1.

A New System of the Universe,

founded upon

A Constant Propulsion and Resistance.

By Orson Pratt, Sen.

Chapter I.

Ethereal Medium.

- – Corpuscular and Undulatory Theories. 2. – Density of the Corpuscular Medium. 3. – Corpuscles can have no Gravity. 4, 5, – Corpuscular Resistance and Momentum – the Corpuscular Theory inconsistant with existing phenomena. 6 – Undulations and Oscillatory Rotations of Ether. 7. – Vast number of Luminous Waves in a Cubic Inch. 8. – Ethereal Resistance. 9. – Ether Intensely Elastic. 10. – Concentration and Dispersion of Ethereal Waves. 11. – Power of Ether over all other Matter. 12, 13. – Ether subject to Gravitation and Repulsion. 14. – Definition. 15. – Language of the Newtonian theory adopted. 16. – The Ethereal Medium of Variable Density. 17. – Does the Velocity of the Ethereal Waves vary with the Density of the Ether? 18. – The question of Variable Velocity can easily be determined by observation – its vast importance in Astronomical Researches.

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- – Corpuscular and Undulatory Theories. Since the days of Newton, until very recently, scientists have been divided, in regard to the manner in which light is transmitted through space. Two theories were propounded; one called the Corpuscular; the other, the Undulatory. The former assumes that light consists of extremely minute particles or corpuscles, projected from luminous bodies, with immense velocity; that such particles impinge upon the optic nerve and produce the sensation of seeing. The undulatory Theory claims that all space is occupied with a substance called ether, extremely rarefied and elastic in its nature; that the molecules of luminous bodies, being themselves in a constant state of tremulous agitation, impart the same to the adjoining molecules of ether; that this jar or vibration is transmitted from molecule to molecule, forming a wave; that the displacements of the molecules in a wave are not that of extention from and compression towards the point of their origin, but in circular forms whose planes are transverse or perpendicular to the line of motion of the wave; that the wave, thus formed, travels through the ethereal space with the immense velocity of 185420 miles per second, which is over 660000 times swifter than the tidel-wave of the ocean, and over 820000 times the velocity of longitudinal sound-waves. It is still further assumed that a white luminous body forms a continuous succession of mixed waves, varying in length, from 37640 to 59750 waves in one inch; that the longer waves are formed more slowly than the shorter ones, but travel with the same velocity; that only 468 millions of millions of the longer waves are successively originated in one second; while 727 millions of millions of the shorter ones are formed in the same time; it is also assumed that the variations of color are merely variations of the wave-lengths, and the rapidity with which they are formed.
- – Density of the Corpuscular Medium. If the Corpuscular theory be assumed as true, the lines of traveling particles must vary in density as they recede from the point of radiation; for lines of radiating corpuscles must diverge from the point of projection; hence, at n times any given distance, such lines will be n times further apart; and the area

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upon which they fall will be represented by the square of n; and the density of the lines will be as the inverse square of n; therefore a traveling Corpuscular medium must decrease in density as the particles recede from the point of radiation.

3.- Corpuscles can have no gravity; for if the projected particles were subject to the law of gravity, they would be continually retarded as they recede from their points of radiation. But light comes to us from the stars, some of which are immensely more remote than others, and yet their light has not been retarded neither accelerated, but reaches us with the same velocity, as the light from nearer worlds: this is demonstrated by the constancy of the angle of aberration, which is the result of the ratio of the velocity of light to the velocity of the earth in its orbit, therefore, radiating particles can have no gravitation.

4.- Corpuscular Resistance. The immense and unceasing shower of corpuscles continually poured upon the enlightened hemisphere of all planets would drive them outwards from their elliptic paths into spiral orbits and bring ruin upon the whole system. But no such phenomena occur, and therefore, the corpuscular molecules have no resistance: but if they have no resistance, all substances must be perfectly transparent to them, and they must continue to move with the same velocity in the interior of all bodies, as when traveling in space; and hence, there could not be any such thing as darkness, or shadow, or variation of intensity. But there is a rebounding, or reflection, or absorption of particles, including a destruction of their velocities. Under any of these last conditions, the particles must suffer a resistance; and the body upon which they infringe, must also suffer an equal amount of resistance: otherwise action and reaction would not be equal: therefore, as all bodies are not transparent, but both reflect and absorb the swiftly flying corpuscles, and destroy their velocities, they must be capable of resisting and being resisted. One class of phenomena proves them to be non-resisting; another class proves them to be capable of resistance, which is absurd; therefore the corpuscular theory is incompati-

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ble with the existing phenomena above referred to.

5.- Momentum of Corpuscles. In the preceding paragraph, it is found that resistance is a property of corpuscles in motion; they must, therefore, have a certain degree of momentum; but momentum is the product of velocity into mass; and mass is universally ascribed to gravitating matter; and therefore, corpuscular matter is subject to gravitation: but it has been proved, (Par. 3.) from another class of phenomena, that it is not subject to gravity: and therefore the radiation and transmission of particles from world to world will not account for existing phenomena, and must be rejected as untrue.

6.- Undulations and Oscillatory Rotations of Ether. If the hypothesis of traveling particles be discarded, the only remaining theories which will account for the phenomena of nature, are either the transmission of motion through space, in the form of vibrations or waves of some highly elastic medium, or in the form of oscillatory rotations, produced by some influence similar to magnetism, on the line of ethereal particles, radiating from the source of light. This latter form of transmission would not require the medium to be possessed of the intense elasticity as is necessarily assumed for the wave theory.

7.- Every Cubic Inch of space must contain, at least Fifty-three millions of millions of Luminous Waves.

It has been demonstrated by numerous and skilful experiments, that one inch in length of the extreme red rays of light, contains 37640 waves. If the depth of each wave measured transversely to its line of motion, is equal to its length, then the number of waves in a cubic inch must be as the (37640)^{3} = 53,327,207,744,000. The extreme violet gives 59750 waves in one linear inch; hence, a cubic inch contains (59750)^{3} = 213,311,234,375,000.

As each wave undoubtedly consists of numerous molecules, how extremely minute must be the pores between these molecules: The spaces unoccupied must be infinitesimally small, and yet all the stellar and planetary bodies of the universe perform all their evolutions in the

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midst of this immensity of substance. Resisted they must be; and without a compensating accelerating force, a universal win must speedily ensue.

8. Waves of Ether, like other matter, are capable of Resisting and being Resisted.

The luminiferous waves are originated by the action of gravitating matter; their course is changed, by other matter, when reflected; they are retarded and their course altered, when passing through water, glass, crystals, and other transparent materials; they are refracted into curvilinear paths, in passing through the atmosphere obliquely to the earth’s surface; they are absorbed and their momentum destroyed by opake materials; they act upon the optic nerve, and impart to it a tremulous motion. That which can exhibit all these complicated phenomena, must, therefore, have the power of resisting and being resisted.

9. – The Ethereal Medium is Intensely Elastic.

When the equilibrium of the ether is, in the least, destroyed, as in the formation of luminiferous waves, the particles almost instantaneously return to their former state of repose. So rapidly are these disturbances and restorations to repose, performed, that over seven-hundred millions of millions of waves of violet light are successively formed and destroyed, in one second of time. And as the return of each wave to its equilibrium, occupies the same time as the disturbance, it follows, that over seven-hundred millions of millions of restorations successively transpire, in one-half of a second. All these restorations are produced by the elastic force; therefore, the ethereal medium is intensely elastic.

10. – The Ethereal Waves can be concentrated or dispersed.

Both the luminiferous and heat waves can be either refracted or reflected to a focus, and vice versa, they can be either refracted or reflected from a focus into a variety of paths.

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11. – The Ethereal Waves can Change the Form of all Matter.

When these waves enter solids in the form of heat, the solids are dilated or expanded, and sometimes converted into liquids, or even into gases. By these ethereal waves, chemical compounds are torn assunder, and their elements are made to appear; by them, mighty ships plow the ocean against both wind and current; by them, lengthy trains, of heavily laden cars, rush with terrible speed, from nation to nation, from ocean to ocean, till distance itself seems almost annihilated; by these almost infintiesimal undulation, lofty mountains are upheaved, and their smoking summits tell of the fiery billows which rage, in awful grandeur, far beneath; by these vibrations of the subtle ether, the earth itself rocks to and fro, and its very foundations tremble as if about to divide assunder. Solids, liquids, gasses, compounds, elements, and all terrestrial phenomena, bow in humble reverence, and submit themselves to the powerful control of this most potent substance – the ethereal medium.

12. – The Ethereal Medium can cause matter to Approach to and Recede from other matter, and is itself subject to gravitation and Repulsions.

Two steel sewing needles, when exposed to the violet rays of the solar spectrum, become magnetized; and when <placed> parallel and near each other on the surface of a liquid, as water or quicksilver, with their opposite poles adjacent, they will gravitate towards each other: when like poles are adjacent they will repel each other; hence, ether is capable of entering into permanent combination with steel; and while thus imprisoned, the steel exhibits the force and phenomena of gravitation; and under a slight change of position, it exhibits the force and phenomena of repulsion. The steel before being magnetized, already possessed common gravity: the superadded force must, therefore, belong to the ether; hence, ether itself is subject to the law of gravitation and repulsion, as well as all other matter.

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13. – A globe of unmagnetized steel needles, of the same mass as the earth, would exert, at the same distance, the same force of gravity as the earth. But if all the needles in such globe were magnetized, and arranged with their opposite poles adjacent, their force of gravitation would, by the superadded magnetism, be greatly increased; hence what is called mass, in this instance, would not depend on an increase of the substance of the steel, but on an increase of force in such substance; and what is usually termed quantity of matter, would be merely quantity of force added to such matter; and this would vary as the force of magnetism varied. The masses of all worlds depend upon the quantity of force they exert: so long as this force is constant, it becomes a definite measure in all calculations, relating to celestial mechanics: but if it were possible for such forces to be increased or diminished to any appreciable extent, it would change, not the quantity of substance, but the weight or movable power in the substance, and we should be left without any definite measure of quantity or mass. The form and dimensions of planetary orbits, and the whole mechanism of the universe would be continually changing.

14. — Definition. The term matter, when applied to ether, will be called, ethereal matter; when applied to other substance it will be called gross matter.

15. – As the particular form of reasoning, founded on the Newtonian theory, is, by constant use, more universally familiar, we shall, in our future discussion, adopt its language, with the express understanding that gravitation is also extended to ethereal matter.

16. – The Ethereal Medium is not of Uniform Density. If the ethereal medium were of uniform density, it must be destitute of the gravitating force: but it has been proved (Par. 12) that its waves can impart magnetism to steel; but magnetism is only the effects of electric waves; manifested in the form of galvanic currents, by which the common force of gravitation

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in iron, can be neutralized, or apparently greatly increased or diminished. Many other substances, besides iron and steel, are similarly affected. The earth itself is a magnet, extending its force over all its surface, and probably for millions of miles into space. The sun, at the distance of over ninety-millions of miles, exerts magnetic powers. As the magnetic or electric waves of ether exert such extended powers, all matter, whether gross or ethereal, must be affected by them – must exhibit the gravitating force; hence, the ethereal medium, by virtue of its gravitation, must collect around all other matter in the form of envelopes. These ethereal envelopes must increase in density, as their respective centers of gravitation are approached.

As gross matter is not impervious to ethereal matter, the latter must infuse itself through all the interior of worlds, its density increasing, till the centers of gravity are reached. Outwards from the surface, the densities will decrease as far as the ethereal envelopes of rotating bodies can extend. Beyond the surfaces of these rotating envelopes, the densities will still continue to decrease, until reaching the limits of equal gravitation, between world and world. Such must be the condition of the ethereal medium, under the force of gravity.

17. – Is it probable that the velocity of the Ethereal Waves is Variable, according to the Density of the Ether through which they are propagated?

