40 RELATIONS PERTAINING SIMPLY [BoOK I. 



I. In the inverse problem, the determination of the time, that is, from the 

 true anomaly, it is requisite to have recourse to a somewhat indirect method, and 

 to derive w from v by trial. In order to meet this inconvenience, the first series 

 should be treated in the same manner as the second : and since it may be readily 

 perceived that v is the same function of v as ?// of iv, so that the table for w 

 might answer for v the sign only being changed, nothing more is required than 

 a table for v&quot;, by which either problem may be solved with equal precision. 



Sometimes, undoubtedly, cases may occur, where the eccentricity differs but 

 little from unity, such that the general methods above explained may not appear 

 to afford sufficient precision, not enough at least, to allow the effect of the third 

 and higher powers of d in the peculiar method just sketched out, to be safely 

 neglected. Cases of this kind are possible in the hyperbolic motion especially, in 

 which, whether the former methods are chosen or the latter one, an error of 

 several seconds is inevitable, if the common tables, constructed to seven places of 

 figures only, are employed. Although, in truth, such cases rarely occur in prac 

 tice, something might appear to be wanting if it were not possible in all cases to 

 determine the true anomaly within 0&quot;.l, or at least 0&quot;.2, without consulting the 

 larger tables, which would require a reference to books of the rarer sort. We 

 hope, therefore, that it will not seem wholly superfluous to proceed to the exposi 

 tion of a peculiar method, which we have long had in use, and which will also 

 commend itself on this account, that it is not limited to eccentricities differing but 

 little from unity, but in this respect admits of general application. 



36. 



Before we proceed to explain this method, it will be proper to observe that 

 the uncertainty of the general methods given above, in orbits approaching the 

 form of the parabola, ceases of itself, when E or F increase to considerable mag 

 nitude, which indeed can take place only in large distances from the sun. To 

 show which, we give to 



3*a_nnv f 206265&quot;, 



the greatest possible error in the ellipse, which we find in article 32, IV., the 

 following form, 



