P'- A 7
These limits will extend throughout the entire duration of the general eclipse, and
form the distorted figure of an 8, the first and last points being the places of begin-
ning and ending on the Earth.
8. When n 90 when n is
10.
for { b a t?} *e time of middle
11.
to be less than 90 and positive
12. Place following,
.,_ tan (S m)
sm/ = cos(S m)cos5 tan h = - r
sm ^
H = Apparent Greenwich time
Longitude E. = h H
h to be in the same semicircle with S m
13. Place advancing,
tan (S + m)
sin / = cos (S + m) cos J tan h = -
Longitude E. = h H
h to be in the same semicircle with S + m
14. FOR A MORE ACCURATE DETERMINATION,
Find the values of D, $, P' A '.
15. Find P' =/> (P ), for a latitude equal to the complement of $ at 6 .
/* sin i' = D!
/* cos i' =; ! cos D + [9*41796] P 7 sin 5
diff. dec. + P' . K *o^ * sin l
k = 7-: = < zw seconds = [3'55030]
cos (i 7 i) /*
16. At the Place,
When diff. dec. and $ have {Afferent 6 } signs ' app * time of true 6 = { ^^l ~ *
which compared with the Greenwich apparent time of the true 6 , will determine the
longitude of the place.
17. k cos i _ A' sin w
A 7 /*
> 4- O *
tan 5
/ to be of the same name as diff. dec.
80 On Eclipses.
IV. PLACES WHERE THE MIDDLE OF THE ECLIPSE IS SEEN WITH THE
SUN IN THE HORIZON.
18. When n < P' (s + P' (5 + s + cr, compute 90 when diff. dec. is negative.
The places to be determined by proceeding with o> as for the beginning and ending
of a phase.
22. FOR A MORE ACCURATE DETERMINATION AT ANY TlME,
Find P' = j> (P T) for the latitude before found.
Find (#), (y), S and A as in No. 14.
For the time of 6 form the constants
(A) = [0-58204] Wl cos D (B) = [0'58204] D,
Compute v from the equations
(A) + O) sin S (B) (y) sin I
\ cos v = } T v ' X sm v = '. ^
P' cos S P' cos I
23. Then
sin
_sin(S + ^
to be + or but less than 90
24. . ; _ tanM
sm / = cos M cos S tan h = : r-
smS
If K 9 K' be both less than s + ff, the angles M, M 7 may be both used in these equa-
tions, and two places determined. If one of the quantities K, K 1 be greater than s+v,
the Corresponding M will be excluded, and only one place determined with the other
value. If P' A 7 only one limit will have place, viz.
26. FIRST AND LAST POINTS OR PLACES OF ENTRANCE AND DEPARTURE.
n+ A 7 /cP'
-=
Places of entrance and departure determined as in Nos. 4 and 5, for the beginning
and ending of a phase, using a = ( t) w and b = ( t) + w.
For the appearance of external contact these determinations are included in No. 18,
and therefore need not be repeated for these limits.
27. PLACES FOR ANY TIMES WITHIN THE LIMITS.
Prepare the following constants, using S at 6 ,
/ TV X ~~ / _ _r_ A ' s * n l
u ~~ A C a ""-*- cosD'
n n+ A 7
n. 7* -t- f\ '
E =
c(n+A')
28. Let t be the time from the middle of the general eclipse
cos w
tan w 1 = t . E
cos w
M=(-0+ w/
29. tan 6 = tan Z cos M
n M tan /
sine sin Z cos M
tan (/i ') = - 1 ^ tan M tan / = tan (6> + D 7 ) cos (h 7 )
"
cos (6> + D') cos (h 7 ) cos /
< 90 and same sign as cos M ; and h ex 1 to be in the same semicircle with M.
On Eclipses,
83
30. FOR A MORE ACCURATE DETERMINATION AT ANY TlME,
Find P'~/> (P T) for the latitude before found.
Also, with Z find the augmented semidiameter s' = s + augmentation, from the
table annexed.
Augmentation of the 3) 's Semidiameter.
Argument True Zenith Distance Z.
For P
Var. for
ForP
Var. for
For P
Var. for
Z
= 54'
10'
in P.
Z
= 54'
10'
in P.
Z
= 54'
10'
in P.
o
ii
n
o
//
/;
n
n
14*0
5-7
30
12 '1
4-9
60
6-9
2-9
1
14 '0
5-7
31
12*0
4-8
61
6'7
2 '8
2
14 '0
5-7
32
11-9
4-8
62
6-5
2-7
3
14-0
5-7
33
11-7
4'7
63
6-2
2-6
4
14 '0
5-7
34
H-6
4'7
64
6-0
2'5
5
13-9
5-7
35
11 '5
4-7
65
5 '8
2 '4
6
13'9
5-7
36
11 '3
4-6
66
5-6
2'3
7
13-9
5-7
37
11 '2
4-6
67
5 -4
2'2
8
13*8
5-7
38
11 '0
4-5
68
5 -2
2 '1
9
13-8
5-7
39
10-8
4-4
69
4-9
2*0
10
13'8
5-6
40
10-7
4.4
70
4'7
1'9
11
13-7
5-6
41
10-5
4*3
71
4-5
1-8
12
13-7
5-6
42
10-3
4'3
72
4-2
1'7
13
13-6
5-6
43
10 '2
4 '2
73
4 '0
1-6
14
13-6
5-5
44
lO'O
4-1
74
3 '8
1 '5
15
13'5
5-6
45
9'8
4-1
75
3-6
1 '4
16
13 '4
5-5
46
9*7
4'0
76
3 '3
1*3
17
13'4
5 '4
47
9*5
3-9
77
3 '1
1'2
18
13*3
5'4
48
9'3
3-9
78
2-8
1 *1
19
13*2
5*4
49
'2
3'8
79
2-6
I'l
20
13'1
5 '4
50
9-0
3-7
80
2 '4
i -o
21
13 '0
5'4
51
8 '8
3-6
81
2'1
0-9
22
12 '9
5 -3
52
8'6
3'5
82
1*9
0-8
23
12-8
5 '3
53
8'4
3 '4
83
l'7
0-7
24
12 '7
5'3
54
8'2
3 '3
84
1 '4
0-6
25
12-6
5-2
55
8'0
3'2
85
1 *2
0-5
26
12 '5
5'1
56
7'8
3*2
86
1 -0
0-4
27
12'4
5-1
57
7-5
3'1
87
0-7
0'3
28
12'3
5 '0
58
7'3
3-1
88
0-5
0'2
29
12 '2
4.9
59
7'1
3-0
89
0-3
O'l
30
12 '1
4-9
60
6-9
2-9
90
o-o
o-o
Then,
f Partial ) ( s' + 90 when n is negative
35. S = ( t) +
On Eclipses.
