14 
MR, DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
But the rise and fall of the tide relative to the nucleus is given by u — cr, and 
a 2 S / v 2 
U — cr= — COS (vt-\-in) —-cr 
g 5 
2 a S 
=-—[cos (vt-\-rj) — cos e cos (vt-r-rj — e)] 
o 2 
2 aS 
= — - — sin e sin (vt-\-in — e). 
5 s v / 
(17) 
Now if the nucleus had been rigid, the rise and fall would have been given by 
2 flS 
Therefore 
cos (vt-\-rj)= H cos (vt-\-7j) suppose. 
u — cr= — H sin e sin (vt-\-rj — e) 
(18) 
Hence the apparent tides on the yielding nucleus are equal to the tides on a 
rigid nucleus reduced in the proportion sin e : 1 ; and since — sin {yt-frj — e) 
IT 1 / 7r\ . . 77 
= cos (vt J r r] J r - — e) they are retarded by -( e — -j. As e is necessarily less than 
I/7r 
this is equivalent to an acceleration of the time of high water equal to — ej. 
It is, however, worthy of notice that this is only an acceleration of phase relatively 
to the nucleus, and there is an absolute retardation of phase equal to arc-tan "\ e C °A 
1 1 3 + 2 cos 3 e 
7. Semidiurnal and fortnightly tides. 
Let the axis of 2 be the earth’s axis of rotation, and let the plane of xz be fixed 
in the earth ; let c be the moon’s distance, and m its mass. 
Suppose the moon to move in the equator with an angular velocity oj relatively to 
the earth, and let the moon’s terrestrial longitude, measured from the plane of xz, at 
the time t be a>t. 
Then at the time t, the gravitation potential of the tide generating force, estimated 
per unit volume of the earth’s mass is 
- -- - -■ wr % -j ^~ — sin 3 6 cos 3 cot) j- 
whicli is equal to 
^ cos 3 + ^ y+d?r 3 {sin 2 6 cos 2<f> cos 2cot-\- sin 2 6 sin 2 <f> sin 2 cot}. 
The first term of this expression is independent of the time, and therefore produces 
an effect on the viscous earth, which will have died out when the motion has become 
steady ; its only effect is slightly to increase the ellipticity of the earth’s surface. 
