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Magnetic Repulsion and the Gyroscopic Force

(Manifestations of Centrifugal Force)

Frederick David Tombe,

Northern Ireland, United Kingdom,

sirius184@hotmail.com

7th December 2014

Abstract. The counterintuitive gravity defying behaviour that is exhibited by a pivoted

gyroscope suggests the involvement of an active spin-induced force, similar in nature to

the magnetic force, F = qv×B. The phenomenon of gyroscopic stability exhibits a strong

spin-induced reactance which cannot be accounted for by the moment of inertia alone.

The physical connection between the inertial forces and magnetic repulsion will be

investigated.

The Spinning Top

I. When an electrically charged particle falls vertically under gravity in a

horizontal magnetic field, it will be deflected horizontally at right angles to the

magnetic field lines. At its lowest point it will be moving horizontally at a speed

which, at the same height, it would have been moving downwards if there had

been no magnetic field. The magnetic force has diverted the effect of gravity

sideways in much the same manner that a concave inclined plane does. The

deflecting force, rather than merely superimposing upon the gravitational force,

has had the effect of actually deflecting the entire gravitational force sideways.

On reaching its lowest point, the particle then begins to rise again. As it

rises it loses speed and the radius of curvature of its path decreases. At

maximum height the particle loops around in a retrograde direction and the

cycle repeats over again, but after each cycle the particle will have advanced in

the horizontal direction. The particle will trace out an average horizontal path

containing a series of loops. If however we block the horizontal advance, the

particle will fall to the ground like a stone since the magnetic force that would

have prevented it from falling will be disengaged.

A spinning top is a gyroscope which is placed on a surface so that the

combination of gravity acting downwards and the upward normal reaction cause

a torque to act. Intuition tells us that this torque should pull the gyroscope right

down, but if the angular speed is fast enough, that’s not what happens. It

follows from Newtonian mechanics that when the gyroscope begins to fall

downwards, its angular momentum vector will move horizontally. That does not

however automatically mean that the spin axis should move sideways to chase

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the angular momentum vector. Whether it does or not and why, is the issue that

lies at the centre of this investigation.

If we physically block the spinning top from moving sideways, then just

like in the case of when we blocked the charged particle in the magnetic field

from moving sideways, it will fall like a stone, and it doesn’t matter that its

angular momentum vector has been displaced sideways from the spin axis.

However, if we do not prevent the top from moving sideways, the spin axis will

counter intuitively deflect sideways to chase the angular momentum vector, and

the top will defy gravity while the spin axis traces out a pattern of loops similar

to that in the case of the charged particle falling through the magnetic field. This

suggests that there is an active spin-induced force at play, similar in nature to

the magnetic force F = qv×B.

The Coriolis Force

II. The Coriolis force has a mathematical structure that is similar to that of the

magnetic force F = qv×B, especially bearing in mind that Maxwell linked the

magnetic force intensity to angular velocity. [1], [2] Despite the conventional

wisdom that a Coriolis force is merely an artefact of making observations from

a rotating frame of reference, it will now be proposed that the Coriolis force is a

real transverse force which follows from Newton’s laws of motion in an inertial

frame of reference. A common example occurs where a person sits on a rotating

stool with outstretched arms. When they retract their arms, the inward motion is

constrained to the radial direction, and a transverse Coriolis force causes the

stool to spin faster. This follows from Newton’s first law of motion. There is

also a third law action-reaction in that the constraining to the radial direction

acts against the Coriolis force while the Coriolis force then reacts against the

constraint causing the angular acceleration.

Another example of a constrained radial motion in a rotating system occurs

in the case of a precessing gyroscope. If we view a spinning gyroscope from the

side, into its equatorial plane, and then apply a precessional torque along an axis

that is pointing towards us, the rim velocity will be radial relative to the axis of

the applied torque. It will be centrifugally directed on one side of the torque axis

and centripetally directed on the other side. A Coriolis force will hence be

induced in the same direction on either side of the torque axis. In conjunction

with an equal and opposite Coriolis force on the far side of the gyroscope, a new

torque will be induced which will be at right angles to the applied torque. This

induced torque has the power to deflect the effects of gravity sideways.

