466 MR. Gh H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
II cosec i must henceforth be taken as the full precession of the earth, and the time 
may be conveniently measured from an eclipse of the sun or moon. Let m /} c / be the 
sun’s mass and distance; S2 t the earth’s angular velocity in a circular orbit; and let 
3 m 
It would be rigorously necessary to introduce a new set of quantities to give the 
heights and lagging of the seven solar tides : but of the three solar semi-diurnal tides, 
one has rigorously the same period as one of the three lunar semi-diurnal tides 
(viz. : the sidereal semi-diurnal with a speed 2 n), and the others have nearly the same 
period; a similar remark applies to the solar diurnal tides. Hence we may, without 
much error, treat E, e, E', e as the same both for lunar and solar tides ; but E"', e" 
must replace E", e", because the semi-annual replaces the fortnightly tide. 
Then if new auxiliary functions <!>„ Tq, X / be introduced, the whole tide-generating 
potential V per unit volume of the earth at the point r£, rr), r£ is given by 
-(r^+r<P)(f ■ —vf) &c - 
If then, as in (10), we put 
c—b=<h e +X e , &c., c — b / =0> /e +X /e , &c., 
the equation to the tidaily-distorted earth is r=a-j- cr-bcr,’ where 
equivalent to tlie action of tlie moon alone. The satellites 1, 2, 3, in fact, give the semi-diurnal or 
4> terms; satellites 4, 5, 6, 7, 8, 9 give the diurnal or terms ; and satellite 10 gives the fortnightly or 
X term. 
This is analogous to “ Gauss’s way of stating the circumstances on which ‘secular’ variations in the 
elements of the solar system depend and the analysis was suggested to me by a passage in Thomson and 
Tait’s ‘Nat. Phil.,’ § 809, who there refer to the annular satellite 10. 
It will appear in Section 22 that the 3rd, 8th, and 9th satellites, which are fixed in the heavens and 
which give the sidereal tides, are equivalent to a distribution of the moon’s mass in the form of a uniform 
circular ring coincident with her orbit. And perhaps some other simpler plan might be given which 
would replace the other repulsive satellites. 
These tides, here called “sidereal,” are known, in the reports of the British Association on tides for 1872 
and 1876, as the K tides. 
In a precisely similar way, it is clear that the sun’s influence may be analysed into the influence of 
nine other satellites and one ring, or else to seven satellites and two rings. Then, with regard to the 
interaction of sun and moon, it is clear that those satellites of each system which are fixed in each system 
(viz.: 3, 8, and 9), or their equivalent rings, will not only exercise an influence on the tides raised by 
themselves, but each will necessarily exercise an influence on the tides raised by the other, so as to produce 
tidal friction. All the other satellites will, of course, attract or repel the tides of all the other satellites 
of the other systems ; but this interaction will necessarily be periodic, and will not cause any interaction in 
the way of tidal friction or change of obliquity, and as such periodic interaction is of no interest in the 
present investigation it may be omitted from consideration. In the analysis of the present section, this 
omission of all but the fixed satellites appears in the form of the omission of all terms involving the moon’s 
or sun’s angular velocity round the earth. 
