182 DETERMINATION OF AN ORBIT FROM [BuOK II. 



resulting from the approximate assumption that n : n&quot; = 6:6&quot; is affected by an 

 error of the fourth order only, and even of the fifth only when the middle is dis 

 tant from the extreme observations by equal intervals. For this error is 



n&quot; _ Off (d c) (if if) 



where the denominator is of the second order, and one factor of the numerator 

 Q6&quot;(d c] of the fourth, the other rj&quot; r\ of the second, or, in that special case, 

 of the third order. The former equation, therefore, being exhibited in this form, 



,/ 7 I c n -4- d n&quot; n -4- n&quot; 

 ao = b-\- . , 



n -j- n n 



it is evident that the defect of the method explained in the preceding article does 

 not arise from the fact that the quantities n, n&quot; have been assumed proportional to 

 6, 6&quot;, but that, in addition to this, n was put proportional to 6 . For, indeed, in this 



way, instead of the factor -Jj , the less exact value 5 = 1 is introduced, 

 from which the true value 



2 jyj/VrV cos/cos/ cos/* 



differs by a quantity of the second order, (article 128). 



133. 



Since the cosines of the angles/,/ ,/&quot;, as also the quantities r/, r&quot; differ from 

 unity by a difference of the second order, it is evident, that if instead of 



n+n&quot; 



7t 



the approximate value 



14- 6ff&amp;gt; 



1 I 2rrV 



is introduced, an error of the fourth order is committed. If, accordingly, in place 

 of the equation, article 114, the following is introduced, 



. Off 



an error of the second order will show itself in the value of the distance $ when 



