298 VII. DYNAMICS OF ROTATING BODIES. 



dS 



Bearing in mind that -Fffr) = . vL> and that -- = r, we see that 



Sill $T O 



the first equation is the integral of equation 78), the second of 79), 

 and the third of 80). 



96. Rotation of the Earth. Precession and Nutation. 



Since the earth is not an exact sphere, it is not centrobaric, that is 

 the direction of the resultant of the attraction of its various parts 

 on a distant point does not pass through its center of mass. Or, in 

 other words, the attraction of a distant mass -point, not passing 

 through the center of mass of the earth, possesses a moment about 

 it, which tends to tilt the earth's axis. The sun and moon are so 



nearly spherical that they may 

 ^.^ be considered as concentrated 

 < r~/\ a ^ their respective centers of 



4<^_J mass. One of them, placed 

 Fig. 106. at M (Fig. 106), attracting the 



nearer portions of the earth 



more strongly than the more distant ones, tends to tip the earth's 

 axis more nearly vertical in the figure, and it is seen that this is 

 the same in whichever side of the earth the body lies. Thus the 

 sun always tends to make the earth's axis more nearly perpendicular 

 to the ecliptic, exept when the sun lies on the earth's equator, that 

 is at the equinoxes. The deflecting moment thus always tends to 

 cause a motion of precession in the same direction, the tendency 

 being greatest at the solstices, and disappearing at the equinoxes. 

 The moon, which moves nearly in the plane of the ecliptic, produces 

 a similar effect. 



It will be shown, in 148, that the potential of a body at a 

 distant point, x, y, 8 is given very approximately by 



J, - -- 



- - 2 - -pr- 



where r 2 = x 2 -f y 2 + 2 , and A, B, C, are the principal moments of 

 inertia of the body. If the distant point is the center of the sun, 

 whose mass is m, the force exerted by the earth on the sun is 



\zr\\ v 2F T/ dV dV 



X = r m-, Y= ym , Z 



