traced upon the Surface of the Sphere. 349 



equation sought : but except in the case of e =. -#, the reduction presents a 

 result too complex to enable us to judge of the nature of the curve. We 

 shall, therefore, for this purpose, take particular values of constants ; that 

 is, instead of taking the origin and direction of co-ordinates arbitrarily, we 

 shall confine them to specific positions. In cases where the intersection of 

 such a locus, arbitrarily situated with respect to other given curves, is sought, 

 we must of course recur to the original equations above given. 



Imo, Let the pole of reference be the extremity of the base. 



FIG. 27. 



Put PM = A, and take the meridian MP as origin of 0. Then, PL = <. 

 Let the given angle MPL L ; then, by a known theorem in spherical tri- 

 gonometry, we have at once the equation 



cot X sin <p cot L sin Q cos (p cos 6 (5.) 



Resolving this with respect to (p, we obtain 



cot L sin 6 cos 6 -+- cot X (cosec 2 A cosec 2 L sin 2 6}* 



cos = = j.i . 



cosec 1 A sin 2 o 



, cot L cot A sin 6 :+: cos 6 {cosec 2 A cosec* L sin 2 6} 3 m \ 



sin = -L IE \ _ s \t') 



cosec 2 A sin* 6 



2do, Taking it now in reference to an origin at the middle of the base, 

 and still as a polar curve, we have, putting RP* = PT = a (the same as A 



* See Figure on page 350. 

 VOL. XII. PART I. Y 



