I06 JUPITER AND SATURN, 



comparisons show that the 3rd and 4th figures of the perturbations- 

 found by Le Verrier are erroneous. So as to make the comparisons 

 just, the only difference made is in the value of a and a' and the 

 resulting value of «. Although Le Verrier started with erroneous 

 values of a and a' , it is to be remarked that his tables contain, 

 the constant perturbations which make the tabular distances 

 from the Sun nearly exact. 



As already stated, the foundation of Le Verrier's theory of Jupiter 

 and Saturn is the value of a, the ratio of the mean distances of 

 the planets. In all the formulae given by Le Verrier allowance 

 for changes, secular and otherwise, in any of the elements is made 

 with the single exception of this ratio. As will be shown at once,, 

 the value for this ratio adopted by Le Verrier is largely in error 

 and his whole theory is thereby/ vitiated. The mean distances of 

 the planets are always determined indirectly through the mean 

 motions. In a purely elliptic theory the equation connecting the 

 quantities is 



a^ n^ = f (i + m) 



but this equation does not hold in the case of three bodies such 

 as the Sun, Jupiter and Saturn. In perturbed motion, constant 

 (or very nearly constant) quantities are added to both a and n. 

 Laplace, in his celebrated Mecaniqiie Celeste, uses the simpler 

 equation a^ rv^=f ; but in the cases of Jupiter and Saturn, the m 

 part is added.* But Laplace in the theory of Jupiter''s Satellites- 

 goes further,t and in fact gives equations by which the true values 

 of the a's can be determined. § This matter is also dealt with 

 very clearly by Tisserand.J Le Verrier was not ignorant of 

 these corrections to the elliptic values, for he states : 



" These corrections which are to be applied to the values of the semi-axes, 

 majores are too small to have an}- influence upon the calculation of the 

 coefficients of the perturbations.'"!! 



On pp. 187 and i8g of the same volume, he gives the " corrections " 

 to the semi-axes deduced from the apparent motion, but it is well 

 to remark that there will still remain the constant parts of the 

 perturbations of the radii-vectores. Newcomb and Hill give the 

 expressions which should be adopted.** 



A tabular exhibit of some of the most important values of log 

 a previously adopted will be interesting. It will be very obvious, 

 that only one of the values given can be correct. 



Log. o. 



XL Laplace .. 97366493. Mec. Cel., IV., p. 86. 



1831. Hansen . . 97367384, Jupiter and Saturn, p. 66> 



1874. Le Verrier . , 97367408. Annales, X., p, 17. 



1890. Hill . . 97365514. Jupiter and Saturn, p. 21. 



1895, Harzer . . 97359957- Sac. Verand, p. 87. 



1898. Newcomb . . 97365514. Tables of Uranus, p. 295, 



* See Tome, III., pp. 139 and 150. 

 t T., IV., p. 86. 

 § See T. IV., pp. 15 and 86. 

 + Mec. Cel. T. IN., p. 18. 

 I! Tome, X., p. 8. 

 ** See Newcomb, Orhit of Uranus, p. jr ; Hill, T. III., Jupiter and Saturn,. 

 p. 20. 



