SECT. 1.] 



TO POSITION IN THE ORBIT. 



37 



5 5 tan i/&amp;gt; (1 + 3 e tan .F ) w bbtsmip(l -\-3 e secF)ia 



if the auxiliary quantity u has been employed ; on the other hand, if F has been 

 used, this effect becomes, 



b b tan i/; (1 + 3 e tan F) to __ to I (1 -)- e cos ) 2 , 3 e sin t&amp;gt;(l -j- e cos t&amp;gt; ) ) 



~ ^ \ tan s if&amp;gt; tan 2 !^ 



If the error is to be expressed in seconds, it is necessary to apply the factor 

 206265&quot;. It is evident that this error can only be considerable when t/; is a small 

 angle, or e a little greater than 1. The following are the greatest values of this 

 third expression, for certain values of e, if seven places of decimals are employed: 



To this error arising from the erroneous value of F or u it is necessary to 

 apply the error determined in V. in order to have the total uncertainty of v. 



VIII. If the equation XL, article 22, is solved by the use of hyperbolic loga 

 rithms, F being employed as an auxiliary quantity, the effect of the possible 

 error in this operation in the determination of v, is found by similar reasoning 

 to be, 



(1 -f- e cos vf ot , 3 e sin v (1 -(- e cos v) w 

 8 &amp;gt; tan 2 1&amp;gt; 



tan 8 



i tan 2 1/&amp;gt; 



where by cu we denote the greatest uncertainty in the tables of hyperbolic loga 

 rithms. The second part of this expression is identical with the second part of 

 the expression given in VII. ; but the first part in the latter is less than the first 

 in the former, in the ratio X w : CD, that is, in the ratio 1 : 23, if it be admissible 

 to assume that the table of Ursin is everywhere exact to eight figures, or 



to = 0.000000005. 