Both of the examples described above involve real physical effects which

can be observed from an inertial frame of reference and which can be analyzed

in terms of an active Coriolis force. While the rotating stool scenario can also be

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analysed in terms of conservation of angular momentum, this is not so in the

case of the gyroscope where the Coriolis force actually causes the angular

momentum to change its direction.

The Coriolis force is very obviously an absolute physical effect and not, as

is commonly taught, merely an artefact arising from making observations in a

rotating frame of reference. It is a real force, every bit as real as its cousin the

magnetic force. Introducing a rotating frame of reference into the analysis is

like introducing a hall of mirrors which merely serves to confuse the whole

issue. Nevertheless, although there are some exceptions [3], [4], it is unusual to

see the Coriolis force being treated as a real force in the literature.

The Magnet and the Gyroscope

III. We’ll consider a closed loop of electric current to be both a magnet and a

gyroscope. We find however, that it doesn’t possess the gravity defying ability

that is possessed by the ordinary gyroscope. Neither the spin associated with the

electric circulation nor the associated magnetic field can stop such an

electromagnetic gyroscope from falling to the ground. A corollary of this is that

two ordinary spinning gyroscopes sitting side by side with their rotation axes

parallel don’t repel each other as do two magnets with parallel magnetic axes.

These two facts combined point to the spin of the individual molecules

within the ordinary gyroscope as being the source of the gyroscopic Coriolis

force, whereas the source of the magnetic force must lie in the magnetic field in

the space beyond atomic and molecular matter. Repulsion between two magnets

must arise due to an effect which occurs at the interface between their two

respective magnetic fields. Maxwell identified magnetic repulsion with

centrifugal force in a sea of molecular vortices that serves as the medium for the

propagation of light (the luminiferous medium). [1]

Centrifugal Force

IV. When two bodies possess a mutual transverse speed, there will be an inertial

centrifugal force acting on each of the bodies relative to their common centre of

mass. The centrifugal forces acting on each body will be equal and opposite,

while the body with the larger mass will have the smaller centrifugal

acceleration. Contrary to what is stated in the mainstream literature, inertial

centrifugal force is a real force that obeys Newton’s laws. If the two bodies are

connected by a string, the string will be pulled taut, hence inducing a centripetal

force pair on the two bodies.

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Inertial centrifugal force does not form part of an action-reaction pair with

centripetal force, and in situations where a centripetal force pair is induced in a

constraint, it is only in the special case where circular motion ensues that it is

equal in magnitude to the inertial centrifugal force pair. This should not

however be confused with the fact that the centrifugal force that actually acts on

the constraint itself is equal and opposite to the induced centripetal force. The

centrifugal force that acts on the constraint is not the same thing as the inertial

centrifugal force that acts on the moving bodies themselves, although they both

form part of a single process with the inertial force being the primary cause.

The Rattleback

V. The physical reality of the centrifugal force can be demonstrated with a

device known as a rattleback. A ratteback is a boat shaped object that sits on a

hard surface and the only officially recognized forces acting on it are gravity,

normal reaction, and friction. The crucial feature of a rattleback is that it is

asymmetrically shaped in relation to rocking it on a horizontal axis, and it is

important that the hard surface provides static friction so as to enable a see-saw

rocking motion to occur. Let’s now assume that centrifugal force is a real force.

When the rattleback rocks, an equal and opposite pair of inertial centrifugal

forces act on either side of the rock axis. The asymmetrical shape ensures that

these centrifugal forces act out of the plane of the rocking motion. On any given

side of the rock axis, the centrifugal force always acts in the same direction,

whether during the upward part of the motion or during the downward part of

the motion. The result is that the centrifugal force pair produces a torque, hence

causing the rattleback to rotate in the horizontal plane in a direction determined

by the chirality of the asymmetry.

When a rattleback rotates in a horizontal plane, there will always be a

certain amount of rocking as well. The centrifugally induced torque will then

act to precess the combined angular momentum of the two superimposed modes

of rotation. If we rotate the rattleback in its preferred direction, the centrifugal

torque will stabilize the rotation, but if we rotate it contrary to its preferred

direction of rotation, the centrifugal torque will act to precess the combined

angular momentum until the rotation has completely reversed its direction. This

is observed initially by an increase in the rocking in conjunction with a decrease

in the initial rotation. At the moment when the rocking amplitude reaches its

maximum, the rotation will change directions. It will then speed up in the new

direction as the rocking fades away. This counter intuitive display would not be

possible if centrifugal force were not a real force.

