24 THEORBITOFTJRANUS. 



from which the terms of ^v of the second order are obtained by multiplying by 

 ri~" and integrating. 



Motion of the Orbital Planes. 



Tlie general theory of the motion of the planes of reference, especially of the 

 motion of the instantaneous orbit, has been so often treated that I can scarcely 

 hope to add anything essentially new to it. 1 sliall, however, endeavor to pre- 

 sent the differential equations of the motion in a simple and general form, and 

 one in wliich the geometrical conceptions of the problem shall be made as clear 

 as possible. 



The orbital plane of each planet being at each moment osculatory to that part 

 of the orbit which the planet is actually describing, its only motion is one of rota- 

 tion around the radius vector of the planet as an instantaneous axis. This rota- 

 tion may be resolved into two others around any pair of rectangular axes fixed in 

 the moving plane. But the rotation produced by any one planet is most simply 

 expressed when referred to axes, one of which coincides with the common node of 

 the two orbits. The rotation produced by each separate planet must, therefore, be 

 first referred to its node on the moving orbit, and then the combined rotations 

 must be resolved into two around axes assumed at pleasure. To effect this, let us 

 suppose positive rotation around an axis to be such that an observer looking from 

 the origin along the positive direction of the axis sees the right hand side of the 

 plane move downwards, and the left hand side upwards. Let us also denote tlie 

 first axis in the order of longitude the principal axis, or tliat of X, and that 90° 

 farther advanced the secondary axis, or that of Y. Let us now put 



dq, the instantaneous rotation around the axis of X; 



dp, the instantaneous rotation around the axis of Y. Let us also put, relatively 

 to any disturbing planet, 



(Z>7, the instantaneous rotation around the ascending node of the disturbing planet 

 on the orbit of the disturbed one. 



dk, that around the corresponding secondary axis. 



Then, from the known equations for the perturbations of the inclination and 

 node of an orbit, we find, that, if any term of the perturbative function be repre- 

 sented, as before, by 



— cos (ix+ a +y;y+y(o), 



the differential rotations ri and k will be given by the equations 

 dn m'h an 



dt ttj cos '^ 



dh m'an dh 



\ (' +i) ^o*- 7 + (*'+/) cosec y > sin JV 



cos N. 



dt tti cos '^ dy 



As R is actually developed, the mutual inclination y docs not explicitly appear, 

 but is replaced by 



a =; sin T, V. 



