302 Mr DAVIES on the Equations of Loci 



Reckoning the angle & to commence at PEP', and the radius-vector 

 PM being denoted by (p, and calling m : n the given ratio, we have 



m(f> = n6 (1.) 



We may examine a few of the cases that arise from giving particular values 

 to the constants m and n. 



I. Let m = n, then (f> = &. 



Here we shall have a curve somewhat resembling the lemniscata of 

 BERNOULLI in its general appearance, and lying wholly upon one hemi- 

 sphere. 



FIG. 9. 



During the first quadrant of longitude *, the point M will be in the 

 spherical octant EOF ; and when one quadrant of longitude has been de- 

 scribed, it will be found at O. 



During the second quadrant, the point (M,,) will be found in the octant 

 OP'Q, ; and at TT of longitude, the point will merge in P'. 



* In these figures, we suppose the sphere orthographically projected on jthe plane 

 of the meridian PEP'Q. As this will require us to represent both the hemispheres on 

 one plane, we shall distinguish that which is between us and the meridian by the name 

 of convex and the other by the name concave hemisphere. We shall trace in full line the 

 parts of the locus that lie on the convex hemisphere, and in dots those which lie on the 

 concave. The same letters are used for corresponding points on both hemispheres ; but 

 those belonging to the concave are accentuated. 



