8 PRECESSION OF THE EQUINOXES AND NUTATION 



The elementary gyration about the line SC will be therefore, (21) 



22 - ~ sin / sin nj dt. 



nr 3 C 



If this rotation about SC is decomposed into components about the lines TC and 

 EC, they will be 



23. , - ~- sin / sin 2 nj dt. 



nr 3 C 



o a f~i _ A 



24. = sin /sin nj cos nj dt. 



mr C 



Th.3 component (23) represents a rotation of the pole about TC, the radius of its 

 motion being PB, or cos I. To obtain the actual value as an arc of a great circle, 

 of this minute displacement, it must be multiplied by cos /; and to refer this to tfie 

 pole of the ecliptic as angular motion, it must be divided by sin I. 1 Performing 

 these operations, integrating, and remembering that by Kepler's laws 



S 4?i 2 



- =, j-=i 2 (T=number of units of time in one year), we get, for precession, 



3 x 2 C A 3 n, C A 



2o. cos I.t _ _ -- cos 1 sin 2n-.t. 



2 n C 4 n C 



And for nutation 



, 

 4 n C 



nr- 3 U-, C A r 



26. , sm /cos 



t 



The first term of (25) is the mean solar precession; making t = , it gives for the 



i 



annual solar precession 



27. 3%*0!~* cos I. 



n C 



Expression (26) is the solar nutation, and the second term of (25) gives the 

 equation of the equinoxes in longitude, or the fluctuating term of the precession cor- 

 responding to the nutation. 



These expressions correspond to those obtained by the ordinary solutions. They 

 differ from most of them, however, in having C in the denominator instead of A, 

 an error of those solutions I have alluded to before, which, however real, analyti- 

 cally, exerts no important influence on the result. 



By the above method the precession is the integral of the components of 

 gyration about a solstitial diameter of the ecliptic, which line itself, by the pro- 

 cess of precession, has an angular motion equal to that precession, the real effect 



In the spherical triangle PP"P in which P" is the pole of the ecliptic 



and PP an arc of a great circle through which the pole P has moved (equal to (23) x cos I), and 



the sides P"P are = /, the angle' PP"P (or the elementary precession) = p -^- 



sin/ 



