26 REPORT — 1855. 



On the Calculation of an Observed Eclipse or Occultation of a Star. 

 By Professor Mossotti. 

 According to the denominations adopted in Dr. Pearson's ' Introduction to Prac- 

 tical Astronomy,' vol. ii. p. 675, and following, the general equation for an eclipse, 

 a passage of a planet over the sun, or an occultation of a star is — 



(m — TT /i)2 + (n — TT v)2 = (rf + D — / — jr 0) tan (D - /))2. 

 This equation, by introducing a new angle |, may be resolved into the following 

 two: — 

 (»» — jT /Ii) cos 1+ (n — 7ri')sin^ = d + D— / — ttu tan (D — /) . . (a) 



— (»» — TT ft) sin ^ + (n •— ff i/) cos ^ ^ 0, (b) 



the last of which gives 



w — i: V 

 tan t = —r " (c) 



The angle | given by this formula may be computed by the values of m, n, and 

 n, deduced from the lunar tables, and the small error by which they may still be 

 affected will not have any sensible influence upon equation (a), because the fluxion 

 for a change in the value of the angle ^ is evidently reduced to nothing in con- 

 sequence of the second equation (b). On this property lies the foundation of the 

 method which we are going to explain. 



If we count the time t from the instant of the observation, the values of the 

 cosines Mq and Mq corresponding to the instant of the true conjunction may be 

 expressed by series* — 



OTj = TO -f- m' < -|- ^ m" <* + 0), 



ng = n -{• n' t ■\- -7i" f + a>: 



but at the moment of the true conjunction, we must have 



wjo = 0, Mq = 6j — Bfl, 

 therefore 



m =: — m' t — -m" t\ 



n = Jo — Bo — »' < — \n" i^; 



and by substituting these expressions in equation (a), and by putting, for sake of an 



easier computation, 



m' = «; cos O, /x = sin ( cos v, m" = w cos e, 



n' ^ V sin O, v = sin f sin v, n" = w sin e, 



we shall have 



— tvcos{i — 0) + {hg — Bo) sin ^ — TT sin f cos (| — i/) — (d + D — /) = 



— TT cos f tan (D — /) + ^ f cos (^ - e). 



The letter t denotes the time elapsed from the instant of the observation to that 

 of conjunction, and its value is negative when the conjunction happens before ; then, 

 if we call T the mean time of observation, as counted at the place, and A the east- 

 ward longitude of the place from the meridian for which the time of conjunction 

 has been computed, which will commonly be the meridian of the lunar tables em- 

 ployed, we must have 



T + ^ = e -f- A, 

 and the preceding equation may assume the form 



G V cos (^— O) -t- A V cos (^— O)— (&o— Bo) sin ^+7r sin f cos i$—v) + d+D='\ 

 Tvcosd— 0)+/— 7rcosftan(D— /)— •if^'wcosC^— e). | ^^ 



The values of the coeflicients v cos (| — O), sin ^, sin f cos (^ — v). as well as those 

 of the last two terms of the second member, may be computed by the elements 

 * See the Note II. at the end. 