The average velocity of light, in passing diametrically across the earth’s orbit, is 185,420 miles per second. But as some portion of its path lies near the sun in close proximity to the densest strata of the ethereal medium, may not the luminiferous waves be continually retarded, for 493.096 seconds, during which they describe the first half of their path? And in the second half of their journey, may they not be continually accelerated, during an equal interval of time? If so, the sum of their retardations will be exactly equal to the sum of their accelerations, and the average velocity will be as stated above.

18. – To those who are anxious to test, by observation, the variable

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velocity of light, if such exist, arising from the gravitating power, and consequent variable density of the ethereal medium, the writer would suggest the following:-

Let the observer, with good instruments, carefully determine, from the eclipses of one of Jupiter’s satellites, the exact time of the passage of light, though different chords of the earth’s orbit. The exact lengths of these chords are easily calculated. It is evident that the shorter the chords, the greater will be the distance from the sun, and the less will be the density of the medium, and the greater may be the velocity of light, and the less may be the time in passing over equal distances. If such variation is found to exist, the exact determination of these data will be of immense importance, in the future development of astronomy. For the data, thus obtained, will determine.

First, the relative densities of the different strata of ether, intervening between the earth’s orbit and the sun:

Second, the relative elastic forces of these strata, as compared with their respective forces of gravitation towards the sun.

These discoveries would very probably develop some law of density, depending on the distance from the sun; if so, such law could be extended into vast distances beyond the present boundaries of our system, till we reach the sphere of equal gravitation, between our system and others.

Such a law would also develop the law of planetary resistance, depending on their respective masses, velocities, distances, and the relative densities of the medium in which they move.

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Chapter II.

Rotating Ethereal Envelopes

19.—Ethereal Surface of Rotation; Definition. 20, 21.- Law of Ethereal Semi-diameters – Examples. 22.- Remarkable Forms of the Ethereal Envelopes. 23.- Ethereal Medium External to the Rotating Envelopes. 24.- 27 Law of Planetary velocity – Table. 28.- Circulating Currents External to the Envelopes – their Periodic times Increase with the Latitude, when the Axial Distances are the same; Examples 29.- Planes of Motion Perpendicular to the Axis. 30, 31. – Tendency to Equilibrium – its effects upon the Rotation of Bodies. 32, 33 – The Author’s Theory Confirmed by Recent Observations – Sun-spot – Periods increase with the Latitude. 34.- Sun’s Rotation less than the Sun-spot Periods. 35.- Rotation Diminished by Ethereal Resistance, unless balanced by a Constant Acceleration.

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19.-Ethereal Surface of Rotation. The Ethereal medium, not only collects around worlds in greater density, but it also partakes of the rotatory motion of the body around which it gathers. This rotation necessarily produces an envelope of definite form and dimensions, depending on the balanced condition, between the centripetal and centrifugal forces. The rotating envelope can only extend to the distance, where the rotating velocity becomes sufficiently great to balance the gravity of the surface particles. The vast ethereal ocean beyond these limits, cannot rotate in the same time as the world within. But if it revolves at all, each particle must have a planetary velocity slower than the surface rotation.

Definition.- The distances from the center of a rotating globe to the limiting surface of the ethereal envelope, which can possibly rotate in the same time as such globe, without being thrown off by the centrifugal force, will be called, Ethereal semi-diameters.

20.- A rotating point on the limiting ethereal surface must have a planetary mean velocity: but planetary mean velocities vary inversely as the square roots of their mean distances; (See par. 24;) hence, the rotating velocity, (which varies as the Axial distance,) of any surface point, on the North or South of the equatorial plane, must be as the inverse square root of its central distance; and consequently, the axial distance, which is as the velocity, varies as the inverse square root of the central distance; therefore, we have the following.

21.- Sean. – The ethereal semi-diameters will vary as the inverse squares of the axial distances of their points of intersection with the limiting surface.

To express this law algebraically, let D and D’ represent the axial distances of any two points of surface intersections with their respective ethereal semi-diameters; let d and d’ represent their corresponding ethereal semi-diameters; then we shall have

1/10^{2 }: 1/10^{2 } :: d : d^{1 }

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d^{1}=D^{2}d/D^{2}…..……………………………….(1)

The equatorial ethereal semi-diameter is equal to the axial distance; therefore, in this case, D is equal to d, and the above formula becomes

=

When the equatorial semi-diameter, D or d, is called unity or 1, we have

d^{1}=D^{3}/D1^{2} …………………………………………….(2)

When D’ = 0, d’ is infinite; hence, the surface of revolution gradually approaches the axis, and finally coalesces with it, at an infinite distance from the center.

In illustration of this law the following problems are given.

- Find the equatorial ethereal semi-diameter of the sun the period of its equatorial rotation being 25 days, and its distance from the earth being 91430000 miles.

Let t = earth’s orbital period.

t’ = 25 days = the solar equatorial rotation.

d = sun’s distance

d’ = the required equatorial ethereal semi-diameter.

By Kepler’s law we have

t^{2} : t’^{2} : : d^{3} : d’^{3};

hence

d’ = 15298548 miles.

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II. – If the equatorial ethereal semi-diameter of the sun, be called unity or 1, what will be the ethereal semi-diameter at any ethereal surface point, where the axial distance is equal to ½ ? and what the solar latitude?

By formula (2) we have

d’ = 1/(1/2)^{2 }=4

hence, when the axial distance is ½ the equatorial, the ethereal semi-diameter will be 4 times that of the equatorial.

The latitude is found by the common method, that is, by finding the angle included between the ethereal semi-diameter and the axial distance, or the hypothenuse and the base.

Lat = 82^{o} 49’ 10”

III. – If the axial distance of a surface point on the sun’s ethereal envelope be ^{.}7937, what will be its ethereal semi-diameter? and what the solar latitude?

Ans. { 1.587401 .

{ Lat. = 60^{o}

IV. = If the axial distance be .1 what will be the length of the ethereal semi-diameter?

Ans. 100 times the ethereal semi-diameter at the equator.

V. – If the axial distance of a surface point is .8908981, what will be the ethereal semi-diameter? and what the solar latitude?

Ans. {1.259920.

{Lat. 45^{o}.

VI. – If the axial distance is .9531843, what the semi-diameter, and what the solar Latitude?

Ans. {1.1006424

{Lat. 30^{o }

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22. – By the solution of these few problems, and the mathematical law which determines the form and dimensions of these ethereal envelopes, we can readily perceive that every sun, planet, asteroid, satellite, and comet which has a rotation on an axis, must have an ethereal envelope of a certain figure, and must be governed by the same law; and that the dimension of each depends upon the mass of the body and period of its rotation.

What appears very remarkable is the sameness and peculiarity of this figure, having no resemblance to any spheroid, ellipsoid, paraboloid, or other hyperboloid or other common known solid. Although it is a figure generated by the mechanical laws of central force, and axial rotation, yet its properties do not seem to have been noticed, so far as the author is aware, by any of the investigators of celestial mechanism.

23. – The Ethereal Medium, Outwards from the Ethereal Rotating Envelopes.

Outwards from these ethereal surfaces of rotation, there must be circulating currents of ether, revolving in the same planes, and in the same directions, as the interior rotating bodies, and their respective envelopes. For a rotating surface imparts its motion to adjoining strata, and these again to others, and so on, to an indefinite distance. The velocities of these successive currents, if not retarded, would necessarily be the same as those of planetary bodies: and, therefore, from the ethereal surface, outwards, the velocities must continually become slower and slower, while the periodic times become greater.

24. – The law of mean velocities, depending on planetary distances, may be derived from Kepler’s law:

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thus

Sect t, t’ represent the periodic times of any two planets in their orbital revolutions: let d, d’ be their respective mean distances from the sun; <then by> Kepler’s law, we have

t^{2} : t’^{2} : : d^{3} : d’^{3};

and t : t’ : : d^{3/2} : d’^{3/2}. . . . . . . . . . .(1)

But t : t’ : : d/v : d’/v’ ;

hence d/v : d’/v’ : : d^{3/2} : d’ ^{3/2};

and 1/v : 1/v’ : : d^{1/2 }: d’^{1/2}; . . . . . . (2)

hence v : v’ : : 1/d^{1/2} : 1/d’^{1/2}; . . . . . (3)

therefore we have the

Law. – Planetary mean velocities vary as the inverse square roots of the planets’ mean distances from the sun.25. – The law of the planetary mean velocities, depending on the periodic times, can easily be derived from (1) and (2) of the last paragraph: thus

By cubing (2), we have

1/v^{3} : 1/v’^{3} : : d ^{3/2} : d’^{3/2};

but (1) gives

t : t’ : : d^{3/2 }: d’^{3/2} ;

hence

1/v^{3} : 1/v’^{3} : : t : t’ ;

and

v^{3} : v’^{3} : : 1/t : 1/t’ ;

therefore

v : v’ : : 1/t^{1/3} : 1/t’^{1/3} ;

therefore, we have the

Law.—Planetary mean velocities vary as the inverse cube roots of the periodic times of the planets in their orbits.

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26. – The law of planetary mean velocities, depending on the intensity of gravitation, can be determined as follows: —

By Newton’s law, we have

g : g’ : : 1/d^{2 }: 1/d^{2} ;

by (3), par. 24, we have

v : v’ : : 1/d^{1/2} : 1/d’^{1/2};

hence

v^{4} : v’^{4} : : 1/d^{2} : 1/d’^{2} ;

therefore

v^{4 }: v’^{4} : : g : g’ ;

hence

v : v’ :: g^{1/4 }: g’^{1/4 };

therefore, we have the

Law. – Planetary mean velocities vary directly as the fourth roots of the gravitating force.

27. – From the preceding laws and proportions, we deduce the formula, in the following

Table.

(1) . . . . . d’ = (t’/t)^{2/3}. d. (7) . . . . .v’ = (d/d’)^{1/2}.v.

(2) . . . . . d’ = (v/v’)^{2} . d. (8) . . . . .v’ = (t/t’)^{1/3 }. v.

(3) . . . . . d’ = (g/g’)^{1/2}. d. (9) . . . . .v’ = (g’/g)^{1/4}. v.

(4) . . . . . t’ = (d’/d)^{3/2}. t . (10) . . . .g’ = (d/d’)^{2}. g .

(5) . . . . . t’ = (v/v’)^{3} . t . (11) . . . .g’ = (t/t’)^{4/3} . g .

(6) . . . . . t’ = (g/g’)^{3/4} . t . (12) . . . . g’ = (v’/v)^{4} . g .

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28. – If tangent lines, touching the equatorial ethereal surface of the sun, be drawn at right angles to the equatorial plane, they will be parallel to the axis; and hence, all particles of ether, situated in those lines, will have equal axial distances; but their central distances will vary as the secant of the latitude, and their planetary velocities will vary as the inverse square roots of these secants; (Par. 24;) and therefore, the periodic times, around the axis in equal circumferences; must vary directly as the square roots of the secants: that is, at 4 times the distance the velocity will be ½, while the periodic time around the axis will be 2. At 9 times the distance, the velocity will be 1/3, and the periodic time 3; and so on. And, therefore, the periodic times of the circulating strata, at equal distances from the axis, will increase as you recede from the equatorial plane. The following examples will still further illustrate.

If the ethereal equatorial surface of the sun rotates in 25 days, at the same axial distance in solar Latitude 37^{o} 8′ 13″ , the planetary velocity of the ether is such that it would require 28 days to perform one complete revolution. In latitude 45^{o}, the axial distance being the same, the periodic time would be 29^{d} 17^{h} 28^{m}, 52^{s}. The axial distance being the same, if we take a distance from the equatorial plane equal to the sine of the latitude 45^{o }, the periodic time of the revolutions of the ether around the axis will be == 27^{d} 16^{h}.

