242 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BOOK II. 



require no further correction, how to judge correctly of which, frequent practice 

 will in time show. When the heliocentric motion is small, the first hypothesis 

 generally supplies those values with sufficient accuracy : but if the motion in 

 cludes a greater arc, if, moreover, the intervals of the times are very unequal, 

 hypotheses several times repeated will be wanted ; but in such a case the first 

 hypotheses do not demand great preciseness of calculation. Finally, in the last 

 hypothesis, the elements themselves will be determined as we have just indicated. 



170. 



It will be necessary in the first hypothesis to make use of the times /, t , t&quot;, t &quot;, 

 uncorrected, because the distances from the earth cannot yet be computed : as 

 soon, however, as the approximate values of the quantities x , x&quot; have become 

 known, we shall be able to determine also those distances approximately. But 

 yet, since the formulas for Q and (&amp;gt; &quot; come out here a little more complicated, it 

 will be well to put off the computation of the correction of the times until the 

 values of the distances ha,ve become correct enough to render a repetition of the 

 work unnecessary. On which account it will be expedient to base this operation 

 on those values of the quantities x , x&quot;, to which the last hypothesis but one leads, 

 so that the last hypothesis may start with corrected values of the times and of 

 the quantities P 1 , P&quot;, Q , Q&quot;. The following are the formulas to be employed 

 for this purpose : 



vn. &amp;lt;/ = / 

 vni. Q&quot;=3f i 



IX. ^008/9 = R cos B cos (a /) 



ttt a \ -I- tf cos J? cos (I a)} 



-p, ((&amp;gt;&quot;cos /3&quot; cos (a&quot; a) -f K r cos B&quot; cos (*&quot; a)), 



X. &amp;lt;&amp;gt; sin /J = R sin B 4- _ 1 +Z1 (,/ s in ft -4- K sin J?) 



^(1 + ^) 

 i r 8 ? y 



p ((/ sin jT+JZ&quot; sin # ) 



