THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 181 



shall have the general precession in right ascension at any time t, equal to z-\-tf-\-&; 

 all of which quantities being taken from the table with the argument t. 



7. Tables I — VIII have been computed as explained in §§ 2 and 3. They show 

 the elements of the planetary orbits at the times given in the first column of each 

 table, and seem to require no explanation as to their uses. 4, and c„ in Tables 

 IX and X, have been computed from the formulae (565) and (566); $•, 4' and e have 

 been computed from equations (567), (570), and (569), by using the values of 0", 

 ft", 4, and £ x given in Tables VIII and IX; and lastly, z, z', and have been com- 

 puted by means of equations (572), by using the values of 4, f, Ei given in Table IX. 



8. Having explained the method of constructing the tables, we will now explain 

 the manner of using Tables IX and X in connection with b" and ft", which are 

 given in Table VIII. The quantities 0", ft", and 4' are useful in reducing the 

 longitudes of the celestial bodies from the mean equinox of 1850 to the mean 

 equinox of any other date, and vice versa. This transformation is effected by 

 means of the following equations, in which 1 and j3 denote the mean longitude 

 and latitude of a celestial body at the epoch of 1850, and X and (3' denote the same 

 co-ordinates referred to the mean equinox of any time I, before or after that epoch. 



cos (3' cos (X — 0" — 4') =cos /3 cos (A — b") ~j 



cos /3' sin (X — 0" — 4')= cos (3 cos ft" sin (^ — 0")-|-sin (3 sin ft" > ^573) 

 sin [3' =sin (3 cos ft" — cos (3 sin ft" sin (X, — 0") J 



For reducing to the equinox of 1850 these equations take the following form: — 

 cos (3 cos Q, — 0")=cos /3' cos (X — 8" — 4') ^ 



cos (3 sin (A — 0")=cos (3' cos ft" sin (X — 6" — 4') — sin ft" sin /3' > (574) 

 sin (3 =sin (3' cos ft"-|-cos (3' sin ft" sin (X — 0" — 4') J 



It is sometimes desirable to find the difference in the longitude or latitude of a 

 star arising from the precession of the equinoxes. This difference may be found 

 by the following formula 3 , the employment of which is perhaps more laborious than 

 that of the preceding from which they were derived; but they may in ordinary 

 eases be managed by the use office-figure logarithms, whereas equations (573) and 

 (574) require seven-figure logarithms to arrive at accurate results. 



X=X-\-^'-\- Tare tan = 



j tan p sin ft" cos (a,— 0" )— 2 sin 2 \ ft " cos (A — 0") sin (A— 0") ) -I g7g . 

 t l^tan^sirT^in (A— 6")— 2sih 2 | ft" sin 2 (3,-0") j J ' W ' ' 



/y__ fl 9 nvc .in \ C0S 3 sin *" sin Ql-0") +2 sin/? sin 2 1 ft" 1 

 (3 -13-2 arc sin j - g _ J^r^y- - | 



7.=X — 4' — arc tan = 



( tan f3' sin ft" cos {X— 0"— 40+2 sin 2 § ft" c os ( A/ — ff — 4') sin (X— 0"— 4') \ ~| ^5^ 

 1 1 —tan /y sin ft" sin (X— 0"^4')— 2 sin 2 A ft" sin 2 (A— $"— 4') I -J 



0=^+2 arc sin I C0S ^ sin * sin f-f^" 8 ^ ^ "'* *" } (577) 



1 2 cos ±Q3-\- (3) ) 



