SECT. 1.] TO POSITION IN THE ORBIT. 39 



r pp&v f r iqqAw , , 



J (T^Iecosw) 2 J (1 + cosw) 2 \P : \^2&amp;gt; 



the integrals commencing from v = and w = 0, or 



r (i+e)^i _ r 2dw 



J (l+ecost&amp;gt;)V 2 ~J (l+cosic) 2 



Denoting - - by a, tan I v by 6, the former integral is found to be 

 1 -\-e 



-j- H 3 ( 1 2 a ) |$ 5 ( 2 a 3a)-)-^(5 7 (3aa^4 3 ) etc.) , 



the latter, tan i w -j- ^tan 3 c. From this equation it is easy to determine to 

 by a and v, and also y by a and w by means of infinite series : instead of a may 

 be introduced, if preferred, 



Since evidently for a = 0, or 8 =. 0, we have f = w, these series will have the 

 following form : 



iv = v -+- d v + (Tdy&quot; + d 3 /&quot; -f- etc. 



= w + d ^ + ^ d w&quot; + d^e/&quot; -f etc. 



where v , v&quot;, v &quot;, etc. will be functions of v, and /, / , ?y w , functions of zp. When 

 d is a very small quantity, these series converge rapidly, and few terms suffice for 

 the determination of w from v, or of v from w. t is derived from w, or w from t, 

 by the method we have explained above for the parabolic motion. 



35. 



Our BESSEL has developed the analytical expressions of the three first coeffi 

 cients of the second series w , ^v&quot;, w &quot;, and at the same time has added a table con 

 structed with a single argument w for the numerical values of the two first w 

 and w&quot;, (Von Zach Momtliche Correspondent, vol. XII, p. 197). A table for the 

 first coefficient w , computed by SIMPSON, was already in existence, and was 

 annexed to the work of the illustrious OLBERS above commended. By the use 

 of this method, with the help of BESSEL S table, it is possible in most cases to 

 determine the true anomaly from the time with sufficient precision; what remains 

 to be desired is reduced to nearly the following particulars: 



