0« the Occultations of Stars by the Moon. 337 



tre of the moon, and denote the distance of both measured on 

 it by S, and the angle formed by this circle and the circle of 

 declination passing through the star to the north pole by P, 

 which is to be counted from O"" to 360°, so that P is between 

 0° and 180°, when «' < A, and between 180° and 360°, when 

 a! >A, we shall have 



{sin X sin P = — cos 8' sin (a' — A) 

 sin ^ cos P = sin S' cos D — cos 8' sin D cos («' — A) 

 cos 5" = sin S' sin D + cos 8' cos D cos (a' — A) 

 The apparent place of the moon is expressed by the true one 

 by means of these formulse : 



A cos 8' sin «' = cos 8 sin a — r cos tp' sin tt sin ^ 

 A cos 8' cos «' = cos 8 cos « — r cos <p' sin -k cos /* 

 A sin 8' = sin 8 — r sin (p' sin tt 



A being the distance of the moon from the place of observa- 

 tion. If we substitute these quantities in (1), we obtain 



' A sin 5 sin P= — cos 8 sin (« — A) + ?-cos4>' sinw sin(p, — A) 

 Asin:; cosP= sin 8 cos D— cos 8 sinD cos (« — A) 

 (2)<j — r sin tt [sin 9' cos D— cos ;p' sin D cos (f*,— A)] 



Acos^ = sin 8 sin D + cos8cosD cos («— A) 



— r sin TT [sin <J)' sin D + cos ^' cos D cos (ft — A) ] 

 which are the formulas given by Lagrange, but referred to 

 the equator. 



3. For the beginning and the end of an occultation, we 

 have S = g' 

 and as A sin g' := sin g, 



we have likewise A sin 5 = sin g ; by which the apparent 

 radius of the moon disappears from the first two of the for- 

 mulas (2), if applied for calculating the occultation or emer- 

 sion. We have, therefore, for these cases 



fsingsin P = — cos8 sin (« — A) + >' cosfp'sinw sin(fA — A) 

 (3)< sin gcos P= sin 8cosD— cos 8 sin D cos (a — A) 



t —rs'imr [simp' cos D— cos <p' sin Dcos (;u. — A)] 



and the third formula is of no farther use, as it only decides 

 whether the distance is g' or 180°— g', which is never doubtful. 

 If we divide these lormula:; by sin tt and assume sin g = 

 /rsin TT, wlierc the constant quantity /f is according to Burck- 

 hardt's I'ablcs =0-272,'5, and its logarithm = 9-4'353665, they 

 will be changed into the following ones: 



, . T» COS 3 sin (a— A) , / ■ / a\ 



^ sm P = — T^ + r COS f sni du- — A) 



(4)^ 



* , ,, sin 3 cos D— cos 3 sin D cos («— A) 

 A COS 1' = : • 



sin ir 



— r [sin 9' COS D— cos f ' sin Dcos (jtt — A)] 

 N. S. Vol.6. No. 35. Nov. 1829. 2 X which 



