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MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
the fluid mass it would be in equilibrium, provided condition (II.) were satisfied; 
and if in addition to these the force (6.) should also act, it would be the only force 
producing motion in the fluid, provided the instantaneous surface of equal density 
should always be identical with that which would be the surface of equal pressure if 
the force (6.) did not act, since such an arrangement of the fluid would be necessary 
in order that the forces (1.) (2.) (3.) (4.) (5.) should produce no motion in the fluid. 
It has been shown (First Series, Art. 12.) that the force Z (= 2 u- a (3 . x) on any par- 
ticle (x y z) may be replaced by a force u 2 a (3 . r, acting perpendicularly to the radius 
vector, and producing motion in the fluid, and another force which would be con- 
sistent with the equilibrium of the fluid. The latter of these may therefore be asso- 
ciated with the above forces, which would produce equilibrium, and the former 
(a 2 a (3 . r) will be the only force producing motion, always supposing condition (II.) 
to be satisfied with reference to all those forces which satisfy condition (I.). Con- 
sequently if these forces were such that the surfaces of equal density should be 
spherical, and if the inner surface of the shell were also accurately spherical, the 
angular velocity generated in the fluid about the axis of y in a unit of time would 
= u 2 a (3, and would be entirely independent of the law according to which the density 
should vary in passing from one spherical surface of equal density to another. It 
would be the same for the heterogeneous as for the homogeneous fluid. 
The action of the force a 2 s (3 . r, and therefore the above result will be modified in 
the case of the earth by the circumstances of the inner surface of the shell and the 
surfaces of equal density being spheroidal, instead of being spherical. It is manifest, 
however, that this modification will be of an order higher than u 2 . a (3, and may there- 
fore be neglected, as before stated with reference to the homogeneous spheroid (First 
Series, Art. 14.). 
Again, in consequence of the fluid motion produced by the force u 2 a f3 . r, the 
condition (II.) will not be accurately satisfied with reference to the forces which 
satisfy (I.), and these forces will therefore produce motion in the fluid. To esti- 
mate this motion, let us suppose no forces to act on the fluid except those which 
satisfy (I.), and let us suppose the surfaces of equal density to be displaced through 
indefinitely small spaces from those positions in which there would be perfect 
equilibrium. The tendency of the forces would be to bring back the instantaneous 
surfaces of equal density to their latter positions, and thus an oscillatory motion 
would arise ; but it is manifest that no continuous angular motion of the whole 
mass could thus be produced, at least of the same order of magnitude as the whole 
oscillatory motion. Now, in our actual case, this motion could not be greater than 
quantities of the order a (3, since the angular motion from which the perturbation we 
are considering arises, is of that order. Consequently the modification of our first 
result due to this cause may be neglected. 
Again, the disturbing forces of the sun and moon will produce an oscillation in the 
surfaces of equal density of the fluid mass, or an internal tide ; but it is manifest that i 
