APPENDIX. 287 



Table la for the Ellipse contains log E v and log E, for the argument A , to 

 gether with the logarithms of their differences corresponding to a change of a unit 

 in the seventh decimal place of the argument. It was computed by Prof. J. S. 

 HUBBAKD, and has been used by him for several years. Since it was in type, a 

 similar table, computed by Mr. A. MARTH, has appeared in the Astronomische Nach- 

 ricMen, Vol. XLIIL, p. 122. The example of article 43 will furnish an illustra 

 tion of its use. 



Formulas expressing the differentials of the true anomaly and radius vector 

 in a very eccentric ellipse, in terms of the differentials of the time of perihelion 

 passage, the perihelion distance and the eccentricity may be obtained from the 

 equations of this article. 



If we put = I, C= 0, we have, article 39, 



tan i iv -|- | tan 3 i w = ^ 

 which, by article 20, gives 



dw a. -., 3 at 7 . t , 



~n = ^dt -^-fr: aq -4- -da. 



4 &amp;gt; 2 



75 2q75 2 75 



We also have, article 40, 



log tan i v = log tan lw Hog(l | (5 tan 2 i iv) -f- log y 

 and, therefore, 



2 sin I v cos ^ w 2 sin w cos 8 w (1 - 



^f &amp;gt; a cos 2 1- ? , , Sat cos 2 \ w , 



sinv 75taniw(l A) ~2 ? 75 tan 1 w (1 f ^) ^ 



w , .rfj&quot;, ^^4 rf|3 



.-*j) flfa &quot;T~T~r-f3rriT 



which, by putting 



rr (t COS 2 1 



75tanlw(l f ^t) 



L 3 



Z - 



