[ BEATTY] AN OUTLINE OF ANALYTICAL SPHERICAL GEOMETRY 261 



We derive from these two, since PF = e . FM, the relation 



e sin~V/ = sin'^A' 

 That is K = sin (e sin"^ //). 



XIII. The roots of //^ and K^ and also the sin"'' operations 

 are here free from ambiguity. This form K = sin (e sin"^ //) is not 

 so interesting as the previous one; however, if we make the sub- 

 stitutions before mentioned and let a finite portion of the sphere 



become a plane surface, we get first 



H' 



since the limit of 



H = o. Later get rid of the one appearing in K^ and the form 

 assumed by the equation is: — 



a? (1 _ é^) -f- ,f = a^ (1 _ e^), 



or 



or 



= 1, 



Y 



where 



XIV. The curve that corresponds to the hyperbola would pre- 

 sent no new features and so the loxodrome will be next considered. 

 This curve is defined to be one which makes a constant angle with 



each meridian it encounters. The meri- 

 dians in the system adopted are all re- 

 presented by tan 6 = c, for various 

 values of c, ranging from — ;» to -|- ps . 

 Let the constant angle be a. Let P and 

 Q be two points on the curve, the co- 

 ordinates of P being and (d, 0) and 

 those of Q as shewn. Let ts = PQ. 



o 



Therefore 



Limit 

 Ad = 



and 

 That is 



T F 



A (f> 



X 



Then 



and 



A d 



A (f> 



_ cos QF 

 sin a 



_ sin (/? -f" a) 

 cos QM 



dcj) 



A ^ d 6 ~~ cos 



cos ^ = sin 6 sin <f), 



A (f) _ cos QF sin (/? -\- a) 



A d cos QM ' sin a 

 cos PN cos (f) = cos PT cos d. 

 d(f> 

 dd 



cos PT 



73-yr cos /? (1 -j- tan /? cot a), 



sin 6 sin cj) cos sec 6 (l ~\- tan ^ cot a). 



