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



the former meridian = r — d— x-f a x. This value for the 

 time compared to another (t') resulting from observations at 

 places whose meridians are known, will give 



x — —r — ——5 or the true difference of meridian 



l — a 



d+X = T-T'+ —^- (T-d-T<). 

 1 — a x ' 



If, as usual, the effect of a change AS of the latitude (or 

 declination) on the time of conjunction has been calculated 

 and found equal to m A 8, and if the change of § in a second 

 of time be denoted by (3, we have a = m /3. It is clear, there- 

 fore, that in the usual manner of calculating in respect to the 

 ecliptic, where /3 may amount to 0""055, it may not unfre- 

 quently happen that the time hitherto neglected may amount 

 to the tenth part of the error of the assumed difference of me- 

 ridians. For nearly central occultations this neglect is of no 

 consequence ; but in cases of occultations of short duration 

 the whole result may be rendered illusive by it. 



The second method, on the contary, does not suppose the 

 difference of meridians to be known, but determines it by the 

 solution of an equation which is obtained by successive ap- 

 proximations, the first of which even may be made independ- 

 ent of an approximate knowledge of the difference of the me- 

 ridians. I shall demonstrate that this advantage is not the 

 only one which the second method possesses over the first, 

 but that it likewise leads to the end proposed by a more easy 

 calculation. This is obtained by introducing as an unknown 

 quantity, besides the corrections of the elements of the calcu- 

 lation, the difference of the meridians of the places of observa- 

 tion itself, instead of taking as such the time of conjunction. 

 Indeed the time of conjunction may be considered as an aux- 

 iliary quantity, the calculation of which is only interesting as 

 far as it affords a means of determining the errors of the tables. 

 If these errors are determined directly, without using the time 

 of conjunction, the knowledge of the latter becomes entirely 

 superfluous: if it be still required, it maybe found without 

 difficult}' after the errors of the tables have been ascertained. 



[3.] In the first place, I shall generally and completely de- 

 velop the equations for the external and internal contacts of 

 two spherical bodies. The symbols which I shall employ are 

 as follows : 



a, 8, q The right ascension, declination, horizontal ra- 

 dius of the nearer bod}\ 

 A, D, R The same for the more distant body. 

 a 'i 8', g' "1 The same quantities as they appear at the place 

 A', D', R'j of observation. 



