JUPITER AND SATURN. IO9 



If these numbers are added to a derived from the observed 

 mean motion by means of the elliptic equation, we have : — 



Log a. 



Mercury 9-58782160 Newcomb, Tables 



Venus 9-85933745 



Earth 0-00000012 



Mars 0-18289616 



Jupiter 0-71623339 Hill, (Laplace's a) 



Saturn 0-97967915 



Uranus i -28309713 Newcomb, Tables 



Neptune 1-47814316 



(2 has been added in the last place of decimals for the action 

 of Mercury in the cases of Earth, Uranus and Neptune.) 



If the correction to log a is neglected so far as concerns the four 

 inferior planets, no harm will be done, because in the case of their 

 largest perturbations, 5 resulting figures will be correct even if 

 Jupiter is the disturbing planet and no perturbation runs into that 

 number ; similarly, the perturbations by Saturn are so small 

 that the error is quite negligible. Again, the corrections to log a 

 for Saturn, Uranus and Neptune are so nearly equal that they 

 practically cancel one another in their mutual perturbations. 

 The only important case is that of Jupiter and Saturn, but there 

 is no reason why accuracy should not be maintained throughout. 



It should be noted that the values of log a for Uranus and Neptune 

 are not the rigorous mean distances, as the theories are not cleared 

 of the effect of the great long period inequality. 



When the above values of log a are used the distances of the 

 planets only deviate from the elhptic values by periodical quantities, 

 and therefore conform to the usual definition of mean elements. 



In order to check some of the perturbations used by Le Verrier, 

 it is necessary to compute the Laplacian b coefficients and their 

 logarithmic derivatives. The new formulae for this purpose which 

 are developed in the 1909 June and December numbers of the 

 Monthly Notices of the R.A.S., make this computation simple, 

 in fact, aU the numbers required can be calculated in one day. 



The value of log a for which the b's are calculated is 97365540 

 but a linear formula is given later which will allow us to pass to 

 the b's for log (o+Aa) quite easily. By the formula on page 

 648 (M.N.R.A.S. LXIX.) we compute &,,, (noting that in this,, 

 as in all cases, the q terms are negligible), we have 



I + v/^ = 1.91559135 



V8(a,i + a;i) = I.91558955 



giving 60.1 = 2-18014144. log = 0-33848467. 



Then if in &i, 1 = "' &o, 1 pi p2 ps Pl 



we stop at bill, we require to find pn by a continued fraction* — 

 here the final terms only are given : — 



* See loc. cit.. p. 640, or Ast. Papers of the American Ephemeris, Vol. III.^ 

 p. 64. 