29. – It should be born in mind, that the planetary revolutions of the ethereal strata, must not be calculated for orbits around the center of gravity of the sun; for these (which would be their natural orbits if left free to move,) they are prevented from describing, by the intervening strata: but they move in circles, parallel to the equatorial plane, and at right angles to the axis.

If lines be drawn parallel to the axis, and exterior to the tangent, the revolving particles, at different points along these lines, will follow the same law as expressed in Par. 28; that is, for the same latitude, the velocities will be less, and the periodic times greater, because of the increased distance of these lines from the axis; and these times will still further

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increase, as the polar regions are approached.

30. – Constant Tendency of the Ethereal Medium to a State of Equilibrium.

The intensely elastic properties of the ethereal medium (Par. 9.) must necessarily operate to preserve an equilibrium; and when such equilibrium is destroyed, it will operate with great force or rapidity to restore the medium to its former state of repose. When a train of cars passes rapidly through the atmosphere at rest, the air is thrown into a violent motion; but as soon as the cause ceases to act, the elastic power of the medium soon restores it to its former condition. The same is true, when the disturbance arises from rotating bodies. As long as the rotation of the body continues, there will be a constant rotation of the air in the immediate vicinity, and also constant effort of the air to counteract the rotation, and to preserve itself in a state of equilibrium; and when the rotation ceases, the circulating currents do not long survive, being overpowered by the friction and elasticity of the surrounding medium.

The ethereal strata, immediately exterior to the rotating ethereal envelopes, when seeking to acquire a planetary velocity, if successful, cannot, for a moment, preserve such velocity; for the still more distant strata, having a slower motion, must, by friction, retard the interior strata, diminishing their centrifugal force, until regaining a small fraction of weight they again slightly press upon the rotating surface, when they begin again to re-acquire the velocity lost by friction. These alternate processes will be repeated, while rotation continues.

31. – The Effects of the Equilibrium Tendency on the Rotation of Worlds.

Besides the ethereal envelopes surrounding worlds, the most of them have also atmospheres of less dimensions, composed of grosser substances, which only extend a comparatively short distance above the solid or liquid portions. These gross atmospheres, have a constant tendency to rotate in the same time as the denser mass beneath. But this they are

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prevented from doing, by the constantly retarding action of the ocean of ether exterior to these rotating atmospheres and ethereal envelopes. This retarding action does not exhibit itself with equal force in different latitudes of the rotating body. At the equator, it is a minimum; in the neighborhood of the poles, it is a maximum, The periodic times of these rotating atmospheric and ethereal strata, will increase with the latitude. The true cause of this, has been explained in paragraphs 28 and 29. It necessarily arises from the excess of gravitating force, acting on the ether, not being so fully counteracted by the centrifugal force as in the equatorial regions. This greater pressure, with greatly retarded velocities of the ethereal strata exterior to the rotatory surforce, must retard the atmosphere with increased force as you recede from the equator.

32.-The Author’s Theory is Confirmed by Recent Observations.

Recently observers have carefully noted the periodic times of rotation of sun-spots in different solar latitudes. At the equator they rotate in about 25 days, as the distance from the equator increases, the periodic time of rotation increases. About 40° of latitude, or nearly half way from the equator to the poles, the periodic time of these spots is a little less than 28 days. Now it is a fact worthy of note, that in latitude 45°, at a distance above the ethereal surface, only amounting to a little less than one-tenth of the equatorial ethereal semi-diameter, the periodic time of the circulating currents, having a planetary velocity, is 28 days.

33.—These sun-spots are probably dark clouds, floating in the denser or grosser atmosphere, and rotate in the same time as the respective strata which they occupy. These sun-spots, as supposed by some, may possibly be the dark body of the sun, far beneath its luminous atmosphere, and seen through openings. If such be the case, these openings must travel around with the rotating current, exhibiting successive portions of the dark body beneath. As you recede outwards from these

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sun-spots, through the chromospheres and coronal regions, into the invisible ethereal envelopes, the ethereal substance must be more and more retarded, until the region of planetary velocity is reached, when the retardations will become much more rapid, until their velocities will finally become so reduced as to be inappreciable, and a comparative equilibrium will be restored.

34—The Real Period of the Sun’s Rotation must be less than 25 days.

For the equatorial sun-spots, have a period of 25 days; and these are retarded by the causes we have named. And though this retardation is reduced at the solar equator to a minimum, yet it must be appreciable, and may vary some hours from the real rotation of the great interior mass.

35.— Will the Real Rotation of the Sun be Diminished by its Ethereal Resistance?

If there is no accelerating force equal to the resistance, the velocity of rotation must be diminished, until it is altogether overcome. And the same is true in regard to all rotating worlds. It is also true in regard to all orbit revolutions. An end must come to all axial and orbital motions, and the universe be reduced to chaos, unless there is a compensating accelerating force, sufficiently powerful to counteract the resisting force of the ethereal medium. Such a force is imperatively called for, and such a force does exist, as we shall proceed, in the next chapter, to show.

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Chapter III.

Gravitation.

36. – Gravitation is not an Instantaneous Force. 37. – Gravitating Force transmitted with the Velocity of Light.38. – Aberration of the Gravitating Force. 39 – 42. – Aberration illustrated by a Right-angled Triangle. 43. – Aberration of Force in Elliptic Orbits the same as that* *of Light. 44. – Law of Aberrating Velocity, expressed in terms of Planetary Distances. 45. – Orbital Accelerations. 46. – Intensity of the Earth’s Gravity towards the Sun. 47. – Intensity of the Earth’s Orbital Acceleration, expressed in terms of Earth’s Gravity. 48. – Orbital Acceleration, expressed in Pounds Weight. 49. – Earth’s Fall towards the Sun in one second. 50, 51. – Excess of Orbital Space, gained in one second, by the Aberrating Force – Space gained in one year – Increase of Periodic Time. 52. – Aberrating Intensities vary with those of Gravity. 53. – Space of Aberrating Forces. 54. – Definition. 55. – Law of Aberrating Forces 55. – Law of Aberrating Forces, expressed in terms of Planetary Distances. 56. – The Law applicable to the Satellites.

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36. – Gravitation is not an Instantaneous Force.

It has been assumed, since the days of Newton, that the force called gravitation is transmitted from world to world instantaneously. This assumption has remained without proof until now. It was undoubtedly made to correspond with another equally erroneous assumption, that there was no resisting medium; and when compelled to admit that all space was filled with an ethereal substance, there was still further a necessity to assume that this ocean of ether had no gravitating or resisting properties. And thus one absurd assumption has been heaped upon another, in order to maintain an assumed hypothesis, namely, that there is no resistance, and therefore no need of an accelerative force to maintain the rotative and orbital motion of the planets. For to admit that gravitation, like light, needs time for its transmission through space, would involve, as we shall presently prove, the necessity of also admitting an accelerating orbital force, which would strike a fatal blow to their other assumptions.

37. – The Gravitating Force is Transmitted with the Velocity of Light. The velocity of light has been very accurately determined. And it is very firmly believed that the heating, magnetic, electric, and chemical rays, as well as the rays of different colors are transmitted with equal velocity. In this respect, the solar radiations are believed to follow the same law as the atmospheric radiations of sound. Sounds of every pitch and intensity are conveyed through the same medium, with the same velocity. Why should the radiations of gravity depart from this law? Why should Neptune receive this solar force as soon as Mercury? How can force be transmitted, or pass through space, without occupying time? Time and space are essential characteristics of all motion; take away either of these, and we can form no conception of motion. An instantaneous motion is inconceivable. In assuming, therefore, that the gravitating force is transmitted with velocity, involving

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time, we only assume that which is analogous to all the motions of nature. In further assuming that it has the same velocity as light and other solar radiations, we are supported by the analogous phenomena of the transmission of sounds.

38. – Aberration of the Gravitating Force. It is evident, that a travelling force, having the same velocity as light, will, when combined with a planet’s orbital velocity, produce an aberration or deviation from the real center of gravity. The amount of this aberration will be the same as the aberration of light, which, in the case of the earth, is equal to the mean deviation of about 20”.25 of an arc. This is called the mean or constant angle of aberration; that is, the earth moves over the arc of 20”.25, during the 8 minutes, and 13.096 seconds in which the force of gravitation is transmitted from the sun to the earth.

39. – The angle of the aberration of light has been calculated on the supposition, that gravitation was transmitted from the sun to the earth instantaneously – that when the earth is in perihelion or aphelion, it moves in a right angle to this line of instantaneous force – that the line connecting the real and apparent centers of the sun, being parallel to the line of the earth’s motion, must also be at right angles to the same line of force, — and that a line, drawn from the earth to the apparent place of the sun, is the hypothenuse of this right-angled triangle. This process of reasoning and the conclusion are correct, so far as the aberration of light is concerned.

40. – But if gravitation moves with the velocity of light, the line of force and the line of light will be identical; both will proceed, not from the real place of the sun at the instant of their reaching us, but from the apparent or aberrating position of that luminary. Hence, when the earth is in aphelion, it does in reality move at right angles to the supposed line of instantaneous force, and consequently must move with an acute angle to the line of the

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traveling or aberrating force, and, therefore, the line joining the centers of the real and ap<p> arent sun, which is parallel to the line of the earth’s motion, must also make an acute angle with the same line of traveling force. These three lines form a right-angled triangle; the right-angle being at the sun’s real center; the apparent sun’s distance is the hypothenuse; the line joining the real and aberrating centers is the perpendicular; and the line connecting the earth with the real center is the base. The base, if the earth’s orbit be considered circular, is less than one-half a mile shorter than the hypothenuse; there being only 2326 feet difference. The acute angle, included between the base and hypothenuse is the angle of aberration, the mean of which is as above stated: the mean of the other acute angle is 89° 59’ 39.75”.

41. – Still considering the earth’s orbit, circular, and its radius equal to 91430000 miles, an arc of 20”.25 would be equal to 8976.1192599286 miles, which would be equal to the perpendicular of the triangle we have just described.

42. – If the sides of this right-angled triangle were considered as consisting of some ridged material, as, for instance, iron-wire, and if the triangle were made to revolve around its right-angle, the greater acute-angle would describe the circle of aberration, and the other acute angle would describe a circular orbit of the earth, with the base as radius. The momentary direction of the earth’s path would be constantly perpendicular to the base;

43. – The effects of aberration in an elliptic orbit, calculated with reference to the traveling force of gravity, will be the same as the phenomena observed in connection with the aberrations of light.

44. – <The> Orbits being considered circular, the Aberrating Velocity at the Different Planets, varies as the Inverse Square Roots of their Distances from the Sun. The aberrating velocities in circular orbits vary directly as the orbital velocities; and these velocities vary as the inverse square roots of their distances;

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(Par. 24), and, therefore, the aberrating velocities of the force of gravitation must vary as the inverse square roots of these same distances.

Hence, the points towards which the planets gravitate, cannot be the real center of the sun, but they are movable points, at a distance from such center equal to the angle of aberration. As one faces the sun, this point for each planet is continually on the right hand of the sun’s center, and circulates around it, in the same orbital period of the planet, and from west to east in the same direction as the orbital motion.

45. – The Aberrating Force of Gravity Accelerates the Orbital Motion of a Planet. The points of gravitating aberration, being on that side of the sun’s center, towards which the planets are moving, must necessarily accelerate their orbital motions.

The perpendicular of the right-angled triangle, described in paragraph 40, represents, both in quantity and direction, the aberrating force of gravity. Complete the parallelogram, by drawing lines parallel to the base and perpendicular; the hypothenuse will be the diagonal of this parallelogram. The force which this diagonal represents may be resolved into two simple forces, represented by the base and perpendicular; but the latter is parallel and equal to the line which the planet describes, during the transmission of the gravitating force from the sun. Of these two simple forces, the base being equal to the Sun’s distance, represents the central force, and the earth’s path, the accelerating orbital force. These forces are in the following proportion (See Par. 41.) central force: orbital force: 91430000:8976.11926.

