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when a compreffing force 27 is employed, the air is compressed into one 27th of it's former bulk, the particles are at of their former distance, and the force is diftributed among 9 times the number of particles; the force on each is therefore 3. In fhort, letbe the diftance of the particles, the number of them in any given vessel, and there fore the denfity will be as x3, and the number preffing by their elafticity on its whole internal furface will be as x2. Experiment fhows, that the compreffing force is as 3, which being diftributed over the number as x2, will give the force on each as x. Now this force is in immediate equilibrium with the elasticity of the particle immediately contiguous to the compreffing furface. This elafticity is therefore as x and it follows from the nature of perfect fluidity, that the particle adjoining to the compreffing furface preffes with an equal force on its adjoining particles on every fide. Hence the corpufcular repulfions exerted by the adjoining particles are inverfely as their diftances from each other; or the adjoining particles tend to recede from each other with forces inverfely proportional to their distances.

Sir ISAAC NEWTON was the firft who reafoned in this manner from the phenomena. Indeed he was the first who had the patience to reflect on the phenomena with any precifion. His difcoveies in gravitation naturally gave his thoughts this turn, and he very early hinted his fufpicions that all the characteristic phenomena of tangible matter were produced by forces which were exerted by the particles at small and infenfible diftances: And he confiders the phenomena of air as affording an excellent example of this investigation, and deduces from them the law which we have now demonftrated; and fays that air confifts of particles which avoid the adjoining particles with forces inverfely proportional to their diftances from each other. From this he deduces (in the 2d book of his Principles) feveral beautiful propofitions, determining the mechanical conftitution of the atmosphere. But he limits this action to the adjoining particles: and this is a remark of immenfe confequence, though not attended to by the numerous experimenters who adopt the

law.

The particles are fuppofed to act at a diftance; this diftance is variable; and the forces diminish as the diftance increases. A very ordinary airpump will rarefy the air 125 times. The diftance of the particles is now 5 times greater than before; and yet they ftill repel each other: for the air of this denfity will ftill fupport the mercury in a fyphon-gage at the height of o 24 of an inch; and a better pump will allow this air to expand twice as much, and fill leave it elaftic. Thus, whatever is the diftance of the particles of common air, they can act five times farther off. The queftion then is, Whether, in the ftate of common air, they really do act five times farther than the distance of the adjoining particles? While the particle a acts on the particle b with the force does it alfo act on the particle e with the force 2'5, on the particle d with the force 1667, on the particle e, with the force 1.25, on the particle, with the force 1, on the particle g, with the force

5,

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08333, &c.? Sir Ifaac Newton fhows plainly, that this is not the cafe; for if it were, the fenfibi phenomena of condenfation would be totally dit ferent from what we obferve. The force necef fary for a quadruple condenfation would be the force must be 27 times greater. Two fpher times greater, and for a nonuple condenfation filled with condenfed air muft repel each othes, and two fpheres containing air that is rarer tha the furrounding air muft attract each other, &c All this will appear very clearly, by applying to air the reasoning which Sir Ifaac Newton has c ployed in deducing the fenfible law of mutual tendency of two fpheres, which confift of pa ticles attracting each other with forces proportional to the fquare of the diftance inverfely.

If we could fuppofe that the particles of air repelled each other with invariable forces at all di tances within fome small and infenfible limit, this would produce a compreffibility and elafticity 6. milar to what we obferve. But this law of corpufcular force is unlike every thing we obferve in nature, and to the laft degree improbable. W muft therefore continue the limitation of this mutual repulfion of the particles of air, and be cotented for the prefent with having established as an experimental fact, that the adjoining particles of air are kept asunder by forces inveriely proportional to their diftances; or perhaps it is better tu abide by the fenfible law, that the density of air 1. proportional to the compreffive force. This law i abundantly fufficient for explaining all the fubordinate phenomena, and for giving us a complet. knowledge of the mechanical conftitution of ou atmosphere.

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SECT. VI. Of the HEIGHT of the ATMOSPHERE.

