368 Astronomical and 'Nmitual CuUeclions. 



same that has been already employed for M, §. 38 ; and the 

 whole equation becomes 



, _M(1+ _-pJe +tangr-.« sin (A" -«'")■ 



§ 59. 



Hence we have the values of v and A in the equation /" = 



(M + y) §' + /*, which determine the influence of p and g', which 



were before neglected, on the value of ^"' ; and the equation might 



be applied to the correction of the elements already calculated. 



But the labour may be much abridged by observing that the 



e' already found can differ but very little from the true value 



which is now to be found ; and if we call the approximate 



Jip' 

 value (§), we shall have, since h is also small, -^=^, and the 



equation for §"' will become ^'"—M (1 +—rP + 7-\ ? and hence 



{' = {U + v)^,v bemg = -p + _. 



*■ (?) 



1.60. 



In order, therefore, to correct the two equations for r"' and 

 h"' , we must multiply all the coefficients which contain M by 



^ "^ " = H, and those which contain M' by H=. The equa- 



tion for r remains unaltered : and since the logarithms of the 

 coefficients are already found, the operation is by no means 

 difficult. 



§. 61. 



It will be convenient in the mean time to collect the equations 

 for the determination of H together, for the sake of employing 

 them more readily. When we have found Q), which is the ap- 

 proximate value off', the time and distance at the perihelium, 

 and \// the true anomaly at the middle observation, as well as the 

 ditterences of the anomalies t and <7-, Ave must compute 



