THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
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sensibly uniform angular velocity on the ecliptic. This is the case at present with 
the earth and moon. Here then dW/clN', as far as concerns the influence of Diana’s 
tides on the moon, is sensibly periodic according to simple harmonic functions of the 
time. From this we conclude that:— 
If the nodes of the satellites orbits revolve uniformly on the plane of reference, then 
the tides raised by any one satellite can produce no secular change in the inclination of 
the orbit of any other satellite. 
There are thus two cases in which the problem is simplified by our being permitted 
to consider only the case of identity between Diana and the moon : 
1st. Where there are two or more satellites, but where the nodes of the perturbed 
satellite’s orbit revolve with sensible uniformity on the plane of reference. 
2nd. Where the planet and satellite.are the only bodies in existence. 
In these two cases, after differentiation of the disturbing function with respect to 
the accented elements, we shall be able to drop the accents. 
There is also a third case in which Diana’s tides will produce a secular effect on the 
inclination of the moon’s orbit, and this is where the nodes of the moon’s orbit either 
revolve irregularly or oscillate. This case is enormously more complicated than the 
others, and forms the subject of Part III. of this paper; I have only attempted to 
solve it on the supposition of the smallness both of the inclination of the orbit, and of 
the obliquity of the ecliptic. 
The first of these three cases is that which actually represents the moon and earth, 
together with solar perturbation of the moon at the present time. 
In tracing the configuration of the lunar orbit backwards from the present state, we 
shall start with the first case; this will graduate into the third, and from this it will 
pass to a state represented to a very close degree of approximation by the second. 
We are not at present concerned to know what are the conditions under which 
there may be approximate uniformity in the motion of the nodes ; this will be 
investigated below. 
We will begin with the first of the three cases, and will find also the rate of change 
of the diurnal rotation and of the obliquity of the planet. 
The second case will then be taken, and afterwards the third case will have to be 
discussed almost ah initio in Part III. 
§ 6. Secidcir change of inclination of the orbit of a satellite, where there is a second 
disturbing body, and where the nodes revolve with sensible uniformity on the fixed 
plane of reference. 
By (13) the equation giving the change of inclination is 
P elf 1 clW , , ..cm 
— 77 —•, Trrfi - tan h i —7 
k dt sm j dH 2 '' de 
5 B 
MDCCCLXXX. 
