THE ORBIT OF URANUS. 185 



Supposing the mass of Neptune to be uncertain by one-fiftieth of its entire 

 amount, which is quite possible, it will be seen the longitude of the mean peri- 

 helion is from this cause uncertain by more than two minutes, the mean longitude 

 of Uranus itself by nearly a minute, and the mean motion by nearly two seconds 

 in a century. 



It will be seen that the logarithm of the mean distance just given does not 

 accurately correspond to that of elements IV plus the constant term of r)» X 0.434:3, 

 as it should. This difterence arises from the rejection of the terms of the second 

 order in 6», which can not affect tlie geocentric longitude of tlie planet by a tenth 

 of a second for a number of centuries. 



It is to be remarked that these mean elements are those to be used in the 

 general theory of the secular variation of the planetary orbits. 



Concluded Theory of Uranus. 



The elliptic longitude and radius vector of Uranus, affected by the secular and 

 long period perturbations of the elements, will be given by the following equations. 

 Put 



l> = »o' + fo^ 



9 = 1 — 7r,j, 



k = e^ -|- tk, 

 e'- = Ir -\- k', 



the zeros mdicating elements IV, and f7*, r^-, and hi being the perturbations of 

 these three elements just given. Thou 

 Elliptic longitude in orbit = I 



+ U - M '' } I ^^' ~ ^'"'^ '"' ''^ ~ ''^'^'' '"" ''^ } 



+ 1 ll ~ 64 "' } { ^'"^ ~ ^^'^'^ "'" '^^ ~ ^^'^'' " ^''^ '"'' ^^ ] 

 -f ^^? I {f-' — 6A-'/r + li ') sin ij — (4/,-7t — 4/./*') cos 4/ I 



+ '^^ff ' I ^^"' ~ ^^^'^'''' + '^^'^''^ ^'" ^^ ~ ^'^'""'^ ~ ^^^''^'' "^ ^' ^°^ ^^ } 

 Neperian logarithm of r = « -j- ig- -|- ^Ve' 



— I 1 — ^e- I |/.-cos(7 + ^sin.<7[- 



_ I ^ _ ^^^^ e- \ I (i' — h") cos 2g + 2hk sin Qj I 



24 May, 1873. 



