[beattt] an outline OF ANALYTICAL SPHERICAL GEOMETRY 259 



IX. If we attempt to see what this becomes, when we make the 

 customary substitutions with R tending to infinity, we shall first get 

 rid of the unity appearing on the left side; the form then assumed is 



r^ = {X - x'Y -f (y - y')\ 



X. We can get equations for 

 curves — not plane curves in this 

 case — which are analogous to the 

 conic sections of analytical plane 

 geometry. We define the curve to 

 be the locus of a point which moves 

 so that its distance from a given 

 point bears always a fixed ratio to 

 its distance from a given great circle. 



XI. The case where this ratio is 

 unity will first be treated. Let the 

 fixed point be F, whose co-ordinates 

 are (a, 0). Let the great circle be 

 tan 6 -[- tan a = o. By a previous 

 theorem we have the relation 



O T F 



n I tnn^ PF\ - (1 + tan' 6 + tan^ 0) (1 + tan^ a) .. 

 (1 + tan FI^) (1 -f tan tan a)\ 



(1 + 

 To get the value of PNwe have that 



sin' P .V = sin' (a + 6) cos' P T | '"^ [iJ'pl 



But 



Therefore 



tan P7' = cos 6 tan ^. 



sm 



, py ^ (sin a cos 6 -\- cos a sin dY sec' d _ sin' a (1 -f- tan' (9 cot-a)* 

 (sec' d tan' (f>) ~ (1 -f- tan' 6 + tan' <^) 



Since here PF = PîVthe equation is 



sin' a (1 -(- tan 6 cot a)' 

 (1 + tan' l9 + tan' (j)) ^ 



sec' g (1 -f tan^ + tan' 0) -(1 + tan^ tan a)' 

 (1 4- tan' d + tan' 0) sec' a 



Sec. III., 1908. 17. 