THE preceding view of the compreffibility air muft give us a very different notion of the height of the atmosphere from what we deduced from our experiments. When the air is of the temperature 32° of Fahrenheit's thermometer, and the mercury in the barometer ftands at 30 inches, it will defcend one roth of an inch if we take it to a place 87 feet higher. Therefore, if the an were equally denfe and heavy throughout, the atmosphere would be 30 X 10 X 87 feet, or 5 mile and 100 yards. But it must be much higher; becaufe every ftratum as we afcend must be fuccefively rarer as it is lefs compreffed by incumben: weight. (See ATMOSPHERE, § 6.) Not knowing to what degree air expanded when the com preffion was diminished, we could not tell the fucceffive diminution of denfity and confequent augmentation of bulk and height; we could only fay, that several atmospheric appearances indicated a much greater height. Clouds have been feen much higher; but the phenomenon of the TWILIGHT is the moft convincing proof of this. There is no doubt that the vifibility of the ky or air is owing to its want of perfect tranfparency. each particle (whether of matter purely aerial or heterogeneous) reflecting a little light.

Let b(fig. 49.) be the laft particle of illuminated air, which can be feen in the horizon by a spectator at A. This must be illuminated by a ray SD b, touching the earth's surface at some point D. Now it is a known fact, that the degree of illumination

llumination called twilight is perceived when the fun is 18° below the horizon of the fpectator, that, when the angle E 6 S or ACD is 18 degrees; therefore bC is the fecant of 9° (it is lefs, viz. about 810, on account of refraction.) We know the earth's radius to be about 3970 miles: hence we conclude B to be about 45 miles; nay, a very fenfible illumination is perceptible much farther from the fun's place than this, perhaps twice as far, and the air is fufficiently denfe for reflect ing a fenfible light at the height of nearly 200 miles.

We have seen that air is prodigiously expanfible. None of our experiments have diftinctly shown us any hint. But it does not follow that it is expanfible without end. It is much more probable that there is a certain diftance of the parts in which they would arrange themselves if they were not heavy. But at the very fummit of the atmosphere they will be a very fmall matter nearer to each other, on account of their gravitation to the earth. Till we know precisely the law of this mutual repulfion, we cannot fay what is the height of the atmosphere. But if the air be an elastic fluid whole denfity is always proportionable to the compreffing force, we can tell what is its denfity at any height above the surface of the earth; and we can compare the denfity fo calculated with the denfity discovered by obfervation: for this last is measured by the height at which it fupports mercury in the barometer. This is the direct meafure of the preffure of the external air; and as we know the law of gravitation, we can tell what would be the preffure of air having the calculated denfity in all its parts.

Suppose a prifmatic or cylindric column of air reaching to the top of the atmosphere. Let this be divided into an indefinite number, of ftrata of very (mall and equal depths or thickness; and let us fuppofe that a particle of air is of the fame weight at all diftances from the centre of the earth. The abfolute weight of any of these ftrata will on these conditions be proportional to the number of particles, or the gravity of air contained mit; and fince the depth of each ftratum is the me, this quantity of air will be as the denfity of the ftratum; but denfity of any ftratum is as the Compreffing force, i. e. as the preffure of the ftrata above it; i. e. as their weight; i. e. as their quantity of matter-therefore the quantity of air in each ftratum is proportional to the quantity of air above it; but the quantity in each ftratum is the difference between the column incumbent on its bottom and on its top: thefe differences are therefore proportional to the quantities of which they are the differences. But when there is a feries of quantites which are proportional to their own differences, both the quantities and their differences are in continual or geometrical progreffion: for let a, b, c, be three fuch quantities that

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creafe, or their depths under the top of the atmofphere decrease, in an arithmetical progreffion, the denfities decrease in a geometrical pro. greffion.

Let ARQ (fig. 50.) represent the section of the earth by a plane through its centre O, and let m OAM be a vertical line, and AE perpendicular to OA will be a horizontal line through A, a point on the earth's furface. Let AE be taken to reprefent the denfity of the air at A; and let DH, parallel to AE, be taken to AE as the denfity at D is to the denfity at A: it is evident, that if a logistick or logarithmic curve EHN be drawn, having AN for its axis, and paffing through the points E and H, the denfity of the air at any other point C, in this vertical line will be reprefented by CG, the ordinate to the curve in that point: for it is the property of this curve, that if portions AB, AC, AD, of its axis be taken in arithmetical progreffion, the ordinates AE, BF, CG, DH, will be in geometrical progreffion. It is another fundamental property of this curve, that if EK or HS touch the curve in E or H, the fubtangent AK or DS is a conftant quantity. A 3d fundamental property is, that the infinitely extended area MAEN is equal to the rectangle KAEL of the ordinate and fubtangent; and in like manner, the area MDHN is equal to SDX DH, or to KA × DH; consequently the area lying beyond any ordinate proportional to that ordinate.

