40 



THE ORBIT OF URANUS. 



the index of the disturbing planet has the same value, arranging the individual 

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

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

 of the disturbed planet, as j<j, will be found in the same series with that of N itself, 

 and j lines above and below. 



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



dv dp 

 anomaly of Uranus, -^71 -JTI 



log r. 



Their values are as follows : 



ndt 



ndt 



r=l 



+.0005507 

 .0468889 cos g 

 .0016494 cos 2*7 

 .0000732 cos 3j 

 .0000035 cos 4 



1.001103 

 +.093933 cos g 

 +.005507 cos 2g 

 +.000336 cos Zg 

 +.000020 cos 4gr 



Considering only those terms which are of the first order, the value of D' t 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 R and 



+.0468889 sin g +.0938294 cos g 



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



+.0002196 sin Bg +.0003357 cos 3g 



+.0000142 sin 4g +- 000 206 cos 4^ 



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



riff 



-. 



dv 



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

 form 



m 



R = h cos N 



will produce in D\R the term 



jy ft m ink s'mN- 



and (2) by forming the expression 



D' t R = 



'dp 



dt 



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

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

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



_,_ d_R_ i >, . . u _ 2.26 123.87 

 m'n ft dt v 



+3470.23 +217.28 +19.125 +2.24 +0.18 



