# Popular Science Monthly/Volume 16/November 1879/Mars and his Moons

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* MARS AND HIS MOONS.
By Professor JOHN LE CONTE,
OF THE UNIVERSITY OF CALIFORNIA.
THERE is no member of the solar system, with the exception of our moon, which can be studied under such favorable circumstances as the planet Mars; for, although Venus, when in inferior conjunction, is nearer to us than Mars in opposition, yet Venus, at this time, turns her darkened hemisphere toward the earth. Moreover, although Mars does not appear so large an object in the telescope as Jupiter, yet he is in reality seen on a much larger scale, not only on account of his much greater proximity to us, but because, being likewise much nearer the sun, his surface is much more brilliantly illuminated, so that a much higher telescopic power can be advantageously ​employed. Accordingly, ever since the invention of the telescope, Mars has been a favorite object of observation. The largest and most powerful instruments have been employed to scrutinize this planet, and the varied physical details of its surface have been most carefully mapped by many astronomers.
When, therefore, it was announced two years ago that the American astronomer, Hall, had discovered two satellites belonging to Mars, we ought not to be surprised at the astonishment with which the news was received by the scientific world. Moreover, there can be no question that for more than two centuries past astronomers have recognized the probability of the existence of satellites to this planet. In fact, analogy would lead us to expect that Mars would be furnished with one or more moons; for, being situated at a greater distance from the sun than the earth, it seems more especially to need such luminaries to cheer its dark nights. Under the influence of these anticipations, the astronomers, who have so carefully studied the physical features of Mars, have doubtless been looking for these satellites. In fact, many of them have contended that the failure to discover them is not by any means a conclusive proof of their non-existence; since, Mars being a very small planet, we might expect his moons to be proportionally small, in which case they might escape detection by the telescope. Thus, for example, the second satellite of Jupiter is only about the forty-second part of the diameter of the planet; and a satellite which would only be the forty-second part of the diameter of Mars would be about one hundred miles in diameter. At the least distance of the earth from Mars a satellite of this dimension would subtend an angle of less than one half of a second; so that, even in the most favorable position of Mars, powerful telescopes might fail to reveal such an object, especially if it do not recede far from the disk of the planet.
Thus, Thomas Dick ("Celestial Scenery," American edition, p. 123, 1838) remarks in relation to this question: "If such a satellite exist, it is highly probable that it will revolve at the nearest possible distance from the planet, in order to afford it the greatest quantity of light; in which case it would never be seen beyond two minutes of a degree from the margin of the planet, and that only in certain favorable positions. If the plane of its orbit lay nearly in a line with our axis of vision, it would frequently be hidden either by the interposition of the body of Mars or by transiting its disk. It is therefore possible, and not at all improbable, that Mars may have a satellite, although it has not yet been discovered. It is no argument for the ​nonexistence of such a body that we have not yet seen it; but it ought to serve as an argument to stimulate us to apply our most powerful instruments to the regions around this planet with more frequency and attention than we have hitherto done, and it is possible that our diligence may be rewarded with the discovery. The long duration of winter in the polar regions of Mars seems to require a moon to cheer them during the long absence of the sun; and, if there be none, the inhabitants of those regions must be in a far more dreary condition than the Laplanders and Greenlanders of our globe."
This state of doubt and uncertainty in relation to the question of the existence of Martial moons afforded legitimate game for the satirical writers of the last century. Thus, Jonathan Swift, in his "Gulliver's Travels," published about 1727, in giving an account of the extraordinary race of abstract philosophers who inhabited the "Floating Island" called Laputa, informs us that "they spend the greater part of their lives in observing the celestial bodies, which they do by the assistance of glasses far excelling ours in goodness; for, although their largest telescopes do not exceed three feet, they magnify much more than those of one hundred with us, and show the stars with greater clearness. This advantage has enabled them to extend their discoveries much farther than our astronomers in Europe; for they have made a catalogue of 10,000 fixed stars, whereas the largest of ours does not contain above one third of that number. They have likewise discovered two lesser stars or satellites, which revolve about Mars; whereof the innermost is distant from the center of the primary planet exactly three of its diameters, and the outermost five; the former revolves in the space of ten hours, and the latter in twenty-one and a half; so that the squares of their periodical times are very near in the same proportion with the cubes of their distances from the center of Mars; which evidently shows them to be governed by the same law of gravitation that influences the other heavenly bodies."