If the orbital force be called unity or 1, then 8976.11926:91430000::1:10185.91635788.

Orbital force = 1 .

Central force 10185.91635788.

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Therefore the earth is accelerated in its orbit, by the aberrating force of gravity, equal to the 1 / 10185.91635788 th part of its central or gravitating force towards the sun.

All the other planets have a similar accelerating orbital force.

46. – To Find the Intensity of the Earth’s Gravity towards the Sun.

The earth’s mean radius is equal to 3955.94943182 miles. Let the earth’s force of gravity towards her center, at the distance of her mean radius, and when freed from the counteracting effect of the centrifugal force of rotation, be equal to unity or 1; let the sun’s mean distance from the earth, (91430000 miles) be expressed in terms of the earth’s mean radius, being equal to

91430000 / 3955.94943182 = 23364.80826992676;

let the mass of the sun be expressed in terms of the earth’s mass, being equal to 314760.n Then, by Newton’s law of universal gravitation, we shall have

sun’s mass / (distance)^{2} = 314760 / (23364.80826992676)^{2}

= G’ = .000576574051819522,

which is equal to the intensity of the earth’s gravitation towards the sun, compared with the intensity of gravity towards the earth’s center, at the distance of her mean radius.

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47. – To Find the Intensity of the Earth’s Accelerating Orbital Force, in Terms of the Intensity of the Earth’s Gravity.

The intensity of the earth’s central force towards the sun, is determined in terms of terrestrial gravity, in the last paragraph. Hence, (Par. 45,) we have

central force : orbital force : : 10185.91635788 : 1; therefore

10185.91635788 : 1 : : 000576574051819522

: .000000056605025170217 = G”.

which is the intensity of the earth’s accelerating orbital force, in terms of the intensity of gravity towards the earth’s center, at the distance of her mean radius.

48. – To Find the Intensity of the Earth’s Accelerating Orbital Force, Expressed in Pounds Weight.

A cubic inch of distilled water weighs 252.458 grains avoirdupois; and 7000 grains make 1 pound; hence, 1 cubic mile of distilled water will weigh 9183125415170 lbs and 4432 grs. The density of the earth is 5.6604 times heavier than water; hence, 1 cubic mile of the earth must be equal to (9183125415170 lbs. 4432 grs.) x 5.6604, which is equal to 51980163100031.8518 lbs. The volume of the earth is 259756014917 cubic miles. These last two numbers, multiplied together, give a product of 13,502,160,021,599,966,659,335,933 lbs, as the weight of the whole earth. This multiplied by the intensity of the earth’s accelerating orbital force, as determined in paragraph 47, will give a product equal to

764,290,107,874,962,825 lbs = p.

This is the pressure, in pounds weight, in the direction of the tangent of the earth’s orbit, when considered circular. This is an accelerating force, constantly acting to increase the velocity of the orbital motion.

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49. – To Find the Earth’s Fall towards the Sun in one Second.

Bodies at the surface of the earth, at the distance of her mean radius from the center, in latitude 45º, fall, in 1 second 16.08538 feet; and were it not for the centrifugal force of rotation, they would fall 16.1131467 feet in the first second.

Let G = the intensity of gravity at the earth’s surface;

G’ = the intensity of gravity of the earth towards the Sun;

f = fall in one second towards the earth’s center;

f’ = fall of the earth in one second towards the Sun.

then we shall have

G : G’ :: f : f’ ;

1 : G’ :: 16.1131467 : f’ ;

substitute the value of G’ (Par. 46.), and we shall find

f’ = .009290422280381360 feet.

The velocity gained at the end of one second is = 2f’

v = 2f’ = .018580844560762720 feet per second.

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50. – To Find the Increased Orbital Space over which the Earth Must Move in one Second by the Action of the Accelerating Aberrating Force of Gravity.

Let f” = the increased space required;

Then

central force : orbital force :: f’ : f”;

or (Par. 47.)

10185.91635688 : 1 :: f’ : f”;

therefore, as f’ is known (Par. 49) we have

f” = .000000912085074525 feet.

This is the excess of orbital space moved over in one second, at the end of which, the velocity gained will be 2 f”;

velocity = 2 f” = .000001824170149050 feet per second.

51. – To Find the Increased Orbital Space over which the Earth must Move in one Sidereal Year, ( = 31558149.6 Seconds;) by the Action of the Accelerating Aberrating Force of Gravity.

Let s = the required increased space, then, by the law of acceleration, we have

s = f” x (31558149.6)² ;

therefore, by multiplying, and reducing to miles, we have

s = 172038.040602598266 miles.

Thus it will be perceived, that the aberrating force of gravity will lengthen the earth’s orbital path, over 172000 miles in one sidereal year, during which the earth will continually recede from the sun in a spiral path. If s be added to the circumference of the earth’s orbit, and the same be considered circular, the semi-diameter will be increased from 91430000 miles to 91457380.704562 miles, making a difference of over 27380 miles. And by Kepler’s law, the

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periodic time would be increased from 1 sidereal year to one sidereal year, 3 hours, 56 minutes, 17.2 seconds.

In like manner, if there is no resisting medium, all the planets must be accelerated in their orbits, by the aberrating force of gravity, and recede in spiral paths from their common center into the far distant regions of space.

52. – The Intensities of the Aberrating Force, acting under the same angle, vary directly as the Intensities of Gravity.

Let i, i’, represent the aberrating intensities, corresponding to any change of mass or intensity towards which a body may be gravitating; let g, g’, represent the gravitating intensities.

Because the angle of aberration is supposed to remain the same, the parallelogram, representing these forces, (Par. 45,) will also be constant; and however much the intensity, represented by the diagonal, may change, the same proportional intensity of change must characterize the two simple forces, into which the compound force has been decomposed, and which are represented by two of the sides of the parallelogram; therefore

i : i’ :: g “ g’ .

53. – The Aberrating Tendencies vary directly as the Intensity of the Force, multiplied into the Aberrating Velocity.

It is seen by the last paragraph, that the intensity of the aberrating force varies as the intensity of gravity; and it is also evident, that the same intensity produces greater or less results in the exact proportion to the velocity of aberration.

Let a, a’, be the velocities of aberration for any two circular orbits; let i, i’, be the corresponding intensities of the aberrating force; and let F, F’, be the aberrating tendencies, resulting from the joint action of the intensities and velocities. For the reasons above mentioned, we shall have

F : F’ :: ia : i’a’ .

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54. – Definition. – Though a, a’, are not properly forces, yet when they, in their combined state with i, i’, represent forces; therefore, i a, i’ a’, or their representatives F, F’, will be called the aberrating forces of gravity.

55. – In the Planetary System, the Orbits being considered circular, the Aberrating Forces of Gravity vary Inversely as the Fifth Powers of the Square Roots of the Distances from the Gravitating Center.

Sect d, d’ represent the respective distances of any two bodies from the sun.

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By Par. 52, we have

i : i’ :: g : g’.

Newton’s law gives

g : g’ :: 1/d^{2} : 1/d’^{2};

hence

i : i’ :: 1/d^{2} : 1/d’^{2}.

By Par. 44

a : a’ : : 1/(d’/2) : 1/(d’^{1/2} );

hence

i a : i¹a¹ : : : ;

but (Par. 53.)

F : F’ :: ia: i’a’;

Therefore

F : F’ : : 1/d^{5/2} : 1/d^{5/2} .

56. – In the Secondary System, the Satellites are governed by the same Laws in relation to their Primaries; as the Planets are in relation to the Sun.

For first. – The circular orbital velocities of the satellites of any one system around their common center of gravity are as the inverse square roots of their respective distances from such center. (Par, 24.)

Second. – The aberrating velocities, being as the orbital velocities, are also as the inverse square roots of the same distances. (Par, 44.)

Third. – Newton’s law of gravity is the same for the secondary systems as for the planetary, and consequently the intensities of the orbital accelerating forces of the satellites must vary as the intensities of gravity; (Par. 52;) and therefore, the aberrating forces of gravity represented by F, F’, (Pars. 54, 55,) must vary as the fifth powers of the square roots of the distances of the satellites, from their common center.

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Chapter IV.

Compound Orbit of the Sun.

57. – Solar Orbit. 58. – Aberrating Intensities vary as the Masses. 59. – Apparent Places of Planets as seen from the Sun. 60. – Law of Aberrating Forces as exerted by Planets on the Sun. 61. – How to find the resultant action of all the Planetary Aberrating Forces. 62. – Acceleration of the Sun in his Orbit, arising from the Aberrating Force of the Ea<r>th. 63. – Pounds Pressure, exerted by the Earth, in Propelling the Sun in his Orbit. 64. – Excess of Orbital Space, gained by the Sun in one year – Enlargement of his orbit, and the Increase of his Periodic Time. 65. – Algebraical expression of the Law, given in Par. 60. 66. – Accelerations of Rotation. 67, 68. – Law of the Aberrating Force of Rotation. 69. – Law, relating to the Rotating Particles of the Envelope.

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57. – Compound Orbit of the Sun.

It is known that the sun revolves from west to east, around the common center of gravity of the system, in a very irregular orbit. If the system consisted of the sun and only one planet, each would revolve around their common center of gravity, in the same plane, and in balancing orbits precisely similar, their motions being parallel, but in opposite directions. The dimensions of these two balancing elliptic orbits would be inversely as the masses, describing them, while the periodic times would be equal, and therefore, their mean velocities would be inversely as the masses. If a second planet be introduced into the system, there will be three orbits, – two planetary and one solar; but the solar will be compounded of its two simple balancing orbits. In like manner, if there be n planets, there will be n + 1 orbits, the solar orbit being compounded of n simple orbits. The sun will describe this irregular compound balancing orbit, in a period equal to that of the most distant planet of the system.

58. – The Aberrating Intensities, mutually existing, between the Earth and Sun, are directly as their Masses.

It has been proved, (Par. 52,) that the aberrating intensities vary as the force of gravity; but gravity, when the distance is the same, varies directly as the mass; therefore the aberrating intensities of the two bodies must vary as their masses.

The same is true in regard to the mutual aberrating intensities, existing between the sun and any other planet.

59. – Planetary bodies, as seen from the Sun, will not appear in their true position, but in the position that they were in at the instant the light left them. During the interval in which light performs its journey from each planet to the sun, each will describe an arc equal to its angle of aberration.

The same is also true in regard to the angles of the aberration of forces as well as light, as the former are being transmitted.

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from the respective planets to the sun.

60. – The Aberrating Force, exerted by any Planet to Accelerate the Sun in his Little Balancing Orbit, is directly as the Mass of the Planet, and inversely as the Fifth Power of the Square Root of its Distance.

This general proposition is demonstrated by a similar process of reasoning, to that in paragraphs 55, 58 and 59.

61. – To find the resultant intensity and direction of all the combined aberrating forces of the planets, exerted upon the sun in any given moment, it is necessary to introduce into the problem, the relative positions of the planets and sun in regard to the common center of gravity of the system; and from these data, connected with the law, expressed in the preceding paragraph, the problem, though tedious, can be solved.

62. – To Find the Intensity of the Earth’s Action in Accelerating the Orbital Motion of the Sun, in terms of the Earth’s Gravity.

Let G” = the aberrating intensity of the solar action in accelerating the orbital motion of the earth; let I = the aberrating intensity of the earth’s action in accelerating the orbital motion of the sun. Then (Par 58) we shall have

sun’s mass: earth’s mass:: G” : I ;

or

314760: 1 :: G” : I ;

As G” is known, Par. 47, ) we have

I = .00000000000017983551.

This is the force, exerted by the mean aberrating gravity of the earth, in accelerating the sun in his balancing orbit, exerted <expressed> in terms of the earth’s gravity, at the distance of her mean radius, the earths gravity being called unity or 1.

