500 MR. G. H. DARWIN ON THE PRECESSION OE A VISCOUS SPHEROID, 
dynamical stability for all values of n between /2 and 2/2, but if n be greater than 2/2, 
this position is unstable.* 
We will now return to the problem regarding the earth. We may here regard — as 
a small fraction, and i as sufficiently small to permit us to neglect sin 4 i ; also 
— — sec i, (-)— sec i will be neglected, 
r n \ t ) n & 
rn \* 
— sec ^ , 
\n ) 
* Added on September 25,1879.—The result in the text applies to the case of evanescent viscosity. 
If the viscosity be infinitely large the sines of twice the angles of lagging will be inversely instead of 
directly proportional to the speeds of the corresponding tides (compare p. 482). Thus we must here invert 
the right-hand sides of the six equations (76). If the obliquity be very small (77), (78), (79) become 
di 
dt 
i r 
— Tsinisin4 6l 1 + 
N 2 V L 
2 ( 1 -*) 
1 — 2 \ 
= 1 jii 
N sV 
.... /1 + 2X-4X 2 \ 
gmt sm46 1 ( T- 2W-J 
> 
(85') 
dN_ df 
dt ^ dt 
—4- sin 4c 
SV 
i 
J 
When 2X= 1, 
di 
dt 
apparently becomes infinite; but in this case the 
viscosity must be infinitely large in 
order to make the tide of speed n— 2Q lag at all, and if it be infinitely large sin 4-q is infinitely small. If 
the viscosity be large but finite, then when 2\ = 1, the slow diurnal tide of speed n— 2Q is no longer a 
true tide, but is a permanent alteration of figure of the spheroid. Thus 6^=0 and ^ depends on 
[sin 4e 1 —sin 2e'] which is equal to sin 4e x [l — 2(1 — X)] when the viscosity is large, and vanishes when 
2X=1. Thus when the viscosity is very large (not infinite) ~ vanishes when 2Q-f-n=l, as it does when 
dt 
the viscosity is very small. 
When 1 + 2X 4X-=0, that is, when X = —=l-f-l~236, — vanishes; and it is negative if X be a little 
greater, and positive if a little less than l-f-1'236. And 1—2X is negative if X be greater than A 
Hence it follows that for large viscosity of the planet, zero obliquity is dynamically unstable, if the satellite’s 
period be less than P236 of the planet's period of rotation; is stable if the satellite's period be between 1-236 
and 2 of the planet's period; and is unstable for longer periods of the satellite. 
but if the viscosity be very small the same 
tp ,i • •, -i t N d t , o i 1 + 2X—4X 3 
It the viscosity be very large-log tan--=-, 
J J d£ ° 2 1 —2X 
1 —2X 
expression = ^ or P os ^i ve values of X, less than 1 and greater than -6910 or l-t-1'447, the former 
is less than the latter, and if X be less than l-f-1'447 and greater than 0 the former is greater than the 
latter. 
Hence if there be only a single satellite, as soon as the month is longer than two days, the obliquity of 
the planet s axis to the plane of the satellite’s orbit will increase more, in the course of evolution, for 
large than for small viscosities. This result is reversed if there be two satellites, as w r e see by comparing 
figs. 2 and 4, Plate 36. 
