14 ON THE PERTURBATIONS OF URANUS. [2 



To these may be added the following, which are of two dimensions in 

 terms of the eccentricities : 



+ 0-57 m' sin 3 {nt -n't + e- e'} 



- 1 -08 mV sin {3 (nt - n't + e - e 7 ) - m + CT'}. 



These expressions may be put under the following form : 

 A, cos (n - ri) t + h, cos 2 (n - n') t + /i 3 cos 3 ( n- n') t 

 + k, sin (n - n') t + k, sin 2 (n - n') t + k, sin 3 ( n- n') t 

 +p 1 cos n't+p, cos (n 2n')t+p t coa (2n 3n')t 

 l sin n't + q..sm (/<-- '2n') t + q, sin (2n-3n')t. 



13. Let the time of the mean opposition in 1810 be taken as the 

 epoch from which t is reckoned ; this date, expressed in decimal parts of 

 a year, will be 1810'328. Also, let 3 synodic periods of Uranus, =3 '0362 

 years, be taken for the unit of time ; then the change of the mean anomaly 

 in an unit of time will be 1 3 0'-5 ; also n = 1 3 0'-6, n = 4 36''0 



.-. -Ji' = 824'-G, Ji-2n'=348'-G, 2n- 3n' = 12 13''2. 

 Hence the equations of condition given by the modern observations will be 

 of the form 



c= 



^cosjl.S 0-5}; + S.r. ; cos{2G 1-0} t 



+ h,cos{ 824-6}<+ h, cos {16 49'2} t + lt 3 cos {25 13'8} t 

 + ^sin{ 824'6}+ L sin {16 49'2} t + k 3 sin {25 13'8}< 



+ p, cos {43 G'O} t + p., cos { 3 4 8 '6} t + p 3 cos {1 2 1 3'2} t 

 + <7iSin{ 436'0}+ </ 3 sin{ 3 48'6} < + 2 3 sin{l2 13'2} 



in which t assumes all integral values from 10 to +10 in succession, 

 and the several values of c" are contained in the table given in Article 9. 



14. The final equations for the corrections of the elliptic elements will 

 be found by multiplying each equation successively by the coefficients of 

 Be, Sn, Saj], and S;/,, which occur in it, and adding the several results. 



Let the equations be treated in a similar manner with reference to 

 the quantities h lt k l} h.,, k t , h.,, k 3 , p.,, q.,, p 3 , q 3 . 



