THE 11 BIT OF URANUS 1-7 



The following are special values of ^j, and 5,, developed to tlic sixth power of 

 the eccentricities, as derived from the preceding formulae : 



Po = — 3(; 



1 3 , , 5 , 7 - 



,3_L _,J> 



1 13,1 



^'= = 2^-3^+16 



_3 . 45 567 



■^^-8^ -128^ +5120' 



1 . 6 ., 



^'^ = 3^-15^ 



125 4 _ 4375 J 

 384 *" ■"9216'' 



P^ = Q«^« ----^' 



27 5 



■^^<' = 8()' 



_ 16807 , 

 -^''- 46080 ' 



, 1 , , 1 , 1 



^' = ^-8^ +192' -9216' 



1 1 , , 2 



48 



2^ = o^-«^ + 



€' 



_3 „ 27 , 243 , 



^=-8'^'-r28' +5r20' 



5^ = 3^-15'^ 



125 , 3125 



38^ 

 27 



^^~384'* 9216''" 



(16)' 



2« = 80'' 



_ 16807 g 

 ^^~ 46080 ' 



Having the developed ^ and r, in terms of time, let us resume the equation (13). 

 As only purely linear operations are performed on Q in this equation, it follows 

 that if we represent its several parts by (?i, Q., etc., and by ^p„ h^^, etc., the values 

 ^p obtained by putting § = §1, <2 = Q^, etc., we shall have 



^p = cipi + cip2 + etc. 

 We have, therefore, only to find the separate values of r^h^ corresponding to the 

 different terms of Q, and to take their sum. Let us then represent, as before, by 



h cos {i'7: -\- iX 4-/cj' +ycj) 



any one term of R. 



3 April, 1873. 



