SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 
9 
of matter <x ; , —-({—1) times that potential, and by adding to it the external disturbing 
O 
potential. 
We have now learnt how to include the surface action in the potential ; and if W, 
be the potential of the external disturbing influence, the effective potential per unit 
volume at a point within the sphere, now free of surface action and of mutual 
gravitation, is W; 
2gw(i — Y)(r\ i ;rp 
2i + i [-) °-<= r ‘Ti suppose. 
The complete solution of our problem is then found by writing UT; in place of W; 
in Thomson’s solution (8).'" 
In order however to apply the solution to the case of the earth, it will be convenient 
to use polar coordinates. For this purpose, write wr'S, for W l} and let r be the radius 
vector ; 6 the colatitude ; <f> the longitude. Let p, to, v be the velocities radially, and 
along and perpendicular to the meridian respectively. Then the expressions for p, to, v 
dr 
will be precisely the same as those for a, fi, y in (8), save that for — we must put ( ~ ; 
„ d d d d 
101 ~ , : njt 5 101 j 
dy rsm ua<p dz rdO 
Then after some reductions we have 
P 2(i-l)[2(t + l) 2 + l]u 
+ ._ 1 dff 
2(i-l)[2(i + l) 3 + l]i; r d6 
i(i + 2)a 2 - (i -1) (i + 3)r 2 d- 1 dff 
2(t —l)[2(i +1) 2 +l]u sin 6dxf) 
(I0)t 
where Tj=tc(S — 2o^7—r -V 
\ ‘ y 2^ + l ay 
These equations for p, to, v give us the state of internal flow corresponding to the 
external disturbing potential * * * § , including the effects of the mutual gravitation of 
the matter constituting the spheroid. 
* The introduction of the effects of gravitation may be also carried out synthetically, as is done by Sir 
W . Thomson (§ 840, ‘ Nat. Phil.’) ; but the effects of the lagging of the tide-wave render this method 
somewhat artificial, and I prefer to exhibit the proof in the manner here given. Conversely, the elastic 
problem may be solved as in the text. 
t There seems to be a misprint as to the signs of the (£§’s in the second and third of equations (13) of 
§ 834 of the ‘Nat. Phil.’ (1867). When this is corrected ju, and v admit of reduction to tolerably simple 
forms. It appears to me also that the differentiation of p in (15) is incorrect; and this falsifies the 
argument in three following lines. The correction is not, however, in any way important. 
MDCCCLXX1X. 
C 