Some rattlebacks work two ways. No matter which way we rotate it, it will

ultimately reverse its direction. This is because there is an additional asymmetry

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with respect to rotation in the horizontal plane. When this situation occurs, the

precession caused by the centrifugal torque never finds a symmetrical axis, and

so the cycle repeats indefinitely until it is damped out by sliding friction.

The Underlying Cause

VI. We have attributed the gravity defying ability of a gyroscope to the Coriolis

force and the spin reversal tendency in the rattleback to the centrifugal force,

but we have so far not identified the underlying physical cause of these inertial

forces. The commonality that underlies both of these inertial forces, and also the

magnetic repulsive force, is the compound centrifugal force principle rooted in

Maxwell’s sea of molecular vortices. The sea of tiny molecular vortices, which

constitutes the luminiferous medium, fills all of space including the interstitial

regions between the atoms and molecules of ponderable matter. The

luminiferous medium should not be confused with the pure aether which is the

actual stuff of which the molecular vortices themselves are comprised. The

luminiferous medium is a sea of tiny aether vortices. [1], [5], [6], [7]

The fundamental principle is that these vortices press against each other

with centrifugal force while striving to dilate. The magnetic force and the

inertial forces arise when motion causes an asymmetry in the fine-grained

centrifugal force that is being exerted on ponderable matter by this background

sea of tiny aethereal vortices. When such asymmetry arises, the fine-grained

centrifugal force presses differentially on each side, resulting in a large scale

‘compound centrifugal force’, alternatively known as a ‘Coriolis force’.

In the case of the magnetic force, the skeleton principle can be illustrated

by considering two molecular vortices sitting side by side and rotating in the

same direction. The centrifugal force that they exert on each other is due to the

tendency to expand, which is in turn related to the mutual circumferential speed

of the electric particles that circulate around the edge of the vortices. If another

particle moves between these two vortices, due to its mutual transverse speed

with respect to the particles circulating around the edge of the vortices, it will

experience a centrifugal force. The mutual transverse speed will be greater on

one side than on the other, and so the particle will deflect at right angles to its

direction of motion. In a magnetic field the tiny vortices of the luminiferous

medium are solenoidally aligned meaning that the individual rotations produce a

similar effect to that of one single large rotation. A particle moving in the

equatorial plane of these solenoidally aligned vortices will experience a

compound centrifugal force, F = qv×B, and be deflected sideways.

In a planetary orbit, the gravitational force of the planet entrains a pocket of

the luminiferous medium along with it in its orbital motion. The entrained

region and the planet move as one like an egg yolk and the rest of the egg. The

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smaller planet’s entrained region moves through the larger planet’s entrained

region like a bubble. A centrifugal repulsion acts between the tiny vortices at

the interface of the two entrained regions as they shear past each other.

In the case of the Coriolis force when it is applied to planetary orbits, the

tiny molecular vortices of the luminiferous medium are precessing due to the

radial gravitational field. In this respect they behave somewhat like compound

turbines, in that where a turbine spins when exposed to a wind, a luminiferous

molecular vortex is already spinning, but then precesses when it is subjected to

an aether wind. The gravitational field constitutes a radial wind of pure

stretched aether. When a planet moves only in the radial direction in a

gravitational field, the effect of the precessing molecular vortices all around it is

symmetrical, and so there will be no deflection. However, in situations where

we have transverse motion occurring in addition to the radial motion, such as in

non-circular orbits, an asymmetry will ensue resulting in a compound

centrifugal force acting in the transverse direction, such as to cause the

conservation of angular momentum.

In the case of rigid body rotations, a variation of the compound centrifugal

force principle involves the spin of the actual molecules of the rigid body itself.

In this respect it is convenient to consider the individual molecules of a

gyroscope to be compound turbines. As the gyroscope rotates, its molecules are

exposed to a circulating wind of luminiferous medium which causes them to

precess with their precession axes parallel to the wind. The faster the motion the

faster will be the precession and the molecules will become aligned such that

their precession axes trace out concentric circles within the gyroscope. The

situation is closely related to Ampère’s circuital law and the solenoidal

alignment of the precession axes is the basis of gyroscopic stability. The

alignment induces a radial compound centrifugal pressure that has the effect of

causing the gyroscope to resist external torque. In a magnetic field, the same

reactive effect is known as inductance. There is no official recognition of the

concept of spin-based inductance in the literature [8], yet when an external

torque is applied to a spinning gyroscope it is quite clear that a strong reactance

can be felt which cannot be accounted for by the moment of inertia. This

reactance, which feels like magnetic repulsion, is a manifestation of Lenz’s law

extended to the gyroscope. It is the opposition which accompanies an induced

effect.