85
36.
A
sill Z =
tan h =r
sin0
check - - - -
cos (0 + d
sin
(S at d )
tan = tan Z cos S
tan S tan / = tan (9 + S) cos h
sin Z cos S
cos (Q -f S) cos h cos /
, same sign as cos S, and less than 90'
, same semicircle with S.
37. FOR A MORE ACCURATE DETERMINATION AT ANY TIME,
Find P', S, A , as in No. 14, and proceed again with these as in No. 36.
38. PLACE WHERE THE ECLIPSE WILL BE CENTRAL AT NOON.
(S at 6 )
cliff, dec.
sin Z = , / = 5 + Z
Apparent Greenwich time of true <5 = Longitude W.
Z < 90 and same sign as diff. dec.
39. For a more accurate determination find the horizontal parallax for the latitude
and with it repeat the operation.
%* All latitudes in the preceding formulae are to be recognized as geocentric, and
will therefore need reducing by the table at page 57.
NAUTICAL ALMANAC, 183G. APPENDIX.
36 On Eclipses.
EXAMPLES.
For an elucidation of the practical application of the preceding formulae we shall
take the Solar Eclipse of May 15, 1836. At the time of New Moon, viz. 2 h 7 m '0,
the Moon's latitude ft is 25' 43", which being less than 1 23' 17" the eclipse is
certain. See the limits at page 55. The elements of this eclipse, as related to the
equator, are
Greenwich Mean Time of 6 in R. A. May 15 2 21 22 -9
}) 's Declination - N. 19 25 9 '8
O's Declination N. 18 57 58 '8
D 's Hourly Motion in R. A. ------- 30 8 '3
O's Hourly Motion in R. A. 2 28 *2
3) 5 s Hourly Motion in Declination - - - - N. 9 58 '7
J s Hourly Motion in Declination - - - - N. 35*1
}) 's Equatorial Horizontal Parallax ---- 5423*9
's Equatorial Horizontal Parallax - - - - 8 '5
3) 's True Semidiameter 14 49 '5
's True Semidiameter 15 49 '9
from which we prepare the following values :
O I II I II
2)'sDec. - - + 19 25 10 }) 's H. M. in R. A. 30 8
0's Dec. - - + 18 57 59 0'sH.M.inR.A. 2 28
Diff. Dec. + 27 11 i 27 40
3) 's H. M. in Dec. + 9 59 }) 's Eq. Hor. Par. 54 24
0's H. M. in Dec. + 35 0's Eq. Hor. Par. 9
D! + 9 24 Rel. Eq. Hor. Par. 54 15 log. - 3 '51255
const. 9'99929
P' - - 54 10 log. - 3-51184
I. BEGINNING AND ENDING ON THE EARTH.
D, + 9' 24" 275128 (1)
! 27 40 3 -22011
9-53117
D + 19 25''2 cos 9 '97456
ftan 9-55661 (2)
e +1Q 49 - - J
jcos - - - 9-97349 (3)
diff. dec. + 27' 11" - 3 -21245
n + 25 34 3-18594
sin t - - 9 '53010 (2) + (3)
const. - - 3-55630
6 -27234 (4)
c 3-52106 (4) (1)
t - - + 19 m 56* - - c tan i - - 3 -07767
(5 - 15 2 21 23
15 2 1 27 - Middle of general eclipse
On Eclipses.
87
p/ 54/ 10" A for Central Phase
+ ff - - 30 39
84 49 = A for Partial Phase
r -
Partial
n - -
+ 3 -18594
A - -
3 70663
fcos -
jtan-
+ 9-47931
-49999
c - -
3 '52106
d h m a
________
2 54 57 -
- 4 -02105
15 2 1 27
14 23 6 30 Beginning
15 4 56 24 Ending
(-0 19 49
w - - - 72 27
92 16
- - - + 52 38
49' |
T -
Central
n - - -
A - -
icos - -
tan -
c - -
h 3 -18594
3 '51184
dog.P')
h 9*67410
o -27109
3-52106
1 43 17 - - 3 '79215
15 2 1 27
15 18 10 Beginning
15 3 44 44 Ending
(__) 19 49
u - - - 61 49
a
---- 81 38
... + 42
PLACE OF PARTIAL BEGINNING.
cos a 8 -59715
cos S - - + 9 -97576
tan a
sin 5
tan h
h
H
HI "40251
- + 9-51191
sin / 8 -57291
. . i -89060
/ - - S. 2 9 7
Reduction 1
89 16 7
347 37
Greenwich time 23 6 30
Equation - - 3 56
Hin
{time - - - 23 10 26.
space - - 347 37'
Latitude S. 2 10 Longitude W. 76 53
In the same manner may the places of Partial Ending, and Central Beginning and
Ending, be calculated, which will come out
o / o /
Partial Ending - Long. E. 28 51 Lat. N. 35 13
Central Beginning Long. W. 98 16 Lat. N. 7 58
Central Ending - Long. E. 52 41 Lat. N. 44 50
II. RISING AND SETTING LIMITS.
P 7 54 10
23 31
84 49
/ //
p?=. 11 46
q = 42 25
Since n > P' A ', these limits will extend throughout the whole duration of the
eclipse ; and we may therefore calculate the position of a place for any time between
the Greenwich times 14 d 23 b 6 m 30 8 and 15 d 4 b 56 m 24 s . As an Example take the
time 15 d O h 30 m .
88
S m
S +771
Assumed time -
Time of Middle
On Eclipses.
d h m a
15 30
- - - - 15 2 1 27
3.*7 o no o
1 I /
o . c .-> i n A
1
19 49
/
{foYl -
O.p i op 7
58 51
u> - . - 58 *) 1
78 40
34 2
3'J.*7 1 O 1
112 42 i
A pi j p
flog A - - -
44 38 "
A p 12 56
A - - - - 17 43
- - 2-88986
Comp. log P 7
- - 6-48816
o /
2)18'93264
%m 17 *9
m 34 2
sin
PLACE FOLLOWING.
h m a
cos (S m) 9*58648 tan (S m) + 0*37850 Greenwich lime 30
cos S - - - + 9*97576 sin 2 - - - + 9 '51 191 Equation - - + 3 56
time 33 56
sin / - - -
9-56224
tan h -
'86659
/
S. 21 24'
h -
82 15 7
Reduction -
8
H -
8 29
1 H in
\space 8 29'
Latitude- - S. 21 32 Longitude- W. 90 44
PLACE ADVANCING.
cos (S-fw) - - + 9 -85225
T
y y/<>jv
9'SOSOl
I
4P i o
Reduction -
11
tan (S+m) 9 '99444
sin $ +9 '51191
tan h -4
h
II
+ o '48253
~108 13
Latitude - - - N. 42 29
Longitude --- W. 116 42
By taking S = ( t) + w instead of ( w, similar computations will give
the places following and advancing for the interval t = l h 31 m 27* after the time of
middle, or for the Greenwich time 15 d 3 h 32 m 54 s . Much time will be saved by taking
the computations two and two in this manner.
