Second Suppofition of the first

Time the Comet should exploy to defcribe the Angle contained by the two vector Rays.

If the Time thus found, does not agree with the obferved Time, another Value is to be affumed, for the curt Distance (Z) corresHypothefis, ponding to the fecond Obfervation, retaining the affumed Distance (Y) correfponding to the firft, and the heliocentric Longitude and Latitude of the Comet from thence deduced, and all the Operations indicated in the foregoing Articles being repeated, another Expreffion will be found for the Interval of Time between the two Obfervations. Which if it approaches nearer the obferved Time, the fecond Value affumed for the Distance (Z) is to be preferred to the firft; if not, a third Value is to be affumed for this Distance, and by the Increase or Decrease of the Errors, the Value to be affumed for it, fo that the Interval of Time calculated may agree with the obferved one, will easily be discovered, and confequently a Parabola will be found, which answers the two first Obfervations, which may be called first Hypothefis.

SECOND

This Parabola anfwering the two firft Obfervations would be the OrHYPOTHE bit fought if it answered likewife the third Obfervation; but as this never happens, another Parabola is to be found which anfwers the two first Obfervations, by increasing or diminishing, at will, the curt Distance (Y) preferved conftant in the firft Hypothefis, and preferving it ftill conftant, but varying the fecond affumed Distance (Z) until this fecond Parabola is obtained.

Paffage
at the
Perihelion.

Place of the
Node.

The third Obfervation calculated in thofe two Parabolas, will fhew which of them approaches nearest the true Orbit fought. To calculate this third Obfervation in each Hypothefis, the Time of the Paffage of the Comet at the Perihelion, the Inclination to the Ecliptic, and the Place of the Nodes of each Parabola is first to be determined.

To determine the Time of the Paffage of the Comet at the Perihelion, find the Number of Days correfponding to one of the two true Anomalies; for Example, to that which correfponds to the first Obfervation in the Parabola whofe perihelion Distance is 1, as before directed, the Logarithm of this Number of Days added to of the Logarithm of the perihelion Distance, will be the Logarithm of the Interval of Time elapfed between the first Observation and the Paffage of the Comet at the Perihelion, which is to be added to or fubtracted from the Time of the Obfervation, according, as it was made before or after the Paffage of the Comet at the Perihelion.

To determine the Place of the Node, fay, As the Sine of the second Arc X' is to the Sine of the first Arc X, fo is the Tangent of the Motion of the Comet in the Ecliptic, to the Tangent of an Angle (R). Then the Radiius, is to the Sine of the leaft Latitude, as the Tangent of the Angle R, to the Tangent of the Distance from the Node. By the Means of this Dif

tance from the Node, and the heliocentric Longitude of the Comet, the heliocentric Longitude of the Node is obtained. With which and the Distance measured on the Orbit of the Comet, the Place of the Periheli- Inclination, on is Determined. To find this Distance fay, As the Sine of Angle R, to Radius, fo is this Distance measured on the Ecliptic, to the Diflance required. To determine the Inclination fay, As the Radius is to the Sine of the Angle R, fo is the Cofine of the leaft Latitude, to the Cofine of the Angle of Inclination.

The Elements of each Parabola being determined, the Place of the Comet feen from the Earth, anfwering to the third Obfervation, is computed in each, by the following Rules.

First, Take the Logarithm of the Difference between the Time of the third Obfervation, and the Time of the Paffage of the Comet at the Perihelion; fubtract from it of the Logarithm of the perihelion Distance, the Remainder will be the Logarithm of the Difference be Rules for tween the Time of the third Obfervation and the Time of the Paffage finding the of the Comet at the Perihelion of the Parabola, whose perihelion Di- heliocentric Longitude ftance is 1. Secondly, Find the true Anomaly correfponding to this and Lati

tude of a

Time, by folving the Equation t3+3t=; (b) in which expreffes comet.

b 27,4038

the Tangent of half the true Anomaly, and b the Time employed in
defcribing it. Thirdly, When the Motion of the Comet is direct, add
this true Anomaly to the Place of the Perihelion, if the third Obferva-
tion was made after the Passage of the Comet at the Perihelion; But
fubtract it from the Place of the Perihelion if the Obfervation was made
before the Paffage at the Perihelion. And when the Motion of the
Comet is retrograde, add the true Anomaly to the Place of the Perihe-
lion, if the Obfervation was made before the Paffage at the Perihelion ;
but fubtract it from the Place of the Perihelion, if the Obfervation was
made after the Paffage at the Perihelion; by this Means, the true he-
liocentric Longitude of the Comet in its Orbit is obtained. Fourthly,
Take the Difference between this Longitude and that of the afcending
Node, which will be the true Argument of the Latitude of the Comet.
Fifthly, fay, As the Radius is to the Cofine of the Inclination, fo is the Tangent
of the Argument of Latitude, to the Tangent of this Argument measured on the
Ecliptic; which added to the true Place of the Node, gives the heliocentric
Longitude reduced to the Ecliptic. Sixthly, fay, As the Radius is to the Sine
of the Argument of Latitude, fo is the Sine of the Inclination of the Orbit
of the Comet, to the Sine of its beliocentric Latitude, which, when the Mo-
b
(b) The Equation 3+3=-
may be folved thus: Make a Right-angled Triangle,
27,4038
one of whofe Sides is expreffed by 1. and the other by

(H), find two mean Proportionals between H+-
ence will be the Value of r.

calculate the Hypotheneuse

b 54,8077

54,8077
and H-

b 54,8077

and their Differ

Rule for finding the

cuit Diftance.

