34 RELATIONS PERTAINING SIMPLY [BOOK I. 



increase to an amount not to be neglected : for this reason formula III., article 8, 

 is less suitable in tbis case. Tbe quantity d may be expressed thus also, 



3 (a r) __ 3 e (cos v-\-e) 



r l ee 



which formula shows still more clearly when the error (1 -\- d) to may be neglected. 



III. In the use of formula X., article 8, for the computation of the true from 



the mean anomaly, the logt/- is liable to the error ( -|- Jd) w, and so the log 



sin | (f sin E \ I - to that of (f -f- \ 8*} to ; hence the greatest possible error in the 

 determination of the angles v E or v is 



or expressed in seconds, if seven places of decimals are employed, 



(0&quot;.166 -f 0&quot;.024 tf) tan l(v E). 



When the eccentricity is not great, S and tan i (v E) will be small quantities, 

 on account of which, this method admits of greater accuracy than that which 

 we have considered in I. : the latter, on the other hand, will be preferable 

 when the eccentricity is very great and approaches nearly to unity, where 8 and 

 tan J (v JE) may acquire very considerable values. It will always be easy to 

 decide, by means of our formulas, which of the two methods is to be preferred. 



IV. In the determination of the mean anomaly from the eccentric by means 

 of formula XII., article 8, the error of the quantity e sin E, computed by the help 

 of logarithms, and therefore of the anomaly itself, M, may amount to 



~T 



which limit of error is to be multiplied by 206265&quot; if wanted expressed in 

 seconds. Hence it is readily inferred, that in the inverse problem where E is to 

 be determined from M by trial, E may be erroneous by the quantity 



. . 206265&quot;=^-^. 206265&quot;, 



X. (1 M lr 



even if the equation E e sin E= M should be satisfied with all the accuracy 

 which the tables admit. 



