CATALOGUE OF POLAK STARS. 237 



We must now find expressions convenient for the numerical computation of 



. . da d'-u d'^ a d* a 



the dilierential coeiiicients -^, -7-, -7-^, -7-, &c. 



dt di' di^ dt*^ 



The form of development given by Bessel is at once the earliest and the most 

 complete published, since it is carried to the fourth power of the time : it is also 

 quite as well adai)ted for logarithmic computation as the more modern forms. 



Introducing into the denominators that power of the radius which will render 

 all the terms homogeneous, we have (see Tab. Iieg., pp. x, xi, and Fundunienta 

 Astronomice, p. 301), after multiplying by the coefficients in the development by 

 Taylor's Theorem, 



du 



-J— = m + « tan d sin u. (IG) 



oP« 71' n m n 



-— = m' + — tan- d sin 2 « + ^7 tan tf cos a 4- --„ 

 dt- li Ji -^li 



= m' + — tan- d sin 2 « + —77 tan tf cos a + 7-7 sin 2 « -I- /i' tan ('i sin n. (17) 



<Fii n^m 2«' Zn'-ni 3 n' / »" v m"\ 



— = — — + -z~ tan' 8 siu 3 a + — ^^ tan" 8cos2a + „ j,., tan 8 sin 3 f4 + 1 ^^ — -„, ) tan il sin a 



8w-??t m' 91 + 2 n' m Sii'n Hii'ii . . ,,„, 



+ — — - cos 2 a + „ tan tS cos n+ —rr- tan- 8 sin 2 a + —77 sin 2 a. (18) 



2^^ Ji A 2Ji 



d*<i Qti* *12w'm Ctii* I2n* lnm'^\ 



7-7 — — - tan* 5 sin 4 a -}- ;r; — tan' 8 cos 3 « -I- 7, tan- 8 sin 4 « + I -7,7 — — 7;~/ tan- 6' sin 2 a 



dt* Ji" JP ii' ^Ji Ji" ^ 



(19) 



9>i"m /2)r' tn v m'\ Sn* /n* 7 7rm''\ 



+ -^^ tan 8cos3 a + \—j,,- - j,, ) tan 8 cos a + ^ ^,3 sin 4 « -t- \~ - -^jf ) sin 2«, 



&C., &C; etc. 



-J- — n cos «. (20) 



d-S «' . ^ nm 



-7-; = — 7, tan 8 sin- u — ~_ sin a + n' cos «. (.^l) 



dt' Ji Ji 



ef"8 3 m' 3;i2m ni(»i- + >r) n" ] 



-rz = — ^77" tan- 5 sin- n cos « — ~ ,., - tan 8 sin a cos « — 7:; cos (4 — , cob' u 



dt" Ji' Ji- L Ji- A- J 



f m'n + 2n'm\ 2n'n 



— ( j sin a - - — tan 8 sin' «. (22) 



* Engolmaun (sec Abliandluugen von Bcsscl, Vol. I. p. 277) gives for lliis k'vtu. 



