SECT. 1.] 



TO POSITION IN THE ORBIT. 



43 



[1] 



By means of the common trigonometrical tables, T B ff E -\- -^ sin E may be com 

 puted with sufficient accuracy, but not E sin E when E is a small angle; in this 

 way therefore it would not be possible to determine correctly enough the quan 

 tities A and B. A remedy for this difficulty would be furnished by an appro 

 priate table, from which we could take out with the argument E, either B or the 

 logarithm of B the means necessary to the construction of such a table will 

 readily present themselves to any one even moderately versed in analysis. By 

 the aid of the equation 



20 B 



\j A can be determined, and hence t by formula [1] with all desirable precision. 



The following is a specimen of such a table, which will show the slow increase 

 of log B ; it would be superfluous to take the trouble to extend this table, for 

 further on we are about to describe tables of a much more convenient form. 



38. 



It will not be useless to illustrate by an example what has been given in the 

 preceding article. Let the proposed true anomaly = 100, the eccentricity 

 = 0.96764567, log q = 9.7656500. The following is the calculation for E, B, 

 A, and t : 



log tan * v 0.0761865 



,1 e 



9.1079927 



log tan 



9.1841792, whence } E= 8 41 19*32, and U = 



