Occultalions of Stars by the Moon. 339 



by the substitution of these values it becomes 



Ic" = vi" sin (M-N)' + \rn cos(M-N) + n i^ 

 and if we suppose 



— sin (M — N) = cos vj/, we have 



^7) .... t=-~ cos (M-N) + — sin 4* 



where the upper sign is to be used for an occultation, and the 

 lower one for an emersion, provided •]> lias been taken be- 

 low < 180°, which may always be done. If we find, however, 

 — sin (M — N) > 1, there will be no occultation, but the moon 

 will pass by the star without occultation. It is evident, how- 

 ever, that this is only necessarily the case after the approxima- 

 tion has been pushed far enough, and that an error in N may 

 produce the appearance of the impossibility of an occultation 

 which really will take place, and vice versa. If cos v^ be found 

 >1, ^ is, notwithstanding, to be calculated by the formula 



t^-J!Lcos (M-N) 



and with this value the approximation is to be continued ; it 

 will then appear whether cos ^ is really greater than ] . In 

 like manner a \1/, which a rough approximation would show 

 to be possible, might prove impossible by a greater approxi- 

 mation. These cases, however, if T is not too distant from 

 the time of occultation, will only occur when the star remains 

 very near the limb of the moon. 



The formula (4) will be converted into" the following one, 

 by introducing the symbols adopted in this section, 

 ^ sin P = — «i sin M — n sin N . ^ 

 k cos P = 7U cos M + « cos N . t 

 and hence by substituting the value of ^ we obtain these: 

 k sin P = — VI sin (M — N) cos N + k sin N sin xj/ 

 k cos P = — m sin (M-N) sin N + k cos N sin ^ 

 and, as ?h sin (M — N) = k cos ^, we have 



sin P= - cos (N±\^); cos P = - sin (N + v^) 

 and 



(8) P = 270° - N + \f/ 



If we choose to describe tlie jilace of occultation or cmc» 

 sion by tiie angle which is inclosed by the great circles drawn 

 from the moon's centre through the star and the north jjole, 

 beginning from the north and coimling to tiie left, we shall 

 vt-ry nearly have this angle Q = J8(r- P = N + vIz-Jm". 

 .5. The quantities j\ rj, ;/, q' which depend on the motion 

 '2X2 of 