About twenty-five years after Swift wrote the foregoing, that is in 1752, the celebrated Voltaire (apparently in imitation of "Gulliver's Travels") cuttingly ridicules the pretensions of the class of reasoners who found their conclusions upon analogy. In one of his satirical tales, Micromegas, an imaginary inhabitant of Sirius, is supposed to make a voyage of discovery through the solar system in company with a denizen of Saturn; they philosophize as they go. Approaching the planet Mars, Micromegas and his companion plainly descried two moons acting as satellites to that body—moons which have certainly escaped the ken of terrestrial astronomers. "I know perfectly well," continues the author of the tale, "that Father Castel" (an astronomer of the time) "will write, and write sufficiently pleasantly, too, against the existence of these two moons; but I appeal against his decision to logicians, who reason from analogy. These excellent philosophers are perfectly aware how difficult it would be for Mars—a planet so far ​removed from the sun—to get on with less than two of these satellites." ("Œuvres de Voltaire"—Micromegas, chapter iii.) How completely the recent discovery of the American astronomer has "turned the tables" on the renowned satirist of the last century! The previsions of those "excellent philosophers" who founded their conclusions upon analogical reasoning, although slumbering in the domains of the unproved for more than two centuries, have at last been verified by direct observation.
As the moons of Mars are very small objects, it is only under the most favorable circumstances that they can be seen by the most powerful telescopes. Mars is nearest to us when his opposition occurs, when he is near his perihelion; and the greatest possible proximity to us occurs when Mars is in opposition in perihelion and the earth is in aphelion at the same time. The oppositions of Mars near perihelion occur at intervals of fifteen and seventeen years successively. A very good opposition occurred in 1862, and a great many distinguished astronomers embraced the opportunity of scrutinizing Mars with the aid of excellent instruments. A still more favorable opportunity was presented in the summer of 1877, when Mars was nearer to us than it has been since 1845. It was at this time that Professor Asaph Hall was fortunate enough, by means of the new 26-inch refractor of the Naval Observatory at Washington, to discover two moons belonging to this planet. It is true that this was probably the first time that so powerful a telescope had ever been directed to the examination of Mars under similar favorable conditions; yet it is a significant fact that, since the announcement of the discovery, the satellites have been detected by means of telescopes of more moderate power. The secret of Professor Hall's discovery seems to have consisted in devising the means of cutting off, from the field of view of the telescope, the glaring light of Mars. In like manner, M. Henry, of the Observatory of Paris, on August 27, 1877, was able to see the satellites when Mars was screened from view. These diminutive moons nestle so closely to the planet that it is difficult to see them in the blaze of light reflected from Mars. Had similar means of screening the planet been employed, it is probable that one or both of these satellites might have been discovered in 1862.
The distance of the inner satellite from the center of the primary is about 2·73 times the radius of Mars; that of the outer one about 6·846 times the same radius. Assuming the diameter of Mars to be about 4,200 miles, these distances become, respectively, 5,733 and 14,376 miles from the center of Mars. The nearest satellite of Jupiter is distant about six times the radius of the primary, and the innermost satellite of Saturn is distant a little more than three times the radius of that planet.
| | Earth. | Mars. | Jupiter. | Saturn. | Uranus. | Neptune. |
| 1 | 60·27 | 2·72 | 5·70 | 2·98 | 7·71 | 14·55 |
| 2 | . . . . . . | 6·82 | 9·07 | 3·83 | 10·75 | . . . . . . |
| 3 | . . . . . . | . . . . . . | 14·46 | 4·75 | 17·63 | . . . . . . |
| 4 | . . . . . . | . . . . . . | 25·44 | 6·08 | 23·57 | . . . . . . |
| 5 | . . . . . . | . . . . . . | . . . . . . | 8·47 | . . . . . . | . . . . . . |
| 6 | . . . . . . | . . . . . . | . . . . . . | 19·67 | . . . . . . | . . . . . . |
| 7 | . . . . . . | . . . . . . | . . . . . . | 24·80 | . . . . . . | . . . . . . |
| 8 | . . . . . . | . . . . . . | . . . . . . | 57·28 | . . . . . . | . . . . . . |
Professor Newcomb gives, for the period of revolution of the inner satellite around Mars, about 7·65 hours, or 7h, 39m., and 30·25 hours, or 30h. 15m., as that of the outer moon. Both of them, like our moon, revolve around the primary from west to east. Mars rotates on its axis from west to east in 24*623 hours, or 24h. 37m. 23s.; this is the duration of the Martial day. We have seen that the period of revolution of the inner satellite is less, while that of the outer is greater, than a Martial day. It is evident, therefore, that, as seen from the surface of the planet, the apparent motion of the satellites will be in opposite directions, the inner rising in the west and setting in the east, the outer (like our moon) rising in the east and setting in the west. This anomalous condition of things must have greatly perplexed the primitive astronomers of Mars, and probably led them to the invention of cycles and epicycles to account for these appearances.