Thefe geometrical properties of this curve are all analogous to the chief circumftances in the conftitution of the atmosphere, on the fuppofition of equal gravity. The area MCGN represents the whole quantity of aerial matter which is above C: for CG is the denfity at C, and CD is the thickness of the ftratum between C and D; and therefore CGHD will be as the quantity of matter or air in it; and in like manner of all the others, and of their fums, or the whole area MCGN; and as each ordinate is proportional to the area above it, fo each denfity, and the quantity of air in each ftratum, is proportional to the quantity of air above it: and as the whole area MAEN is equal to the rectangle KAEL, fo the whole air of variable denfity above A might be contained in a column KA, if, inftead of being compreffed by its own weight, it were without weight, and compreffed by an external force equal to the preffure of the air at the furface of the earth. In this cafe, it would be of the uniform denfity AE, which it has at the furface of the earth, making what we have repeatedly called the homogeneous atmosphere.

Hence we derive this important circumstance, that the height of the homogeneous atmosphere is the fubtangent of that curve whose ordinates are as the denfities of the air at different heights, on the fuppofition of equal gravity. This curve may with propriety be called the ATMOSPHERICAL LOGARITHMIC; and as the different logarithmics are all characterised by their fubfequents, it is of importance to determine this one. It may be done by comparing the denfities of mercury and air. For a column of air of uniform denfity, reaching to the top of the homogeneous atmosphere, is in equilibrio with the mercury in the barometer. Now it is found, by the best experiments, that when mercury and air are of the temperature 32° 0.

A 2

Fahrenheit's

Fahrenheit's thermometer, and the barometer ftands at 30 inches, the mercury is nearly 10440 times denfer than air. Therefore the height of the homogeneous atmosphere is 10440 times 30 inches, or 26100 feet, or 8700 yards, or 4350 fathoms, or 5 miles wanting 100 yards.

Or it may be found by obfervations on the barometer. It is found, that when the mercury and air are of the above temperature, and the barometer on the fea-fhore stands at 30 inches, if we carry it to a place 884 feet higher, it will fall to 29 inches. Now, in all logarithmic curves having equal ordinates, the portions of the axes intercepted between the correfponding pairs of ordinates are proportional to the fubtangents. And the subtangents of the curve belonging to our common tables is 0'4342945, and the difference of the logarithms of 30 and 29 (which is the portion of the axis intercepted between the ordinates 30 and 29), or o'0147233, is to o'4342945 as 883 is to 26058 feet, or 8686 yards, or 4343 fathoms, or 5 miles, wanting 114 yards. This determination is 14 yards lefs than the other, and it is uncertain which is, the most exact. It is extremely difficult to measure the respective denfities of mercury and air; and in measuring the elevation which produces a fall of one inch in the barometer, an error of one 20th of an inch would produce all the difference. We prefer the laft, as depending on fewer circumftances. But all this investigation proceeds on the fuppofition of equal gravity, whereas we know that the weight of a particle of air decreases as the fquare of its diftance from the centre of the earth increases. In order, therefore, that a fuperior ftratum may produce an equal preffure at the furface of the earth, it must be denfer, because a particle of it gravitates lefs. The denfity, therefore, at equal elevations, muft be greater than on the fuppofition of equal gravity, and the law of diminution of denfity must be different.

Make OD: OA≈OA ; Od ;..

