264 
NATORE 
[ Fuly 28, 1870 
HOPKINS DELAUNAY 
have received from Archdeacon Pratt a copy of a paper 
Versus 
W E 
nay’s objection to Hopkins’s method of determining the thickness 
of the earth’s crust by the precession and nutation of the earth’s 
axis. : 
The archdeacon, on this most important question, states :— 
“J am ready to allow, and so would Mr. Hopkins have 
allowed, that if the crust of the earth revolved round a steady 
axis, always parallel to itself in space, and if at some particular 
epoch a difference existed between the rate of movement of the 
crust and of the fluid within it, the resulting friction would gra- 
dually destroy this difference and bring about a conformity in the 
motien of both parts. I will even go further, and allow that the 
effect of the internal friction and viscidity of the fluid may be such 
that the resulting rotary motion may be the same as that which 
the whole mass would have had at the epoch if it had suddenly 
become one solid body and thereby suddenly retarded the rota- 
tion.” ; A, 
He thus illustrates his position :— 
*¢ Suppose a spherical shell or crust of mass C to have 
within it a solid spherical nucleus, of radius 2 and mass N, fit- 
ting it exactly ; and the crust to receive an angular velocity of 
rotation around an axis fixed in the crust, the nucleus at that 
moment having no angular velocity ; but suppose that a slight 
force of friction between the surfaces gradually generates a rotary 
motion in the nucleus; and suppose this force to vary as the dif- 
ference between the angular velocities of the crust and nucleus— 
that is, of the surfaces in contact. Let 7) and w be the angular 
velocities at the time 4, & and # the radii of gyration of the two 
bodies, F (w—«’) the force at the equator of the nucleus which 
represents the friction between it and the crust. Then the equa- 
tions of motion are 
oe Fs es ere en 
Fa CH (w—0'), > NeF (w-w’). . » (1) 
Suppose also that 8 would have been the angular velocity, when 
the primitive impulse was given, on the hypothesis of the crust 
and nucleus being rigid'y connected so asto be one mass. Then 
RiCEEENEA) aC Ane cy eal. eitenee (2) 
Subtracting the second of equations (1) from the first, putting 
=~ I I = vs 
F(a + Nee =C, . «ye one (3) 
and integrating, we have 
w — w’ = const. x e—¢t, 
When /=0, w=a and o’ 
2. @—@! =ae—ct. 
Hence, by the first of equations (1), 
dow Bite Neca aid 
a Ce Caan © by (3)5 
NZ”? 3 
OS amare ad 
and also 
eon? er 
=a( I+ CE ect), by (2) 
The paper then continues :—‘* The first of these expressions 
shows that the angular velocity of the crust begins with a; and 
when c¢ becomes very large indeed, it is reduced to 8. Hence 
the effect of the constant friction of the nucleus against the 
inner surface of the crust is a¢ /ast to reduce the velocity of the 
crust to what it would have been at first if the crust and nu- 
cleus had been one solid mass. 
“‘We may conclude perhaps that the same effect would be pro- 
duced, though ina much longer time, if the interior were not a 
solid sphere, but a fluid mass. 
“The above reasoning shows that if the disturbing force pro- 
ducing precession and nutation did not exist, and the interior of 
the earth were fluid (whatever the thickness of the crust), it may 
be fairly assumed that the motion of rotation of the crust would 
now, the earth having existed so many ages, be exactly what it 
would have been had the earth been one solid mass, all difference 
of motion having been long ago annihilated by the internal fric- 
tion and viscidity. 
‘* But the disturbing force producing precession and nutation 
does exist. It consists of two parts, one constant and the other 
variable and periodical. The constant part is that which produces 
communicated to the PAilosophical Magazine on Delau- | 
| is actually produced. 
the steady precession of the axis (and which I will call for con- 
venience the precessional force) ; the other produces the nutation. 
