THE ORBIT OF URANUS. 27 



The corresponding terms of 8 . and 6- , and may be obtained in tlie same wav 



ow d^ ' ■' 



by substituting -- ~ and — — for E in (30) and continuing the corresnondino- sub- 

 ov op 1 O 



stitutions of the general terms of the derivatives of R as given on page 9. 



The equation (31), besides being of the second order with respect to the disturb- 

 ing forces, is also of the second order with respect to the mutual inclinations. For 

 ^k, hk', Sy;, and br;' are of the first order with respect to both quantities, and, when- 

 ever t is not zero, 7i is a quantity of the second order, containing a" as a factor. It 

 is, therefore, only in exceptional cases that the terms of tlie second order depend- 

 ing on the motion of the orbital planes can become sensible. 



Reduction of the longltiuJe in the orlit to I<> gilude on the ecVqjtic. 



The integration of (23) gives a value of hv, which, added to the longitude in 

 orbit corresponding to tlie pure elliptic motion gives the longitude in the disturbed 

 orbit, counted from a fixed point in the moving plane of that orbit. The position 

 of this fixed point is coTnpletely determined by the condition that the instanta- 

 neous rotation of the plane in question around the axis perpendicular to itself is 

 always zero, so that the motion of the point of reference is always perpendicular to 

 tlie direction of the plane. But, although this instantaneous rotation is zero, the 

 integrated rotation is not rigorously zero when we consider the terms of the second 

 order. It follows that the value of c, the longitude in orbit, and the position of 

 the plane of the orbit do not rigorously determine the position of the planet: Ave 

 must also know how the fixed point of reference has changed its position in con- 

 sequence of the motions which tlie plane has undergone. Let us consider the 

 relative positions of this plane at two epochs. If the fixed point were equally 

 distant from the common node of the two planes, the integrated rotation of the 

 plane around its own axis would be zero. But, these distances not being equal, 

 their difference is a correction to be applied to the longitude of the planet in its 

 orbit. Suppose, now, that at the end of any time the inclination of the actual 

 orbit to the primitive orbit is (|), and the distance of its ascending node from the 

 present position of the moving axis of a; is 0. A rotation around the line of nodes 

 will not change the quantity sought. But, if we represent the infinitesimal rota- 

 tion around an axis perpendicular to it by (//• we shall have 



cos dp — sin dq = dr, 



dq and die being the instantaneous rotations around the respective axes of x and y. 

 By this rotation it is easy to see that the relative distance of any two fixed points, 

 one on each plane, from the node, will be altered by tlie quantity, 



dr (cosec cp — cot ^) = dr tan \ ^, 



the relative longitude of the fixed point on the moving plane being increased by 

 this amount. The correction to the longitude in orbit from this cause is, therefore, 



dl = dr tan 5 <?j = tan g <p (cos 6 dp — sin d dq). 