1II.-PLACE WHERE THE RISING AND SETTING LINES INTERSECT.
o /
90 o
- - 18 58
/ - - 71 2
On Eclipses. 89
p - - 9 '99872
P v - - 3 '51255
P'- - 54' 5" 3 '51127
sin - + 9*51191
const. - - 9'41796
+ 4 36 +2 '44114
^ . . - . 3 -22011
cos D - - 9 '97456
26 6 3 -19467
30 42 r= /i cost' - - + 3/26529
.- + 2-75128
- +~9~'48599
- + 9 "98056
- . - . +3 -28473
+ 26 54 - + 3'20790
cos (t'-~ c) - - 9 "99948
k \- 3 -20842 +3 -20842
cost- - +9'97349 sin i h 9 '53010
3 -55630
+ 3 '18191
3 '26458 + 6 -29482
cos w - - 9-91733 log; - - - +3'01009
sinu, - - 9 75036 h~~^-
3-01494 ---
3 '28473 12
T- 9-73021 App. Time true c$ 1 1 42 56 at the place
15 2-95424 ~~h m 7
2 21 23
8 4' - - _2j68445_ Equation 3 56
cos 9^99568 App. Time true 6 2 25 19 at Greenwich
f time 9 h 17 m 3T ) T
tan / - - -45953 Long, in { r } E.
\space 139 84' J
/ - - N. 70 5l'
Reduction 7
Latitude N. 70 58
Thus we find the required place to be in Longitude E. 139 24 r and Latitude
N. 70 58', where simple contact will have place at sunset and again at sunrise; also
the middle of the eclipse would be seen at midnight if it were not intercepted by the
opacity of the Earth. The duration of the eclipse will correspond with the duration
of the night, and therefore no portion of it will be visible.
90 On Eclipses
IV. PLACES WHERE THE MIDDLE OF THE ECLIPSE HAS THE SUN IN
THE HORIZON.
In the present case n is >P' (-J+<0 and <.$+&. We must therefore proceed
as in N 19.
1 . For the extreme Points,
c - - 3-52106
P'- - 3 '51184
+ 25 34 3 '18594
cF
30 39 - 3-84696 (1)
5 5 2 -48430
P'- - 3 -51184
. cos 8 '97246
- - 95 23'
sin 9 "99808 (2)
(0 19 49 7 3 - - 1 56 39 3 -84504 (1) + (2)
o> 8 95 23 21 27 time of middle
a 115 12 0448 time of beginning
b + 75 34 3 58 6 time of ending
PLACE OF BEGINNING, OR FIRST EXTREME PLACE.
h m s
cos a 9 '62918 tan a - - + '32738 Greenwich time - 4 48
cos $ + 9-97576 sin $ - - + 9-51191 Equation- - - - 3 56
time 8 44
sin / 9 -60494
tan h -
h -
H -
'81547
/ S.23 45
Reduction 8
o /
81 18
- + 2 11
. /ti
in {
I S
space 2 11'
Latitude S.23 53 Longitude W. 83 29
PLACE OF ENDING, OR LAST EXTREME PLACE.
cos b +9 -39664 tan b - - + '58943 Greenwich time -
cosS + 9*97576 sin $ - - + 9*51191 Equation- - - -
sin/ + 9-37240 tan h 1*07752 f time
H in
/ N.lsV h - - + 94V
Reduction 5 H - - 60 31
Latitude N. 13 43 Longitude E. 34 16
On Eclipses,
91
2. For the extreme Times,
cP'
the value of r l taken out from the preceding logarithm of is l h 57 m 10*.
n
2 1 27 time of middle
1 57 10 - - r l
4 17 first appearance
3 58 37 last appearance
PLACE OF FIRST APPEARANCE.
sin i +9 -53010
cos +9 -97576
sin I 9 '50586
I S. 18 42
Reduction 7
cot t - - + -44339
sin S - - + 9 -51191
0-93148
Greenwich time -
Equation - - - -
time
o ;
h - - 83 19
H - - + 2 3
f ti
H in I
I si
space
Latitude S. 18 49 Longitude W. 85 22
PLACE OF LAST APPEARANCE.
Latitude N.I 8 49
h-
H
o /
83 19
180
96 41
60 38
Longitude E. 36 3
Greenwich time -
Equation - - - -
time
H in I
space
3 58 37
3 56
For the computation of places in this line, we have therefore the whole range
between the Greenwich mean times O h 4 m 17* and 3 h 58 h 37 s . As an example,
take the time l h 30 m .
Time of Middle 2 l
1 30
cos a - - + 9*91132
cos S - - + <
sin / . - 4. 9 -88708
/ - -' - N. 50 27
Reduction - 11
o 31 27 :
<-.)- -"-19 49 ?- 3 ' 846 9 6
15 34 - - sin - 9 '42881
a . . . 35 23
b 4 15
tana 9 '85140
sin 5 - - + 9 -51191
tan h - -
h m s
Greenwich time 1 30
Equation - - - 3 56
0-33949
h - - 114 35
H - - + 23 29
Hinf
l s l
time 1 33
space 23 29'
Latitude - N. 50 38 Longitude W. 138 4
)2 On Eclipses,
By similarly using the angle b we shall find the position for the interval 31 m 27'
after the time of middle, or for the time 2 h 3~2 ra 54 s ; thus,
Greenwich time 2 32 54
Equation - - - 3 56
cos b - -
gin / - -
+ 9 "99880
+ 9'97576
tan b -
sin -
tan h -
8 -871D6
- + 9-51191
+ 9-97456
- + 9-35915
/ - - - N. 70 35'
Reduction - 7
Latitude - N. /O 42
h 167 7
H - - + 39 13
206 20
153 40
H in
time 2 36 50
space 39 13'
{W
F
The places may be computed by two together in this way ; and it will perhaps be a
little more convenient to assume a value of t in the first instance. We may take any
value which does not exceed r t or l h 57 m 10 s . In the present example we should
take =:31 m 2/ s , and begin as under:
(-0
IV
o ;
- 19 49
- 15 34
35 23
4 15
log t - - 3 ^7577
rP'
3 '84696
n
sn
- - .9 '42881
and then proceed for the places as above.
