208 



Xdt'^df-—^-^ {\) yde = d{—-^-—)l (2) 



\\ + x^ -^ y^J ^ W -{- X- + yV 



in which x and y are used to denote the rectangular spheri- 

 cal coordinates of the moving point (vid. page 127), and x, y, 

 the moments, in the planes of the x and y arcs of reference, 

 of the resultant of the forces acting upon the point. The re- 

 action of the surface being taken into account, this resultant 

 is tangential to the sphere, and so may be conceived to act 

 along a great circle passing through the point. 

 From equations (1) and (2) we derive a third, 



(^y-v-)*' = "(ff^) (3) 



which leads to important consequences. 



It appears from the second formula in p. 129 that, if the 

 equations (I) (2) and (3) be multiphed respectively by 



2dx 9,dy 2(ydx—xdy) 



they will give for the velocity, v, of the moving point, 

 ^^ xldx + yjydx-xdyy] -\-y[dy + x(xdy-ydx)] _ .^ 



Now, if the resultant tangential force R act always along 

 a great circle which passes through the origin, that is, if we 

 consider the case analogous to that of a central force in the 

 dynamics of the plane, 



X 



X = R , „ — 2-T, and Y = R 



In this case, therefore, which for simplicity we may call that 

 of a central force, equation (3) gives 



ydx-xdy ^ 



\+x^ + if ' ^ ^ 



and equation (4) is reduced to 



