TRANSACTIONS OP THE SECTIONS. 



27 



drawn from lunar tables without any sensible error arising in our equation ; T 

 and/ are given by observation, only the quantities ©, A, bg — Bg, tt and d + D, or 

 some of them may form the queries of the problem, according to circumstances ; 

 and as all these quantities are contained under a linear form, their determination 

 can be directly obtained by the resolution of equations of first degree, without having 

 recourse to the method of corrections by supposed errors, which is an analytical and 

 practical advantage of the formula we propose. 



If we refer to a construction upon the usual plane of projection, as seen in the 

 annexed figure, it will be easily seen that the angle $ which we have employed is 



the angle, S H A, which the line, S L, uniting the apparent centres, S and L, of the 

 occulting and occulted bodies, makes with A B, the perpendicular to the projection, 

 C M, of the circle of declination ; O is the angle, L' Q B, which the relative orbit, 

 Li L L', of the occulting body makes with the same perpendicular, v the relative 

 velocity of this body in its orbit, ( the zenith distance, C S, of the occulted body, 

 and V the complement, S C A, of its angle of variation. This being understood, 

 it is clear that the leading idea of our method consists in valuing the abscissae 

 CI, C s of the projected centres of the said bodies, at the instant of the observa- 

 tion, and in a direction parallel to the line which unites them, and to make the sum 

 of their projected semidiameters, diminished by the phasis, equal to the difference of 

 those abscissae, upon the length of which difference a small error on the angle $ 

 has no important influence. 



Note I. — In the expressions of m and n, given at page 635 of the quoted work, 

 it is supposed that A, a, B and b denote the right ascensions and declinations of 

 the occulting and occulted bodies, but we may suppose as well that they represent 

 their longitudes and latitudes. In this case, calUng P the angle of position of the 

 occulted body* at the instant of the observation, we must compute | by formula 



♦o^ t jre sin P + M cos P — n v ^, 



tan f = — — ! ; — — . (c)i 



m cos P — wsmP — ir fi' ' ' ' ' ' 



and we must substitute in formula (A) | — P and v — P instead of ^ and v. Then 6 

 will be the time of conjunction in longitude, and bg — Bq the difference of latitude 

 of the two bodies ; but in the expressions of /i, v and a>, the letters a and b will con- 

 tinue to represent the right ascension and declination of the occulted body, and 



* The angle of position P is to be taken positive in the ascending signs of the ecliptic 

 T2o, and negative in the descending signs 2&£iK. 