Time of middle - - -
/
h m s
2 1 27
Q 1 27
O 1 & j
Time before middle -
Time after middle -
1 30
2 32 54
V. NORTHERN AND SOUTHERN LIMITS.
1. FOR THE PARTIAL PHASE, we have only Southern Line of Simple Contact,
Constants E, cos w, D', '.
i //
s + 6" 14 56
ff 15 50
A' - --- 30 46
n - - - + 25 34 - - - + 3 '18594
n _ A' - ~512~ 2*49415
0-69179
c - - - 3 -52106
- - 2 '49415
P' 3 '51184
E 7 -17073
cos w 8 '98231
A' 3 -26623
cos i +9 ^7349
log u -f 3 -23972
u - - 28 57
S +18 57 59
D' +19 26 56
- - - 3 -26623
sin i +9 -53010
+ T-79633
cosD'-f 9 '9/448
log ' 2 '82185
o! 11' 4 /;
On Eclipses. 93
The extreme places will be the same as those which have the middle of the eclipse
with the Sun in the horizon, page 90 ; and we may compute for any time between the
corresponding times of beginning and ending, viz. O h 4 m 48 s and 3 1 ' 58 m 6 s ; or we
may take any value of t less than l h 56 m 39 s . For an example take < = h 58 m 33".
h m s 01 t - - 3 '54568
Time of middle - - 2 1 27 ( t) 19 49 E 7 '17073
u/ - 100 53 tan u' 71641
Before middle 1 2 54 M 120 42 cos u>' 9'2757l
After middle - - - 3 - - - M + 81 4 cos w 8 '98231
7 4- )
+ 23 6"
+ 3-14176
p _
3 -51255
(*) - -
+ 35
cos(D)
+ 9-97419
P .
9-99901
log (a-) -
sin J - -
+ 3 -32222
4- Q '51 208
log (y) -
+ 3 -11595
+ 9 '51208
P/
ros ^ -
3-51156
Q -Q7574
J 3 j <* /*
+ 2 -83430
+ 2 -62803
P'cosS- -
3 -48730
(a;) sin
(A) - -
+ 11' 23"
+ 1 39 40
(y) sin S
(B) - -
+ .7' 5"
+ 35 54
cos Z - - -
2 \ cos i' -
9 -80331
o -63740
+ 1 51 3
f
H- 28 49
sin 2 0- - -
9-16591
log
+ 3 -82367
3*487^0
{log
+ 3-23779
3*48730
sin - - -
9 '58296
2 30' 4
4 >. f-~0 35 48
M - , - - + 30 57' -1
Z - - - - + 50 27' '9 -
a 1
a - -- -
fa a' - - + 22 47
\log + 3-13577
-cos - - + 9'97419
tan - - -
cos - - -
P'
3 -10996
3 -33203
9 '77793
9 '93328
+ 3 -39875
+ 3 -51156
sin -
cos -
- - + 9-88719
h 9 -80383
. . _ 4.
D/ - - +
+ D' +
o
46
65
5-8
+ 45 42 -3
tan Z
cos M - - - -
+ -08336
'+ 9 "93328
tan Q - - - -
+ o -01664
sin e
+ 9 -85764
+ O f\(> A QQ
y o^yy
lanM-
+ -23265
+ 977793
tan - - - -
+ -01058
+ O *S44 1 P
tan (6 + D')
y OTi-i i <
+ '33249
tan / - - - -
+ 0-1766I
/
NKAO on/ .e
Reduction - -
D(J ~U O
10 '4
Latitude - -
N. 56 30 '9
Sin Z +9 '88719
cosM h 9 '93328
+ 9 '82047
comp.cos(A *') + -15588
comp. cos / - - + '25630
- check + -23265
Greenwich time
Equation - - -
[time - - + 3 15-23
Hin
[space - - + 4850'-8
45 42 '3
Longitude - - - W. 3 8 -5
98
On Eclipses.
VI. CENTRAL LINE.
We have, at page 87, found the semiduration of the central appearance on the Earth
to he l h 43 m 17", which is therefore the greatest value of t for this phase. As an
example for a time within the limits take the same value of t as in the two preceding
examples.
h m . o / *- - - - - 3-62325
c 3-52106
Time of Middle
/ _
2
I
I
1
27
n
(-0
19
Cl
49
A -I
Before Middle -
After Mirl flip .
*
51
11
27
97
- s - -
. a .
_i_
71
31
30
19
tanw - - - '10219
cos w - - - 9 '79246
rc 3-18594
A 3 -39348
p/ 3 -51184
J -
Remaining computation for the time 3 b
11
m
T *?
27.
\ tan Z -
0994
tan Z - - - -
+
06994
sin Z - - - -
+
9
88164
cosS
+
9
92905
cos S - - - -
+
9
92905.
1
A
-
- - + 44
56'0
tan0 - - - -
+
9
^99899
+
9
81069
1 10
*C ()
"* ^ ""I"" 1
*J O U
sin 6 - - - -
-f
9
84898
comp. cos A - -
+
15009
4-
a - + 63
54 -0
9
64339
comp. cos / - -
+
24480
4.
20559
- - check - - -
+
20558
tan S - - - -
+
9
79354
ftan A - - - -
+
9
'99913
4- 44
t;A A.
1
I
i m s
-f t 1 *
l)\J V
\cos A - - - -
+
9
84991
Greenwich time
3
n 27
tan (0+5) -
+
30990
Equation - - -
3 56
tan / - - - -
+
15981
Hin| time " "
3
15 23
N.55 18''7
n in<
[space - -
+
48 51
T? p rln /> +1 on _
i n A
/,
i
44 57
XvcClU.CtJ.Ull
Latitude - - -
M. \J \-f
N.55 29'-
Longitude - -
W.
3 54
A MORE ACCURATE CALCULATION.
/ //
ex. corr.
-7- * J ou */ i f
IP
cosD - - -
"P fc 1.1 M. 1 \J
+ 9-97419
^ - -
+ 18 58 28
()----
+ 3-11595
(X) -
+ 35 -
<*)
+ 3 '32222
P-*
3'51255
s... + 3io 52 '. 7 {^ s s ;::
+ 9 79373
+ 9*92899
p - -
9 '99903
A - - - -
+ 3'39323
F. .
3 '51 158
Z ... + 4 9 35'-6{f ;
f 9 -88165
+ 0-06994
On Eclipses.
99
tanZ - -
\. o '06994
sin Z - - - - H
h 9 '88165
e 1
u 44
cos S - -
/
55 '8 tan - -
- - + 9'92899
cosS - - - - H
H
h 9-92899
- - + 9 '99893
h 9 '81064
& - - - H
+ $ - H
- 18
- 63
58-5 sine - -
. _ + 9 -84895
+ Q *6433 1
comp. cos h - - J
comp. cos / - - H
- - check - - - H
h 0-15020
- '24480
+ '20564
h '20564
tan S - -
- - + 9-79373
h - - - H
h 44
/ Ttan h - -
[cos h - -
tan (0 + <
tan 1 - -
i
- - + 9'99937
_ . + 9 -84980
5) . + -31000
Greenwich time
Equation - - -
{time - - -
space - - -
h m
3 11 27
3 56
- - + 0-15980
Nr>50 10 .7
h 3 15 23
h 48 50''8
1-44 5*7 '5
1O 'ft
Latitude - - - N. 55 29 *3 Longitude - - W. 3 53 '3
CENTRAL ECLIPSE AT NOON.
