CATALOGUE OF POLAK STARS. 



237 



We must now find expressions convenient for the numerical computation of 



da </-(c d*a d* a 



the differential coefficients — , — , -7-,, — . &c 



dt dt 1 dt 1 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 adapted 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. Reg., pp. x, xl, and Fundamental 

 Astronomic, p. 301), after multiplying by the coefficients in t he development I 

 Taylor's Theorem, 



< 



la 



dt 



m + n tan 3 sin a. 



(10) 



•: 



cfr a n 2 nm n 



— = m f + — tan 2 8 sin 2 a + — rr tan 8 cos a + vT> Rln - a + Jff t:m $ sin «• 0") 



at- jB It lit 



d*a ?r?n 



+ ~— tan 8 8 sin 3 a + ~ 7?2 ' tan 2 8 cos 2 a + 7>2 tan 8 sm 8 (< + 1 5 />2 — ~>>j y tan * sin « 



e^ 3 2-R* J2* i? 



3 w 2 w m'n + 'ln'm Sn'n m , 8n'n . _ /to . 



+ -~^7 cos 2 a + tan 3 cos « + — — tan 2 3 sm 2 a + — — «» 2 «. (is) 



2i? 2 .ft A 2 It 



d*a 6/? 4 *12/, 3 m 6w 4 . . . f'2n 4 7» ? m s 



rfF = iZ 3 



tan 4 3 sin 4 a + — — — tan 8 3 cos 3 a + — tan 2 3 sin 4 « + [— - Ji3 J tan 2 8 sin 2 a 



(19) 



9>/ 3 »i ' /2w*m ??w 3 \ 3» 4 . />/ 4 T;/ 2 m 2 \ , 



+ "k* - tan 5 cos 3a+ \p*~ ~ ~tf ) tan 5 cos rt + 4^ 3 S1U 4 rt + vj» ~ YiF' "' 



etc., *tc, *fcc. 



dS 

 dt 



n cos «. 



(20) 



# 8 rc 



2 



n m 



7r2 = — ~V> ^ au $ s * n a "" ~ 7 > sin a + n' cos a. 



(21) 



d*d 3 » 



«' t* It* 



t s 3« 3 m r«(m a + n*) u* ] 



- tan 2 3 sin 2 a cos « — — — tan 3 sin « cos « - ™ cos a - ,, .. cos' « 



/m'n + 2n'»i\ > 2?<'» 



( ^.^ ) sin « — — ~ tan 3 sin 2 «. 



(22) 



* Engelmann (see Abhandlungen von Bessel, Vol. I. p. 277) gives — for this term. 





