SECT. 3.] PLACES IN ORBIT. 141 



in article 94 for the ellipse. In such a case, it is admissible to suppose the 

 elements of the orbit, roughly at least, known : and then an approximate value 

 of n is immediately had by the formula 



s mf\/rr / 



tan 2 n = 4^ 7 .. 

 a^(e e 1 ) 



which readily follows from equation 6, article 99. z also will be had from n by 

 the formula 



cos2n sin 2 n 





2 cos 2 n cos 2 n 



and from the approximate value of z, that value will be deduced with a few 

 trials which exactly satisfies the equation 13, 13*. These equations can also be 

 exhibited in this form, 



( tan 2 ra , , / &amp;lt; i-o i \ 



i i a~ t hyp. log tan (45 +) 



,. sm*n x 8 I fy,j sm- n , J Jcos2ra 



I 



tan 2 n 



and thus, a being neglected, the true value of n can be deduced. 



104. 



It remains to determine the elements themselves from z, n, or c. Putting 

 a \j (ee 1) = (5, we shall have from equation 6, article 99, 



Mo sin/^r/ 

 P = o 

 tan 2 ?i 



combining this formula with 12, 12*, article 99, we derive, 



PI m / / i \ an an n 



[19] y/ (* 1) = tan y = -|^_ g) , 



n 9*1 tan ty - - tan / tan 2 n 







whence the eccentricity is conveniently and accurately computed ; a will result 

 from ft and ^ (ee 1) by division, and p by multiplication, so that we have, 



