IXXXVJ EI'IIKMEKIUES AND AUXILIARY TABLES 



The defect of illumination in right-ascension and declination may be readily obtained upon 

 the admissible assumptions that the planet is spherical and illuminated by parallel solar rays. 

 We are to compute the corrections to the measurements by a micrometer, the threads of which 

 represent hour and declination-circles, and are tangent to the defective limbs at the points h 

 and d, respectively. The illuminated portion of the planet is separated from the other half by 

 a plane perpendicular to the line from the sun to the planet, and its visible boundary will be 

 projected upon the plane of the apparent disc as a semi-ellipse. The plane is horned or gibbous, 

 according as the planetocentric angle between earth and sun is more or less than 90; and the 

 unilluminated portion of the solar disc is, in each case, upon that side of the planet from which 

 the sun is more than 180 distant this geocentric angular distance being counted from the 

 centre of the planet toward either side. 



Employing r, as heretofore, for the apparent semidiameter of the planet, and denoting the 

 semiaxes of the ellipse by a and b, we have r = a. And representing, in the plane triangle 

 Sun-Earth-Planet, the two first angles by the initial letters of the respective bodies, we may 

 assume, without appreciable error, 



b = r cos (S + E) 



an expression which gives to the minor semiaxis a negative sign when the visible ellipse is 

 unilluminated i. e., the planet horned. 



Denoting now the geocentric longitude, right-ascension, and declination of the sun, by L,A,D, 

 the geocentric right-ascension and declination of the planet by a, 8 

 the heliocentric longitude and latitude " " ^, /9 



all of which quantities may be directly taken from the ephemerides we have the equations 

 cos S=. cos ,9 cos (A L) 

 cos E = sin d sin D -\- cos d cos D cos (a A) 

 sin E co&p = cos d sin D -\- sin d cos D cos (a A) 

 sin E sin p = cos D sin (a A) 



It is evident that the angle p is equal to the angle made with the semiaxis a by the tangent 

 at d, or to the complement of that made by the tangent at h. The first equation may also be 

 written 



S=I80 + L /I + 2 sin 2 ^ cotg (L /) 



and of the three angles S, E, p, the first two determine the magnitude of the ellipse, and never 

 exceed 180, while the last, counted like other angles of position from north through east round 

 to the semiaxis b, fixes its position. 



A convenient mode of computing the angles E &udp is afforded by the employment of auxili- 

 ary quantities g and G ; so that 



g sin G = cos D cos (a A) 

 g cos G = sin D 



te _ cos D sin (a A) 



cotg E = tan (G + d) cos p. 



The general expression for the tangent of the angle included between a tangent to the ellipse 

 and its major axis is 



/, V* 



D *y = -*r 



o a y 

 and the distance between the centre of the ellipse and the point where this tangent intersects 



the major axis is Hence we have 

 x 



7 2 nj 



at the point h ................. cotgp = . - 



o a i/ 



at the point d... tan p = 



a s y" 



