Connected tuith the Earth's Field of Force. 147 



Let Z be the component force acting on the sphere 

 through its center normal to the parallel surface. From 

 the manner in which the latter is constructed it is easily 

 seen that the line of action of Z is also normal to the 

 level surface, and let X be the component of force acting 

 perpendicular to Z in the plane of the meridian, and posi- 

 tive when toward the equator. The attraction of the 

 earth and the centrifugal force of rotation, both of which 

 act on the sphere as if its mass were concentrated at the 

 center, are included in X and Z. Let / denote the force 

 of friction at the point of contact ; / evidently is directed 

 along a tangent to the level surface and parallel to X; 

 it is the only force having a moment about the center of 

 the sphere. Let R denote the reaction of the level sur- 

 face at the point of contact. 



Resolving the forces along tangent and normal, we 

 find^^ 



--g=X+/ (1) 



z+ a = 



and taking moments about an axis through the center and 

 perpendicular to the meridian 



-mF^^g=/a (2) 



where m is the mass of the sphere and k its radius of 

 gyration ='v/| ^- Eliminating between (1) and (2) gives 



-'''df^ = V+¥^' (^) 



Let ge be the intensity of gravity at sea level at the 

 equator ; then g, the intensity in latitude <^, may be written 



g = ge(l -^ (3 sia' <^), (4) 



p being a constant, and it may easily be shown-^ that a, the 

 change in the direction of gravity for elevation h above 

 the level surface, is given by 



a = — /3 sin <^, (5) 



r being the radius vector of the earth. The direction 



" The general treatment of a sphere roUing on any surface is given in 

 Eouth, Advanced Eigid Dynamics (5th Ed.), p. 143. ^^ , ^^ ^o 



2« Clarke, Geodesy, p. 101. Helmert, Hohere Geodasie, Vol. II, p. 98. 



