412 Prof. Bessel on the Calculations requisite Jbr predicting 



9, Such an accurate calculation is, however, not required 

 when the circumstances of the occultation are only wanted for 

 the purpose of making the observation ; in that case an error 

 of one minute is of no consequence; and if every thing requi- 

 site for this purpose only is wanted, it will be sufficient, pro- 

 vided t falls below, or at least does not much exceed one hour* 

 to apply the following much shorter calculation. In place of 



cos S. sin fa— A) , sin S. cos D— cos 3. sin D cos (a— A) 



.-^ and rr : 



sin fT sin TT 



a-A ^ J 3-D 



we nut cos d and 



^ IT T 



and neglecting the variations of cos S and tt, so as to designate 

 the right ascension and declinatioH of the moon at the time T 

 by a and 8 and their hourly variations by Ju and J5, we have 



a— A . , Aa 5, 



p =r cos 6; J)' =z cos 6 ; 



q = ; q' = - 



We next neglect t in the expressions for li and t) and obtain 

 u — r cos<p' . sin (ju.' — A) 



V = r sin ip' . cos D— r cos ip' . sin D . cos (/x'— A) 

 li = r cos p' . K cos (]«,' — A) 

 Tjf = r cos f' . X sin (j«,'— A) sin D 

 For the place to which the calculations refer, the logarithms 

 of r cos p' and r sin ip' are to be considered as known. If we 

 write a for r cos <p' sin (ju.' — A) j 6 for r cos (p' cos (p-' — A) ; 

 c for r sin ^' cos D, we have 



u = a; u' = b . X ; v = c — b sin D 

 t/ = a . A sin D : where log A = 9*4192 and c may 

 be taken from a small table which exhibits the value of this 

 quantity for every degree of D with sufficient accuracy to four 

 figures of decimals. The solution of equation (6) remains the 

 same. In order to have a clear view of the calculation, I sub- 

 join here the whole detail of it : 



1830, April 5. 82 Leonis. 



T= 7^; ft' = 118° 32'; It'- A = -50° 42'; tt = 54'-18 



a = 168° 37'-93 ; 8 = 4° 44'-00 ; Ja = 28'-50 



A = 169 14-11 D = 4 14-08 J8 = — 9*08 



— 36-18 +29-92 



* Should this supposition not be justified at the end of the calculation, 

 or for other reasons a greater accuracy be desired, it may be obtained by 

 a second approximation for u' and i/ without changing p' and q'. It is 

 easily seen that the error of the first approxifliation increases with the di- 

 stance of the star from the path of the moon's centre, and that its limits 

 cannot be unconditionally assigned. 



