traced upon the Surface of the Sphere. 



321 



Hence, 



cos 2 cos'K sin 2 <r 2cos0 sin< cos<r sin<rcos/c cos0 + cosVsin 8 ^) cos t d = (cos* -cos* K) Bai'<f> . , 

 Or, subtracting cos 2 Q sin 2 $ + cos 2 K cos 1 $ from both sides, and changing 

 all the signs of the result, we obtain, by extraction, 



cos (f> cos K cos + sin (f) sin a- cos = cos K ... (9.) 



7% is the equation of a circle, whose centre K , \ and radius (>, are de- 

 termined by the following : 



tan X, = tan <r sec K 



-+- COSK 



COS ^ = 



; sin 2 K cos* <r . 



.. (10.) 



.(8.) 



XXVII. 



SPHERICAL EPICYCLOID. 



Let SN be the directrix, NM the generatrix, and R the tracing point. 

 Join RO cutting the circle NM in M, and let S be that point of the di- 

 rectrix which was in contact with M. Put PN = R, NO = r, and 

 OR = p ; put also SPR = & , SPN = % , and PR = <. 



FIG. 



Then, since SN = NM, and the radii of the circles of which these are 

 segments are as sinR to sinr, we have SN = MN = %sinR; and when 