It follows that the phenomenon of two moons meeting in mid-heavens will be no unusual occurrence to the observers on the surface of Mars. The apparent motion of the fixed stars from east to west, produced by the rotation of the planet upon its axis, is at the rate of 14∙62° per hour. The real motion of the inner satellite among the stars from west to east is at the rate of about 47∙06° per hour, while that of the outer one is at the rate of 11∙90° per hour. Hence it follows that the apparent motion of the inner satellite from west to east across the heavens, to an observer on Mars, will be at the rate of about 32∙44° per hour, while the apparent motion of the outer moon from east to west will be at the rate of nearly 2∙72° per hour.
It likewise follows from the preceding calculations that the time elapsing between two successive meridian passages of the inner satellite will be about 11∙09 hours, and the time elapsing between two successive conjunctions of the inner with the outer moon will be about 10∙24 hours; consequently two conjunctions will occur in less time than it takes for Mars to rotate on its axis, or than a Martial day. This satellite completes more than three orbital revolutions in a Martial day.
As the apparent motion of the outer satellite from east to west is at the rate of only about 2∙72° per hour, it is obvious that the time elapsing between two successive meridian passages of this moon will be about 132∙35 hours: so that there will be no less than twelve ​junctions with the inner moon in the course of its lunar day. It is likewise evident that the outer satellite will frequently be above the horizon of Mars more than sixty hours, during which period six conjunctions with the inner may occur. Moreover, as the outer moon will go through its cycle of phases in a little more than thirty hours, all of these changes may be accomplished while it is above the horizon of the observer on the surface of Mars.
The apparent diameter of Mars, as seen by an observer on the inner satellite, would be no less than 41·8°, or about seventy-eight and a half times the apparent diameter of the sun as seen from the earth; and from the outer moon the diameter of Mars would subtend an angle of 16·7°, or about 31·3 times the apparent diameter of the sun as seen by us. Of course the apparent areas of the disk of Mars, as seen from his two satellites, would be in the ratio of the squares of these numbers, that is, the apparent area of the disk of Mars, as seen from his inner moon, would be 6,167, and from the outer 980 times the apparent area of the solar disk, as seen from the earth.
From the innermost satellite of Saturn, the diameter of the primary would subtend an angle of 35·8°; from the nearest satellite of Jupiter, the diameter of that planet would subtend an angle of 18·6°; and from our moon the earth's diameter would subtend an angle of less than 2°.
Astronomers are, as yet, ignorant of the real magnitude of the Martial satellites; but, assuming each of them to be one hundred miles in diameter, it is easy to calculate their apparent magnitudes as seen by an observer on Mars. The inner moon being 5,733 miles distant from the center of Mars, would, when in the zenith of the observer, be only 3,633 miles distant from the surface of the planet. Hence it appears that, when this satellite is seen in the horizon of the observer on the surface of Mars, its diameter would subtend an angle of about 60', or nearly twice the apparent diameter which our moon presents to us; but, when it is in the zenith of the observer, it would subtend an angle of 94·3', or more than three times the apparent diameter presented by our moon. In other terms, in rising from the western horizon to the zenith, the apparent diameter of this moon would be increased nearly in the ratio of two to three; and, of course, its apparent area would be augmented nearly in the ratio of four to nine.
​The outer satellite would, under like positions, present apparent diameters, respectively, of 24' and 28', or considerably less than the apparent diameter of our moon. The nearest satellite of Jupiter (having a diameter of 2,310 miles) would, in like positions, present to an observer on the surface of that planet apparent diameters, respectively, of 31' and 37'.
