464 MR. G. H. DARWIN ON THE PRECESSION OP A VISCOUS SPHEROID 
Adding (3 2') to (32), the whole precession is 
dp re ( 
dt n 
cos i 
(32") 
We thus see that the effect of the non-periodic part of the tide-generating potential, 
which may be conveniently called a permanent tide, is just such as to neutralise the 
effects of the tidal action. The result (32") may be expressed as follows: — 
The precession of a fluid spheroid is the same as that of a rigid one which lias an 
ellipticity equal to that due to the rotation of the spheroid. 
From this it follows that the precession of a fluid spheroid will differ by little from 
that of a rigid one of the same ellipticity, if the additional ellipticity due to the non¬ 
periodic part of the tide-generating influence is small compared with the whole 
ellipticity. 
Sir William Thomson has already expressed himself to somewhat the same effect 
in an address to the British Association at Glasgow.* 
• 1$ T 
Since c n =4—, the criterion is the smallness of —. 
u * g n 2 
T TC 
It may be expressed in a different form; for — is small when — -f- n is small compared 
with e, and -- -P n is the reciprocal of the precessional period expressed in days. Hence 
'lb 
the criterion may be stated thus : The precession of a fluid spheroid differs by little 
from that of a rigid one of the same ellipticity , when the precessional period of the 
spheroid expressed in terms of its rotation is large compared with the reciprocal of 
its ellipticity. 
In his address, Sir William Thomson did not give a criterion for the case of a fluid 
spheroid without any confining shell, but for the case of a thin rigid spheroidal shell 
enclosing fluid he gave a statement which involves the above criterion, save that the 
ellipticity referred to is that of the shell itself; for he says, ‘‘The amount of this 
difference (in precession and nutation) bears the same proportion to the actual precession 
or nutation as the fraction measuring the periodic speed of the disturbance (in terms 
of the period of rotation as unity) bears to the fraction measuring the interior ellipticity 
of the shell.'’ 
This is, in fact, almost the same result as mine. 
This subject is again referred to in Part III. of the succeeding paper. 
* See ‘Nature,’ September 14, 1876, p. 429. The above statement of results, and the comparison with 
Sir William Thomson’s criterion was added to the paper on September 17, 1879. 