DiflT sin /
x' z 1 1
Therefore, as cot h 1 = , tan D ; - sin A', = sin P, we find
NAUTICAL ALMANAC, 1836, APPENDIX. h
102
On Eclipses.
cot h 1 = cot h
p sin P cos /
- - ; -
cos D sm /i
Or,
-~, / p sin P sin A sin A'
tan D' = ( 1 - - - - tan D
\ sm D J sin h
* A -**'=/ e^L*
\cos D sm
tanD _ tan D'/ j> sin P \
sin h sin h' ~~ \cos D sin h) ^
(1)
which present a direct method of calculating the apparent position of the Moon, at
any time, from that of the true. The former of these equations is evidently subser-
vient to the other, and must necessarily be computed first. As the calculation of these
expressions will, in general, require seven places of figures, it will be more convenient
to determine the simple effects of the parallax, or the small differences A.R. A.R.',
D D', for which other expressions may be derived from them. Let A.R. A.R.'
==: h' h = A h y and D D' = A D ; then by multiplying the equation
i. L M. n p sin P cos /
cot h cot h' = -
cos D sm h
by sin h sin h', the left-hand member will become sin (h 1 h) or sin A h.
Again we have
. p sin P cos / .
. * . sin A h = sm h 1
cos D
tan D tan D'
sin h
sin h 1
. f
sin P sin I
cos D
But
tanD tanD' tan D tan D'
sin h
sn
sin h
sin (D
(J__JL\ anI y
\ sin h . sin h 1 J
sin /*' sin h
sin h cos D cos D' sin h sin h 1
sin A D 2 sin % A h cos (h
tanD'
A/*)
sin A cos D cos D'
Equate this with - =- : - , and we find
cos D sm h
sm h sin h'
tan
sin AD
cos D cos D'
j> sin P sin
cos D
2 sin A A cos(/i + i A A) sin D'
sin h'
sin A A p sin P cos / sin h 1
But 2 sin % A A = = * - - - r
cos is A n cos D cos A A
Substitute this value and multiply by cos D cos D', and we deduce
cosD'
sin A D = p sin P ! sin / cos D' cos / sin D 7
cos (//
(A + iA/0 1
cos \ A A J
On Eclipses.
103
We shall therefore have, for the parallax of the hour angle, and that of the de-
clination.
(P cos/) sinP
sin A ft = 1 -^ sin h'
cosD
sin A D = sin P \( f sin 1) cos D' - (o cos /) sin D'
cos A
(2)
These are still however not adapted for direct calculation, since they involve the
apparent quantities ft', D', which it is our object to determine. The only use that
can be made of them is first to use the true quantities, in order to get the parallaxes
and apparent values approximately, and then to repeat the operation. To avoid this
difficulty, substitute in the former ft + A ft instead of ft', and in the latter put D A D
instead of D' and we get, by expansion,
sin A ft = - (sin ft cos A ft -f cos ft sin A ft)
sin AD rr^sinP cos AD jsin/ cosD cos/ sinD COB (* + **&) 1
I cos i A ft J
+ f sin P sin A D \ sin / sin D + cos / cos D . * . >
I cos ^ A ft J
Divide these by cos Aft, cos AD, respectively, and solve for tan Aft and tan AD, and
we find
tan Aft ==
tan A D =
p cos / sin P \
1 Sin ft
cos D J
/ P cos / sin P \
1 cos ft
\ cos D J
f sin P | sin / cos D cos / sii
(3)
cos(ft + _Aft)
cos 4 A ft
. -r, ( . . COS (ft+ T Aft)
1 f sm P < sm / sm D + cos / cos D
r, tanD
(P sin / sm P) cos D \ 1
tan/
cos i A/i
(/J + TA/0
COS
i A ^
I
1 +
tan/tanD
cos (ft + A ft)
cos T A ft
(4)
These expressions are all of them perfectly rigorous, and better suited to calcu-
lation than they would appear at first sight. The process of the calculation, in which
five places of figures will be sufficient, is more detailed in the following equations :
(/>cos/) sinP
cosD
n sin h
tanA/i= j-
1 n cos/4
h 2
104
On Ecli
c =r (y sin /) sin P
\\ k tanD
n. 2 =
cos A h
_k__
tan D
c cos D (1 n t )
1 c sin D ( 1
tan A D
The expression (4) for tan A D may, however, be neatly resolved by
spherical triangle as follows :
Assume
cos (h + i A/0
(6)
means of a
cos (A) =
cos -j A h
(a)
(/O being very nearly equal to h + T A A. And let N be the
North Pole, Z the central zenith, and M the Moon; then
NM = 90 D, NZ = 90 / and the Z N = h. Without
changing these values of NM, N Z, let us suppose the hour angle N
to become increased to the value of (/O; and with the triangle
so constituted suppose the altitude of the Moon to be , so that
Z M = 90 e then the spherical relations
sin ZM cos M = cos NZ sin NM sin NZ cos NM cos N
cosZM = cosNZ cosNM + sinNZ sinNM cosN
gve
cos cos M = sin / cos D cos / sin D cos (A)
.
= sin / cos D cos / sm D
cos $ A ft
sin c = sin / sin D + cos / cos D cos (A)
cos A +
_
= sin / sm D 4- cos / cos D
cos k A h
Comparing these with the former expression of (4) we have therefore
_
cos M .....
_ (p sin P) cos e
tan A D = t r , . '^
1 (^sin P) sm
Before this can be used the angles M and must be determined.
Draw ZD perpendicular -to MN, and by spherics,
tanND = tanNZcosN
sin MD tan M = tan ZD sin N D tan N
sin ND
. * . tan M = - - tanN
Also by (c)
sinMD
tanMZ ~ l ,- , or cot MZ = cot MDcosM -
cos M
sin ND
BinMD
tan 1\I cos N sin M cos N sin NZ
(d)
tanN
cos M sin N
cos M sin MZ
On Eclipses 105
Let now ND = 0, and MD = MN e = 90 (0 + D) ; and the equations
0> (ft)? ( c )> ( ( 0> ( {> )> (/)) will give the following
cos (A + A A) f ,
cos A A
tan r cot / cos (A)
tanM= ^7?TD) tail(/ '
tan rr tan (0 + D) cos M
sin cos (A) cos /
cos (0 + D) cos M cos c
(p sin P) cos e
tan A D = ^ --^ cos M
1 ( p sin P) sine
in which the equation (e) is used as a check on the preceding compulations. This
check affords a good security to the accuracy of the work, and gives to these equations a
decided preference over those of (6), although a trifle more perhaps in point of calcu-
lation. They have also another advantage, inasmuch as M may be considered as the
parallactic angle and e the altitude of the Moon; the former of these is useful in deter-
mining the position of the line joining the centres of the two bodies in relation to the
vertical, and the other is useful in finding the augmentation of the Moon's semi-
diameter, which we shall now consider.