As we have seen, the inner satellite of Mars completes three orbital revolutions in less than a Martial day. "This anomalous fact in the planetary system would seem, at first view, to be utterly inconsistent with the nebular hypothesis." According to this hypothesis, the orbital-periods of the satellites should be approximately equal to the rotation-periods of the primary at the epochs when the satellites were thrown off from it. The acceleration of the rotation-period of the primary, in consequence of its subsequent contraction, would necessarily render its time of rotation less than the orbital-period of any satellite. As far as yet known, the inner satellite of Mars affords the only instance in which the rotation-period of the primary is greater than the orbital-period of the secondary.
It must be remembered, however, that if we regard the rings of Saturn as composed of clouds of independently revolving minute satellites, those constituting the innermost portions of the inner ring must revolve in less time than the rotation-period of that planet. Under this view, therefore, the case of the inner satellite of Mars is not unique.
There are, however, several methods by which the apparently anomalous fact may be accounted for consistently with the nebular hypothesis:
1. In the first place, it has been suggested that Mars may not have obtained his satellites by means of the usual process of moon-formation, but by the appropriation to himself of a couple of the numerous asteroids or planetoids, some of which, in their perihelion excursions, approach comparatively near to Mars in his aphelion positions. Thus, the planetoid called Phocea, when it is at its least distance and Mars at his greatest distance from the sun, would only be about 11,000,000 miles from each other. It is, therefore, possible that some of the planetoids, moving in orbits of greater eccentricity than any yet discovered, may at some former period, have approached so near Mars as to have become permanently attached to it as satellites.
2. In the second place, it is possible that these Martial moons may have originally revolved in larger orbits, and therefore in longer periods than at present, but that the retarding influence of a resisting medium, on such small masses might, in the course of myriads of ages, have contracted their orbits and consequently shortened their orbital periods. In this connection it must be borne in mind that, according to the nebular hypothesis. Mars must be a vastly older planet than the earth; so that this retardation may have been in progress for an ​incalculable number of centuries before the earth became a separate planet.
Until quite recently, it was generally conceded that two comets of short period have revealed the existence of a resisting medium in the celestial spaces. It is well known that the celebrated Encke inferred the existence of a resisting medium from the fact that the periodic times of the comet which bears his name were progressively diminishing.
Thus he found the following values of these times:
| 1786-1795, | periodic | time | $=$ | 1208·112 | days. |
| 1795-1805, | " | " | $=$ | 1207·879 | " |
| 1805-1819, | " | " | $=$ | 1207·424 | " |
| 1845-1855, | " | " | $=$ | 1205·250 | " |
In this view he was sustained by Olbers and most contemporary astronomers, although Bessel and some others dissented from it. But Encke continued steadfast in his theory of a resisting medium in space for more than forty years; in fact, up to the period of his death in 1865.
There are two other periodical comets which were expected to furnish important evidence on this question. These are Faye's and Winnecke's comets, which have periods of seven and a half and five and a half years respectively. The orbit of the former has been carefully determined by Professor Axel Möller, of Lund, Sweden. At first his calculations indicated that the period of this comet was shortened at each revolution by about seventeen hours; and Encke, in his declining years, thought that this fact was a complete proof of his hypothesis of a resisting medium. But, in 1865, Professor Möller revised his calculations, and found that it was possible to harmonize all of the facts without the assumption of the resisting medium.
With regard to Winnecke's comet, it seems that, according to the computations of Professor Oppolzer, of Vienna, it is scarcely necessary to call in the assistance of a resisting medium to account for its motions. It thus appears that, up to the present time, Encke's comet stands alone in demanding the existence of a resisting medium to explain its motions. Nevertheless, it must be recollected that such investigations involve the computing of complex planetary perturbations, and that, consequently, more accurate data and better mathematical methods may, in the future, place these two comets in the same category, in relation to a resisting medium, as that of Encke.
In the mean time, divers physical considerations press upon us the inherent probability of the existence of a resisting medium in the celestial spaces. The connection between our organs of sense and remote bodies necessarily implies the existence of some intervening medium; and, moreover, to convey physical impression to the organ of sense, this medium must be material. Whatever theory of light we ​adopt, we are equally driven to the conception of the existence of some form of matter in the celestial spaces. The fact that light and heat are propagated from one part of space to another in time demands that the medium of communication should possess inertia—an essential property of matter. According to the wave theory, the celestial bodies move in an attenuated and subtile ethereal medium; according to the corpuscular theory, they move in a perpetual shower of corpuscles emitted by the sun and stars. In both cases matter exists—inertia exists—therefore resistance must be encountered. The smallness of the resistance, however small we choose to suppose it, does not allow us to escape this certainty. There is resistance, and therefore the movements of satellites cannot escape its influence. Nevertheless, such attenuated and bulky masses as comets are best adapted to test the existence of a resisting medium.
