 the same path on the surface of the sphere traced out. The figure whose 

 orthographic projection is PM, OM /y P'M /A , OM IV is the complete locus, 

 therefore, of the equation 



2. Let 7ra = 2w, or < = i 0: then the figure (10.) is the orthographic 



representation ; and if m n, or < = 20, the figure (11.) is the result, In 

 this there is to be understood a dotted branch lying under the branch PRP'. 

 We have therefore also traced in Fig. 12. the two branches visible, by ta- 

 king POP'O' as the convex hemisphere, instead of PEP'Q. 



a 



3. Let (p = -, which is the particular case considered by PAPPUS. Here 

 the curve cuts the equator after a complete revolution of 0. A second revo- 