The induced effect is the Coriolis force and it can be explained by

considering a large gyroscope spinning clockwise with the cardinal points

marked on the clock face in the inertial frame. If we force precess the gyroscope

about a north-south axis in its plane of spin, there will be a change of the angle

of attack of the ‘electric wind’ of the luminiferous medium that is circulating

inside the gyroscope, and so the instantaneous effect will be for the precession

axes of the individual molecules to become realigned within the gyroscope. The

compound centrifugal force will hence act out of the gyroscope’s plane of

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rotation causing a large scale torque to act at right angles to the forced

precession, while also restoring the solenoidal alignment of the molecules

within the gyroscope. A forced precession on a north-south axis, east

downwards, will therefore result in the gyroscope tilting upwards at the north

point and downwards at the south point. A large scale Coriolis torque will have

been induced about the east-west axis, caused by a small scale Coriolis force

that is acting on the individual constituent molecules. Meanwhile the energy

from the forced precession will be converted into simple spin in the new plane

of rotation. Gyroscopic stability is a kind of sponge for spin.

This should not be seen as a case of Coriolis force being explained in terms

of itself on a smaller scale, because on the smaller scale the circumstances are

different and a distinct cause has been identified. On the smaller scale the

molecules of the gyroscope are actually moving through the sea of even smaller

molecular vortices that make up the luminiferous medium. It is this motion of a

spinning molecule through the luminiferous medium that generates a

differential centrifugal force on either side of the molecule, just like when a

spinning cricket ball moves through the air.

In the case of the rattleback, the apparent simple centrifugal force on the

large scale is being caused by a Coriolis force on the molecular scale.

Polar Coordinates in the Inertial Frame of Reference

VII. The mainstream literature contains two common methods for deriving the

inertial forces. One method invokes the use of polar coordinates in an inertial

frame of reference while the other method invokes a rotating frame of reference.

Close examination of these two methods however exposes them to be identical.

Let’s first of all look at the derivation using polar coordinates without invoking

a rotating frame of reference.

Consider a particle in motion in an inertial frame of reference. We write the

position vector of this particle relative to any arbitrarily chosen polar origin as,

ˆ

rrr

(1)

where the unit vector

ˆ

r

is in the radial direction and where

r

is the radial

distance. Taking the time derivative and using the product rule, we obtain the

velocity term,

ˆ

ˆ

rr rr θ

(2)

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where

ˆ

θ

is the unit vector in the transverse direction and where

is the

angular speed about the polar origin. Taking the time derivative for a second

time, we obtain the expression for acceleration in the inertial frame,

2

ˆˆˆ

ˆˆ

r r r r r rr θ θ θ r

(3)

which can be rearranged as,

2ˆ

ˆ

( ) (2 )r r r r rr θ

(4)

The centrifugal force and the Coriolis force appear as the first and the third

terms on the right hand side of equation (4). Note that no rotating frame of

reference is needed, and that all that is necessary is to identify a centre of

rotation. Contrary to popular belief, centrifugal force is a product of absolute

rotation and not of circular motion. In the case of uniform straight line motion,

the total acceleration will be zero, and hence we can deduce that the centrifugal

force takes on the same mathematical form as the second (centripetal term) term

on the right hand side of equation (4). It should also be noted that while the

centrifugal force is specifically a radial force, the Coriolis force is specifically a

transverse force. Uniform straight line motion corresponds to the inertial path,

and so we can conclude that the inertial forces are the underlying cause of the

inertial path, Newton’s first law of motion, and the law of conservation of

angular momentum.

