ASTRONOMY.
32V
Table of the Equations of the Moon's Latitude , Kith their respective Arguments for September 2, 1830, at 10 j Hours, whence hertrue Latitude at that time is ascertained.
(Calculated from De La Lande’s Tables.)
No.
Argument.
Quantity of Argum.
Equat
on.
Powt
r.
6.
0
r
»
-
fL
1
P’s true place in orbit -f- complement node equated...
5
29
38
8,4
+ »
57,5
O
Double true distance p from 0 — Argument 1.
6
0
14
33,6
—
2,3
3
Argument 1 — O’s mean Anomaly.
Argument 1 — P ’s mean Anomaly.
3
28
5
26,5
+
2,7
4
1
19
38
34,5
13,4
5
Argument 4 — P’s mean Anomaly...
9
9
39
0,6
+
24,7
6
Argument 5 — p’s mean Anomaly.
4
29
39
26,7
+
1,0
7
Argument 2 4* © s mean Anomaly...
8
1
47
15,5
6,0
8
Argument 2 — Q's mean Anomaly.
3
28
41
5 i,7
3,2
9
Argument 2 4- p’s mean Anomaly...
10
10
14
7,5
+
1,7
10
Argument 2 — P’s mean Anomaly.
1
20
14
59,7
+
12,2
11
Argument 10 — P’s mean Anomaly...
9
10
15
25,8
+
5,1
Sum of the above affirmative equations 2' 52",9; sum of the re-maining negative ones 18',9; difiercnce, or true latitude of Moon north descending, 2' 34".
Determination of the Moon ’s horary motion in her orbit fromthe same arguments which w ere made use ot to reduce her meanto her true place in the zodiac.
Equation of Moon ’s horary motion in longitude resulting fromArg. 1, + O',22; Arg. 2, - 0",42; Arg. 3, - 0",7l3 ; Arg. 4,4- 0",72 ; Arg. 5, + 22",61 ; Arg. 0, + 0",4S ; Arg. 7, - 0",40 ;Arg. 8,+0",ll; Arg. 9,-0",09; Arg. 10,-0",55; Arg. 11 — 0", 11 ;Arg. 17, + 0",01 ; Arg. 13, — 0'',01; Arg. 14, — 0", 13 ; Arg. 10,
— O',05; Arg. 17.-f-0,03; Are;. ID, —[— 3A y 1 / ,09: Arg. 20, -j-41",46; and Arg. 21, — O’,57. Sum of the affirmative equations,36' 22",73; sum of the negative, 3",09; difference, or Moon ’strue horary motion in her orbit, 30’ 19",64. The equation — 8",67,resulting from the 22nd argument of longitude, applied to theMoon ’s horary motion in her orbit, gives 36' 10",97, her horarymotion in the ecliptic at the same time. In order to obtain theMoon ’s horary motion in longitude for an hour preceding or sub-sequent to the time for which the I) ’s true place is calculated,recourse must be had to the equations of the second order arisingfrom Arguments 5, 19, and 20, of the Moon ’s longitude. In thisexample, the first of these gives — 0", 14; the second-)- 0",92, andthe last is insensible. Their sum -)- 0",78 being added, according toits sign, to 36' 10",97, gives 36' 11*,75, the Moon ’s horary motion inecliptic at half past eleven, but added with a contrary sign gives36' 10", 19 , her horary motion at half past nine, equal time atGreenwich.
To obtain the Moon ’s horary orbitical motion in latitude, the1st, 2d, and 5th arguments of latitude must be made use of. Inthe example before'us the corresponding equations are — 2' 58",08,
— 4”,29, and -)- 0",04: their sum 3' 2",33 is the Moon ’s horarymotion in latitude in her orbit, and is exact enough for the pre-ceding and subsequent hour on account of her great proximity toone ot her nodes at the middle time. The equation — 19", 18’ap-plied to — 3' 2",33 gives the Moon ’s eeliptical horary motion inlatitude — 3' 21“,51. The sign —being prefixed to the latitude shewsthat it is decreasing, i. c. that the Moon is moving towards theecliptic. The Moon ’s horizontal parallax is determined from thearguments of longitude in the same way as the Moon ’s horary mo-tion. From Arg. 1, the first equation of her horizontal parallax is-f 0",2; Arg. 2. — 0”,3; Arg. 3, — 0”,5 ; Arg. 4, — 0",1 ; Arg.5, -f- 20", 1 ; Arg. 6, + 0“,5 ; Arg. 7, — 0”,6 ; Arg. 8, -)- 0",1 ;Arg. 9, + 0",4. Avg. 10, — 0",5 ; Arg. 19, -f 59' 1*,4; Arg. 20,-)- 27",2 ; and Arg. 21, — 0",6. The sum of the affirmative equa-tions is 59' 49",9, and the sum of the negative 2", 6; their difference59' 47",3 is the true lunar horizontal parallax at Paris or London ,the difference for either city being insensible.
