50G MR, G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
C-A 
di 
d\[r 
divided by —-— n-\-n- F2/2 and give rise to terms in — and - r sin i of speed 2 fl. When 
A clt (It 
2/2 is neglected compared with n, we obtain the formula, given in any work on physical 
astronomy, for the fortnightly nutation. 
C—A 
Now it is obvious that if —-—n+w = 2/2, the former of these two terms becomes 
A 
C-A 
infinite. Since in our case the spheroid in homogeneous —— = e the ellipticity of the 
spheroid ; and since the spheroid is viscous e~\-. Therefore the critical relationship 
is 2/2. 
g 
When this condition is satisfied the ordinary solution is nugatory, and the true 
solution represents a nutation the amplitude of which increases with the time. 
2/2 
The critical point where the above integration stops is given by — =rcos i, and this 
2/2 
n 
critical point by —=1 + f - ; it follows therefore that — is little larger in the second 
case than in the first. Therefore this critical point has not been already reached where 
the integration stops, but will occur shortly afterwards. 
It is obvious that the amplitude of the nutation cannot increase for an indefinite 
time, because the critical relationship is only exactly satisfied for a single instant. 
In fact, the problem is one of far greater complexity than that of ordinary disturbed 
rotation. The system is disturbed periodically, but the periodic time of the disturb¬ 
ance slowly increases, passing through a phase of equality to the free periodic time; 
the problem is to find the amplitude of the oscillations when they are at their maximum, 
and to find the mean configuration of the system some time before and some time 
after the maximum, when the oscillations are small. This problem does not seem to 
be soluble, unless we take into account the slow variation of the argument in the 
periodic disturbing term ; and when the argument varies, the disturbing term is not 
strictly a simple time harmonic. 
In the case of the viscous spheroid, the question would be further complicated by 
the fact that when the nutation becomes large, a new series of bodily tides is set up 
by the effects of inertia. 
I have been unable to make a satisfactory examination of this problem, but as far as 
I have gone it appeared to me probable that the mean obliquity of the axis of the 
spheroid would not be affected by the passage of the system through a phase of large 
nutation ; and although I cannot pretend to say how large the nutation might be, yet 
I consider it probable that the amplitude would not have time to increase to a very 
wide extent." 
* I believe that I shall be able to show in an investigation, as yet incomplete, that when this critical 
phase is reached, the plane of the lunar orbit is nearly coincident with the equator of the earth. As 
the amplitude of this nutation depends on the sine of the obliquity of the equator to the lunar orbit, it 
seems probable that the nutation would not become considerable.—June 30, 1879, 
