12 



ON THE PERTURBATIONS OF URANUS. [2 



10. Let Se, Sa, Se, and Sw denote the corrections to be applied to 

 the tabular elements of Uranus, then the correction of the mean longitude 

 at any time t is 



f ('- 1 



= Se + 2(rSnr + 1 Sft - 2 cos (nt + e - w) + - cos 2 (nt + e - TST) V e Sra- 



I ^ J 



+ -{ 2 sin (n + e - nr) + - sin 2 (n* + e nr) - Se. 



If we include the small term Sr-'Sra- in the quantity Se, this correction 

 may be put under the following form : 



Se + t S/i + cos nt 8x, + sin nt 8y 1 + cos 2nt 8x, + sin 2n< S_y, 



in which expression 



j 

 Sx.,= -e {cos (e CT) 00!, + sin (e 



Sy., = R {sin (e w) Sx, + cos (e 



11. Also, adopting the notation of Pontecoulant's Theorie Analytique, 

 the perturbations of mean longitude 



+ m'e 1G, sin {i (nt - n't + e - e') - (nt + e - OT)} 

 + /uV S//i sin {i (r^ - n't + e - e') - (n^ + e - CT')}. 



Where the accented letters belong to the disturbing planet, i takes all 

 integral values, positive and negative, except zero, and if we put i(n n') = z, 

 the values of F t , G t and H { are the following : 



,-, 

 Fi = 



3 in 4 in" \ 2n 3 a dA f 



2 /- - ~^\ + i r aA i + n ~\ a ~r- > 

 z- (z- n-) z'-ri'} z (z 2 n-) da ' 



,-, _ . 



~ ~ + + 



\ ~ (z-nyz(z-2n) ~ z(z-2n) ^Tr z(z 



3in 3 \ 

 -n)(z-2n)j 



J 3 (i-l)n 1 1 (i-l)n 1 n" 2in 3 \ ^dA,- 



h \ ~ 2 (z -nyz(z-2n) ~2z (z- 2n) ~ 2 z 2 ^ 2 ~ z (z -n)(z- 2n)j " da 