To find the equal time of the ecliptic opposition of the Sun andMoon at the time of the eclipse, first subtract the Sun ’s true place,5s. 9° 53' 8",5, at 10 J. hours, from that of the Moon , 11s. 9° 49'34',7; the remainder, 5s. 29° 55'26",2, also subtracted from sixsigns gives 3' 33",8, the distance oflhe luminaries from the sizygv.Then w ith the relative hourly motion of the Sun and Moon, 33'-VOL, 1 .— NO. 21.
45",54, (which is the dilference of their respective horary motionsin the ecliptic,) make the following proportion: As 33' 45",54, isto 1 hour; so is 3' 33",8, the quantity that the Moon is short of; the opposition, to 6' 19',9, which added to 10’- hours (because thei Moon is not yet at the full) the sum is 10b. 36m. 19,9s. the exact| equal time of the ecliptic opposition at Greenwich. 'I o obtain, the true latitude of the Moon at tire time of the ecliptic opposition,
! say, 'as 1 hour is to 3' 21",51, the Moon ’s horary ecliptic motionj hi latitude ; so is 6m. 19,9s. the time of the opposition after half1 past ten, to 21”,26, which subtracted from 2' 34", the calculatedlatitude, because ttie Moon is approaching the ecliptic, gives A12",7 the latitude sought.
Sect. VII. Or the Calculation of Lunar . Eclipses.
If the Moon , at the instant of the true opposition, i. e. when sheis 180° from the Sun , be far enough from her nodes for her latitude toexceed 30', the eclipse will not he total; and if the latitude be-greater than 63'| there w ill be no eclipse, because the semidiamc-ter of the Earth’s shadow, in the orbit ot the Moon , never occu-pies more than 46' 43", and the semidiameter of the Moon neverm ire than 16'48"; so that, in order that the Moon ’s limb may! toach the Earth 's shadow, it necessary that the distance ofi their centres, or the latitude of the Moon , exceed not 63'£, andthat she he not more than 12° 34' from one of her nodes. As thelunar motion is measured by the celestial ares which she appearsto describe, it is also neces-ary to measure m the same way theshadow which she passes through in the time of an eclipse, i. e. thewidth of the Earth's dark conical shadow in the orbit of the Moon .LetS he the centre of the Sun , (Hate XXV*. tig. 1,) T the centreot the Earth, L that of the Moon in opposition, AO the diameterof the Sun, BG the diameter of the Earth , APO the cone oi theshadow that the Moon passes through from C to E, LC the dia-meter of the shadow in the orbit of the Moon ; this lii..‘ LC is theradius of a circle whence is formed the section perpendicular tothe axis of the cone of the shadow in the region of the Moon . Theangle CTL, which has the Earth ’s centre for its angular point;and for a base the side C L, is what is called the semidiameter ofthe shadow, that is to say, the arc which the Moon describes inher orbit during the semiduration of a central eclipse. The sideAT of the right-angled triangle CAT, continued to D, has its ex-ternal angle CTD equal to the two internal opposite angles, takentogether, i. e. to the angles BAT and BCT, one of winch is thehorizontal parallax of the Sun, the other that of the Moon . Hencethe ijmgle.CTD is equal to the sum of the parallaxes. If we takefrom it the angle LTD, there w ill remain the angle CTL, or thesemidiameter of the shadow : but the angle LTD is equal to theopposite angle ATS, which is the Sun ’s apparent semidiaineter ;consequently, if from the sum of the parallaxes we subtract thesemidiameter of the Sun , the remainder w ill be the semidiameterof the Earth’s shadow in the orbit of the Moon .
Example. The horizontal parallax of the Moon , at the instantI of the opposition of September 2, 1830, will be 59'47“,3 ; the| horizontal parallax of the Sun is constantly 8",6; their sum thenI is 59’ 55",9 : if we take awav from this sum the semidiameter of4P the