OC: OA OA: Oc;

OB: OA=OA: Ob, &c.; fo that Od, Oc, Ob, OA, may be reciprocals to OD, OC, OB, OA; and through the points A, b, c, d, draw the perpendiculars AE, bf, cg, db, making them proportional to the denfities in A, B, C, D; and let us fuppofe CD to be exceeding ly fmall, fo that the denfity may be fuppofed uniform through the whole ftratum. Thus we have OD X Od OA2, =OCX Oc

and Oc: Od≈OD: OC:

and Oc: Oc-Od=OD: OD-OC,

or Oc: cd=OD: DC;

and cd CD=Oc: OD;

vitation of each particle. Therefore, cd X cg is as the preffure on C arifing from the weight of the ftratum DC; but edXcg is evidently the element of the curvilineal area AmnF, formed by the curve Efghn and the ordinates AE, bf, cg, ab, &c. mn. Therefore the fum of all the elements, fuch as cabg, that is, the area cmng below cg, will be as the whole preffure on C, arifing from the gravita tion of all the air above it; but, by the nature of air, this whole preffure is as the denfity which it produces, that is, as cg. Therefore the curve Ege is of fuch a nature that the area lying below or beyond any ordinate cg is proportional to that ordinate. This is the property of the logarithmic curve, and Egn is a logarithmic curve.

But farther, this curve is the fame with EGN. For let B continually approach to A, and ultimately coincide with it. It is evident that the ultimate ratio of BA to Ab, and of BF to bf, is that of equality; and if EFK, Efk, be drawn, they will contain equal angles with the ordinate AE, and will cut off equal fubtangents AK, Ak. The curves EGN, Egn are therefore the fame, but in oppofite pofitions. Laftly, if OA, Ob, Oc, Od, &c. be taken in arithmetical progreffion decreas ing, their reciprocals OA, OB, OC, OD, &c. will be in harmonical progreffion increafing, as is well known; but, from the nature of the logarithmic curve, when OA, Ob, Oc, Od, &c. are in arithmetical progreffion, the ordinates AE, bf, cg, dh, &c. are in geometrical progreffion. Therefore when OA, OB, OC, OD, &c. are in harmonical progreffion, the denfities of the air at A, B, C, D, &c. are in geometrical progreffion; and thus may the density of the air at all elevations be difcovered. Thus to find the denfity of the air at K, the top of the homogeneous atmofphere, make OK: ÔA=OA: OL, and draw the ordinate LT, LT is the denfity at K.

Dr HALLEY was the firft who observed the relation between the denfity of the air and the ordinates of the logarithmic curve, or common logarithms. This he did on the fuppofition of equal gravity; and his discovery is acknowledged by Sir Ifaac Newton in Princip. ii. prop. 22. Schol. His differtation on the subject is in N° 185. of the Phil. Tranf. Newton extended the fame relation to the true fate of the cafe, where gravity is as the fquare of the diftance inverfely; and fhowed, that when the distance from the earth's centre are in harmonic progreffion, the denfities are in geometric progreffion. He fhows indeed, in general, what progreffion of the diftance, on any fuppofition of gravity, will produce a geometrical progreffion of the denfities, fo as to obtain a fet of lines OA, Ob,

or, because OC and OD are ultimately in the ra- Oc, Od, &c. which will be logarithms of the den

tio of equality, we have

cd: CD=Oc: OC=OA2: OC2, and cd CDX

OA2 OC

anded Xcg CDXcgX;

AO2 OC2;

but CDXcg X. OA is the preffure at C arifing from

OC2

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infinity of matter in the universe, and that it is inconfiftent with the phenomena of the planetary motions, which appear to be performed in a space void of all refiftance, and therefore of all matter. But this fluid must be fo rare at great, diftances, that the refiftance will be infenfible, even though the retardation occafioned by it has been accumulated for ages. This being the cafe, it is reafon able to fuppofe the vifible univerfe occupied by air, which, by its gravitation, will accumulate itfelf round every body in it, in a proportion depending on their quantities of matter, the larger bodies attracting more of it than the fmaller ones, and thus forming an atmosphere about each. And many appearances warrant this fuppofition. Jupiter, Mars, Saturn, and Venus, are evidently furrounded by atmospheres. The conftitution of thefe atmospheres may differ exceedingly from other caufes. If the planet has nothing on its furface which can be diffolved by the air or volatilised by heat, the atmosphere will be continually clear and transparent, like that of the moon.

