209 



C xdx + Ydi/ _ 



It is easy to show that these two latter equations are equiva- 

 lent to the two following, 



sin^pdb) = hdf, (7) 



2lRdpzzv\ (8) 



in which p is the vector arc drawn from the origin to the 

 moving point, and w is the angle between it and the a; arc of 

 reference. Let us now describe the circle of the sphere 

 which osculates the trajectory along which the point p moves, 

 and let c be the arc of the great circle passing through p 

 and the origin, and intercepted within the osculating circle ; 

 then it may be shown that 



(9) 



tan I c 



Up denote the arc of the great circle drawn from the origin 

 perpendicular to the arc touching the trajectory at p, we may 

 deduce from (7) that 



v' - -J^. (10) 



sm p 



By the help of equation (9) it may be proved that " A 

 material point may be made to describe a spherical conic if it 

 be urged by a force, acting along the arc of a great circle 

 drawn from the J'ocus to the point, and varying inversely as 

 the square of the sine of the vector arc p." 



Also: " A material point may be made to describe a 

 spherical conic by the agency of a force, acting along the arc 

 of a great circle drawn from the centre to the point, and vary- 

 ing as tanp sec^p." 



In the dynamics of a point constrained to move on the 

 surface of a sphere, we have, for the discussion of the inverse 

 problem of central forces, the following equation, 



s 2 



