40 



THE ORBIT OF URANUS. 



the index of the disturbing phmet has the same vahie, arranging the individual 

 terms of each scries according to the index of the disturbed planet. Thus, the 

 index of the product of any term, as 7t cos N, by any multiple of the mean anomaly 

 of the disturbed planet, asjy, will be found in the same series witli that of iV itself, 

 and j lines above and below. 



The next process will be the formations of the required functions of the mean 



anomaly of Uranus, -jr^ -j-, -3, ^^S ''• '^'lieir values are as follows : — 



po^ 



» 



ndt ~ 



dvp 



ndt 



1.001103 

 +.09:3933 cos g 

 -J-.005507 cos 2(/ 

 4-.00()336 cos Sj 

 -I-.000020 cos 4<j 



+.0005507 

 —.0468889 cos g 

 -.0016494 cos 2g 

 —.0000732 cos 3y 

 —.0000035 cos 4g 



+.0468889 sin g +.0938294 cos g 



+.0032988 sin 2g +.0055012 cos 2g 



+.0002196 sin Sg +.0003357 cos Zg 



+.0000142 sin 4g +.0000206 cos ig 



Considering only those terms which are of the first order, the value of D\R may 

 be found in two ways, the agreement of which will afford a check upon the entire 

 development of the perturbative function, and upon the computations of E and 



These are (1) by direct differentiation, witli respect to the time as con- 



6v 



tained in the mean anomaly of a single planet, whereby each term in E of thb 



form 



m 



E ^ — h cos N 



will produce in D\E the term 



D',E 

 and (2) by forming the expression 



''^Jnh sin N- 



r._ c')E dvg dE (7p(, 

 ' ^v dt'^ dp dt' 



As several " mechanical multiplications," like those indicated in this last 

 expression, are to be performed, the following example of the form of com- 

 putation is presented. It exhibits the formation of the product of those terms 



2 3 4 5 



+ 54.42 + 6.988 +1.18 +0.10 



+ 163.32 + 2.553 +0.33 +0.06 



+ 0.33 + 0.056 



— 0.(9 + 9.576 +0.15 + .02 







— .048 + .58 



+217.28 +19.125 +2.24 +0.18 



