EPHEMEREDES AND AUXILIARY TABLES. 



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



a a 2 



yi = ' ' Sm P ' 



or eliminating x 1 and x 1 ' by the equation of the ellipse a 2 6 2 = 6 2 a? -\- a 2 y a 



xf = a 2 cos 2 p + i 2 sin 2 j>. 

 2/, 2 a 2 sin 2 ^) -f- 5 2 cos 2 p. 



in which the substitution of the values a = r, b = cos ($ + E) give 



! = rVl sin 2 J9 sin 2 (S + E) 

 y, = rVl cos 2 _p sin 2 (S + ) 



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

 Ja cos 3 = r X L = r (1 V 1 sin 2 p sin 2 ( + E), 

 Jd=r y l = r(l V I cos 2 p sin 2 ( + #), 

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



a!- a =-^.P cos< f f . sin(0-) 

 A cos<5 



< TT . , &m(8 ) 



ff o = 5 . /) sin tc . i -- i.' 



A sin C / 



in which of and ^ denote the geocentric places, and is the auxiliary angle 



tan =r tan ^ sec (6 a) 

 The second equation may be written, 



ff 3 = ^? p sin ^ (sin 5 cotg cos 8) = ^? | /> cos y! sin 5 cos (tf a) p sin <p' cos 5 > 



x cos d x 1 sin ^ v 



___ ____cos(^-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 a! = A sin (6 a) 

 d ff = D + E cos (0 a) 



and construct tables of A =. - - -: D = - : E= - -. and for meridian observa- 



A cos d A A 



tions D + E= !"JL sin (/ d) 



