190 MASSES OF JUPITER'S SATELLITES [8 



We also take Laplace's m'" as an approximation. We then find the equations 

 between the residuals : 



Laplace's (5), 0= 804-05 - 970'68 -- 31'45 + 1'05 ~ + 1590'24^r 



//. m m" m' 



Laplace's (2), = - 298'48 - 2963'45 -^ - 56'90 - 3807'27 ^r 



p m m" 



Finally the annual sidereal motion of the node of II as derived from 

 Damoiseau's p. iii, is 134207"'06. Another value is given on p. i, namely 

 12 4' 40"'4 = 134198 VV 76, but the former appears to be that employed in 

 constructing the tables. Substitute this in the second equation for p of 

 Ch. vii., and we get an equation corresponding to Laplace's (6) of Ch. ix. ; 

 and the quantities I must be evaluated from Damoiseau's coefficients. I is 

 the coefficient of a term in * which refers to the node of II. And 

 disregarding inequalities the semi duration of an eclipse is, by Ch. vm. 



/ * 



T 7 /I (l+p') 2 -~r,, where we may also take /8 = T x synodic motion. The 



value of p' is given in the same chapter. Hence by comparison with 

 Damoiseau we find ( 1 + p') ~ = Damoiseau's M; also 



I II III IV 



Value of T l h 7 m 52 s I 1 ' 25 m 49 3 l h 46 m 50 s 2 h 22 m 42 s 



f} -' ( 



/^ /v r 



3 T>A -0-001097 -0-083957 0'004656 0'000222 



lerm in M\ 



Concerning p' there is some doubt as to the value used by Damoiseau as 

 his coefficients are not consistent with each othei'. We take 1 +p'= 1'0796, 

 which is consistent with several of his coefficients. 



We then find 



?=-108"'235, Z'=-5216"-0, Z"=178"-59, 



7 7" 7'" 



whence 



^='020750, ^=-'034239, y= -'000933, 



differing very little from the values in the Mecanique Celeste. With these 

 constants the equation for p yields 



jv (\ 



= 859-8 - 109613-2 -^ - 48477 -- - 17908'8 - - 770'3 



" 



m m m 