Let’s now take a look at the alternative derivation of the inertial forces

which is prominent in the literature. It begins in the same way by considering

the position vector of a particle, but this time the particle is specified to be

undergoing circular motion. However, after establishing the velocity equation

for the case of circular motion, the general case is then considered and the

velocity equation is extended to,

(dr/dt)S = (δr/δt)R + ω×r (5)

where (dr/dt)S is the velocity of the particle relative to the inertial frame,

and ω is the angular velocity of the rotating frame. It is assumed that the

velocity of the particle in the rotating frame, (δr/δt)R, can be in any direction,

but if that is so, then r cannot be the same vector throughout the equation, since

the origin of the latter will have to be the fixed point in the rotating frame which

has the transverse speed ω×r. It’s a simple question of vector addition of

velocities, and so a serious error has been made. Equation (5) can only make

sense if r is the same vector throughout the equation, but in that case it becomes

equivalent in every respect to equation (2), and therefore the meaning changes

and the rotating frame of reference at the beginning of the derivation becomes

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irrelevant and misleading. The (δr/δt)R term therefore cannot have any

transverse component, and since the Coriolis force term takes on the vector

cross product format, 2ω×(δr/δt)R, the Coriolis force must be strictly a

transverse force.

The consequence of confusing the situation by the introduction of a rotating

frame of reference is that there prevails a further misinformed belief that the

centrifugal force is merely a product of the rotation of the frame of reference

itself rather than a product of the absolute rotation of the particle relative to the

inertial frame. The latter error leads to the bizarre notion that a particle at rest,

when observed from a rotating frame of reference, experiences a fictitious

outward centrifugal force in the radial direction, even though circular motion

can only induce transverse artefacts. This discrepancy is then patched up with

an even more bizarre argument involving a radial Coriolis force. The argument

runs that since the stationary object, as observed from the rotating frame of

reference, is seen due to its inertia to trace out a circular path, there must exist a

fictitious centripetal force acting upon it which can be justified as being the

resultant of the outward radial fictitious centrifugal force and an inward radial

fictitious Coriolis force.

While this is clearly wrong mathematically as well as being arrant nonsense

in its own right, this is the argument which is nevertheless used in modern

physics in order to mask the fact that both the centrifugal force and the Coriolis

force are real forces. The centrifugal force, rather than being a real outward

force that can pull a string taut or reverse the direction of a rotating rattleback, is

reduced to a mere artefact of making observations from a rotating frame of

reference. The active outward physical effect of centrifugal force, which is the

very essence of the common understanding of the concept, is being denied by

using a mathematical conjuring trick. Attributing the cause of the inertial forces

to a rotating coordinate system is a fraudulent way of thinking which seems to

be part and partial of the modern relativity culture where there are no absolutes,

and it appears to be inspired with the intention of denying the existence of the

absolute motion that is clearly exposed by Newton’s bucket. This in turn seems

to be aimed at denying the existence of the luminiferous medium.

In the parts of the mainstream literature where centrifugal force is derived

from polar coordinates in the inertial frame without the involvement of a

rotating frame of reference, the ensuing cognitive dissonance is typified by this

quote from Marion [9], which was made in the context of the centrifugal term in

the radial planetary orbit equation,

This quantity is traditionally called the centrifugal force, although it is not a “force” in the

ordinary sense of the word. We shall, however, continue to use this unfortunate terminology

since it is customary and convenient. Jerry B. Marion, 1965

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So on the one hand they teach that centrifugal force is merely an artefact of

making observations from a rotating frame of reference, yet in situations where

no rotating frame of reference is involved and where the centrifugal force is real

enough to hold a planet up or to snap a string, they choose to apologize for

using the name ‘centrifugal force’, considering it to be an unfortunate

terminology despite its convenience. Indeed more often they simply avoid using

the term ‘centrifugal force’ altogether. The unpalatable truth however is that we

can perceive and feel the outward effect that is induced by rotation, without the

need to introduce a rotating coordinate system.

Conclusion and Discussion

VIII. It has been suggested by the late Professor Eric Laithwaite that there must

exist an active spin-induced force and a spin-induced inductance, both of which

remain unrecognized in classical mechanics. [8]

There is no doubt about it that when we apply a precessional torque to a

spinning gyroscope we feel a very distinct reactance. It feels like magnetic

repulsion and it is stronger than any reactance that could be accounted for by the

moment of inertia alone. This reactance is the basis of gyroscopic stability.

