7] THEORY OF JUPITER'S SATELLITES. 179 



n =825826010-423, 

 n' =411412421-501, 

 n" =204205627-040, 

 n'"= 87542597-800, 

 M= 337212-092; 



these satisfy the condition 



n 



- 3n' + 2n" = 0, 



making w-2w' = n'-2n" = 3001167-421. 



Laplace gives the following values for the masses * : 



m =0-0000173281, 

 m! =0-0000232355, 

 m" =0-0000884972, 

 m'" = 0-0000426591, 



the unit being the mass of Jupiter ; 



also /a- g^= 0-0219013, 



,. a ' 



^- = 5-698491, .=9:066548; 



*. -^J- 



so that 



/= (p - - <fr\ ^ = 0-0006744505, /' = (p - - <f> j -7, = 0'0002664317, 



(1 + m)f= 0-000674462, ( 1 + m')f = 0'000266438. 



First from the equations 



let us find the exact values of a and a' that satisfy them. 



[* To elicit these values from the observations a more elaborate theory is required than 

 the one here developed.] 



232 