63. – To Find the number of Pounds pressure, exerted by the Aberrating Force of the Earth’s Gravity, in Accelerating

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the Sun in his Balancing Orbit.

Let p’ = the required number of pounds.

By Par. 58, we have

sun’s mass: earth’s mass : : p : p’;

or

314760 : 1 : : p : p’;

As p is known, (Par 48,) we have

p ‘ = 2,428,167,835,414 lbs.

64. To Find the Increased Space, over which the Sun must move in his Balancing Orbit, in one Sidereal Year, by the Action of the Accelerating Aberrating Force of the Earth’s Gravity.

Let S’ = the increased space required.

By Par. 58, we have

sun’s mass : earth’s mass : : S : S’;

As S is known, (Par. 51,) we have

S’ = . 54656894333 mile.

Thus, it will be perceived, that by the action of the earth’s aberrating force, the solar balancing orbit will, in one sidereal year, be increased over one-half a mile, which would increase his periodic time in his increased balancing orbit, from one sidereal year to 1^{y }3^{h} 56^{m} 17.2 ^{S}. (See Par. 51.)

It will also be perceived, that the sun’s balancing orbit with that of the earth’s, will no longer be an ellipse but a spiral of the same form as the larger spiral orbit of the earth, each receding from the common center of gravity between the two bodies. Such must necessarily be the phenomena, under the mutual action of aberrating forces, unless counteracted by a resisting medium whose opposing forces shall exactly balance the aberrating forces.

65. – To find, in terms of the earth’s gravity, the action of the aberrating force of any other planet, in accelerating the sun in the balancing orbit, existing between him and such plan-

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el, the law, expressed in Par. 60, must be used.

To express this law algebraically,

Let m = earth’s mass;

m’ = planet’s mass, expressed in terms of the earth’s mass;

d = the earth’s distance from the sun;

d’ = the planet’s distance from the sun;

F” = the aberrating force, exerted by the earth on the sun;

F”‘ = the required aberrating force, exerted by the planet on the sun;

then

m/d^{5/2} : m’/d’^{5/2} : : F” : F”‘

F”‘ = (d/d’)^{5/2 .} m’F”/m .

If m and d be each made equal to unity or 1, then the formula becomes

F'” = m’/d’^{5/2} . F”.

66. – Acceleration of Rotation. In a rotating world of a globular form, every particle without the axis will be accelerated in the direction of the tangent to its circular path. These accelerations are caused by the aberrating forces of gravity. The line of motion of every particle in a rotating body makes an angle with the central lines of traveling force; and the velocity of force must necessarily have a certain ratio to the velocity of each particle, giving use to the phenomena of force aberration.

67. – When the density, throughout a rotating globe, is the same, we have the following: –

Law. – The Aberrating Force of Gravity, upon any Particle, situated either on the Surface or in the Interior, will be jointly as its Central Distance multiplied into its Axial Distance.

For in such a globe, the intensity of the force of gravity of any particle towards the center, is directly as the distance, and its

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rotating velocity is as its axial distance: but the aberrating force (Par. 5 4,) of gravity upon any particle as its gravitating intensity, multiplied into its rotating velocity.

Let i, i’, represent the gravitating intensities of any two particles towards the center; d, d’, their respective central distances; v, v’, their rotating velocities: D, D’, their axial distances; A, A’, their aberrating forces.

Then

i : i’ :: d : d’;

and

v : v’ :: D : D’;

hence

iv : i’v’ :: dD : d’D’;

but

A : A’ :: iv : i’v’;

therefore

A : A’ :: dD : d’D’;

or

A’ = d’D’/dD . A .

68. – It should be remembered, that the expressions iv, i’v’, in the last paragraph, are not true representations of the aberrating forces, only when the momentary spaces, described with the velocities v, v’, are at right angles to the lines of gravitating force, or those radiating from the center of gravity, which is always the case in rotating bodies.

69. – The mass of the ethereal envelope, enclosing a rotating globe, is extremely small, compared with the interior mass of gross matter; therefore, the aberrating or accelerating force of rotation, exerted by such interior mass, upon any ethereal particle of the envelope, will be directly as its axial distance, and inversely as the square of its distance from the center.

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Chapter V.

Aberrations in Elliptic Orbits.

70. – Elliptic Orbits. 71. – Law of Velocity in an Elliptic Orbit. 72. – Law of Angular Velocity. 73, 74. – Aberrating Velocity – its Law in Elliptic Orbits 75. – Law of Aberrating Forces in Elliptic Orbits. 76. – Angular and Aberrating Velocity at the mean Distance. 77. – A particle at rest must describe an Orbit around a body in Motion. 78. – How to estimate the joint Aberrating Forces of two bodies in Motion. 79. – When the Aberrating Forces of two Bodies become neutralized or zero. 80. – General Theorem.

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70. – Elliptic Orbits. Thus for our investigations have been confined to circular motions. But the planets and the most of comets, revolve in elliptic orbits, in each of which, the angle of aberration, the velocity of the body, and the intensity of the focal force, are constantly changing. At the extremity of the minor-axis, as the body approaches the perihelion, the angle, between the radius vector and the line of motion, is acute, being then at its minimum value: from this point the angle increases, becoming a right angle at the perihelion, and still opening out into an obtuse angle, it attains its maximum value at the other extremity of the mirror axis: it now begins to decrease, passing through a right angle at the aphelion, still decreasing until reaching again the minimum point.

The velocity of the body, and the intensity of the focal force, increase from the aphelion where they are the least, to the perihelion where they are the greatest, and decrease from the latter to the former.

71. – The velocity of a planetary body, moving in an elliptic orbit, around a focal force, is expressed by the following,

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Law. — If a body move in an elliptic orbit, under the influence of a focal force, varying inversely as the squares of its distances from the focus, the squares of its velocities in any two points of its orbit, will vary directly as its distances from the upper focus, and inversely as its distances from the lower focus.

Let d, d’ be any two distances of the body from the lower focus; let k, k’, the distances from the upper focus: h, h’, the distances of the apsidul points from the lower focus; p,p’, the perpendiculars let fall from the lower focus on the two tangents, drawn from the points in the orbit at the distances d and d’.

From the First Math. Tract of Dr. Matthew Stewart, Prop. 21, Cor., we have

p² : h x h’ : : d : k ;

hence

p² = h x h’ x d/k ;

also

p’ : h x h’ : : d’ k’ ;

hence

p’² = h x h’ x d’/k’ ;

therefore

p² : p’² : h x h’ x d/k : h x h’ x d’/k’ : d/k : d’/k’ ;

hence

1/p^{2} : 1/p’^{2} : : k/d : k’/d’ ;

But (see James Adams Ellipse, Centrip. Forces, Cor. 1, Prop. 1.)

v² : v’² : : 1/p^{2} : 1/p’^{2} : : k/d : k’/d’

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72. – In an Elliptic Orbit, the Angular Velocity of a body around the focus varies inversely as the Square of its distance from the focus. (See Dr. Stewart’s First Math, Tract, Prop. VI.)

73. – The real velocity in an Elipse may be resolved into two velocities; one in the direction of the radius vector; the other in a direction perpendicular to the radius vector: the former has no aberrating effect; the latter gives rise to the aberrating velocity of gravity.

74. – In an Elliptic Orbit, the aberrating velocity, when considered independently of the changing intensity of gravity, varies inversely as the distance of the moving body from the focal force.

Let d and d’ be any two points in an ellipse; let V and V’ be the angular velocities of a body at those two points; and let a and a’ be the aberrating velocities, or those parts of the velocities which are at right angles to the radius vector.

The actual velocities at right angles to the radius vector, must be as the angular velocities multiplied into the respective distances : hence

a : a’ :: Vd : V’d’ ;

but (72)

V : V’ : : 1/d^{2} : 1/d’^{2} ;

hence

Vd : V’d’ :: 1/d : 1/d’ ;

Therefore

a : a’ :: 1/d : 1/d’ .

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43

75. – The Aberrating Forces in any two points in an Elliptic Orbit, vary inversely as the cubes of their distances from the focal force.

Since we have (74)

a : a’ : : 1/d : 1/d’ ;

also (53 and 54)

i : i¹ : : 1/d’ : 1/d’^{2} ;

hence

ia : I’a’ : : 1/d^{3} : 1/d’^{3} ;

but

F : F’ : : ia : I’a’

therefore

F : F’ : : 1/d^{3} : 1/d’^{3} .

Cor. If the mean distance in an ellipse be represented by d, and the intensity of gravity at such distance be represented by i, and if both d and i be each taken as unity or 1, then F’ = a/d’^{3} .

For

a : I’a’ : : 1 : 1/d’^{3} ;

therefore

i’a’ = F’ = a/d’^{3} .

76. – To Find the Angular Velocity in an Ellipse, at the mean distance from the focal force.

Let d be the mean distance; let c represent the semi-conjugate axis; let v be the angular velocity in a circle at the mean distance; and V, the required angular velocity in the ellipse.

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44.

The velocity in an ellipse at the mean distance from the focus, is the same as the velocity in a circle at the same distance. Also the mean distance is to the conjugate semi-axis, as the angular velocity in the circle to the angular velocity in the ellipse. (See Centripital Forces, by James Adams, Prop. 8, Cor. 2.)

Hence

d : c : : v : : V ;

Therefore

Therefore

V = cv/d .

Cor. 1. – The Angular Velocity, at the mean distance in an Elliptic Orbit, is equal to the Aberrating velocity

For the momentary arc described, at right angles to the radius vector, is the same as the momentary angular arc :

V – cv/d = a.

Cor. 2. – When c = d, the ellipse becomes a circle; and we have

V = v = a.

When c =0, V and a each equal zero, and the ellipse is resolved into a straight line.

Cor. 3. – If v and d be each assumed as unity or 1, we have

V = c = a .

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45.

77. – A body, moving at an angle with a line, connecting it with a particle at rest, exerts upon the latter an aberrating force, causing it to revolve in a spiral curve around the moving body. This spiral orbit will continually increase in magnitude, but decrease in its eccentricity.

For the appraarent place of the moving body, as viewed from the particle at rest, will always be behind its true place. The amount of displacement the first moment will be equal to the aberrating velocity of the body, during the interval in which its force is transmitted to the particle. Both the apparent and true places continue moving. The particle at rest will, for the first moment, commence falling, not towards the real place of the body but towards its apparent place. The next moment, the apparent place is changed, but the course of the falling particle is not changed in an equal degree; its centrifugal force, gained the first moment, causes it to fall towards a point, behind the apparent place. The third moment, the apparent position of the moving body is still further changed, and the velocity and centrifugal force of the particle are much greater, and the tangential line makes a still greater angle with the line to the apparent center. This angle, centrifugal force, and velocity continue, from, moment to moment, to increase, urging the particle still more and more away from the apparent central line. And when the particle has fallen to a point whose distance from the apparent center is equal to nearly one-half its original distance, it will have acquired a velocity sufficiently great to maintain it in a circular, or any elliptic orbit at such mean distance. Let us now suppose the aberrating force to cease, and the moving body to become stationary, it is evident that the particle would necessarily revolve in an elliptic orbit, whose mean distance and eccentricity would depend upon the mass and angular velocity of the moving body, and the original distance of the particle when at

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46.

rest.

78. – In estimating the aberrating force of two bodies in motion, if their aberrating velocities are opposite their sum must be taken: but if they are in the same direction, or nearly so, their difference will be the amount. In the falling of the particle which we have been considering, the sum of the relative aberrating velocities must be used so long as the aberrating angle increases: but as the particle recedes from the perihelion, the angle which its path makes with the radius vector continues increasing, but the aberrating angle does not increase in the same proportion, because both motions are now inclined in the same direction; and when their relative velocities are equal, the aberrating angle ceases: but as the particle now has the swiftest velocity, and its direction being momentarily changed, its aberration will soon gain the ascendancy, and recede towards the aphelion, under the retarding influence of gravity. But this retardation is not so great as it would be if gravity alone acted. The angle of aberration begins to increase, diminishing more and more the retardation, until the particle arrives at its aphelion. In consequence of the retardation being lessened, the particle will reach its aphelion at a greater distance than it originally had when at rest. Thus the major axis is lengthened and the dimentions of the spiral curve increased. At the aphelion the path makes an angle with the radius vector of 90º.

