8 PRECESSION OF THE EQUINOXES AND NUTATION 



The elementary gyration about the line .^C will be therefore, (21) 

 09 M ^^Z:A sin / sin «,/ dt. 



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

 EC, they will be 



23_ ^^_ ^A sin / sin- v,t dt. 



«/•" C 



01 ^ sin /sin nd cos nd dt. 



^"*- . m^ G 



Th3 component (23) represents a rotation of the pole about TO, the radius of its 

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

 of this minute displacement, it nuist be multiplied by cos /; and to refer this to the 

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

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



_ =i^=Mj- (r=number of units of time in one year), we get, for precession, 

 jj 2'" 



Zn-rC—A J, 3 w, C—A j ■ ^ , 

 25 — I cos I.t ~ cos 1 sm Ind. 



In G ^n G 



And for nutation 

 26. 



3 ru G — A • T n t 

 i sni i cos Ind. 



\n G ' 



lit 

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



annual solar precession 



07 Q "1 G—A J- 



li. 6—n — --. — cos /. 



11 G 



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

 equation (f the equinoxes -in longitude, or the fluctuating terra 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 tlie ecliptic, which line itself, by the pro- 

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



-'p 



In the spherical triangle PP'T in wliich P" is the pole of the ecliptic 

 and PP an arc of a great circle throusrh which the pole P has moTcd (equal to (23) x cos /), and 



the sides P"P are = /, the angle PP"P (or the elementary precession) =£^. 



sin / 