If s' denote the Moon's apparent semidiameter, and s her true semidiameter as seen
from the centre of the Earth, the actual semidiameter of the Moon will be represented
by both r sin s and r' sin s' ; also, if a perpendicular be drawn from the centre of the
Moon upon the radius f produced, this perpendicular will be represented by both
r sin Z and r 1 sin Z'. We must therefore have - =
sin s sin L
Let M be the true position of the Moon, in the preceding figure, and
sin ZM sin ZNZM = sin NM sin N will be sin Z sin ZNZM ~ cos D sin A; for
the apparent position of the Moon the angle NZM will remain the same, and
sin Z' sin ZNZM = cos D 1 sin h'.
sin Z' cos D' sin A'
sin Z cos D sin A
Also, by means of the equations (8), (9), page 53,
sin Z' _ p sin P sin Z' sin z cos z cos 2
sin Z p sin P sin Z p sin P sin Z p sin P sin Z 1 p sin P cos Z
sin s f sin Z ; cos D 7 sin h 1 cos z , .
sin s sin Z cos D sin A ' " 1 p sin P cos Z
All the preceding formulae are strict in theory. It now remains to consider
what allowances may be made and what facilities given in their actual calculation.
In the first place the value of cos 1 A A may be safely assumed equal to unity, and
may therefore be rejected in the equations (2), (4), (6), and (7), so that
(A) = A + ^ A A : it may be shown that this supposition cannot involve an error of
more than 0" '03 in the value of A D.
106
On Eclipses.
Also, as the arcs P, A A, AD are small, we must have very nearly
an A h tan A D
AD
= tan 1" = [4 -68557], where P,
A /t, A D, denote respectively the numbers of seconds they contain. These equations
may be made more exact, for the limits between which the angles are always com-
prised, by adopting numbers differing a little from sin \" and tan 1"; thus, by assuming
sin P
= [4 '68555]
tan A h
= [4 -68561]
P A h
The first supposition will not in any case involve an error exceeding that of 0" *05 in
the value of P, nor the second an error of more than 0" *1 in the value of A/I, and
these are much too small to merit attention ; the latter assumption applies equally the
same to AD.
Thus we shall have (/i) = h -f i A/t, sin P r [4 '68555] P, A h = [5 '31439]
tan A h t A D = [5 '31439] tan A D ; also A h = A or, the parallax in right ascension.
The equations (3) and (7) may therefore be commodiously arranged as follows:
c==[4'68555]/>
m = A cos
cosD
n = k cos h
(9)
In
By taking h less than 180, positively or negatively, A will have the same sign
as h.
(/i) ~ h + ^ A oc
G~ cos
tan Q =r cos (A) cot I
sin e
B COS M COS
n v = A sin
tan(/i)
cos I
tan = tan (0 + D) cos M
sin0 G
check - -
cos (0 + D)
AD = [5 -31439]
AB
(10)
The auxiliary arc may be taken out in the first quadrant, + or ; calling to 180
the first semicircle, and 180 to 360 or to 180 the second semicircle, the paral-
lactic angle M must be taken out in the same semicircle with h ; and A D will have
the same sign as cos M.
It will appear by the preceding investigations that the values of A , A D, so
deduced, are the quantities to be subtracted from the true values of A. R., D, to get
the apparent.
As the number n is always very small, the values of comp. log. ( 1 n) to the
fifth place of figures may be comprised in the following useful Table under the title
of Correction of Log. Parallax, and conveniently taken out with the nearest third
figure of the argument.
On Eclipses.
107
Correction of Log. Parallax.
Argument : log. n.
Log
Corr.
Log
Corr.
Logra
Corr.
Log n
Corr.
Log n
Corr.
5 '00
7 'ioo
54
7 '400
109
7-/00
218
8 -000
436
10
no
55
410
112
710
223
010
447
20
1
120
57
420
114
720
229
020
457
30
1
130
58
430
117
730
234
030
468
40
1
140
60
440
120
740
240
040
479
50
1
150
6l
450
123
'750
245
050
490
60
2
160
63
460
125
760
251
060
501
70
2
170
64
470
128
770
257
070
513
80
2
180
66
480
131
780
263
080
525
90
3
190
68
490
134
790
269
090
537
6 '00
4
200
69
500
137
800
275
100
550
10
6
210
71
510
141
810
281
110
563
20
7
220
72
520
144
820
288
120
576
30
9
230
74
530
148
830
294
130
590
40
ll
240
76
540
151
840
302
140
604
50
14
250
77
550
155
850
308
150
618
60
17
260
79
560
158
860
315
160
632
70
22
270
81
570
162
870
323
170
647
80
27
280
83
580
165
880
331
180
663
90
34
290
85
590
169
890
338
190
678
7 4 oo
43
300
87
600
173
900
346
200
694
7 *ooo
43
310
89
610
177
910
355
210
710
010
44
320
91
620
181
*920
363
220
727
020
46
330
93
630
186
'930
371
230
744
030
47
340
95
640
191
940
379
240
761
040
48
350
98
650
195
950
388
250
779
050
49
360
100
660
199
960
398
8 -260
798
060
50
370
102
670
204
970
407
070
51
380
104
680
209
980
417
080
52
390
107
690
213
7 '990
427
090
53
7 '400
109
7-700
218
8 '000
436
7*100
54
This correction is additive when n is positive, and subtractive when n is negative.
For the parallax in declination it will always be additive if the Moon be above the
horizon.
For the augmentation of the Moon's semidiameter we may assume cos z = I and
Z = 90 e, so that
_ _ 1 _ l
s I f sin P sin e 1 n^
n t being the number which enters into the computation of A D. Hence
s _ [9 -43537] P
s ~
1 w,
171,
- - - - (11)
108 On Eclipses.
This and tlie last formulae, for A c*, A D, entirely preclude tlie necessity of Laving'
recourse to a table of the sines and tangents of small arcs, and possess much unifor-
mity and simplicity in their application.
To get the relative parallax of the Moon with respect to the Sun, we must use
P IT, instead of P. If, therefore, P' denote the value of j> (P TT), or the relative
horizontal parallax reduced to the latitude of the place, we must use sin P', instead
of p sin P, in the preceding formulae.
