THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 179 



The coefficients in these equations are logarithms of seconds of arc ; and/,/!,/ 2 , 

 &c. have the following values : 



/=45".225870 Z=/ 4 =50".439525 



/ 1= 43 .770258 / 6= 49 .777863 



/ 2 =32 .812Q39 / 6 =47 .523448 



/ 3 =31 .503029 / 7= 24 .504515 



In all cases t denotes Julian years of 365J days. 



5. The general precession of the equinoxes in longitude is very nearly the same 

 as the precession on the apparent ecliptic, which is denoted by $, and is given by 

 equation (552). But as the apparent ecliptic is continually shifting its position in 

 space, the motion of precession on such an assumed plane becomes the same as it 

 would be along a warped surface, and very imperfectly represents the general pre- 

 cession at times only a few hundred years from the epoch, although its maximum 

 deviation from the truth can never exceed one-fourth of a degree. But on account 

 of the importance of the subject we shall determine in a rigorous manner the gene- 

 ral precession of the equinoxes in both longitude and right ascension. For this 

 purpose we shall consider the spherical triangle formed by the fixed ecliptic of 

 1850, and the apparent ecliptic and equator of any time t. In this triangle there are 

 known the two angles and the included side ; namely, the angle of inclination of 

 the apparent ecliptic to the fixed ecliptic of 1850, which is denoted by <>", and the 

 inclination of the equator to the same plane, which is denoted by c 1} and the 

 included side which is equal to ^-\-6". The three remaining parts of the triangle 

 are the distances from the extremities of the known side of the triangle to the point 

 of intersection of the apparent ecliptic and equator, and the angle included by 

 these sides. We shall denote these quantities by ^'-\-Q", $, and e. $ denotes the 

 general precession in longitude, 3 denotes the planetary precession, which is the 

 distance between the fixed and apparent ecliptics measured on the apparent equator, 

 and e denotes the apparent obliquity of the ecliptic. The fundamental equations 

 of spherical trigonometry will therefore furnish the following formulae for the 

 determination of $, 3, and e : 



sn s sn = sn ty sn 



sin E cos 3=sin <?>" cos e l cos (^-j-0") +cos q>" sin 



sin e sin (4/+0")=sin EX sin 



sin E cos (4/-j-0")=sin c a cos $" cos (4-|-0")-|-cos Sl sin 



The negative sign is given to the first equation because a forward motion of the 

 equinox is a diminution of precession. 



The equations (567) give the values of ^ and e ; and either one of the equations 

 (568) will give the value $ when e has been determined. Since $ is always a very 

 small arc, it is determinable with all desirable precision by means of its tangent ; 

 but this is not the case with e. This quantity cannot be determined with extreme 

 precision by means of its sine, without using logarithms to more than seven places 



