320 The Orbit and Motion of 



HENCE may be derived the following rule for approximating 

 to the true ratios of AS and ES to CS : 



Make CS : aS =.fm. u :Jin.v, 

 CS : eS fin.y :Jln.x t 

 eS : S fin. ia :fin. z, 

 aS : iS Jin. z '.fin. w. 



Then make AS = aS + -^-, and ES = eS -^-. Then the 



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points A, C, E, will be in the circumference of an ellipfe, of 

 which S is the focus, and O the centre, and having the fedlors 

 ASC, CSE, very nearly equal. 



THE approximation will be much eafier, and almoft as ac- 

 curate, if of the difference of the logarithms of aS and *S 

 be added to the logarithm of aS, for the logarithm of AS, and 



of the difference of the logarithms of t S and eS be fubtradl- 

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ed from the logarithm of eS for the logarithm of ES. 



IT may even be fufficient to add of the difference of the 



logarithms of eS and S to the logarithms of aS, and to fub- 

 tracl it from the logarithm of eS. 



THE following Theorem may be of ufe for conftrucling the 

 ellipfe, and, I believe, is new : 



LET DAP be an ellipfe, (fig. 2.) of which O is the centre, 

 S the focus, and ap the directrix ; from any three points A, C, 

 E, draw lines Aa, Cc, Ee, perpendicular to the directrix ; draw 

 the radii AS, CS, ES ; draw AK, xCH, and E, perpendicular 

 to Aa, and AG, CF, perpendicular to ES, and Sp perpendicular 

 to ap. 



LET 