Rules for

tion of the Comet is direct, is North or South, according as the Argument of Latitude is lefs or greater than fix Signs; and when the Motion of the Comet is retrograde, it is North or South according as the Argument of Latitude is greater or lefs than fix Signs. Seventhly, Add the Logarithm of the Cofine of the heliocentric Latitude to the Log, of the perihelion Distance, and fubtract from this Sum the Log. of double of the Cofine of half the true Anomaly, the Remainder will be the Logarithm of the curt Diflance correfponding to the third Obervation. Eighthly, Take the Difference between the Logarithm of the curt Distance, and that of the Distance of the Earth from the Sun, add 10 to the Characteristic of this Difference, and it will be the Logarithm of the finding the Tangent of an Angle; from which fubtract 45. and to the Logarithm geocentric of the Tangent of the Remainder, add the Logarithm of the Tangent Longitude of the Complement of half the Angle of Commutation, the Sum will be the Logarithm of the Tangent of an Arc, which add to this Complement, if the curt Distance of the Comet from the Sun exceeds the Distance of the Earth from the Sun, but fubtract from this Complement if the Distance of the Comet be less than that of the Earth; in order to obtain the Angle of Elongation, which added to or fubtracted from the true Place of the Sun, according as the Comet feen from the Earth, is to the Eaft or to the Weft of the Sun, will give the geocentric Longitude of the Comet. Ninthly, and laftly fay, As the Sine of the Angle of Commutation, is to the Sine of the Angle of Elongation, fo is the Tangent of the heliocentric Latitude of the Comet to the Tangent of its geocentric Latitude. The Longitude and Latitude thus found ought to agree with the observed ones, if the Parabola obtained was really the Orbit described by the Comet.

and Lati

tude.

VII.

Example. Let it be proposed to find the Elements of the Parabola defcribed by the Comet which was obferved in Europe; the beginning of March 1742, with a very remarkable Tail,coming with extraordinary Rapidity from the southern Hemisphere, and afterwards advancing towards the North Pole, its heliocentric Motion being retrograde, and its Velocity and Splendor decreasing to the 6th of May, when it disappeared.

1 Suppofition, Y=0,879, Z =0,957 of the mean Distance of the FIRST HYEarth from the Sun, then Angie C=105° 42′ 48′′, C=61° 31′ 0′′, POTHESIS. C+A=164° 9′ 52", and C+A=118° 9' 17', wherefore Angle D Heliocen 15° 50' 8", and Angle D=61° 50' 43", consequently the heliocentric and LongiLatitudes, E-12° 31' 42" North and E520 3' 38", and the Log. of tude of the the vector Rays, V=9,954455 V'=0,192159.

tric Latitude

Comet.

The Angle of Commutation D=15° 50' 8", being added to 5 14°, 27' 44", and Angle D'=61° 50' 43" fubtracted from 7s 4° 27' 16", the correfponding Longitudes of the Earth, gives the heliocentric Longitudes of the Comet, L=6° 0° 17′ 52", and L'=5° 2° 36′ 33′′; their Difference Angle contained by F 27° 41' 19" is the Motion of the Comet in the Ecliptic, the Arc the two vecX will be found 34° 37' 11", and Arc X=42° 51' 7"; confequently tor Rays. the Angle contained by the two vector Rays 45° 22' 8".