Another counterintuitive feature of spinning gyroscopes is that when they are

placed on a pivot, they defy gravity in a manner which parallels the gravity

defying effect that a horizontal magnetic field produces on a falling charged

particle. Spin therefore induces real physical effects that must involve a

mechanism similar in principle to that which lies behind electromagnetic

phenomena. The medium for the propagation of light (luminiferous medium),

which is the cause of magnetic phenomena, can be shown to account for a spin-

induced compound centrifugal force, F = qv×B, that has a similar mathematical

structure to the Coriolis force.

It is proposed therefore, that contrary to what it says in the literature, the

Coriolis force is a real transverse force which follows directly from Newton’s

laws of motion, as can be clearly seen when they are expressed in polar

coordinates in an inertial frame of reference. The Coriolis force is observable

from an inertial frame of reference and often disguised in the uniform straight

line inertial path, or in the law of conservation of angular momentum. The

mainstream literature wrongly presents the Coriolis force as being merely an

artefact of making observations from a rotating frame of reference. This error is

what has led to all the confusion.

In the case of north-south air currents in the atmosphere, the east-west

deflection is not fictitious because it interacts physically with the un-deflected

air that is at rest relative to the Earth. In applications involving Newton’s third

law of motion, such as constrained radial motion within a rotating system, the

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Coriolis force causes the entire rotating system to angularly accelerate (or

decelerate). This could be a marble constrained to a radial groove on a rotating

platform, or it could be a gyroscope being subjected to a forced precession.

Artefacts do arise in a rotating frame of reference, always in the transverse

direction since we are merely imposing a circular motion on top of an already

existing situation, but such transverse artefacts are not the thing that is being

described by the Coriolis force formula.

References

[1] Clerk-Maxwell, J., “On Physical Lines of Force”, Philosophical

Magazine, Volume XXI, Fourth Series, London, (1861)

http://vacuum-physics.com/Maxwell/maxwell_oplf.pdf

[2] Tombe, F.D., “The Coriolis Force in Maxwell’s Equations”, (2010)

Galilean Electrodynamics, Volume 25, Number 2, p.22, (March/April 2014)

http://gsjournal.net/Science-Journals/Research%20Papers-Astrophysics/Download/3161

[3] Teodorescu, P.P., “Mechanical Systems, Classical Models”, Volume 2, Mechanics of Discrete and

Continuous Systems, Chapter 16.2.1.7, page 420, (2002)

[4] De Mees, Thierry, “Is the Differential Rotation of the Sun Caused

by a Coriolis Graviton Engine?”, Section 3, (2010)

http://www.mrelativity.net/Papers/14/tdm35.pdf

[5] O’Neill, John J., “PRODIGAL GENIUS, Biography of Nikola Tesla”, Long Island, New York, 15th July

1944, quoting Tesla,

“Long ago he (mankind) recognized that all perceptible matter comes from a primary substance, of a tenuity

beyond conception and filling all space - the Akasha or luminiferous ether - which is acted upon by the life-

giving Prana or creative force, calling into existence, in never ending cycles, all things and phenomena. The

primary substance, thrown into infinitesimal whirls of prodigious velocity, becomes gross matter; the force

subsiding, the motion ceases and matter disappears, reverting to the primary substance”.

http://www.rastko.rs/istorija/tesla/oniell-tesla.html

[6] Whittaker, E.T., “A History of the Theories of Aether and Electricity”, Chapter 4, pages 100-102, (1910)

“All space, according to the younger Bernoulli, is permeated by a fluid aether, containing an immense number

of excessively small whirlpools. The elasticity which the aether appears to possess, and in virtue of which it is

able to transmit vibrations, is really due to the presence of these whirlpools; for, owing to centrifugal force,

each whirlpool is continually striving to dilate, and so presses against the neighbouring whirlpools.”

[7] Tombe, F.D., “The Double Helix Theory of the Magnetic Field” (2006)

Galilean Electrodynamics, Volume 24, Number 2, p.34, (March/April 2013)

http://www.wbabin.net/Science-Journals/Research%20Papers-

Mechanics%20/%20Electrodynamics/Download/252

[8] Laithwaite, Eric R., “The bigger they are, the harder they fall”, Electrical Review, pages 188-189, (14

February 1975)

http://www.gyroscopes.org.uk/papers/Bigger%20they%20are%20the%20harder%20they%20fall.pdf

[9] Marion, Jerry B., “Classical Dynamics of Particles and Systems”, Chapter 10.6, page 275, (1965)

5th October 2015 revision