AN ELLIPTICAL ORBIT. 



Nat. cofine of a zr .4127292.5 log. zr — i. 61 56652 

 Eccent, -|- .0167923. 



.4295215.5 log. =— 1.6329849 



Diff. log. rz .0173197 

 Comp. of conj. femidiam. log. zr . 613 



Cotang. of ^, 6^'^ 2)7' 24". 96 =: 9.6562166 



Cotang. of true anomaly 64° 45' o".8 = g-62T,^gy6 



Log. fine «= 9.9594487 

 Log. fine 64° 45' o".8 comp =: .0436122 



10.0030609 

 Comp. of log. conj. — 613 



Log. diftance =r 1 0.002 c 



Hence the equation is i''44' 59".2. In Zach's tables it is 

 1° 44' 59". 25, 



The feries above given converge flowly when the mean ano- 

 maly is near 3 S. or 9 S. In this cafe the true anomaly may be 

 obtained with great accuracy by a feries derived from that which 

 expreffes the cofine in terms of its correfpondent arch, as follows, 



Subtraft the eccentricity from the mean anomaly and call 

 the remainder R. Let the difference between R and 90" be = z. 



Then R + 1 zz ± f s' —^ZZl^z' x '^-Zfl z' u. 

 ^ ^ 2 24 8 +88^ 



720 24 16 



And a will be the true anom. counted from the upper apfis. 

 For the Earth's Orbit. 



azzR -j 3.9240802. zz The uppermoft of the figns pre- 



+ — 4. 1 49 1 904. z3 fixed to the 3d 5th and 7th 



4.8430581. z* powers of z mufl be ufed when 



+ — 5.5462722. z5 R exceeds 90° and -the lower 



-j 5.3413007. z^ fign when R is lefs than 90". 



+ — 6.4890406. z'' 



D 2 Example^ 