The Log. of the greatest vector Ray, 0,192159 lefs the Log. of the least, 9,954455=0,237704, and its Half 10,118852, 10 being added to its Characteristic, is the Tangent of 52° 44′ 38', from which 45° being fubtracted, and to the Log. of the Tangent of the Remainder 7° 44' 38", the Log. of Cotangent of 11° 20' 32", the of the Motion (45° 22' 8",) of the Comet in its Orbit being added, the Sum will be the Logarithm of the Tangent of 34° 8' 5", whereby the Halfs of the two true Anomalies are found to be 22° 47′ 33′′, and 45° 28' 37" 1 True Ano.. confequently the leaft true Anomaly 45° 35' 7", and the greateft 90°, malies. 57' 15"; and their Difference being equal to the Motion of the Comet in its Orbit, thofe two Anomalies are on the fame Side of the Perihelion. The Log. of the perihelion Distance will be found =9,883835. Perihelion To determine the Time the Comet employed to defcribe the Angle Distance. contained by the two vector Rays, to the conftant Log. 1,9149328 adding 0,007233 Log. of the Tangent of 45°. 28'. 37", and to the conftant Log. 1,438112 adding 0,021699 Triple of the Log. of this fame Tangent. I find 83,592 and 28,808 for the Numbers correfponding to 1,922166 and 1,459512 Sums of thofe Lagarithms, confequently 112,400 Days is the Time correfponding to the true Anomaly 90°. 57 Interval of 15", in a Parabola whofe perihelion Diftance is 1. By a like Procefs, I Time be find the Number of Days 36,579 correfponding to the true Anomaly tween the 45° 35' 7", in the fame Parabola, I take the Difference 75,821 of vations calthofe Times, because the two Anomalies are fituated on the fame Side culated. of the Perihelion, whofe Logarithm 1,879789 added to 9,825752 the of the Log. of the perihelion Distance, is the Log. 1,705541, to which correfponds 50,762 Days, Time employed by the Comet to defcribe the Angle contained by the two vector Rays.

Comparing this Time with the Interval 50,728 between the two Obfervations, I find it exceeds it by 0,033, I therefore make a Variation of 0,001 in the Distance (Z), in order to difcover which Way,

two Obfer

Second Suppofition of

pothefis.

and by how much the Elements of the corrfponding Parabola will be changed.

11 Suppofition, Y=0,879, Z=0,956, and repeating the fame the firft Hy- Calculations as in the firft Suppofition, I find the heliocentric Latitudes E 120 31 42", E' = 52° 1' 54", the Log. of the vector Rays, V=9,954455, V=0,191424, the heliocentric Longitudes, L=6 0° 17′ 52", L'=5 2° 43' 11". The Motion of the Comet in the Ecliptic =27° 34' 41", and the Motion of the Comet in its Orbit =45° 18' 13" the true Anomalies 45° 32′ 3′′, and 90° 50' 16", the correfponding Days 36.529 and 112,056, the Log. of the perihelion Distance =9,883997; finally the reduced Time employed in defcribing the Angle contained by the two vector Rays 50,594 Days. From whence I find that by increasing Z by the Quantity 0,001, I diminish the Time by 0,168 And I fay, 0,168: 0,001::0,033 0,0002. I diminish therefore Z by 0,0002 to obtain a Parabola anfwering the Conditions required.

III Suppofition, Y=0,879, Z=0,9568, and I find the heliocentric Latitudes, E=12° 31′ 42′′, E'=52° 3′ 16′′, the Log. of the vector Rays, V=9,954455, and V'=0,192009; the heliocentric Longitudes, L=6$ 0° 17′ 52", and L'=5$ 20 37' 53"; the Motion of the Comet in the Ecliptic, 27° 39′ 59′′; and the Motion in its Orbit 45° 21′ 22′′; the true Anomalies 45° 34′ 28", and 90° 55' 50"; the correfponding Times 36,568, and 112,330 Days: The Log. of the perihelion Distance 9,883870, and the Time reduced employed in defcribing the Angle contained by the two vector Rays, 50,728 Days, agreeable to Obfervation. Having found a Parabola answering the two first Obfervations, I fearch SECOND HY for another, answering the fame Observations, by making a Variation POTHESIS. in the Distance (Y) preferved conftant in the first Hypothefis.

IV Suppofition, Y=0,878, Z=0,957, and I find the heliocentric Firft Suppo- Latitudes, E=12° 42′ 11′′, E=52° 3' 38", the Log. of the vector fition of the Rays, V=9,954257, V=0,192159, the heliocentric Longitude,s L= fecond Hy- 60° 31' 54", and L'=5 2 36' 33"; the Motion of the Comet in the pothefis.

Ecliptic270 55' 21", the Angle contained by the two vector Rays =45° 17′ 56", the true Anomalies 45° 44′ 56′′ and 910 2′ 52", the correfponding Times 36,743 and 112,680, the Log. of the perihelion Diftance 9,883115, the reduced Time employed in defcribing the Angle formed by the two vector Rays 50,714, which differs by 0,014 from the obferved Interval, confequently by diminishing Y by 0,001, the Time is diminished by 0,048. I fay, 0,048 0,001 0,014: 0,0003. v Suppofition, Y=0,8783 Z=0,957, I find the heliocentric LatiSecond Sup. tudes, E-12° 39' 2" E=52° 3' 38" the Log. of the vector Rays, pofition of V=9,954316 V=0,192159, the heliocentric Longitudes, L=60° the fecond 27 40", L'=5° 2° 36′ 33", the Motion of the Comet in the Ecliptic Hypothefis. 27° 51' 7" the Angle contained by the two vector Rays 45° 19′ 20′′, the true Anomalies 45°41' 45" and 91° 1' 5" the correfponding Times 36,689,