MARS has an atmosphere which appears precife ly like our own, carrying clouds, or depofiting fnows: for when, by the obliquity of his axis to the plane of his ecliptic, he turns his north pole towards the fun, it is obferved to be occupied by a broad white spot. As the fummer of that region advances, this fpat gradually wastes, and fometimes vanishes, and then the fouth pole comes in fight, furrounded in like manner with a white spot, which undergoes fimilar changes. This is precife ly the appearance which the fnowy circumpolar regions of this earth will exhibit to an aftronomer on Mars.

The atmosphere of JUPITER is alfo very fimilar to our own. It is diverfified by fireaks or belts parallel to his equator, which frequently change their appearance and dimenfions, in the fame manner as thofe tracks of fimilar sky which belong to different regions of this globe. But the moft remarkable fimilarity is in the motion of the clouds on Jupiter. They have, plainly a motion from E. to W. relative to the body of the planet: for there is a remarkable spot on the furface of the planet, which is obferved to turn round the axis in gb. 51′ 16′′; and there frequently appear variable and perishing spots in the belts, which fometimes laft for several revolutions. These are ob. ferved to circulate in 9h. 55' 05". Thefe numbers are the results of a long series of observations by Dr Herschel. This indicates a general current of the clouds weftward, precisely fimilar to what a spectator in the moon must observe in our atmof phere arifing from the trade-winds. Mr Schroeter has made the atmosphere of Jupiter a study for many years; and deduces from his obfervations that the motions of the variable spots is fubject to great variations, but is always from E. to W. This indicates variable winds.

The atmosphere of VENUS appears alfo to be like ours, loaded with vapours, and in a state of continual change of abforption and precipitation. About the middle of the 17th century the furface of Venus was pretty diftinctly seen for many years chequered with irregular spots, which are defcribed by Campani, Bianchini, and other aftronomers in the fouth of Europe, and also by Caffini at Paris,

and Hooke and Townley in England. But the fpots became gradually more faint and indistinct; and, for near a century, have difappeared. The whole furface appears now of one uniform brilliant white. The atmosphere is probably filled with a reflecting vapour, thinly diffufed through it, like water faintly tinged with milk. It appears to be of a very great depth, and to be refractive like our air. For Dr Herfchel obferved, by the help of his fine telescopes, that the illuminated part of Venus is confiderably more than a hemif phere, and that the light dies gradually away to the bounding margin. Venus may therefore be inhabited by beings like ourselves.

The atmosphere of COMETS feems of a nature totally different. This feems to be of inconceiva ble rarity, even when it reflects a very fenfible light. The tail is always turned nearly away from the fun. It is thought that this is by the impulfe of the folar rays. If this be the cafe, we think it might be difcovered by the aberration and the refraction of the light by which we fee the tail: for this light must come to our eye with a much smaller velocity than the fun's light, if it be reflected by repulfive or elaftic forces, which there is every reafon in the world to believe; and therefore the velocity of the reflected light will be diminished by all the velocity communicated to the reflecting particles. This is almoft inconceivably great. The comet of 1680 went half round the fun in ten hours, and had a tail at least a hundred millions of miles long, which turned round at the fame time, keeping nearly in the direction oppofite to the fun. The velocity neceffary for this is prodigious, approaching to that of light. SECT. VII. Of the MEASUREMENT of HEIGHTS by the BAROMETER.

We have shown how to determine a priori the denfity of the air at different elevations above the furface of the earth. But the denfities may be difcovered in all acceffible elevations by experiments; namely, by obferving the heights of the mercury in the barometer. This is a direct measure of the preffure of the incumbent atmosphere; and this is proportional to the density which it produces. Therefore, by means of the relation fubfifting between the denfities and the elevations, we can difcover, the elevations by obfervations made on the denfities by the barometer; and thus we may measure elevations by means of the barometer, and, with very little trouble, take the level of any extenfive tract of country. See BAROMETER, 24: and Plate XXXVI.

