416 Prof. Encke on the Calculations requisite 



From the auxiliary tables for occultations of stars we have 

 t, A A and AD; and from the catalogue of stars, A and D. 

 We have then to calculate 



a' = a + a sec 8 + a! 



8' = 8 + b sin(y-8) + U , ? J = r + Ar 



A a' = ?» + wi' sec 8, A 8' = w 4- n f sin 8 



A' = A + AA, D'=D + AD 



If we now put 



(A' — a') cos 8' = N sin >j A a' cos 8' = M sin £ 



(D'-8') =Ncos)j A8' = M cos $ 



where the different signs determine the quadrants in which *j 



and £ are, (N and M being always considered as positive,) and 



if we likewise determine i by this equation: — J* = sin i, 



we have for the mean time of the beginning and end of the 

 occultation T + t -f Z, the double value of t in minutes of time 

 being derived from this equation : 





60 -|- At 



5 N cos (>j — ?) -f- I s cos i > 



M 



The upper sign is for the beginning, the lower for the end of 

 the occultation. 



If N . sin (>] — ?)> ^, no occultation takes place : the moon 

 passes to the north of the star if sin yj — £ is positive, to the 

 south if sin (>} — ?) is negative. For the place of the star's dis- 

 appearance and reappearance on the moon's limb, the angle u, 

 counted from the northernmost point of the moon's disk 

 through east, south, and west, is found by these equations : 



r 1 sec 8' sin u = A'-a'-Aa' (-^r, — ^ 



^ V 60 + At J 



^cosw = D'-S'-AS'Arr 1 -") 



where, with rare exceptions, u must be taken for the disappear- 

 ance in the two first, for the reappearance in the two last qua- 

 drants. As an approximate value of u is only required, the 

 second formula will be sufficiently accurate. 



Among the various values of t, it is most advantageous to 

 take the one which lies nearest to the true conjunction of the 

 star with the moon in right ascension. The choice of this 

 value is facilitated by the consideration, that for positive values 

 of t the apparent right ascension is smaller than the true one; 

 for negative ones greater; or that the parallax in right ascen- 

 sion has a contrary sign to t. If, however, the true conjunc- 

 tion should happen in the middle of a lunar hour, it would be 

 necessary, in order to attain all the accuracy of which the 

 method admits, to make the calculation for the two hours be- 

 tween 



