5^*«^^ lit^iv relaivng to the Mobie^ OrlM^ -^ "^ -^ 33 

 following equation for finding tlie kpsidal distances, 'P:^/^^^. • 



Wo'(?^2:?M the mean distance, and e the eccentricity, the apsidal 

 3istarices'are a[\-\-e) and «(1— e). 



Substituting these values for r in the above equation, and de- 

 veloping the small term to quantities of the fourth order, we 

 obtain 



A h^^2fJM{l + e) -h Ca%l 4- 2^ + e^) - £^ a\l -f 4e + 6e^) = ^ - 

 -riQfi g^xioofiT j')di 'to muln/ b-; 



whence it follows that . MroRgsfe^ 



iu ai Ji ^^ai^-.2/>tfl + C«2(l + e2) _ ^a4(i^gg2)^0 ^ ^^ 



These equations give the relations between the arbitrary con- 

 stants h and C, and the new constants a and e by which the 

 former may be replaced. ^ 



From the second of them, we find '^ 



'C'lq aim to , 3 . ri <( jf wov^A 



siUmiO m&t>n. . ■ a= y^+^7t; - :.. ^r 4 ^^^^ 



(0) itoiimfp^}M L a L , ^^^ ^5 



or, putting for a in the small term its first approximate value ^, 



which agrees with Professor Challis^s expression in p. 281. ' ' 

 Now apply a similar process to the equation -i'^i^M^ 



which diifers from the equation (C) in having a put for r in the 

 smalj term. In this case, we find 



and ^ « , m' ^ | 



from the latter of which equations it follows that v - * 



C 2«'^ G^ 

 PAi/. ilfa^. S. 4. Vol. 8. No. 49. July 1854. D 



