SECT. 1.] TO POSITION IN THE ORBIT. 15 



e s in e =_ 29543&quot;.18 = 812 23&quot;.18 

 Jf+esine .... 324 16 31 .59 

 differing from e . . . 8 .41. 



1 90 95 



This difference being multiplied by -^ri = n775 S ives 2 &quot;09&amp;gt; whence, finally, the 

 corrected value of E 32416 31&quot;.59 2&quot;.09 = 32416 29&quot;.50, which is exact 

 within 0&quot;.01. 



14. 



The equations of article 8 furnish several methods for deriving the true 

 anomaly and the radius vector from the eccentric anomaly, the best of which we 

 will explain. 



I. By the common method v is determined by equation VII, and afterwards 

 r by equation II. ; the example of the preceding article treated in this way 

 is as follows, retaining for p the value given in article 10. 



i^=162 8 / 14&quot;.75 log e ..... 9.3897262 



log tan IE. . . . 9.5082198w log cos v .... 9.8496597 



log tan (45 $9) . 9.8912427 9.2393859 



log tan 40 .... 9.6169771w ecosv =0.1735345 



i0 = 15730 41&quot;.50 logp ..... 0.3954837 



123.00 log (1 + ecosv) . . 0.0694959 



logr ..... 0.3259878. 



II. The following method is shorter if several places are to be computed, 

 for which the constant logarithms of the quantities y/a(l -4- e), y/ a(l e) should 

 be computed once for all. By equations V. and VI. we have 



sin v y/ r = sin E y/ a (1-4-e) 

 cos i v \J r = cos J E y / (l e) 



from which J v and log y/ r are easily determined. It is true in general that if we 

 have P sin Q = A, P cos Q = B, Q is obtained by means of the formula tan 



-A. A 7? 



Q = -j,, and then P by this, P = ^^, or by P = =. : it is preferable to use 

 ft sin Q J cos Q 



