CATALOGUE OF POLAR KTAUS. 239 



^-— = * t- [8.867G08G] tan'' S sin 4 « + [9.5299797] tair'' 8 cos ;:5 u + [8.867608G] tan- S sin 4 u 



+ [9.6330799?i] tan= d sin 2 a + [0.4050410] tan d cos 3 u + [8.9GG7501/t] tan 8 cos u (27) 



+ [7.9645186] sin 4 « + [9.3320498/*] sin 2 a. 



— = 4- [1.3022060] cos a. (28) 

 dt 



d-d 



— = + [7.2899869«] tan d am' a + [7.6513280m] sin a + [5.9360108;i] cos a. (29) 



-— .=: + [3.7548891«] tan^ 8 sin'' a cos a + [4.1162302;/] tan 8 sin n cos « + [4.0757367«] cos « 

 a t 



+ [3.2777678] cos' u + [2.0359378] sin u + [2.4009130] sin'' a tan 5. (30) 



— ^ = + [9.7426700] tan' 8 sin^ a + [0.8447299h] tan' 8 sin= a cos^ a + [0.4050411] tan^ 5 sin a 

 dt* 



+ [0.8821623] tan^ 8 sin « cos^ a + [0.8333289] tan 8 sin'-' a + [0.4653522rt] tan 8 (31) 



+ [9.2655487] tan 5 sin^ a 4- [0.1686387rt] tan 5 sin- u cos^ a + [0.4248824] sin « 



+ [0.4050411*1] sin a cos^ a. 



. d'a - f?'S ,„ 

 Hill has given a slightly different form of development lor -— and -^. (bee 



Star Tables of the American Ephemeris, p. xvii.) 



Introducing the required power of the radius, we have : 



<^^« "■'' "'"■ . "^ ■ „ o V /DON 



— -, = m' + — = sin 2 u + n' sin a tan t( + ~. cos a tan 8 + -, sm 2 « tan- 8. (rf-) 



J' « ?» n^ 3 m w2 3 ?! «/ 2 71^ - TO- ;/ . „ 3 >f' . „ ^ , 



2wn' + Hm' 3w?r 3 » ?;' 



+ cos a tan 8 + „, cos 2 « tan^ (5 4- ^ ^^ sin 2 n tan- 5 (3^) 



2 «' 4 n' 



■f -^ sin a tan' (5 -f- " cos 2 a tan' 8. 



d^8 mn n^ . .oji 



-7—, = ~ ~^ sin a 4- «' cos « — — sin''' « tan 5. V''^) 



dt^ It li 



* One dash over the characteristic of the logarithm inJicatcs one coniplelion of the cycle, and two dashe.i 

 indicate two completions of a cycle. 



