338 Prof. Bessel on the Calculations requisite for predicting 



which consist of two separate parts; one of which deyiends 

 only on the place of the moon, wliile the other depends on 

 the place of observation only. The sum of the squares of both 

 gives this equation : 



,,. ,3 ( COS S sill (a— A) , . , A \ ■) ^ 



(5).../-= ! r- — rcos a' sm (a — A) J 



^ ' I sin T r \i / ^ 



( sin S cos D — cos S sin D cos (a— A) 

 i sin ?r 



— r [sin cj>' cos D— cos f' sin D cos (j«. — A)] | 



As the parts which are to be squared may be considered as 

 functions of the time, these formula; contain no other unknown 

 quantity but the time of occultation or emersion. 



4. The times of innumerable occultations and emersions 

 will be contained in this equation if taken without restriction, 

 and it is consequently a transcendental one and cannot be 

 solved by a direct process ; it is only to be solved by trials or 

 by successive approximations. The latter proceeding appears 

 to me to be more convenient. 1 assume, therefore, «, 8, tt, /a 

 as known for a time T, which is so near to the time of occul- 

 tation or emersion T+ ^ which is required, that the terms on 

 the right of the sign of equality may be converted into rapidly 

 converging series. On this supposition we assume 



cos S sin fa— A) , , . 



^ ' = p -\- p t 



sin tr 



sin 8 cos D — cos S sin D cos {a. — A) .... = q + q' t 



r cos ip' sin (jx— A) = zt + u' t 



r sin ip' cos D — r cos 4>' sin D cos (/x. — A) = u + u' ^ 

 and pt q, u, v are the values corresponding to the time T ; while 

 p', q', u', -d are functions of t, in which, however, agreeably to 

 our supposition, the terms dependent on t and its higher powers 

 are very small. If we suppose that t is approximately known 

 as far as it has influence on the value of these quantities, the 

 solution of equation (5), or what it will be after making the 

 above subtitutions, viz. 



(6)...F = [p- u + {p'-u)tY' + {q-v + {q'-^)tY 



will produce a greater approximation for t ; by means of which 

 values for p' — u' and y' — t'', more accurate than those assumed 

 in the calculation, will be obtained, which substituted in the 

 formula will again lead to a closer approximation to the value 

 of t, and so on. 



The solution of equation (6) will be facilitated by making 



p — u = m sin M, ■ p' — iJ = n sin N 

 q — V = 771 cos M, q' — r/ = n cos N; 



by 



