10 
MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
3. The form of the free surface at any time. 
If p be the surface value of p, then 
i(2i+l) 
yZ + 1 
9 2(i —l)[2(i +1) 2 +1] v Ti 
Hence after a short interval of time 8 1, the equation to the hounding surface of 
the spheroid becomes r=a-\-cr l J r p'&t; but during this same interval, or,- has become 
f a ’ St, whence 
clt 
or 
do-j 
dt ~ p ~~ 
da-, 
i(2i + 1) wa i+l 
2(i-l)[2(t + l) 2 + l] v 
S 
gvxt 
2(i+iy+i v 
(Ti 
gwa 
cr; — 
i(2i + Y) 
%oaj 
i+l 
dt 1 2(i +1) 3 + 1 v 1 2(i —l)[2(*i+1) 2 +1] v 
Si .... (11) 
This differential equation gives the manner in which the surface changes, under the 
influence of the external potential r‘\ S,. 
If S,- be not a function of the time, and if Si be the value of cr,- when t = 0, 
9)' 
i + l « ! ’S; 
cr,— 
2(<-l) 9 
1 —expij-r: 
—gwait 
[2(i + l) 3 + l]u 
/ —await 
+Si eXp ( [2(i+l)»+l> . 
( 12 )* 
When t is infinite 
_ 2i + l- ffSi 
a ' , '“2(t-l) 7 
(13) 
and there is no further state of flow, for the fluid has assumed the form which it 
would have done if it had not been viscous. This result is of course in accordance 
with the equilibrium theory of tides. 
If Si be zero, the equation shows how the inequalities on the surface of a viscous 
globe would gradually subside under the influence of simple gravity. We see how 
much more slowly the change takes place if i be large; that is to say, inequalities of 
small extent die out much more slowly than wide-spread inequalities. Is it not 
possible that this solution may throw some light on the laws of geological subsidence 
and upheaval ? 
4. Digression on the adjustments of the earth to a form of equilibrium. 
In a former paper I bad occasion to refer to some points touching the precession of 
a viscous spheroid, and to consider its rate of adjustment to a new form of equilibrium, 
* I write “ exp.” for “ e to tlie power of.” 
