10 



PRECESSION OF THE EQUINOXES AND NUTATION 



In which the line of greatest declination is regarded as stationary during the single 

 revolution, and taking a consecutive position for the next; but it will be in har- 

 mony with the fact, and allowable, to regard the line as in continuous motion and 



2.-i 



the above amount of gyration to be uniformly spread over the time /' , of the revo- 

 lution, producing thus an elementary gyration, in the time dt, of 



^ 3 



2 



Let 



n, 



- sin i dt. 



be a great circle in the plane of the ecliptic ; f the line of equinoxes, 



NON' tlfe line of moon's nodes, eF^ the equator, 

 and Nin'MN' the moon's orbit crossing the equa- 

 tor at m'. The line of the moon's maximum 

 declination, OM, will be 90 from the line Om. 

 The pole E of the earth is supposed to undergo 

 a displacement by gyration about OJ/represented 

 by EE'; the precession produced will be the 

 f '] m angle EOE'; the nutation, the angle E^E'. 



In the spherical triangle Nm'f the angle at N 

 is = /', the inclination of moon's orbit to ecliptic; 

 the angle at f is the supplement of / (inclination 

 of the equator) and the angle at m' is i (or the 

 variable inclination of the moon's orbit to the 



equator), and the side TiV is = nj (calling the angular velocity of the moon's 

 node 3 ); therefore, 



32. cos i = cos /' cos / -j- sin /' sin / cos nj, 



and 



33. tangm'r=. _ si ?_ 



sin /cot /' cos /cos ttj, 



In the spherical triangle mm'f 



I O 



34. tang mf = cos / tang m'r 



OS is the line of maximum declination of the sun, or the solstitial diameter of the 

 ecliptic about which the annual gyration produced by the sun is made. As the 

 inclination / of the moon's orbit is small, the arc MM', drawn through M^=, is 

 approximately equal to mf, and the angle MO^ differs immaterially from the com- 

 plement of mf; hence by (33) and (34} 



35. tang MM'=^ cos /sin *** cos /sin P sin n 3 t 



sin /cot /' cos /cos n 3 t sin /cos / cos /sin /' cos n 3 t 



If the gyration about OM (31) is decomposed into components about OM 1 and 0^ 

 we shall have for the first (calling the coefficient of sin i.dt, K) 



36. Ksln i ros MM'dt, 

 and for the second 



37. K&mi sin MM'dt. 



