19] CONTINUATION OF DAMOISE ATI'S TABLES OF JUPITER'S SATELLITES. 117 



Let v denote the longitude and r the radius vector, calculated from 

 the mean longitude of Jupiter corrected by the secular term in Le Verrier's 

 Table V., and the term 8L in Table IX., and the longitude of the Peri- 

 helion corrected only by the secular term in Table V., employing the constant 

 eccentricity 



e = 0-0480767, loge = 8'6819346, 



and the constant value of the mean distance 



a=5'2025605, loga = 07162171. 

 Also JF=9916":53, logE= 3'9963597, 



log V^ = 0-0208955. 



1 e 



These constant logarithms may be used when r is found by passing 

 through the eccentric anomaly. If we employ series and call A the mean 

 anomaly we shall have 



v, = L + SL + 1 9827"'3 sin A + 595"'4 sin 2 A + 24 // '8 sin 3 A + l"'2 sin iA, 



and then r, = - 



I + e cos (v () - CT) 



where log a (I -e 2 ) = 07152121. 



Next, let r denote the longitude in the orbit and r the radius vector, 

 as calculated from Le Verrier's Tables, and we shall have 



The value thus found for <j> is to be used instead of <f> + SE, and the 

 value found for <^ is to be used instead of <f> 1 + Sr, in Damoiseau's formula 

 for Table III. of each Satellite. For J in the same formula, Le Verrier's 

 value of 8L in his Table IX. is to be used. 



It should be remarked that in forming the complete arguments given 

 in Table I. of each Satellite, wherever <f>, or $ multiplied by a constant, 

 occurs in Damoiseau's formula, J+<f> must be substituted instead of <. 



