JUPITER AND SATURN. I07 



The only approximately correct value is that given by Hill and 

 adopted later by Xewcomb. Hansen introduces the correction to a in 

 his computation of terms depending on the second and higher powers 

 of the masses and Le Verrier could also have done so, but did not . 

 Harzer's valne is evidently derived from corrected mean motions 

 but it overlooks the partially compensating correction to the 

 mean distances. Xewcomb is not ever5^vhere consistent, — thus 

 in computing the secular perturbations of the inner planets he 

 uses values of the mean distances which practically reproduce 

 Le Verrier's value of o ; but it should be added that the errors 

 so introduced are very small. The greatly erroneous value of a 

 used by Harzer, seriously damages his work and will account, in 

 part at least, for the discordance to which Hill has already drawn 

 attention.* It is to be remarked that the interesting conclusions 

 as to the motions of the perihelia of Jupiter and Saturn, drawn by 

 Hill in the just-quoted paper, being based on Le Verrier's value 

 of o will also require considerable modification. In a paper 

 re-published in his Collected Works, Hill draws attention to the 

 differences between the mean distances of Jupiter and Saturn 

 according to the methods of Laplace and Hansen (T. 11, pp. 85-86). 

 But I quote from a letter which Dr. Hill kindly wrote me (30th 

 April, 1907) : — 



" My exact values for Hansen are given in the Introduction to my tables. 

 They are : 



Hansen, log. a = 0'7i623737i6. log. a' = o-9794957io3. 



" Apply to these the corrections to Laplace's formulae and they become : 

 Laplace, log. a = o-7i62374665. log. a' = 0-9794965 529. 



" If the constant of precession is changed these numbers also undergo a 

 change, but this is minute. 



" I hardly think anyone would care for Hansen's a at present. Laplace's 

 form is to be preferred in almost all investigations. These values of the a are 

 derived from the mean motions which prevail at epoch 1S50. But as far as 

 the Laplace a is concerned, I do not think the values for 1900 would differ 

 more than a unit in the loth decimal." 



The Laplacian a resulting from Hill's figures is 



log (ft) = 97367409136 10 which we add the con- 

 stant part of the per- 

 turbations of the radii- 

 vectores -0001866830 from Hih's tables, pp. 28 



and 172. 



Hence log o = 97365542306 



If a has been calculated from the equation a^ »-=/ (i-l-;/z) the 

 reduction to the true a may be found as follows : — 



If F denotes the secular part of the elliptic perturbative 

 function then 



{a) Action of outer planet on inner 



Hog. a = -\ — — -MaDF 



{b) Action of inner planet on outer 



c^log.a'=-fi^^^,M(i+D)F 



♦Eccentricities . , of Jupiter and Saturn, CoU. Works. T. IV., p. 135. 



