9. 



REPETITION OF SOME NUMERICAL CALCULATIONS IN THE MECANIQUE 

 CELESTE RELATING TO THE THEORY OF JUPITER'S SATELLITES. 



THE calculations in question are those of Livre vin., Nos. 20, 22, 23. 

 In No. 20 Laplace calculates the ratios of the mean distances of the 

 Satellites by help of a formula of No. 3, but neglects the corrections 

 to those ratios that are produced by the mutual perturbations of the 

 Satellites. Employing the exact formula, with Damoiseau's values of the 

 masses, and p $(f>= '0220021, which results from substituting Damoiseau's 

 value of p for Satellite II, together with his values of the masses and the 

 values of I : V, &c. found by Adams from Damoiseau's constants (see p. 

 188) in the second equation for p of Livre vin., No. 23, we find 



a = 5-698464, 

 a' = 9-066548, 

 a" =14-462403, 

 a'" = 25-437328, 



where the value found by Laplace is that ascribed to a', for the following 

 reason : that the compression of Jupiter is found from the motion of the 

 node of Satellite II, and the form under which this compression enters the 

 perturbations of that Satellite is (p ^fy/a'*, and this is the only con- 

 siderable term in which the absolute value of the distance appears (see 

 the formal equation for p in No. 9) ; hence as one of the four distances 

 is indeterminate by our method, we may save a correction to the equation 

 for p that we employ, by choosing the unit of length so that a' has the 

 value Laplace ascribes to it. 



