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+1.7',

Taking the nucleus as an origin of rectangular co-ordinates, the axis of the comet's figure being that of abscissas, positive towards the tail, negative towards the head or front, Mr. Bond locates any point on the curve of the comet's figure by the co-ordinates (a, b). For the vertex of the head, b=0, and on Sept. 2nd a =— - 1.5'; on Oct. 5th, a—— 1.0', b 0, these values being reduced to distance unity. One of these series of measurements, that on Oct 2nd, is printed a which I take to be an evident mistake as to the algebraic sign. Rejecting it, however, and using the other eleven values which I have taken from Mr. Bond's data, I find the average value of a=— - 1.164', or in seconds of arc, and changing the algebraic sign for my purpose, a 69.84". But on Sept. 30th, Envelope A was 40.9" from the nucleus, according to Mr. Bond. Hence it had 28.94" yet to travel in order to reach the vertex of the comet's head, or its front. At the rate of 2.94" per day, determined by Mr. Bond, Envelope A would therefore reach the point where it would begin to separate from the comet's front in about ten days, or ten days after the perihelion passage.

I have computed the value of the comet's radius-vector at this date, Oct. 10th, from Mr. George Searle's Elements of the Orbit, given in Dr. Gould's Astronomical Journal. These elements are :—

T

1858 Sept. 294.75230, Washington Mean Time.
36° 12' 21.4"

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I find, from Mr. Searle's Elements, the logarithm of the comet's radius-vector, ten days after perihelion passage, to be log. r= 1.792684, and from Mr. Bond's value of a 69.84", I find log. = log. sin a = 4.529679.

Hence log. 2 ()=10.512015= log. ♫ nearly.

Hence 0.0000000003251

22. In order to have a better standard with which to compare such small masses as those of the comets, I have computed very carefully the value of the mass of the Earth's atmosphere in

terms of the sun's mass as the unit. Denoting the mass of the Earth's atmosphere by A, in terms of the sun's mass, I find

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Hence the mass of Donati's comet, the great comet of 1858, was about one hundred and thirty times as great as the mass of the Earth's atmosphere.

23. A limit to this mass may also be obtained simply from the measurement of Po at any time, by means of the approximate formula > 2()'.

Thus on Sept. 30th, very nearly the date of the perihelion passage, Mr. Bond gives the two measurements a—— -1.4', b=0, and a 0.8', b= 0, the former made at Liverpool, the latter at Poulkova. The mean of these measurements, changing the algebraic sign, is 1.1' or 66" for the angular value of fo. Taking Mr. Searle's q for r we have

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That is, the mass of Donati's comet, by this computation, was more than one hundred and thirty-five times as great as the mass of the Earth's atmosphere. Although this limit is somewhat greater than the former result, yet it is perhaps sufficiently near to give confidence in the general method.

24. I shall next apply the formula for the limit to the first comet of 1770, or the famous comet of Lexell, which passed very close to the planet Jupiter, and for which La Place assigned the limits of the Earth's mass. This limit is equivalent to

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I shall employ the elements of Lexell's comet computed by Leverrier, as given in Cooper's "Cometic Orbits." They are as follows:

Elements of Lexell's comet by Leverrier.

T= 1770, Aug. 13.54085, Greenwich Mean Time of Per. Pass. 356° 16′ 51′′ Longitude of Perihelion.

2131° 58′56′′

66 "Ascending Node.

i= 1° 34' 28" Inclination to the Ecliptic. log. q9.8289491, log. of perihelion distance.

e=0.786119, Eccentricity.

Motion Direct.

The comet was discovered by Messier, June 14, 1770. "He stated," says Mr. Cooper, "that on the night of his discovery, the comet showed a very feeble nebulosity, occupying little space; the centre was brilliant, but it was difficult to decide whether the object was a nebula or a comet. He could observe no relative change of its place in comparing it with fixed stars during two hours. On the night of 17th-18th June, the diameter of the nucleus 0'22"; that of the nebulosity = 5′ 23′′.” "On the 29th-30th, the former 1'22", the latter 54', without any sign of tail."

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From the rapid increase of these angles, as the comet approached the Earth, I judge that they have not been reduced to the unit of distance. Hence it is necessary to compute the Earth's distance from the comet at the date of the observation on June 17th-18th, in order to find f。 or P1 or p.

From Leverrier's elements of Lexell's comet, I have found for June 17.54085, Gr. M. Time, log. r 0.0731141, and for D, the comet's distance from the Earth at that date, log. D=1.2380469. Hence D sin (2' 41.5").

=

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The mass of Lexell's comet is therefore greater than that of the Earth's atmosphere, but appears to be far within the superior limit assigned by La Place.

25. I shall next apply the formula for the limit to Encke's

comet.

The comet of 1795, discovered by Miss Caroline Herschel, was subsequently recognized as Encke's comet. It was discovered on Nov. 7, 1795, and the discoverer observed that the diameter on that evening was "about 5'." "It had no nucleus, and had the appearance of an ill-defined haziness, which was rather strongest about the middle." I quote from the Notes of Cooper's "Cometic Orbits." On Nov. 21st, Olbers observed the comet, and found its diameter to be "about 3'." I shall therefore take the mean of these determinations, or 4', for the diameter on Nov. 14, 1795. Olbers states that the comet was "round, badly-defined, and without a distinct nucleus."

The Elements of this comet, for 1795, were subsequently computed by Encke, and are given by Cooper, as follows:

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Perihelion Passage 1795, December 21a.44098, Gr. M. T.

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From these elements and the Earth's place in her orbit, I have found for Nov. 14, 1795,

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Encke's comet has therefore more than three times the mass of

the Earth's atmosphere.

26. I shall next apply the formula for the mass to Halley's

comet.

Arago says that on January 25, 1836, Sir J. Herschel made two measurements of the head of Halley's comet, as follows:

Diameter of the head in the direction of right-ascension, 229.4" "declination, 237.3"

66 66

66

66 66

Two hours after, the measurements were respectively 196.7" in R. A. and 252.0" " Dec.

Whether these measurements were reduced to the unit of distance, or not, is not stated. I shall therefore compute the limit of the mass, first on the supposition that they have not been so reduced, and again by supposing that they have been reduced to the unit.

I shall take the mean of the first two measurements, or 233.4′′ for the apparent diameter, which differs only about 41" from the mean of the four.

From Cooper's "Cometic Orbits," the elements of Halley's comet are taken, at its perihelion passage in 1835, being the second set of elements computed by H. Westphalen. The inclination, however, has been changed to its supplement, because the motion is retrograde, in order to use Gauss's general formulæ.

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Halley's comet, perihelion passage Nov. 154.93927, 1835, Gr. M. T. 304° 31' 34"

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From these elements, and the Earth's position at the date, 1836, January 25a.5, I have found log. 2 (+) = 11.9331398

Hence 34.331

log. A

12.3974508
1.5356890

If, therefore, Herschel's observations were not reduced to the unit of distance, Halley's comet had a mass more than thirty-four times as great as that of the Earth's atmosphere.

But if, as is probable, Herschel reduced his measurements to the unit of distance, then I find that Halley's comet had a mass somewhat more than forty-five times that of the Earth's atmosphere; or 45.098.

μ

A

Regarding a comet as comparatively a small mass of gaseous matter, which largely changes its distance from the sun, the preceding analysis shows that the sun exercises a great and constantly varying influence over the comet's figure of equilibrium. As long as any completely closed figure of the comet is maintained,

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