PREDICTION OF OCCULTATIONS. 29 



very close to the star, when log cos 4^ will result very near 0. In these cases, a re- 

 calculation should be made according to the method which follows, using 



t = ——cos(M—m, 

 n 



which may give log msm{M—JV) less than log k, when the star will be occulted. 

 On the other hand, it may happen that in these cases of very near approach, a first 

 determination may give a cos 4- less than 1, which a re-calculation will show to 

 impossible. The angle 4- is then to be considered = 0° when msxa{M—N) is positive, 

 and we shall have Q = 270°— iV. When msm{M—]Sf) is negative, 4 = 180°, or 

 Q = 270° — iV-*- 180°. "We shall also have, at the time of nearest approach, 



star's distance from moon's limb = 57' x(msin(if— iV) — 2725), nearly, 



the error in this computed distance increasing with the distance. 



By Angle from North Point, is to be understood the arc included between the star 

 when in contact, and the point where the limb is intersected by an arc of a great circle 

 passing from the moon's centre to the North Pole ; and by Angle from Vertex, the arc 

 between the star at contact, and the point where the limb is intersected by an arc of 

 a great circle passing from the moon's centre to the zenith. These angles are reckoned 

 from the North point and from the vertex, towards the right hand round the circum- 

 ference of the moon's disc, as seen with an inverting telescope. For direct vision, add 

 180° to the angles given by the equations. 



The results obtained by the above equations are only approximate, yet the computed 



times of immersion and emersion will usually be within one or two minutes of the truth. 



The error generally increases with the star's distance from the apparent path of the 



moon's centre, and may, in some cases, amount to several minutes. For an immersion 



this error is not of much consequence ; but for an emersion, especially of a small star, the 



time should be determined with greater precision. For this purpose, u' and v' must be 



computed with 



h' — d = h — d-+-i y. 



f* being the symbol by which we express the sidereal equivalent of t in these equations. 



u' = r cos cp' A cos (h' — cZ) 



v' = r cos $' A sin iji' — d) sin D. 



Then with these values of u' and v', recompute N, n, 4, and t, by means of 



nsinN— p' — u' 



n cos N= q' — v' 



, m sin (M— 2^ 



cos 4' = ^ ' 



k 



m ", ,_ T,n ^ sin 4* 

 t= cos (M- JSf) =F 



