

SECT. 1.] TO POSITION IN THE ORBIT. 47 



in which the law of progression is obvious. Hence is deduced, by the inversion 

 of the series, 



7 = 1 -M + jHJH. Th A 3 + ^nh ^ + T^Wiln, # + etc. ; 

 Putting, therefore, 



C will be a quantity of the fourth order, which being included in our table, we 

 can pass directly to v from A by means of the formula, 



if _ - ____ 



denoting by y the constant 



i + 5 



In this way we gain at the same time a very convenient computation for the 

 radius vector. It becomes, in fact, (article 8, VI.), 



Nothing now remains but to reduce the inverse problem also, that is, the 

 determination of the time from the true anomaly, to a more expeditious form of 

 computation : for this purpose we have added to our table a new column for T. 

 T, therefore, will be computed first from v by means of the formula 



then A and log.B are taken from our table with the argument T, or, (which is 

 more accurate, and even more convenient also), and log B, and hence A by 

 the formula 



finally t is derived from A and B by formula [1], article 37. If it is desired to 

 call into use the Barkerian table here also, which however in this inverse problem 



