and the Methods of calcidating their Restdts. 271 



fjt,, $' The right ascension and declination of the cor- 



rected zenith. 

 v, 7r' The equatorial parallaxes of the two bodies. 



A, A', r The distances of both bodies, and of the centre 

 of the earth, from the place of observation. 

 I have here adopted the relations to the equator ; those to 

 the ecliptic might as well have been taken, and, more generally., 

 any point of the sphere may be chosen, and the great circle 

 of which that point is the pole, may be substituted for the equa- 

 tor, and any point of that great circle may be substituted for 

 the point from which the right ascensions are counted. The 

 calculation being the same for all assumptions, it is unneces- 

 sary separately to develop it for each case. 



We have the following well-known equations : 



"A cos 8' sin a' = cos 8 sin a — r cos $'. sin it . sin [x, 

 A cos 8' cos a' = cos 8 cos a — r cos <$>'. sin % . cos p 

 / , \ 1 A sin 8' = sin 8 — r sin <j/. sin it 



*• ] A'cosD'sinA'= cosD sin A— r cos <p'. sin tt' . sin jt*. 



A' cos D'cos A'= cos D cos A— r. cos <f>'. sin if . cos /* 

 _ A ' sin D' = sin D — r sin <p'. sin if 



If we denote the first three of these quantities by a, b, c f 

 and the latter three by a', #, c', we have directly 

 A A ' [cos V cos D' cos («'— A') + sin 8' sin D'] = a a! + b b' + c c\ 

 and as the expression multiplied by A A ' is the cosine of the 

 apparent distance of the centres ( if), we have 



A A' cos S = a a! + b b' + c c'. 

 For the commencement and the end of the occultation we 

 have X = §' + R', where the upper sign refers to an ex- 

 ternal, the lower one to an internal contact of the limbs. We 

 have, therefore, likewise, 



A A' cos X — A A' cos §' cos R' + A A' sin g' sin R'; 

 whence in conjunction with the following equations 

 A sin g' = sin g, A cos g' = a/ (a 2 + b 2 + c 2 — sin g 2 ) 

 A' sin R' = sin R, A' cos R' = yV*+ # 3 + c n - sin R 2 ) 

 we derive this equation : 



A A ' cos X = V (a 1 + b 2 + c 1 - sin f) \/(a' 2 + b' 2 + c' 2 - sin R 2 ) 

 + sin g sin R. 



This being made equal to a a! + b b' + c c', and the equation 

 transformed so as to do away the sign of the square-root, we 

 shall arrive at this expression : 



{a 2 + b 2 + c 2 ) (a' 2 + b' 2 + c'-) -{a a! + bb< + c c') 9 = 

 sin f {a! 2 + b' 2 + c' 2 ) + sin R 3 (a 2 + b 2 + c 2 ) ± 



2 sin g sin R {a a! + b b' + c c'), or by an easy transformation, 



(2)... 



