SECT. 1.] 



TO POSITION IN THE ORBIT. 



41 



^&quot;Ml-TcosV) &quot;&quot;- 206265&quot;; 



from which is evident of itself that the error is always circumscribed within 

 narrow limits when E acquires considerable value, or when cos E recedes further 

 from unity, however great the eccentricity may be. This will appear still more 

 distinctly from the following table, in which we have computed the greatest 

 numerical value of that formula for certain given values of E, for seven decimal 

 places. 



E= 10 maximum error = 3&quot;.04 



20 .76 



30 .34 



40 .19 



50 .12 



60 .08 



The same thing takes place in the hyperbola, as is immediately apparent, if the 

 expression obtained in article 32, VII., is put into this form, 



w cos F (cos F-\- 3 e sin F) y (e e 1) 



The following table exhibits the greatest values of this expression for certain 

 given values of F. 



When, therefore, E or F exceeds 40 or 50 (which nevertheless does not easily 

 occur in orbits differing but little from the parabola, because heavenly bodies 

 moving in such orbits at such great distances from the sun are for the most part 

 withdrawn from our sight) there will be no reason for forsaking the general 

 method. For the rest, in such a case even the series which we treated in article 



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