344 Prof. Bessel's Additions to the Theory of Eclipses, 



ForT'= -2 b-2c + ±£d-l 



6 " S 



e 



1 h- c+ ±d - i 



b— ±d 



3 «* T2 



e 



= + i & + * + i <* + j* * 



= + 2 $ + 2c + -v-<* + &* 



The remainder of the calculation is as above. 



This second manner of conducting the calculation, supposes, 

 therefore, the determination of p and q by interpolation, while 

 the former one constantly proceeds from the same values of 

 these quantities. But it has the advantage of a more easy 



calculation of the term (6) — -S--^ — — -r— - — - which is the 



v ' n cos •v£ 



correction of the assumed difference of meridians, and which 

 is, of course, commonly very small; its convergency to the 

 truth is likewise the greatest possible, and the error of the 

 first approximation arises only from the moon's motion during 

 the. interval between the supposed and the real difference of 

 meridians being taken as it would be at the beginning, or at 

 the end of this interval of time, while it ought to have been 

 taken for the middle of it. For the observatories of Europe, 

 whose meridians are very nearly known, the square of 

 n (T'+ i) might even be neglected, and the equation to be re- 

 solved might thus be reduced to one of the first degree. But 

 all these advantages of the second method of calculating ap- 

 pear to me to be insignificant in comparison to the trouble of 

 the interpolation for finding p and q. I therefore prefer the 

 first. In the application of these formulae, however, the in- 

 terpolation will never require to be carried beyond the se- 

 cond differences, and consequently three values of P and Q 

 will be sufficient. 



[8.] The method here explained leaves it to the choice of 

 the calculator, whether the quantities a, S, shall refer to the 

 equator or to the ecliptic. In the result of the calculation 

 there is nothing referring to either of these great circles, and 

 they serve only to denote the relative situation of the various 

 points of the celestial sphere referred to in the problem. The 

 former is, however, always more easy, if the places of the 

 moon in relation to the equator are contained in an ephe- 

 meris ; and even if this should not be the case, whenever se- 

 veral observations are to be calculated at the same time. But 

 when there are few observations, and when the places of the 

 moon are to be derived from the tables themselves, or are to be 

 be taken from an ephemeris containing the moon's longitude and 

 latitude, the preparatory calculations required in finding the 



quantities 



