SECT. 3.] PLACES IN ORBIT. 125 



where v v is greater than 360 we must conceive the area of the whole ellipse 

 added to the area of the sector and the area of the segment just as many times 

 as the motion comprises entire revolutions. 



Moreover, since b = a cos (f&amp;gt; , from the combination of equations 1, 10, 10*, 

 follow 



sin q tan f 



[191 COS CD = aT/i^siX 

 J--sin a r 



ri n-fc-i sin q tan f 



[19*] cos 01 = 5-77 . ,/., 



2(L sin ^gy 



whence, by substituting for I, L, their values from article 89, we have 



s n fsin q 



This formula is not adapted to the exact computation of the eccentricity 

 when the latter is not great : but from it is easily deduced the more suitable 

 formula 



roil f.nn 2 4 m sin ^ (/ ff) + *an 2 2 M 

 ^- 



to which the following form can likewise be given (by multiplying the numerator 

 and denominator by cos 2 2 o&amp;gt;) 



T221 tin 2 i ro sini&amp;gt; 2 (f~ff) + cos2 i (/ 9) sin2 2 &amp;lt; 

 *&amp;lt;? - - _ -- 



The angle y can always be determined with all accuracy by either formula, using, 

 if thought proper, the auxiliary angles of which the tangents are 



tan 2 to tan 2 w 



*(/-?) sin 



for the former, or 



sin 2 to sin 2 



for the latter. 



The following formula can be used for the determination of the angle G, 

 which rea,dily results from the combination of equations 5, 7, and the following 

 one not numbered, 



T9T1 fin a - 



I -io i tan cr -T-- 



- - - j. 

 2oo8/yrr 



from which, by introducing w,is easily derived 



