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



immersion, the lower one to an emersion ; but it is more con- 

 venient to leave out the double signs and to remove the un- 

 certainty which remains in finding \j/ by its cosine by the rule, 

 that this angle is taken in the two first quadrants if the obser- 

 vation is an immersion, and in the two last ones, if it is an 

 emersion. If this rule be adopted, we have for both cases 



(c) TV_ - m,cos (M— N - ■jj . •■ i> Ak 



n.coa-^ tang $ w sin ^ 



In order to apply this solution, T' and thence p' and q' are 

 assumed as accurately as the existing knowledge of the dif- 

 ference of the meridian of the place of observation allows ; 

 the approximate determination of T' thereby obtained, is then 

 employed for obtaining a second approximation, and so on. 

 I observe, however, with a view to prove the adequacy of the 

 method, that it is not necessary to suppose an approximate 

 knowledge of the difference of the meridians, but that in the 

 beginning the value of T may be supposed equal to the time 

 of the conjunction as given by the tables, this supposition be- 

 ing always in unison with the supposition of the convergency 

 of the results on which the calculation is founded. When T' 

 has been found, T + T' will be equal to the time of observa- 

 tion less the difference of meridians t— d, to be taken positive 

 if easterly, consequently d = t — T — T', or 



(6).. t rf = ;-T + -^. C03(M -?-^ + ;+ * +-4* 



y ' n cos -v£ tang y n sin y> 



[6.] I shall now further develop the calculations required 

 in the application of this method. Although it converges 

 quickly to the result even if T' amounts to some hours, yet 

 the smaller the value of T' the more convenient will be 

 calculation, for the convergency itself will approach the 

 nearer to its maximum, and a small T' may be more accu- 

 rately determined by its logarithm, than a greater one. It is 

 therefore advisable to adopt for T a time which is as near 

 to the mean of the times of the observations which are to 

 be calculated, as this can be effected by multiples of tens of 

 minutes of time, or generally by such parts of the day as ad- 

 mit of a more easy calculation from the tables or ephemerides 

 of a, 8, 7r than such as are expressed by higher numbers. The 

 sort of time in which T is to be expressed, ought to be that 

 for which «, 8, n can be most easily found, consequently mean 

 time, if these values are calculated from the tables, or from 

 Encke's Ephemeris, which perfectly supplies their place; and, 

 on the contrary, apparent time, if the columns for the moon 

 in the Connaissance des Tems or the Nautical Almanac are 

 to be applied. The values of «, 8, tt are to be calculated for 



the 