3. In the last place, it is possible that Mars may have originally rotated on his axis in five or six hours, but that the tidal rotation-retardation produced by the action of his moons might have brought about its present rotation-period. It is evident that the solar tides, on a planet so small and so remote from the sun, must be inappreciable; and, at first sight, the lunar tides produced by such small masses might be supposed to be equally insignificant. But it must be recollected that the tide-generating power of a moon is (other things being equal) inversely proportional to the cube of its distance; so that nearness might more than compensate for smallness of mass. To be more specific: In the mathematical language, the tide-generating power is in proportion to the
* Diameter of Primary × Mass of Satellite
* (Distance of Satellite)$^3$
Thus, for example, let us suppose the diameter of our moon to be twenty times the diameter of the inner satellite of Mars, and both moons to be equally dense; then the mass of our moon would be 8,000 times that of the Martial satellite. Taking the diameter of the earth as equal to twice the diameter of Mars (and it is not so great), and the distance of our moon from the center of the earth to be forty-one and a half times the distance of the inner satellite from the center of Mars, we then have the tide-generating power of our moon acting on the earth, will be to that of the inner satellite acting on Mars as
| 2 × 8000 | to 1, or as | 16000 | to 1, or as | 1 | to 1, or as 1 to 4½. |
| (41½)$^3$ | to 1, or as | 71743 | to 1, or as | 4½ | to 1, or as 1 to 4½. |
Hence, the tide-generating power of this small satellite would, in consequence of its nearness to Mars, be about four and a half times as great as the tide-generating power of our moon on the earth.
This view, however, is not free from the most serious physical difficulties. For it is evident that the tidal rotation-retardation produced by the moons would be limited by the final condition, that the ​rotation-period of the primary becomes exactly the same as the orbital-period of the satellite. When this condition is attained, the tides can no longer retard the rotation-period of the planet. So far, therefore, as the inner moon of Mars is concerned, it must long ago have ceased to retard the rotation of the primary. For, the orbital-period of this satellite being far shorter than the present rotation-period of Mars, its tidal action would tend to accelerate instead of retarding the time of rotation of the planet. So far as the outer moon is concerned, it is evident that its tidal action must tend to retard the rotation-period of Mars; but, in consequence of its greater remoteness, the magnitude of its influence must be small compared with that of the inner satellite. It is, therefore, difficult to conceive how the tidal influences of the moons of this planet can explain the anomalous fact that its rotation-period is longer than the orbital-period of one of its satellites.
In connection with the idea of the rotation-period of Mars having, at some former time, been much shorter than it is at present, it may be noticed that the great compression or ellipticity of this planet is totally inconsistent with its observed rotation-period.
In 1784 Sir William Herschel estimated the ellipticity of Mars at $^1$⁄$_{16}$. Schröter refused to admit this result; he contended that, if the ellipticity existed, it would not exceed $^1$⁄$_{80}$. Bessel failed to discover any appreciable ellipticity of Mars, even with the celebrated heliometer of Königsberg, On the other hand, Arago's measurements, executed at the Observatory of Paris, from 1811 down to 1847, all confirm the existence of an ellipticity in this planet of about $^1$⁄$_{30}$. ("Astronomic Populaire," tome iv., p. 130. Paris, 1867.) More recent observations give somewhat contradictory results. Professor Kaiser, of Leyden, makes the ellipticity $^1$⁄$_{114}$ Main, of the Radcliffe Observatory, deduced $^1$⁄$_{39}$; and Dawes's measurements give negative results.
