SOS 



^at' = d(-~~--^ {\) Ydt' = d(-—^-—)i (2) 



\1 H- a;^ + i/J ^ W + x^ + y^J 



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

 cal coordinates of the moving point (vid. page 1S7), and x, y, 

 the moments, in the planes of the a; 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, 



(x,-..)*' = .(l^=^) (3) 



which leads to important consequences. 



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

 equations (1) (2) and (3) be multiplied respectively by 



2dx 2dy . 2(ydx — xdy) 



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



^C x[dx + y(ydx - xdy)] + y [Jy + a; (xdy—ydx)] _ ^^ 

 J l+x^ + y'^ 



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, 



and Y = R 



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

 oi a. central force, equation (3) gives 



ydx — xdy _ 



l+x^' + y 

 and equation (4) is reduced to 



hdt, ip) 