The next revolution, under the influence of still greater aberrating angles, the eccentricity will be diminished, and the orbit enlarged: this will continue each succeeding revolution, until the spiral curve becomes nearly or quite circular, unless the process is sooner arrested by a resisting medium.

79. – When two bodies gravitate directly towards or recede directly from each other, they exert no aberrating forces; also when they are moving with equal velocities, in the same di-

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section, parallel to each other, or when their relative aberrating velocities in, or nearly in, the same direction, are equal, the aberrating forces are neutralized.

80. From the foregoing considerations, and from the numerous mechanical laws known to exist, we are warranted in adding the following

General Theorem.

Every Particle of Matter in the Universe transmits its Force to every other particle with the Velocity of Light, by Virtue of which every Moving Particle exerts an Aberrating Force upon every other particle, Directly as its Mass, multiplied into its Velocity of Aberration, and Inversely as the square of its Distance from each.

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48.

Chapter VI

Resisting Medium.

81. Ethereal Resistance, unlike that of Gross Matter 82. Transfused Resistance. 83. Mass Resistance. 84. Velocity Resistance of one particle. 85. Resistance of a given number of particles. 86. In a Medium of Uniform Density, Resistance is as the Square of the Velocity. 87. Density Resistance. 88. General Law of Resistance. 89. Law of Density of the Ethereal Medium, in terms of the Sun’s Distance. 90. Law of Resistance, expressed in terms of the Sun’s Distance. 91. Resistances of the planets vary as their Orbital Accelerations —Under what condition, the two Antagonistic Forces Balance each other.

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81. – In Chapter I, we have dwelt upon the existence and some of the properties of the Ethereal Medium: but in the present Chapter, we propose to investigate more fully the single property of its resistance to moving planetary matter. All gaseous substances of a gross nature offer a resistance to the passage of all other gross substances through them. These resistances are not altogether proportional to the quantity of matter resisted, but depend also upon the form and magnitude of the surfaces passing through the gas. This is principally owing to the impervious nature of these substances: the gas cannot penetrate freely the surfaces and interior of bodies, and therefore, the resistance is almost wholly confined to the surface. If the transfusion were perfect, the moving body would have no resistance, whatever might be its form or magnitude. But perfect transfusion of one substance through another, without resistance, is not known in nature.

82. – Transfused Resistance. Some of the waves of light are resisted and reflected; some are transmitted through transparent substances, with a slight degree of resistance in velocity; others are absorbed and destroyed. The waves of heat are not resisted as much as those of light, but they slowly penetrate bodies where light cannot follow. The electric and magnetic waves are more generally diffused, traveling by the aid of good conductors with great velocities. It may be said that such bodies offer but slight resistance to their transfusion. The ether, itself, which exhibits all these varieties of waves, is no doubt more transfusible than any of its waves or tremulous agitations. It exists in all space; it is transfused through all worlds; it enters largely into the composition of all substance. Its resistance is a transfused resistance, not effected in the least by the form or magnitude of surfaces of gross matter, but only by the quantity of matter which moving bodies contain.

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50.

83. – When the Velocities of Bodies, moving in Ether of the same density, are the same, the Resistance is as the Quantity of Matter in the Bodies Resisted.

As ethereal resistance does not depend upon form nor surfaces, (Par. 82,) but upon masses, it is quite evident, that each atom of every substance will be equally resisted; and hence, the whole resistance, offered to a body, will be the sum of the resistances of its several particles; and consequently, the resistance, offered to different bodies, will be as the number of atoms which each contains; and therefore, as their respective quantities of matter.

84. – The Resistance, offered by an Ethereal Particle is as the Velocity with which it is struck.

If the velocity be doubled or threbled, the resistance will be doubled or threbled. If the ethereal particle be struck with n times the velocity, its resistance will be n times greater.

When the Velocity of contact is equal, the Resistance will be as the Number of Ethereal Particles struck.

It is evident that if two particles of ether be struck by a moving body, the resistance will be doubled; if n particles be impinged upon, the resistance will be n times increased.

86. – The Resistance of an Ethereal Medium of Uniform Density is as the Square of the Velocity of a body moving therein.

If a body move with n times the velocity, it will go, in the same time, n times as far, and will meet with n times as many ethereal particles, and will impinge upon each particle with n times greater velocity; and therefore, the resistance will be equal to n times the particles multiplied into n times the velocity, which is equal to either the square of the number of particles, or the square of the velocity.

87. – When the Velocity of a body is the same, the Resistance of the Resistance of the Ethereal Medium is as its Density.

If the ethereal medium contain twice, or thrice, or n

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51.

times the number of particles in a given volume, its density will be twice or thrice, or n times greater. And as the resistance varies as the number of particles, (Par. 85,) it, therefore, must vary as the density of the medium.

88. – The Resistance of the Ethereal Medium is as its Density, multiplied into the Square of the Velocity of a body moving therein.

It was proved (Par. 87,) that with equal velocity, the resistance varies as the density. It was also proved, (Par. 86,) that the resistance varies as the square of the velocity; therefore, the resistance of the ethereal medium varies directly and jointly as its density, multiplied into the square of the velocity of a body moving therein.

89. – The Density of the Ethereal Medium varies Inversely as the Cube of the Square Root of the Distance from the Sun.

The law of density must be such, that the resistance to planetary orbital circular motion will be exactly equal to, and balance the planetary orbital circular acceleration, arising from the aberrating force of gravity. In the next two paragraphs, it will be proved, that the law of density, just expressed, does fulfil these requirements, and maintain the stability of the system, so far as circular orbits are concerned.

90. – In an Ethereal Medium whose Density varies Inversely as the Cube of the Square Roots of the respective Distances of the Planets from the Sun, the Resistances will vary Inversely as the Fifth Powers of the Square Roots of such Distances.

Let d, d’, be the distances of two planets from the Sun;

v, v’, their orbital velocities;

D, D’, the densities of the medium at the respective distances,

r, r’, the resistances at the respective distances.

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52.

Then (Par. 89,) we have

D : D’ : : 1/d^{3/2} : 1/d’^{3/2} ; …………………….…………..(1)

and (Par. 88) r : r’ : : DV^{2 : }D’V’2; . . . . . . . . . . . . (2)

combining (1) and (2) r : r’ : : V^{2}/d^{3/2} : V’^{2}/d’^{3/2} ; ………..….(3)

By Par. 24 V² : V’² : : 1/d : 1/d^{3} ; ……………………(4)

combining (3) and (4)

r : r’ : : 1/d^{5/2} : 1/d’^{5/2} .

This law of resistance is similar to the law of acceleration, imparted by the aberrating force of gravitation. (See Par. 55.)

91. – The Resistance of the Ethereal Medium to the Orbital Motion of the Planets, considered as circular, varies as their Orbital Accelerations, under the influence of the Aberrating Force of Gravity.

For (Par. 55,) we have

F : F¹ : : 1/d^{5/2} : 1/d’^{5/2} ;

and (Par. 90.)

r : r’ : : 1/d^{5/2} : 1/d’^{5/2} ;

therefore

F : F’ : : r : r’ .

Cor. – If F = r, then F’ = r’; therefore, under these last conditions, the two forces will exactly balance each other, and the circular orbits and periodic times will remain unchanged. Any other law of density of the ethereal medium, than that expressed in Par. 89, would destroy the stability of the system.

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53.

CHAPTER VII.

Resistances in Elliptic Orbits.

92. General Law. 93. To find the Point in an Ellipse where the two Antagonistic Forces Balance – Balancing Point when the Ellipse becomes a Circle, or a Straight Line – Discussion of the Formula for Ellipses of any Eccentricity – Elliptic Perturbations – Minute Excess of Resistance – Decrease of Eccentricity and of the Periodic Times – Instability of hyperbolic and Parabolic Orbits. Example 1. Balancing Point in the Earth’s Orbit – Example 2. Difference of the two Forces at the Perihelion – Example 3. Difference at Aphelion – Examples 4 and 5. Difference at two intermediate points – Example 6. Balancing Point in the Orbit of Venus. 94. – How to find the Radius Vector for any given time – Tables I. and II. – The Sum of the two Forces very nearly Balanced in the whole of the Earth’s Orbit. 95. – Excess of Resistance in one year, expressed in terms of the Earth’s Gravity. 96. – Retardation of the Earth in one Second. 97. –Retardation in one year. 98. – Retardation in One thousand years – Diminution of the Mean Distance and Periodic Time. 99. The calculations in Table II. only Approximative.

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92. – If a body move in an elliptic orbit, under the influence of a focal force, varying inversely as the squares of its distances from the focus, and if it be resisted by the ethereal medium, whose density varies inversely as the cube of the square root of the distances from the lower focus, the resistance will vary directly as the distance to the upper focus, and inversely as the fifth power of the square root of the distance of the lower focus.

Let d, d’, be the distances of the body at any two points of its orbit, from the lower focus; let k, k’, be the corresponding distances to the upper focus; let v, v’, be the corresponding velocities; D, D’, the densities of the resisting medium; R, R’, the corresponding resistances.

Then, by (71,) we have

v² : v ‘ ² : : k/d : k’/d’ ;

we also have (by hypothesis)

D : D¹ : : 1/d^{3/2} : 1/d’^{3/2} ;

hence

V² D : V’ ² D’ : : k/d^{5/2} : k’/d’^{5/2} ;

but (88)

R : R’ : : V² D : V’ ² D’ ;

therefore

R : R’ : : k/d^{5/2} : k’/d’^{5/2} ;

93. – To find the distance from the lower focus, in any given ellipse, where the resisting and aberrating forces balance each other, the density of the resisting medium being the same as in Par. 92, and d, i, k, and R, being, at the mean distance each equal to unity.

Let the angular velocity in a circle at the mean distance = 1; let the angular velocity in the ellipse of the mean distance = a. Then we shall have (See James Adams Ellipse, Centrip.

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55.

forces, Prop. 8, Cor. 2.)

d : conjugate semi-axis : : Ang. vel.in a cir. : a;

or

1 : conju. Semi-axis : : 1 : a;

Therefore

Conjugate Semi-axis = a.

At the mean distance, the angular velocity in the ellipse is equal to the aberrating velocity. (Par. 76, Cor. 1.) Therefore, by Par. 75, Cor. we have

a : a’i’ : : 1 : 1/d^{3} ;

hence

a’i’ = a/d’^{3}

By par. 92, we have

1 : K’ : : 1 : : k’/d’^{5/2} ;

hence

K’ = k’/d’^{5/2} .

Because (by hypothesis) the resisting and aberrating forces are required to balance each other, we have

a/d’^{3} = k’/d’^{5/2} . In this equation, let d’ = x,

Then k’ = 2 – x = the distance to the upper focus; and we shall have

2 – x = a . x^{5/2}/ x’^{5/2}= a/√x

squaring both sides, and reducing, we have

x³ – 4x² + 4x – a² = 0 ………………… (1)

In this equation, when a² = 1, one of the value of x will equal 1, the ellipse resolving itself into a circle. When a² = 0, one value of x = o; the other two roots will each = 2; hence, the lower focus will be on the ellipse at the lower axis, and the upper focus will be on the ellipse of the upper axis; hence, the ellipse will be resolved into a straight line whose length will equal 2; that is, twice the mean distance. As a² cannot be greater than 1, nor less than nothing, it follows, that one of the values of x, expressing the required distance of the point where the two forces balance, can never be less than 1, nor greater than 2. Therefore, by finding the required root from the equation (1), we shall have the distance of the point,

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where the resisting and aberrating forces balance each other.