To show the discordance of these results with what may be deduced from the theory of gravitation, it must be recollected that the ellipticity of a rotating planet depends upon the ratio of the centrifugal force at its equator to the force of gravity at the same place. Thus, to compare the earth and Mars—
| Let | r | and | r' | = | equatorial radii | of | earth | and | Mars | respectively. |
| " | t | " | t’ | = | time of rotation | " | " | " | " | " |
| " | Q | " | Q’ | = | mass | " | " | " | " | " |
| " | f | " | f’ | = | centrifugal force at equator | centrifugal force at equator | centrifugal force at equator | centrifugal force at equator | " | " |
| " | g | " | g’ | = | force of gravity | " | " | " | " | " |
​Then, by dynamical principles, we have—
| | f | : | f' | :: | r | : | r' |
| | f | : | f' | :: | t$^2$ | : | t'$^2$ |
| and | g | : | g' | :: | Q | : | Q' |
| and | g | : | g' | :: | r$^2$ | : | r'$^2$ |
| Now, for these two planets we have— | Now, for these two planets we have— | Now, for these two planets we have— | Now, for these two planets we have— | Now, for these two planets we have— | Now, for these two planets we have— | Now, for these two planets we have— | Now, for these two planets we have— | Now, for these two planets we have— |
| r | = | 3962∙8 miles, | 3962∙8 miles, | and | r' | = | 2100 miles. | 2100 miles. |
| t | = | 86164 seconds, | 86164 seconds, | " | t' | = | 88643 seconds. | 88643 seconds. |
| Q | = | 1 | | and | Q' | = | 1 | of mass of the sun. |
| | | 326690 | | | | | 3090000 | |
Substituting these numbers in foregoing proportions, and performing the arithmetical operations, and we have—
| f | : | f' | :: | 1 | : | 0∙500704, | and |
| g | : | g' | :: | 1 | : | 0∙376482 | |
| Hence we have | f | : | f' | :: | 1 | : | 0∙500704 | or 1 : 1∙32996. But for the earth, |
| | g' | | g' | | | | 0∙376482 | or 1 : 1∙32996. But for the earth, |
| f | = | 1 | ; hence we have | 1 | : | 1 | : | f' | :: | 1 | : | 1∙32996. Consequently for |
| g | = | 289 | ; hence we have | 289 | : | 1 | : | g' | :: | 1 | : | 1∙32996. Consequently for |
| Mars we have | f' | = | 1∙32996 | = | 1 | Now, according to the elegant |
| Mars we have | g' | = | 289 | = | 217 | Now, according to the elegant |
In the case of the earth, we know that it is much smaller; being about $^1$⁄$_{300}$ instead of $^1$⁄$_{231}$. Hence, for Mars also, we should expect an ellipticity smaller than $^1$⁄$_{174}$; whereas, as we have seen, nearly all the measurements indicate a much greater ellipticity.
It is evident that a more rapid rotation of the planet would augment its ellipticity; hence the question naturally suggests itself: Might not this great ellipticity of Mars have been the result of solidification having taken place when his rotation-period was much shorter than it is at present? This explanation is not free from serious difficulties. For, if aqueous and aërial agencies were in action after solidification took place, they would have tended to make the shape of the planet conform to its new rotation-period.
* ↑ It was on the memorable night of the 11th of August, 1877, that Professor Asaph Hall, of the Naval Observatory at Washington, caught the first glimpse of these diminutive companions of Mars. The intervention of unfavorable weather kept him in a state of anxious suspense, and postponed, for a period of five days, the complete verification of his great discovery.
* ↑ The following table exhibits the mean distances of the satellites from the centers of the primaries, expressed in equatorial radii of the latter. ("Nature," December 13, 1877, p. 129.)
* ↑ Professor E. C. Pickering, of the Harvard College Observatory, has attempted to determine the real magnitude of the satellites of Mars, by comparing the intensity of the light reflected from the primary with that reflected from each of his satellites. He is thus led to estimate the diameter of the inner satellite to be about seven miles, and that of the outer one to be about six miles! ("Annual Report of the Director of Harvard College Observatory," November, 1877, page 17.) It is very questionable whether estimates, founded on photometrical comparisons in which the relative reflecting powers of the bodies compared are unknown, can inspire the confidence of astronomers in relation to the accuracy of the deduced diameters.
* ↑ The oblateness or compression or ellipticity of an oblate spheroid is the difference of its equatorial and polar radii, divided by its equatorial radius. Thus, if a and b are the equatorial and polar radii respectively, then ellipticity = a $-$ b a
* ↑ That the values of ellipticity deduced from the assumption of an homogeneons liquid mass in the rotating planet must be maxima is evident from the consideration that, if the density augmented from the surface toward the center of the planet (which must, from the compressibility of matter, be the real condition of things), it would render the computed ellipticity smaller. The problem of the theoretical figure of a rotating planet is greatly complicated as soon as we abandon the assumption of homogeneousness.