180 SECULAR VARIATIONS OF THE ELEMENTS OF 



of decimals. But as e never differs greatly from f,, we may compute the difference 

 between e and ti by eliminating $ from equations (567), and we shall find, 



sin (e EI)= 



sin 2 ft" cos 2 e l sin 3 ft" sin 2 e, cos 3 (^+^')+ 2 cos sm ft" cos c i sin c ' cos C^+fl") (5(59) 



sin (E+CI) 



Altliough f appears in the denominator of this equation it is readily determinate 

 from its sine with sufficient precision to be used in finding sin (e ej with accuracy. 



Since $ never differs greatly from 4, we may readily transform equations (568) 

 so as to give the difference of these quantities, and we shall find 



sin E sin (4> 40= cos f i sm ft" s i n C^+0") \ ( 



-2 sin 2 1 tf sin f , sin (4>+0") cos (^+6") j l 



This equation determines 4 1 4 1 ' w i tn a ^ desirable precision, E having been 

 previously determined. 



6. We shall now consider the spherical triangle formed by the fixed ecliptic of 

 1850, the fixed equator of 1850, and the apparent equator of any time t. Since 

 the inclination of the equator to the fixed ecliptic of 1850 is given by equation 

 (566), we may suppose this quantity to be known for the given time. Then calling 

 fj the inclination of the fixed equator and ecliptic E/, the inclination of the apparent 

 equator to the fixed ecliptic, and ^ the total luni-solar precession during the time t; 

 the two angles and the included side of the proposed triangle will be known, and 

 the three remaining parts may readily be determined in the following manner: 

 Let the distances from the intersection of the fixed and apparent equators to 

 the fixed ecliptic be denoted by 90 z and 90-|-z', and the angle of intersection 

 of the two equators be denoted by 0, we shall have the following equations for 

 determining these last named quantities: 



sin cos z =sin 4- sin EI 



sin sin z =sin BI cos e/ cos 4/ sin e/ cos e 1 



cos =cos E! cos ^'-(-sin e, sin e/ cos ^ (571) 



sin cos z^sin 4- sin ei 

 sin sin z'=cos 4- sin s l cos e/ cos e l sin e, 



These, equations determine z, z', and rigorously ; but since e i is nearly equal 

 to e ,', they are under a very inconvenient form for accurate computation when 4- is 

 a small angle. They may, however, be readily put under the following form for 

 very accurate and convenient computation: 



sin cos z =sin 4> sin e/ 



sin sin z =2 sin 2 \ 4/ sin f/ cos ^-j-sin (e, e/) 



sin 2 \ =sin 2 \ (e l e^+sin^^sinfisinf/ } (572) 



sin cos z'=sin ^ sin ei 

 sin sin z'=2 sin 2 \ 4- sin e, cos e/ sin (e, E/] 



The sum of the quantities z and z* might very properly be called the luni-solar 

 precession in right ascension ; and if to this we add the planetary precession we 