I will consider the precession first. Suppose now, for the sake 
of argument, that at the present moment, as M. Delaunay 
imagines, the crust and the fluid are revolving precisely as one 
mass, all previous differences of motion, even under the action of 
the disturbing force which produces precession and nutation, 
having been annihilated by friction and viscidity. I ask—-What 
will be the action of the precessional force from this moment ? 
It tends to draw the pole of the crust towards the pole ofthe 
ecliptic: and this tendency, as mathematical physicists well 
understand, combined with the rotary motion of the crust, pro- 
duces this singular result, viz., the pole does not move towards 
the pole of the ecliptic, but shifts in a direction at right angles 
to the line joining the poles towards the west ; so that the incli- 
nation of the axis to the ecliptic remains constant, but the axis 
shifts towards the west. The space through which it shifts in 
an infinitesimal portion of time varies as the length of the time 
and the force directly, and as the inertia of the mass to be moved 
inversely. The inertia of the mass depends upon the thickness 
of the crust only ; for the friction of the fluid against the inner 
surface of the crust (which might, as I have shown, in the course 
of years, produce a sensible effect) cannot do so during the infi- 
nitesimal portion of time I am considering before the precession 
The precessional force has its full effect 
in producing the precession of the solid crust, the fluid not 
having time to diminish that effect before the axis has assumed 
a new position ; and in this new position of the axis the preces- 
sional force is precisely the same in amount as before, to go on 
causing the precession as before. The precessional force is, in 
fact, ever alive and active, and shows this in incessantly produc- 
ing the effect I have ceseribed ; and the precession gues on 
steadily, the amount of it depending upon the mass of the crust 
thus moved, which the fluid has not time to retard or lessen. M. 
Delaunay says that ‘the additional motion due to the aboye- 
mentioned causes (the disturbing forces which give rise to pre- 
cession and nutation) is of such slowness, that the fluid mass 
which constitutes the interior of the globe must follow along 
with the crust which confines it, exactly as if the whole formed 
one solid mass throughout.” In reply to this, I say that it is 
not the slowness of the motion, but the want of solid connection 
between the crust and the fluid in contact with it that affects the 
problem. The motion, whatever its amount, is incessantly being 
generated by the disturbing force, and owing to this want of 
solid connection, the friction of the fluid has not time during the 
successive moments during which the precession is generated, to 
stop or even sensibly to check it. 
-**Tt-will thus be seen that at every instant the precessional 
force proceeding from the action of the sun and moon on the 
protuberant part of the earth’s mass will, if the earth be a solid 
mass, haye to move the whole mass; and if the earth have a 
solid crust only with a fluid interior, the force will have to move 
only the crust against the evanescent resistance of the fluid within 
during so short a space of time as it takes to produce precession. 
The resulting precessional motion will be different in the two 
cases ; and therefore the actual amount of the precession which 
the earth’s axis has (and which is a matter of observation) is a 
good test of the solidity or fluidity of the interior, This is Mr. 
Hopkins’s method. 
“The force producing nutation is much smaller, even at its 
maximum, than the precessional force. Its effect, however, is 
precisely the same in this respect—that it depends upon the mass 
of the solid crust, and in no respect upon the friction of the 
fluid within it, which has not time to influence the nutation 
before the nutation is actually produced. 
**T do not here undertake to go into Mr. Hopkins’s numerical 
calculations ; I simply vindicate his method. 1 do not here con- 
sider what modification the elasticity of the solid material of the 
earth may have upon his numerical results. I conceive that it 
would have vo effect, if the disturbing force were constant and 
there were no nutation. For, under the dragging influence (if 
I may so call it) of the constant precessional force, the solid 
material would be under a steady strain, and would communicate 
the effect of the force, continuously acting, from particle to par- 
ticle of the solid part as if it were really rigid ; and the resulting 
precessional motion would be greater or less as the mass of the 
solid part may be smaller or larger—that is, the solid crust 
thinner or thicker. But as the disturbing force is not constant, 
but variable, and there is constantly nutation of the axis as well as 
precession, the action aboye described will be somewhat modified; 