The determination of the apparent relative positions of the centres of the two
bodies, as well as the augmentation of the semidiameter of the Moon, at any time, has
now been reduced to a practical and expeditious set of formulae. A series of these
apparent positions of the Moon, with respect to that of the Sun, will trace out her
apparent relative orbit; and the contact of limbs will evidently take place when the
apparent distance of the centres becomes equal to the sum or difference of the semi-
diameter of the Sun and the augmented semidiameter of the Moon. For a distance
equal to the sum of these semidiameters we shall have partial beginning or ending ;
for a distance equal to their difference we shall have
{ annu/ar } be S innin g or ^ing, when s' {>} a
Since the hour angle of the bodies is subject to the rapid variation of nearly 15 per
hour, the effect produced by parallax will be of so irregular a nature as to give a
decided curvature to the apparent relative orbit of the Moon. This curvature will be
more strongly characterized when the eclipse takes place at some distance from the
meridian or near to tbe horizon ; and the apparent relative hourly motion of the
Moon, even during the short interval of the duration of the eclipse, will, through the
same irregular influence, experience considerable variation. These circumstances
will, in some measure, vitiate any results deduced in the usual manner, by supposing
the portion of the orbit described, during the eclipse, to be a straight line, and using
the relative motion, at the time of apparent conjunction, as a uniform quantity. The
method we are about to pursue is very simple, and consists in assuming any time
within the eclipse, and computing for this time the relative positions and motion of the
bodies, and thence finding, without any reference whatever, either to the time of the
middle of the eclipse, or to the time of conjunction, the times of beginning, greatest
phase and ending, and the relative positions of the bodies at these times. The nearer
the assumed time is to the time of the greatest phase, the more accurately will the
time of that phase be determined ; and, similarly, the nearer that time is to the time
of beginning or ending, the more certainty will attach to the determination.
To find the apparent relative motion of the Moon, we must first determine the vari-
ation which takes in the parallax. For this, take the equations (2), page 103, viz.:
, sin P 7 cos/ .
sin A = sin A h = sin h 1
ccsD
Bin A D = sin P' { sin I cos D' - cos / sin D' cos ( ; * +
I cos^ A
or, substituting small arcs instead of their sines,
A , = p,2ll s ; n/l <
cos D
AD = F j sin / COB D< - cos / sin D' 2iii^^il
\ cos * A h
On Eclipses'. 109
Since a portion of the apparent disc of the Moon is projected on that of the Sun,
the apparent declination D' can differ very little from c. As the hourly variations of
these small quantities are only required approximately, we may therefore use 5
instead of D', and neglect A /t, so as to have
/ cos /
A a = P' - = sin h
cosD
A D ~ P' (sin I cos 5 cos / sin cos A)
which expressions, though rough values of A , A D, will give their hourly vari-
ations pretty accurately. For these, observing that h is the only quantity which,
by its rapid variation, has any sensible influence on these values, we have by differen-
tiation
d ( A ) / _,. d/i . . A cos
-V
dt dt
But by the equations (9)
m [4 '68555] P'cos/
n = [4 '68555] P' =r cos/i
L cos D
Substitute, therefore,
cosD
P'cos/ =[5 '3 1445] 7?i
and we get
rf(AD)
dt
= [5 -3 1445] (-^ sin I'M m sin c^ sin
If we adopt 14 29 7 as a mean value of-y, we shall have -j- sin 1" = [9 "40274], and
[5 '31445]( ^ sin 1" J [4 71719] or [4 7172]. Therefore, if (5), the value
of the Sun's decimation at the time of the middle of the eclipse, be adopted in the
value of j - -, we may form the constants
= [4 7172] 1
and then, using A ,, A D! in place of C , -- , we shall have
A* =Q l n
AD 1 = Q 2 si
which offer a simple calculation.
110
On Eclipses.
Let now, at any assumed time within the duration of the eclipse, S and M be the
apparent positions of the centres of the Sun and
Moon ; and B M E an arc of a great circle coin-
ciding with the relative direction of the Moon's
motion at that time, which arc we shall first adopt
in place of the curvilinear orbit actually described.
On the circle of declination, S N, demit the great
circle perpendicular M d, and suppose B and E to
be the positions of the Moon at the respective times of partial beginning and ending
of the eclipse, and n the middle point. Assume SB = SE = ,$ / -j- is the apparent difference
of the right ascensions of the bodies, and that D' D A D, is the apparent declina-
tion of the Moon ; and that
x = { D' + ( A a) corr. }
y = { A } cos D'
and consequently also
a; 1 =D 1 ~AD 1
cos D'
(H)
(15)
Moreover, the figure occupying so small a portion of the sphere, and being com-
posed of arcs of great circles, we may, without any appreciable error, treat these arcs
as straight lines ; thence we shall obviously have
a;
cot i =
sin S cos S
Hourly motion in the orbit =
cos i
71= Wcos(S +
Again, in the triangles B S M, E S M,
- - - (16)
and consequently, by plane trigonometry,
112 On Eclipses.
0} EM =
COS (U COS W
With the above hourly motion in the orbit we shall therefore have
Time of describing
BM = ZUL* s in { ,, + (S + ) }
?/! cos o>
W cos e .
n M = sin ( S + i)
W cos t
EM =r sm { w (S + t) }
?/, COS W
Let now, t 2 be corrections to be applied to the time assumed to get the times of
beginning and ending, and (0 the correction for the time of the greatest phase. Then
we have evidently
{*i ] [ BM ] f negative 1
(t) \ = the time of describing i n M > with a < negative [sign.
e,J I EM J [ positive J
To have these times expressed in seconds, assume
c = Wcos ' x 36oo _ w cos ' . t
yi cos a> 7/j cos
and then we shall derive
/j = csm { (S-f ">} * g :=csiii { (S + t) + *>}
(/) =r ccos w sin { (S + t)}
and hence
{beginning j f c sin { (S + t) w} 1
greatest phase > = Assumed time + \ ccoswslnf (S + t)} >-(18)
ending [ csin { (S + + a '} J
It has been observed, that any one of these values will be the more to be depended
on the more nearly it approximates to the assumed time. Thus, if the assumed time
be within ten minutes or so of the end of the eclipse, the point M will approximate so
closely to the point E, that no sensible error can arise by supposing the small portion
M E of the orbit to be a straight line, and to be passed over by the Moon with an
uniform motion. This circumstance renders it advisable, in the first instance, to take
the assumed time near to the time of the middle of the eclipse, so as to give, a good
result for the time of the greatest phase, and results for the times of beginning and
ending, which may be nearly equally relied on. Such a computation will be suf-
ficiently exact for the usual purposes of prediction. When the time of beginning or
ending is wanted to great minuteness to compare with observation, it will only be
necessary to repeat the operation for a time assumed as near as convenient to the first
determination, which will mostly give within a fractional part of a second of the true
theoretical result ; a degree of accuracy, however, seldom wished for, and quite unsup-
ported by the present state of the lunar theory.
