10 PRECESSION OF THE EQUINOXES AND NUTATION 



1,1 wliich the lino of greatest deelination is regarded as stationary during the single 

 revoUition, and taking a eonsecutive position for the next; but it will be m hai- 

 inonv witli the fact, and allowable, to regard the line as in contniuous motion and 

 the above amount of gyration to be uniformly spread over the time ~^, of the revo- 

 lution, producing thus an elementary gyration, in the time (//, of 

 ■ o, ^ "•^"' £""^- sin i dt. 



Let fS^ be a great circle in the plane of the ecliptic; T O: the line of equinoxes, 



NON' the line of moon's nodes, Ti^ii the equator, 

 and NinMN' 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/representcd 

 ■^ by EE'\ the precession produced will be the 

 ^ 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 X is the supplement of /(inclination 

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

 variable inclination of the moon's orbit to the 

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

 node ».,y, therefore, 



32. cos i = cos /' cos / -|- sin /' sin I cos 71/ 



and 



83. 



tang ?7/t ^ 



sni n^t 



sin /cot /'—cos /cos n/ 



In the spherical triangle mvi'°f 



'^^- tang vif = 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 th(> moon's orbit is small, the arc 3131', drawn through il/- is 

 approximately (Hpial to mr, and the angle 310^ difi'ers immaterially from the com- 

 plement of «JT; lience by (3:3) and (3-ij 



35. tang ilL1/'=^ ^:^^ /^si" ^h* cos /sin F sin n^t 



sm /cot /'—cos /cos nJ~miTcos T—cos /siiT/ cos n/ 



If the gyration about 031 (31) is decomposed into components about 031' and 0-, 

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



^^- A'sini cos il/J/W/, 



and for the second 



^"^^ K sin i sin 313rdL 