Cor. 1. – The less the eccentricity of an ellipse, the nearer the point of the balancing forces approaches the mean distance. And when becoming a circle, the balancing forces remain equal throughout the entire orbit.

Cor. 2. – When a² = 0, 2 – x = k’ = 0

hence, k’ = a’i’ = o; but as i’ is greater than nothing, therefore, a’ = 0 ; therefore, the resisting and aberrating forces each equal nothing.

Scholium. By solving numerically equation (1), for all possible values of a², between its limits of one and nothing, it will be perceived that two of the values of x never touch the elliptic curve, except when the ellipse becomes a straight line; one value being at some point within the curve, the other at some point without the curve; the former always being less than the perihelion distance, and the latter always greater than 2, or the aphelion distance: while the third value of x will always be between the limits of 1 and 2, and will be the true distance sought for, where the resisting and aberrating forces balance. It is evident that there will be a point on each side of the major axis, at equal distances from the lower focus, where the two forces will balance. Between these two points, by the way of the perihelion, the sum of the resistances exceed the sum of the aberrating forces: but between these two points, by the way of the aphelion, the sum of the aberrating forces exceed the sum of the resistances. When the resistances preponderate, there will result a slight perturbation from the elliptic curve in an inward elliptic spiral: on the other hand, when the aberrating forces preponderate, the result will be a minute perturbation in an outward elliptic spiral. These two forces, when their action in the whole orbit is considered, become partially restorative to each other, leaving a minute balance in favor of resistance.

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57.

This residual resistance will be greater, where the eccentricity is greater: but its tendency is to decrease the eccentricity, and reduce elliptic to circular orbits. When an elliptic orbit, under the influence of the residual force, has closely approximated the circular form, as is the case with all our planetary orbits, the excess of resistance becomes inappreciably small, requiring immense periods to attain to a perfect equilibrium in a circular orbit, where the resistance and aberrating forces, will, throughout the whole orbit, be exactly equal.

As another consequence, attending the propelling and resisting forces, all cometary bodies which may assume, for a moment, parabolic or hyperbolic orbits, cannot maintain themselves in such orbits; for the residual resisting force will compel them into orbits of the elliptic form, and afterwards still further reduce them to the planetary form of small eccentricity, from which, after the lapse of ages, they will reach their final destiny in circular orbits of different dimensions.

Another tendency of these two antagonistic forces is to continually correct, in a measure, any derangements which may happen in a system of bodies. If from some extraneous cause, the eccentricity of a cometary orbit should suddenly be increased, as has been the case in some rare instances, the residual force begins, slowly but surely, to work a restoration, so far as the gradual diminution of the eccentricity is concerned.

When the changes upon the whole orbit are taken into the calculation, the residual force, except in circular orbits, is always in favor of resistence; therefore the result will always be an inward elliptic spiral, which will, not only diminish the eccentricity, but shorten the transverse axis, and diminish [page break]

58.

the mean distance, and consequently lessen the periodic time. These minute perturbations of cometary orbits may, by close observation, be detected, in cases, where the eccentricity is very great. It is in the diminution of the periodic time, that the phenomena, alluded to, will more readily and satisfactorily develop themselves.

Example 1. – Let the mean distance of the earth from the sun = 1; let its semi-minor axis = .99985578 = a; let the density of the resisting medium be the same as in Par. 92; let i, k, and K, be, at the mean distance, each = 1; let a’, i’, k’, d’, K’, represent the same quantities, as in the former propositions: at what distance from the sun will the earth, in its elliptic orbit, be equally acted upon by the propelling and resisting forces?

By Par. 93, the angular velocity a = semi-minor-axis; hence formula (1) becomes

x³ – 4x² + 4x – (.99985578)² = 0.

In this equation, find the value of x between 1 and 2 which will be the distance required.

x = d’ = 1.00028834.

From this example, it will be seen, that the radius vector, x, is only a small fraction greater than the mean distance, therefore, the two points, where the antagonistic forces balance each other, are situated near the extremities of the minor axis, on the aphelion side of the same.

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59.

Example 2. – Find the values of the resisting and aberrating forces at the perihelion point of the earth’s orbit, the data being the same as in Example 1.

K’ = k’/d’^{5/2} = dis. of upper focus/(dis. of perihelion)^{5/2} = 1.01678880/(.98321120)^{5/2} ;

therefore

K’ = 1.06075164.

a’i’ =a/d’^{3} = .99985578/(.98321120)^{3} = 1.05195439.

K’ – a’i’ = .00879725 = excess of resistance at the perihelion = B.

Example 3. Find the values of the aberrating and resisting forces, at the aphelion point of the earth’s orbit, the data being the same as in Example 1.

a’i’ = a/d’^{3} = .99985578/(1.01678880)^{3} = .95114144.

K’ = k’/d’^{5/2} = dis. of upper focus/ (dis. of Aphelion)^{5/2} = 98.321120/(1.01678880)^{5/2 }=.94312647.

a’i’ – K’ = .00501497 = excess of aberrating force at the Aphelion = A.

Therefore

B – A = .00879725 – .00801497 = .00078228 = residual force of resistance at the perihelion, above the excess of aberrating force at the aphelion.

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60.

Example 4. – What is the difference between the resisting and aberrating forces, where the latus – rectum of the earth’s orbit cuts the ellipse?

1/2 latus-rectum = (Semi-minor axis)^{2}/semi-transverse axis = .99971158.

R’ = k’/d’^{5/2} = 1.00028842/(.99971158)^{5/2} = 1.00101004.

a’i’ = a/d’^{3} = .99985578/(.99971158)^{5/2} = 1.00072141.

_____________________

Required difference = .00028863.

Example 5. – What is the difference between the aberrating and resisting forces, where the ordinate, passing through the upper focus of the earth’s orbit perpendicularly to the transverse axis, cuts the ellipse?

The ordinate in this case is equal to the latus-rectum.

a’i’ = a/d’^{3} = .99985578/(1.00028842)^{3} = .99899114.

R’ = k’/d’^{5/2} = .99971158/(1.00028842)^{5/2} = .99899110.

___________

Required difference = .00000004.

It will be seen by this example, that the two antagonistic forces very nearly balance. The balance would have been complete, if the radius vector d’ had been about 7 miles less. (See example 1.)

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61.

Example 6. – If the elements of the orbit of Venus be represented by the same symbols as those of the Earth, and those at the mean distance of Venus from the sun, be considered as unity, a, representing the angular velocity = the semi-conjugate axis, at what distance from the sun, will the two antagonistic forces balance each other?

Equation (1), Par. 93, becomes

x^{3} – 4x^{2} + 4x – .999952661705 = 0

x= 1.000047336054

2 – a^{2} = 1.000047338295

When the mean distance is 1, a^{2} = half the latus-rectum. The value of 2 – a^{2}, in this instance, does not differ from x to eight places of decimals. The cause of this very small difference is the near approximation of the orbit of Venus to a circle. When a circular orbit is reached, 2 – a^{2} = x= 1; but in all other conditions 2 – a^{2 }> x; that is, 2 – a^{2} is the distance from the lower focus to that point in the ellipse from which <an> ordinate, let fall perpendicularly to the transverse axis, would pass through the upper focus.

94. – The sum of the aberrating forces throughout the whole of an elliptic orbit, deducted from the sum of the resisting forces, will, as stated in the preceding paragraph, give a minute remainder, which alone becomes effective in gradually, (and almost imperceptibly in an orbit of small eccentricity,) changing its elements. To calculate approximately the sum of each of these opposing forces, it is first necessary to determine the length of the radius vector, for equal intervals of time, between the aphelion and perihelion points. The necessary data which enter into this calculation, are derived from Kepler’s law of the equable description of areas in equal times, by which, first, the true anomaly of the sun

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62.

is deduced from the mean anomaly, for any given time from the aphelion; and second, the length of the radius vector is derived from the true anomaly. (See Robison’s Mechanical Philosophy, p. 191; also Fig. 37, Plate 9, p. 236.)

The following tables, I have calculated for every ten days, from the aphelion to the perihelion of the earth’s orbit. The distance from the upper focus of the ellipse is obtained, by merely subtracting the radius vector from 2. In the calculation, Lockyer’s Table of Elements is used.

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63.

Table 1

Days from the aphelion of the Earth’s Orbit. |
Mean Anomaly reckoned from the aphelion. |
Radius Vector = d¹ (Mean distance = 1) |
Distance from Upper Focus = k’ |

0 |
0º00’00”.00 |
1.01678880 |
.98321120 |

10 |
9º51’21”.93 |
1.01654898 |
.9835102 |

20 |
19 42 43.86 |
1.01583592 |
.98416408 |

30 |
29 34 05.79 |
1.01466876 |
.98533124 |

40 |
39 25 27.72 |
1.01307896 |
.98692104 |

50 |
49 16 49.65 |
1.01110975 |
.98889025 |

60 |
59 08 11.58 |
1.00881521 |
.99118479 |

70 |
68 59 33.51 |
1.00626169 |
.99373837 |

80 |
78 50 55.44 |
1.00351380 |
.99648620 |

90 |
88 42 77.37 |
1.00065772 |
.99934228 |

90 |
89 00 57.4 |
1.00028834 |
.99971166 |

100 |
98 33 39.30 |
.99777419 |
1.00222581 |

110 |
108 25 01.23 |
.99494863 |
1.00505137 |

120 |
118 16 23.16 |
.99226620 |
1.00773380 |

130 |
128 07 45.09 |
.98980963 |
1.01019097 |

140 |
137 59 07.02 |
.98765352 |
1.01234648 |

150 |
147 50 28.95 |
.98586572 |
1.01413428 |

160 |
157 41 50.88 |
.98450819 |
1.01549181 |

170 |
167 33 12.81 |
.98361903 |
1.01638097 |

180 |
177 24 34.74 |
.98322894 |
1.01677106 |

182 |
180 00 00.00 |
.98321120 |
1.01678880 |

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64.

Table II

Days from the Aphelion of the Earth’s Orbit. |
Aberrating Force, (that in a circular orbit, at the Earth’s mean distance =1.) F’ = a/d’ |
Resisting Force, (that at the Earth’s mean distance =1.) R’ = k/d’ |

0 |
.95114144 |
.94312647 |

10 |
.95181477 |
.94391300 |

20 |
.95382053 |
.94625589 |

30 |
.95711582 |
.95010484 |

40 |
.96162882 |
.95537565 |

50 |
.96725830 |
.96194967 |

60 |
.97387338 |
.96967360 |

70 |
.98130621 |
.97835098 |

80 |
.98938954 |
.98778610 |

90 |
.99788550 |
.99770095 |

90 |
.99899139 |
.99899139 |

100 |
1.00656208 |
1.00782452 |

110 |
1.01516208 |
1.01785663 |

120 |
1.02341733 |
1.02748462 |

130 |
1.03105807 |
1.03639415 |

140 |
1.03782353 |
1.04428152 |

150 |
1.04347984 |
1.05087508 |

160 |
1.04780232 |
1.05591288 |

170 |
1.05064644 |
1.05922743 |

178 |
.84890924 |
.85600126 |

Sum |
19.78908663 |
19.78908663 |

180 |
1.05189745 |
1.06068529 |

½ Sid. year |
7.05195439 |
1.06075164 |

The following proportions give the additional forces corresponding to 178d 1h 41m 11 sec |
||

10 |
8 |
1.05189745 : .84890924 |

10 |
8 |
1.06068529 : .85600126 |

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65.

The following are the approximate aberrating and resisting forces at the points of the orbit from 178^{d} 1^{h} 41^{m} 11^{s} to one half of a sidereal year, from the aphelion.

