EPHEMEEIDES AND AUXILIARY TABLES. 



and transforming to oblique coordinates x l and y l parallel to the hour and declination circles, 



a 2 a 2 



P- 



or eliminating ccf and cd by the equation of the ellipse a 2 6 2 = 6 2 x~ -f~ a 2 7/ 2 



a^ 2 = a 2 cos 2 &amp;gt; -j- & 2 sin 2 ^&amp;gt;. 

 2/j 2 =r a 2 sin 2 ^&amp;gt; -f- 6 2 cos 2 ^?. 



in which the substitution of the values a = r, 6 = cos ($ -j- #) give 



sn 



y l = r V I cos 2 p sin 2 (/S -f- E) 

 which, being subtracted from the semidiameter, give the defect of illumination 



Aa. cos d = r x l =. r (1 V 1 sin 2 ^&amp;gt; sin 2 (/S + E], 

 Ad = r y l = r (1 VI cos 2 p sin 2 (S + ^), 

 For the preliminary correction in parallax we have, as usual, 



t ^n P cos rf i a \ 



d =. 9 c - 3E_. sm(0 a) 



A cos o 



&amp;lt;M w n / sin(&amp;lt;? } 



o 3 = _. p sin &amp;lt;p. i -- 2/ 



A sin C 



in which d and $ denote the geocentric places, and is the auxiliary angle 



tan tan tp sec (^ a) 

 The second equation may be written, 



o o = _? p sin (p 1 (sin cotg cos ) ] p cos (f/ sin 8 cos (# a) p sin y cos d [ 



x cos ^ x sin o /n 



==-2- -- -- ma (8-a) 



It is very convenient to make use of auxiliary tables for these values, at least in the case of 

 series so extended as the Santiago observations; we, therefore, write 



a d =. A sin (Q a) 

 8 3 = D + E cos (6 a) 



and construct tables of A= - -; D &quot; c &quot; s ; E= S A in . and for meridian observa- 



A cos d A A 



tions 



