114 SOME METHODS OF COMPUTING THE RATIO OF 



In which X, Y', and Z are the sun's geocentric coordinates ; 



M — ?^ r M — ^7; M — ^° 2 TV — ?!s r" TV — ?!• 11" nnrl TV — ^ t" 



are known constant quantities ; r and t" are the known variables, 

 having the same signification as in (6) ; and ^ and y, the unknown 

 quantities to be found from (108), 



f 108) ^ = 1 — -r -JT ^-y^) H«' 



yT can therefore be found accurate to terms of the second order, 



(109) when the error of tan./" is of the fiist order. When v'^ — v^ is small, 

 the error of the first approximation to tan./", as found by (87), will 

 be of the second order, and consequently that of ^ of the third. 



Assuming for a first approximation |- ^ 1 , or ^^^ = ^ ^ = -^,j 

 we may find (v" — v') and (v' — v), as in (87), by trial from the 

 equations fi^^^ = ^ [^ = L, and «_.o) = K-«) +K- f); 



(110) or from the expression, tan. (v' — v) = ^ +° oM'j' - ».) • 



Substituting the value thus obtained of tan. 1 («' — «) = tan./" 

 in (108), there results a corrected value of ^. If the elements are 

 elliptical, G" is used, and its logarithm is taken from Table I. ^vith 



(111) the argument smJ' g" = ^, '""' •{- . 



In the hyperbola H" is employed, and its logarithm is found from 

 Tables I. and XL, with the argument tan.= h" = — 4 '^^• 



If they are parabolic G" =H" = 1. 



Having from (108) found corrected values of ^ and -, they 

 are to be used for new values of tan. f" and tan. /. The smaller 

 the latter quantities are, the more rapid (109) will be the con- 

 vergence to the true values of ^ and j ; these may commonly be 

 found with accuracy to seven places of decimals vnth five-figure 

 logarithms. The amount of allowable error in y, and — may be 

 estimated from considerations similar to those pointed out in (107 a). 



Table II. contains values of the logarithms of HG=l-\-^^ 



