474 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
other two give e= 18 C 1 o' and e= 41° 37'. This result therefore confirms the geometrical 
construction fairly well. 
It is proper to mention that the expressions of dynamical stability and instability 
are only used in a modified sense, for it will be seen when the effects of tidal friction 
come to be included, that these positions are continually shifting, so that they may be 
rather described as positions of instantaneous stability and instability. 
* I will now illustrate the case where there is only one satellite to the planet, and 
in order to change the point of view, I will suppose that the periodic time of the satellite 
is so short that we cannot classify the semi-diurnal and diurnal terms together, but 
must keep them all separate. 
Suppose that n = 5f2 ; then the speeds of the seven tides are proportional to the 
following numbers, 8 , 10, 12 (semi-diurnal); 3, 5, 7 (diurnal) ; 2 (fortnightly). 
These are all the data which are necessary to draw a family of curves similar to 
those in Plate 36, figs. 2 and 3, because the scale, to which the figure is drawn, is 
determined by the mass of the satellite, the mass and density of the planet, and the 
actual velocity of rotation of the planet. 
Then by (16) and (29) we have 
u!=~LaP 7 2 sin 4 e i-p 3 2 3 (p 3 -r) sin 4e-lpp sin 4e 3 —|sin 4e" 
clt (pi 
+l>p 5 q{p'+3q~) sin 2e\—^pq(p 2 —q 2 ) 3 sin 2e' —±pq 5 (3p 2 +q 2 ) sin 2+] 
where p— cos - and q— sin 
i 
9 
This equation may be easily reduced to the form 
— = ' sin i\ I 10 sin 4e, —10 sin 4e 2 +16 sin 2e\ —16 sin 2eh —12 sin 4e" 1 
dt 12 8 1 L 1 * 1 
-|- cos i [15 sin 4e x — 4 sin 4e+15 sin 4e. : +18 sin 2e\ — 24 sin 2e'+18 sin 2eh] 
+ cos 2f[6 sin 4eq —6 sin 4e 2 +12 sin 4e"] 
+ cos 3t[sin 46^4 sin 4e+ sin 4e 2 — 2 sin 2e\ —8 sin 2e' —2 sin 2eV]j 
which is convenient for the computation of the ordinates of the family of curves which 
illustrate the various values of ~ for various obliquities and viscosities. 
In Plate 36, fig. 4, the lag (e) of the sidereal semi-diurnal tide is taken as the 
standard of viscosity. The abscissae represent the various obliquities of the planet’s 
cLz 
equator to the plane of the satellite’s orbit; the ordinates represent the values of — 
^the actual scale depending on the value of ^J ; and each curve represents one degree 
of viscosity, viz.: when e=10°, 20°, 30, 40° and 44 c . 
* From here to the end of the section was added July 8, 1879. 
