28 



THE ORBIT OF URANUS. 



Counting the integrated values of p and q in a direction perpendicular to the 



moving plane we have 



. tan 



sm Q = - 



tan q 

 cos = - 



tan<|> 



which, being substituted in the expression for dl, gives 



cos $> ,, 7 , , 



dl = - - (tan qa p tan paqj. 

 1 -f- cos <p 



The approximate value of the integrated correction is therefore 



(32) 



For every pair of periodic terms in p and q, such as 



q s sin 



will contain the secular term \ s z 



, p = s cos ;z, 



, which will be confounded with the mean 



motion, and, if it were not so confounded, would in few or none of the larger 

 planets amount to a second in a thousand years. If the secular terms in p and q be 



q = st ; p = s't 



H will vanish. We hence conclude that these terms are entirely unimportant in 

 the present state of astronomy, and that, if we consider the positions of the plane 

 of the orbit at two epochs, we may consider the points of departure in them to be 

 equally distant from their common node. 



We have therefore only to consider the motion of the inclination and node due 

 to the change of the position of the orbit and of the ecliptic. If we put 



$, the inclination of the orbit of the planet to the ecliptic, 



0, the longitude of its node counted on the ecliptic, 



T, the longitude of the same node counted from the same fixed point in the 

 moving plane of the orbit from which v is counted, 



Then, the longitude of the planet on the ecliptic, or Z, will be given by the 

 equation 



tan (L 0) = cos <p tan (v r), 



or, when developed in powers of <>, 



i = r _l_0 T + Z>, (33) 



where D is the reduction to the ecliptic, the value of which is 



D = tan 2 $ sin 2 (v T) -f- J tan 4 \ <> sin 4 (v T) etc. 



Let us refer the instantaneous rotations of the orbit and of the ecliptic to the 

 fixed points of reference in the two planes; q being the rotation around an axis 

 passing through the sun and the fixed point, and p that around an axis in 90 

 greater longitude, and the accented quantities referring to the ecliptic. We then 

 have 



