36 RELATIONS PERTAINING SIMPLY [BoOK I. 



rithms, assuming e and u or e and F to be known exactly, the first part will be 

 liable to the error 



5 (uu l)e&amp;lt;o 



if it has been computed in the form 



or to the error 



3(M-fl). 



if computed in the form 



A. G it &quot;-&quot; ~ 

 2 u 



or to the error 3 e (a tan F if computed in the form X e tan F, provided we neglect 

 the error committed in log X or log i k. In the first case the error can be 

 expressed also by Sew tan F, in the second by - whence it is apparent that 

 the error is the least of all in the third case, but will be greater in the first or 

 second, according as u or - ]&amp;gt; 2 or &amp;lt; 2, or according as + _F&amp;gt;- 36 52 or &amp;lt; 36 52 . 



But, in any case, the second part of N will be liable to the error w. 



VII. On the other hand, it is evident that if u or F is derived from JV by 

 trial, u would be liable to the error 



( w + 5 e o tan F) -r^=, 



or to 



, . Beta . du 



according as the first term in the value of .ZV is used separated into factors, or into 

 terms ; F, however, is liable to the error 



dF 



(w + 3 e at 



The upper signs serve after perihelion, the lower before perihelion. Now if 

 ^ is substituted here for -r-^ or for ^, the effect of this error appears in 

 the determination of v, which therefore will be 



