518 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
Now by comparing this with the small term due to the secular change of figure of 
the eart h, we see that it is fairly negligeable, being of the same order of magnitude as 
that term. As far as it goes, however, it tends to increase the obliquity of the ecliptic. 
§ 22. The change of obliquity and tided friction due to an annular satellite. 
Conceive the ring to be rotating round the planet with an angular velocity fl, let its 
radius be c, and its mass per unit length of its arc -y—y so that its mass is m. Let cl be 
the length of the arc measured from some point fixed in the ring up to the element 
cS l ; and let fit be the longitude of the fixed point in the ring at the time t. Let SV 
be the tide-generating potential due to the element —S?. Then we have by (5) 
Where the suffixes to the functions indicate that fl-\-l is to be written for fl. Then 
integrating all round the ring from 1—0 to l—Hu it is clear that 
V 
wrr~ 
■p~cf sin 2 0 cos 2 (<% — n)-\-2pq(p 2 —f l ) sin 0 cos 0 cos ( <f> — n ) 
+ (i-cos 2 0)U 1 ~ 6 PY) 
which is the tide-generating potential of the ring. 
Hence, as in Section 2, the form of the tidally-distorted spheroid is given by (9), 
save that E x , E. 2> E\, Eh, E" are all zero. Also, as in that section, the moments of 
the forces which the tidally-distorted spheroid exerts on the element of ring are 
f ^^ C '’ W ^ ere ^ r> h r > C r are P u f equal to the rectangular 
coordinates of the element of ring, whose annular coordinate is l. 
Now if x, y, z are the direction cosines of the element, equations (7) are simply 
modified by fl being written fl-\-l.. Hence the couples due to one element of ring- 
may be found just as the whole couples were found before, and the integrals of the 
elementary couples from l— 0 to 277 are the desired couples due to the whole ring. 
Now a little consideration shows that the results of this integration may be written 
down at once by putting E x , E 2 , E\, Eh, E" zero in (15), (16), and (21). Thus in 
order to determine the change of obliquity and the tidal friction due to an annular 
satellite, we have simply the expressions (33) and (34), save that tt / must be replaced 
b y 
It thus appears that an annular satellite causes tidal friction in its planet, and that 
the obliquity of the planet’s axis to the ring tends to diminish, but both these 