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If the mercury in the barometer ftands at 30 inches, and if the air and mercury be of the temperature 32° in Fahrenheit's thermometer, a column of air 87 feet thick has the fame weight with a column of mercury one 10th of an inch thick. Therefore, if we carry the barometer to a higher place, fo that the mercury finks to 29'9, we have afcended 87 feet. Suppofe we carry it ftill higher, and that the mercury ftands at 29°8; it is required to know what height we have now got to? We have evidently ascended through another ftratum of equal weight with the former: but it must be of greater thickness, because the air in it is rarer, being lefs compreffed. We may call the den

fity

fity of the firft ftratum 300, measuring the denfity by the number of tenths of an inch of mercury which its elafticity proportional to its denfity enables it to fupport. For the fame reason, the denfity of the fecond ftratum must be 299: but when the weights are equal, the bulks are inverfely as the denfities; and when the bafes of the ftrata are equal, the bulks are as the thickneffes. Therefore, to obtain the thickness of this fecond ftratum, fay 299 30087: 87°29; and this fourth term is the thicknefs of the fecond ftratum, and we have afcended in all 174'29 feet. In like manner we may rife till the barometer fhows the denfity to be 298: then say, 298: 30=87: 87'584 for the thickness of the third ftratum, and 261875 or 261 for the whole afcent; and we may proceed in the fame way for any number of mercurial heights, and make a table of the corresponding elements as follows: where the first column is the height of the mercury in the barometer, the fecond column is the thickness of the ftratum, or the elevation above the preceding ftation; and the third column is the whole elevation above the first station. Bar. Strat. Elev.

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feen that, upon the fuppofition of equal gravity, the denfities of the air are as the ordinates of a logarithmic curve, having the line of elevations for its axis. We have also seen that, in the true theory of gravity, if the distances from the centre of the earth increase in a harmonic progreffion, the logarithm of the denfities will decrease in an arithmetical progreffion; but if the greatest elevation above the furface be but a few miles, this harmonic progreffion will hardly differ from an arithmetical one. Thus, if Ab, Ac, Ad, are 1, 2, and 3 miles, we shall find that the correfponding elevations AB, AC, AD, are sensibly in arithmetical progreffion alfo: for the earth's radius AC is nearly 4000 miles. Hence it plainly follows, that BC— of a mile, or 4000+4001' 16004000

I

or

I

I 250

AB is of an inch; a quantity quite infignificant. We may therefore affirm, that in all acceffible places, the elevations increase in an arithmetical progreffion, while the denfities decrease in a geometrical progreffion. Therefore the ordinates are proportional to the numbers which are taken to measure the denfities, and the portions of the axis are proportional to the logarithms of thefe numbers. It follows, therefore, that we may take such a scale for measuring the denfities that the logarithms of the numbers of this fcale fhall be the very portions of the axis; that is, of the vertical line in feet, yards, fathoms, or what measure we pleafe: and we may, on the other hand, choose fuch a fcale for measuring our elevations, that the logarithms of our scale of denfities fhall be parts of this scale of elevations; and we may find either of these fcales fcientifically. For it is a known property of the logarithmic curves, that when the ordinates are the fame, the intercepted portions of the abfciffæ are proportional to their fubtangents. Now we know the fubtangent of the atmospherical logarithmic: it is the height of the homogeneous atmosphere in any measure we pleafe, fuppose fathoms: we find this height by comparing the gra. vities of air and mercury, when both are of fome determined denfity. Thus, in the temperature of 32° of Fahrenheit's thermometer, when the barometer stands at 30 inches, it is known (by many experiments) that mercury is 10423,068 times heavier than air; therefore the height of the balancing column of homogeneous air will be 10423,068 times 30 inches; that is, 4342,945 English fathoms. Again, it is known that the fubtangent of our common logarithmic tables, where I is the logarithm of the number 10, is 0,4341945. Therefore the number 0,4342945 is to the difference D of the logarithms of any two barometric heights as 4342,945 fathoms are to the fathoms F contained in the portion of the axis of the atmospherical logarithmic, which is intercepted between the ordinates equal to these barometrical heights; or that 0,4342945: D=4342,945: F, and 0,4342,945 : 4342,945=D: F; but 0,4342,945 is the ten-thoufandth part of 4342,945, and therefore D is the ten-thoufandth part of F.

Thus the logarithms of the denfities, measured by the inches of mercury which their elafticity supports in the barometer, are juft the 10,000th part of the fathoms contained in the corresponding por

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