F’ = a’I’ = 1.05165604 K’ = 1.06040396

180^{d} 1.05189745 1.06068529

½ Sid.year 1.05195439 1.06075164

3)3.15550788 3)3.18184089

Average 1.05183596 1.06061363 Average

1.05183596

Average Excess of Resistance = .00877767

This average excess of resistance acts for 4^{d} 13^{h} 23^{m} 23.8^{s} before the earth arrives at the perihelion, and also for the same period after leaving the perihelion. Throughout the remainder of the orbit the sum of the aberrating and resisting forces balance each other. Therefore it is only for the short period of 9^{d} 2^{h} 46^{m} 47.6^{s}, when nearest the perihelion, that the effects of the resisting force becomes permanent in decreasing the eccentricity and diminishing the transverse axis, and consequently the periodic time. These effects will become less and less each century, until the orbit becomes circular, when they will entirely disappear.

95. If the Excess of the Resisting Force in the Earth’s Elliptic Orbit, average .00877767, during the 4^{d} 13^{h} 23^{m} 23.8^{s}, both before and after the perihelion passage, (the resistance at the mean distance being =1) what will be the intensity of this Excess of resistance, expressed in terms of the Earth’s Gravity, taken as unity or 1?

Because the resistance at the mean distance in an elliptic orbit is equal to the aberrating force at the same distance in a circular orbit, we therefore have (47).

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66.

1 : .00877767 : : y’’ : Intensity required

= .000000000496860231286 = r,

which is the intensity of the Earth’s surplus Resisting force in the elliptic path, during the time above expressed, in terms of the intensity of gravity towards the Earth’s center, at the distance of its mean radius.

96. – To find how much the Earth is retarded in its Elliptic Orbit, in one second, by the action of the surplus resisting force, averaging .00877767, as expressed in the last example.

Let the required retardation = r’;

we then have (Par. 49.)

Earth’s gravity : r :: 16.1131467 feet : r’;

or

1 : 000000000496860231286

:: 16.1131467 feet : .000000008005981796107 feet = r’.

97. – To find how much the Earth will be retarded in its Elliptic Orbit, in one sidereal year, by the action of the surplus resisting force, during the 4^{d} 13^{h} 23^{m} 23.8^{s}, both before and after the perihelion passage.

It is only for the short period of 9^{d} 2^{h} 46^{m} 47.6^{s} = 787607.6 seconds in a sidereal year, that the retardations are not compensated by the accelerating forces. Let s represent the space diminished from the Earth’s orbit by the retardation. Then, by the laws of retarding forces which are the same as those for accelerating forces, we have

s = r’ x (787607.6 sec.)² = 4966.316515 feet;

Hence the mean distance will be diminished } .14989 miles, or 791 feet.

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67.

By this last example, it will be perceived, that the dimensions of the Earth’s elliptic orbit is, under its present eccentricity, being diminished at a rate less than one mile per year. But the diminution of the perihelion velocity, also must diminish the elliptic eccentricity, and thus the surplus resistance necessarily becomes less each year.

98. – If the surplus resistance were to remain uniform, at each perihelion passage of the Earth, how much would the mean distance and periodic time be diminished in one -thousand years?

Answer. The mean distance would be diminished 149.87 miles, and the periodic time would be diminished about 1^{m}18^{s}; or at the rate of .078 Sec. per year; which is about 1/13^{th} of a sec. per year.

99. – It is generally known from the records of ancient eclipses, that there has been no appreciable diminution of the periodic time for the last two or three thousand years; and therefore that the mean distance has remained nearly constant during that period. These historical facts seem to contradict the foregoing demonstrations, and if they can not be reconciled may prove a serious objection to the theory. But it should be remembered that the antagonistic forces, as inserted in Table II, were calculated on the supposition that the Earth was strictly confined to its elliptic path, having an invariable elliptic velocity. Such, however, is not the case: for nearly one-half the orbit the aberrating forces preponderate, producing perturbations or variations from the elliptic path; this in turn produces a slight change from the elliptic velocity; and this again still further changes, both in direction and intensity of the aberrating and resisting forces. For the remaining portion of the orbit, the resisting forces preponderate, giving rise to somewhat similar, but oposite, perturbations, wherein the directions, intensities, velocities, aberrating and resisting forces are mutually changed, in a slight degree, from those acting upon a body confined to a strictly elliptic path.

By inspecting the table of forces, referred to, it will be seen, that a change in the velocity produces a more rapid change in the

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68.

resistance, than in the aberrating force; hence, when near the perihelion, the retardation of velocity is in a measure compensated by the slower change in the aberrating force. And thus throughout the whole orbit, any perturbing changes are almost entirely if not wholly remedied by the self-adjusting tendencies of the two antagonistic forces. The diminution of nearly one mile in the Earth’s orbit per year, when calculated in reference to a strictly elliptic path, is probably a hundred fold greater than what actually does take place, under the action of the self-adjusting forces, alluded to.

It should also be borne in mind that the forces in Table II, were calculated for intervals of ten days: the approximation to the sum of each force would have been much nearer, if the intervals had been only one day, instead of ten. The immense labor, involved in these calculations, is the author’s apology for not giving more accurate approximations, founded on the daily variations of the radius vector.

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Chapter VIII

Effects of the Aberrating and Resisting Forces upon the Earth’s Diurnal Rotation

100. – Application of Force near the Earth’s Center. 101 – 103. – Rotation retarded by a minute excess of Resistance, compared with Solar Aberration. 104. – Rotation Accelerated by Terrestrial Aberration. 105. – Rotation is also effected by Lunar Aberration. – When the sum of the Solar, Lunar, and Terrestrial Aberrating Forces, is compared with the Resistance, the present Axial Rotation will be found nearly Constant – Minute Perturbations of the Diurnal Period.

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100. – If the earth were a perfect sphere whose radius = 3955.94943182 miles, a particle on the surface at the equator, would have a rotating velocity of .2884724175 of a mile per second; while the progressive orbital motion of the earth’s center, at the mean distance of 91430000 miles from the sun, is 18.203590512 miles per second; at what distance from the earth’s center of gravity must a force be impressed in the equatorial plane and parallel to the tangent of the orbit to produce both the rotator and progressive velocities?

Let R = Earth’s radius;

V = Rotatory velocity per second;

v= Progressive velocity per second;

r = Required distance from the earth’s center.

Then (See J. R. Young’s Dynamics, fr. 204.)

r = 2VR/5v = R/157.75850140 = 25.0759825728 miles.

101. – If there ever no resisting medium, and the mean distance of the center of the earth from the sun be taken as unity or 1, what effect will the solar aberrating force, exerted upon points of the earth at the distances of 1 + r and 1 – r, have, in retarding or accelerating its progressive motion and diurnal rotation?

Let 1 + r = d’’ ; 1 – r = d’

let a = the aberrating velocity of the center of the earth in its elliptic orbit, at its mean distance from the sun. (That in a circular orbit being = 1.)

Let a’’ = the aberrating velocity at the distance of 1 + r;

let a’ = the aberrating velocity at the distance of 1 – r;

let i’’ i’ be the corresponding intensities of the aberrating forces at the distance of 1 + r and 1 – r.

Let F’’, F’ be the corresponding forces for the same distances.

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71.

Then we have

1+r = d’’ = 1.00000027426;

1-r = d’ = .99999972574;

(1+r)^{2 }_{=} d’’^{2 }= 1/00000054852;

(1-r)^{2 }= d’^{2 }= .99999945148;

We also have (76. Cor. 3.)

a= .999855780 = semi-minor axis.

The rotating velocity of a point at the distance r from the center in one second of time = .0018285063 of a mile.

therefore

a’’ = a – Rot. Vel./v = 999956228;

and

a’ = a – Rot. Vel./v = 999755332.

By Par. 53, the aberrating force is equal to the aberrating velocity multiplied into the intensity.

Therefore

F’’ = a’’i’’ = a”/d”^{2} = .999955681;

F = a i = a/d^{2
}F’ a’i’= d’d’^{2
}F’’-F’ = .000199801.

Thus it will be seen that when the earth is at its mean distance in its elliptic orbit, the aberrating force, acting upon the two points at the distances 1+r and 1-r, is greater than at the latter point; and therefore, if there were no resistance, and the aberrating force of the sun alone acted upon the two given points, the rotating and progressive velocities would be accelerated by a force equal to F’’-F’.

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72.

102. – If there were no aberrating force of gravity, and there existed a resisting medium varying in density inversely as the cube of the square root of the distance from the sun, what effect, when the earth is at its mean distance from the sun, will such resistance, exerted upon two points of the earth at the distances of 1 + r and 1 – r, have, in retarding or accelerating its dimnal rotation?

Let u = elliptic velocity of the earth’s center;

u’’ = 1 + Rot. Vel./v , at the distance 1 + r ;

u’ = 1 – Rot. Vel./v , at the distance 1 – r;

u’’ = 1.0001004476

u = 1.

u’ = .9998995524

By Par. 88, resistance varies as the square of the velocity multiplied into the density; therefore

R’’ = u”^{2}/d”^{3/2} = 1.000200494;

R =u^{2}/d^{3/2} = 1. ;

R’ = u’^{2}/d’^{3/2} = .999799526 ;

hence

R’’ – R’ = .000400968 ;

therefore, the dimual rotation will be retarded at the distance 1 + r, with a resisting force equal to R’’ – R’.

From this and the preceding examples we have

(R’’ – R’) – (F’’ – F’) = .000201167.

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73.

103. – What will be the value of (R” – R’) – (F” – F’) = .000201167 in terms of the earth’s gravity?

By Par. 47 we have

1 : .000201167 : : .000000056605025 : .000000000011387.

This last term is equal to the resistance to the diurnal rotation at the distance 1 + r, expressed in a decimal of the earth’s gravity, considered as unity.

104. – What is the aberrating force arising from the earth’s rotation, at the distance r from the center, in the plane of the equator expressed in terms of the earth’s gravity, assumed = 1 at the distance of the mean radius?

We have

vel. of light : Rot. vel. at r : : r : Rot. vel. at r x r/Vel. of light ;

or

185420 : .0018285063 : : 25.0759825728 : .0000002472850.

This last term is equal to the rotation at r, while the force of gravity travels from the center to r; hence, the two last terms of the proportion have the same ratio as the central force at r to the aberrating force at r. The central force at r is equal to the th part of the central force at the surface which is assumed as unity; therefore we have

25.0759825728 : .0000002472850 : : : .000000000062510.

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74.

105. – Thus we see that the solar aberrating force is insufficient to balance the resistance at the two given points, the latter preponderating, producing a retardation of the rotating velocity.

In like manner, if the algebraic sum of the two antagonistic forces, acting upon every particle in the nearest hemisphere of the earth to the sun, be compared with the like sum acting upon the most distant hemisphere, it will be found that the resisting force, at the earth’s mean distance, preponderates. But the aberrating force of the earth’s matter in rotating must be added to that of the solar, before we can exhibit an approximate balance. Also the effects of the lunar aberrations upon the rotatory velocity must not be forgotten. When all these forces are accurately calculated, for all the different points in the earth’s elliptic orbit, as well as for all the points in the moon’s elliptic path, the balance of the two forces will be nearly completed, and the resulting axial rotation will be nearly constant. But a perfect balance cannot be attained, until these two great forces, have greatly reduced the eccentricity of the terrestrial and lunar orbits. But even as it is now, the purturbations of the diurnal period are so minute as to be scarcely detected by the, most refined and careful observations.

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[Transcribed by Nora Fowers, Becca Staker, Doratha Young, Brandan Hull, Portia Williams, Erin McAllister, Heather Hoyt, Shannon Devenport, Verlie Brown, Dick Grigg, Marlene Peine, Pat Reynolds, Kathy Ladle, Vincy Stringham, Mauri Pratt; Feb. 2011]