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Analytical and numerical theory of the motions of the orbital planes of Jupiter's satellites

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79

Since the expressions (61) and (65) are the same, we have:

+ 8 ~ÏT

The relation (58) can be used to check the values of previously found. Souillart has computed the Bji with his values of the a's (* = "-). In the notation of Laplace, employed by Souillart,

a' BjtM = «cW

The values of 4™ are given in Souillart II p. 11—14. These values have been corrected, — as Souillarts x differs slightly from ours, — By means of the formula:

S , h dcW m = & 4. -x

where — is of Souillart, also tabulated by him. (See above, p. 52). dx

We found the following values:

Sat. log \ [*] H '**A

I— II 0-6886949 I-S34 56I 1*534 556

I III 0-2104968 0-319890 0-3198896

I_IV 9-8693788—10 0-0829241 0-0829241

II— III 0-6848129 1-517083 1-517083 H__IV 0-1400352 0-2460512 0-2460517

III— IV 0-552 2932 1-014051 1-014114

Remembering that Jj&i are computed so as to be accurate to 6 decimal places and to 5, the numerical agreement of dj^ and Bjt* as required by (58) is all that can be wished, except in the case of In order to investigate the disagreement of d3^ and \ a'B3^ we have computed directly

the c<0 by means of table IV, add. ire of Leverrier {Annales de l'Obs. de Paris, Tome II) and then i « <<*>. We found for the latter 1-014115, so that in this case also (58) is numerically satisfied. At the same time we can conclude that e<'> of Souillart is erroneous. In the computation of the fts it appeared that in the combination III—IV W and JU) did not agree with Souillart's values. Considering that the o's have been computed by Souillart by means of the (see Souillart II p. 11) it is clear that #ie è's could not entirely agree with ours.

In the case of the Sun Souillart has not used Leverrier's development but, since in the theory

of the satellites - is small, he has preferred the development employed by Puiseux in his lunar theory. *)

rs

In this development everything is neglected that is of an order higher than (-^) , or, which is the same,

*) Puiseux, Sur les principales inégalités du mouvement de la Lune, Annales de l'Ecole normale supérieure p. 45-