Ball — On Laplace's Perturbations of Jvpiter^s Satellites. 551 



XXXI. — I^OTES ON- Laplace's Anaxttical Theory of the Pertueba- 



TIONS OF JuPITEIi's SATELLITES. By SiR .EoBERT S. BaLL, LL.D., 



P.R.S. 



[Read, February 22, 1886.] 



The matliematical difficulties of that portion of the Mecanique Celeste 

 AV^hich relates to the Perturbations of Jupiter's Satellites are well 

 known. Any aid, however slight, towards the elucidatii.ii of the obscure 

 points of Laplace's analysis may therefore be worthy of the attention 

 of the Academy. The laborious Bowditch has not removed the diffi- 

 culties I have felt. 



In the present communication I have rewritten the theory of the 

 perturbations in radius vector and longitude which are independent of 

 the eccentricities and the inclinations. The results are those of 

 Laplace, but the methods of demonstration have been modified. 



I employ throughout the notation of Laplace as used in vol. iv. of 

 the Mecanique Celeste. I commence with a theorem which Laplace 

 does not use, though if he had done so the first part of his theory 

 would have been greatly simplified. I find the theorem in Ponte- 

 coulant, vol. i., p. 511. 



Let n be the real mean motion of the disturbed satellite. 



a be the mean distance which in an undisturbed orbit would cor- 

 respond to the mean motion n. 



« + S« + Sr be the distance of a satellite from the centre of Jupiter, 

 the periodic terms being represented by Sr. 



i^be the portion of the disturbing function R which does not con- 

 tain periodic terms. 



Then the fundamental theorem is, that 



3 da 



For we have accurately 



di" 1 dh' [x 1 dP 



dt~ r dt~ r- r ' dr ' 



R. I. A. PROC, SER. II., VOL. IV. — SCIENCE. 



3G 



