10 KELATIOXS PERTAINING SIMPLY [BOOK I. 



9. 



If a perpendicular let fall from any point whatever of the ellipse upon the 

 line of apsides is extended in the opposite direction until it meets the circle 

 described with the radius a about the centre of the ellipse, then the inclination to 

 the line of apsides of that radius which corresponds to the point of intersection 

 (understood in the same way as above, in the case of the true anomaly), will 

 be equal to the eccentric anomaly, as is inferred without difficulty from equation 

 IX. of the preceding article. Further, it is evident that r sin v is the distance of 

 any point of the ellipse from the line of apsides, which, since by equation VIII. it 

 = a cosy sin E, will be greatest for E= 90, that is in the centre of the ellipse. 



This grecitest distance, which =acos(p = - = \jap, is called the semi-axis minor. 

 In the focus of the ellipse, that is for v = 90, this distance is evidently =p, or 

 equal the semi-parameter. 



10. 



The equations of article 8 comprise all that is requisite for the computation 

 of the eccentric and mean anomalies from the true, or of the eccentric and true 

 from the mean. Formula VII. is commonly employed for deriving the eccentric 

 from the true ; nevertheless it is for the most part preferable to make use of 

 equation X. for this purpose, especially when the eccentricity is not too great, in 

 which case E can be computed with greater accuracy by means of X. than of 

 VII. Moreover, if X. is employed, the logarithm of sine E required in XII. is 

 had immediately by means of VIII. : if VII. were used, it would be neces 

 sary to take it out from the tables; if, therefore, this logarithm is also taken 

 from the tables in the latter method, a proof is at once obtained that the calcula 

 tion has been correctly made. Tests and proofs of this sort are always to be 

 highly valued, and therefore it will be an object of constant attention with us to 

 provide for them in all the methods delivered in this work, where indeed it can 

 be conveniently done. We annex an example completely calculated as a more 

 perfect illustration. 



