744 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
§ 9. Application to the case where the planet is viscous. 
As in 8 7 
1 d£ T 2 
/ (ft = q s ^ n — sin 4f 2 +Ti sin 2g x — r 3 sin 2g 3 —A sin 4h} . . (68) 
If j be put equal to zero this equation will be found to be the same as that used as 
the equation of tidal reaction in the previous paper on ct Precession.” 
If the viscosity be small, with the same notation as before 
l?t = ’q sin 4 4*i—®2+i(U—U)—M’J’i+^+U+ryi-W]' ■ • <#») 
Now 
and 
Therefore 
^i-^2+i( r i- r 2)=i cos i cos i 
^i + To + Tj + ro+A— \ 
lft = 2 g sin 4f IL cos i cos j-\]. 
(70) 
We see that the rate of tidal reaction diminishes as the inclination of the orbit 
increases. 
§10. Secular change in the inclination of the orbit of a single satellite to the invariable 
plane, where there is no other disturbing body than the planet. 
This is the second of the two cases into which the problem subdivides itself. 
If there be only two bodies, then the fixed plane of reference, which was called the 
ecliptic, may be taken as the invariable plane of the system. It follows from the 
principle of the composition of moments of momentum that the planet’s axis of rotation, 
the normal to the satellite’s orbit and the normal to the invariable plane, necessarily lie 
in one plane. Whence it follows that the orbit and the equator necessarily intersect 
in the invariable plane. From this principle it would of course be possible either to 
determine the motion of the node from the precession of the planet or vice versa, and 
the change of obliquity of the planet’s axis (if any) from the change in the plane of 
the orbit or vice versd ; this principle will be applied later. 
We have found it convenient to measure longitudes from a line in the fixed plane, 
which is instantaneously coincident with the descending node of the equator on the 
fixed plane. Hence it follows that where there are only two bodies we shall after 
differentiation have to put N=N'= 0. 