To fix on a time near to the middle of the eclipse for the radical computation, one
of the most simple expedients will be to determine roughly the time of the apparent
conjunction.
On Eclipses. 1 13
We shall now briefly consider the apparent positions of the Moon, as related to
the Sun's centre.
It is clear that S is the angle of position of the Moon's centre from the North
towards the East, at the time assumed ; also, that the angle N S B = to + t is the
similar angle of position from the North towards the West at the time of begin-
ning j and that the angle N S E = w i is the angle of position from the North
towards the East at the time of ending; and that the angle N Sn = t is the same
angle towards the West at the time of the greatest phase. Therefore, by estimating
all these angles towards the East we shall have
{beginning | f ( _ 4 ) _ w J
greatest phase > Z of 3) 's centre from N. towards E. = |( _ t ) [-(19)
ending j (( _ t ) + w J
In the computation of the parallax in declination we find an angle M, which in
practice may be supposed to be the angle N S Z for the assumed time, the zenith Z
being reckoned towards the East ; consequently, at this time we shall have S M for
the angle of position of the Moon's centre from the Zenith towards the East. At any
other time the parallactic angle M for the latitude of Greenwich may be taken from
the following table, arguments the corresponding apparent time and the Sun's decli-
nation. This table, for any other place, may be computed by formulae, such as at
page 105, viz.
tan = cot / cos h tan M = 7^ r N tan h
cos(0+c>)
h being the angle answering to the apparent time.
Those who may be engaged in the computation of Eclipses, for any particular places,
\vill find considerable facility in the formation of similar tables.
For an Occultation of a Star by the Moon the argument, instead of the apparent
time, will be the star's hour angle, or the sidereal time minus the star's right ascen-
sion. In this case the required positions will be those of the star with respect to the
Moon's centre, which will therefore be different from the angles of position for a solar
eclipse, in which the Moon's centre is referred to that of the Sun. The angular posi-
tions of the contacts at immersion arid emersion will consequently be determined in
the same way as for an eclipse of the Sun, and will be estimated in the opposite direc-
tions. Thus, for an Occultation,
A . f immersion 1 , ~ , f XT . , -r, f(180 t ) w}
At 1 emersion } Z of * from N " towards E ' {(!*- ,) + .)
And so must 180 be applied to the other angles of position, as expressed for a solar
eclipse : this will make the expressions for the direct images of occultations the same
as those for the inverted images of eclipses of the Sun, in estimating the contacts
either from the north point or from the vertex.
114
On Eclipses.
Parallactic Angles for the Latitude of Greenwich.
{same sign as 7i)
Arguments: Apparent Hour Angle and Declination.
Dec.
Hour Angle h.
North.
o
o
o
o
o
o
o
o
o
10
20
30
40
50
60
70
80
90
100
110
120
130
140
o
o
o
o
o
o
o
o
o
o
o
o
o
o
8
15
22
27
31
35
37
38
39
38
37
35
31
27
1
8
15
22
27
32
35
37
38
39
38
37
34
31
27
2
8
16
22
28
32
35
37
38
39
38
37
34
31
27
3
a
16
22
28
32
35
37
38
39
38
36
34
31
26
4
8
16
23
28
32
35
37
38
39
38
36
34
31
26
5
9
16
23
28
33
36
38
39
39
38
36
34
30
26
6
9
17
23
29
33
36
38
39
39
38
36
34
30
26
7
9
17
24
29
33
36
38
39
39
38
36
34
30
26
8
9
17
24
29
34
36
38
39
39
38
36
33
30
25
9
9
17
24
30
34
37
38
39
39
38
36
33
30
25
10
9
18
25
30
34
37
39
39
39
38
36
33
30
25
11
9
18
25
31
35
37
39
39
39
38
36
33
29
25
12
10
18
25
31
35
38
39
40
39
38
36
33
29
25
13
10
19
26
31
35
38
39
40
39
38
36
33
29
25
14
10
19
26
32
36
38
40
40
39
38
36
33
29
25.
15 .
10
19
27
32
36
39
40
40
39
38
36
33
29
24
16
n
20
27
32
37
39
40
40
40
38
36
33
29
24
17
11
20
28
33
37
39
40
41
40
38
36
33
29
24
18
11
21
28
34
38
40
41
41
40
38
36
33
29
24
19
11
21
29
34
38
40
41
41
40
38
36
33
29
24
20
12
22
29
35
39
41
41
41
40
38
36
33
29
24
21
12
22
30
36
39
41
42
42
40
39
36
33
29
24
22
12
23
30
36
40
42
42
42
41
39
36
33
29
24
23
13
23
31
37
40
42
43
42
41
39
36
33
29
24
24
13
24
32
38
41
43
43
42
41
39
36
33
29
24
25
14
25
33
38
42
43
43
43
41
39
36
33
29
24
26
14
26
34
39
42
44
44
43
42
39
36
33
29
24
27
14
26
35
40
43
44
44
43
42
39
36
33
29
24
28
15
27
35
41
43
45
45
44
42
40
37
33
29
24
29
16
28
36
41
44
45
45
44
42
40
37
33
29
24
By .subtracting the parallactic angle, for the respective times of beginning, greatest
phase, and ending, from the foregoing angles of position of the Moon's centre from the
North towards the East, we shall evidently obtain the same angles from the Zenith or
Vertex towards the East.
If, however, the operation be repeated for the accurate determination of the times of
beginning and ending, we shall have in the calculations the angle M, also at these
times. Let i l9 u^ M l be the angles appertaining to the beginning, and t s , u^ M 2
those for the ending, and we shall evidently have the following values, which will be
more accurate than the preceding :
On Eclipses.
115
Parallactic Angles for the Latitude of Greenwich.
(same sign as /*)
Arguments: Apparent Hour Angle and Declination.
Hour Angle h.
Dec.
South.
o
o
o
o
o
o
o
o
o
a
10
20
30
40
50
60
70
80
90
100
110
120
130
140
o
o
o
o
o
o
o
o
8
15
22
27
31
35
37
38
39
38
37
35
31
27
1
8
15
21
27
31
34
37
88
39
33
37
35
32
27
2
8
15
21
27
31
34
37
38
39
38
37
35
32
28
3
8
15
21
26
31
34
36
38
39
38
37
35
32
28
4
7
15
21
26
31
34
36
38
39
38
37
35
32
28
5
7
15
21
26
30
34
36
38
39
39
38
36
33
28
6
7
14
20
26
30
34
36
38
39
39
38
36
33
29'
7
7
14
20
26
30
34
36
38
39
39
38
36
33
29
8
7
14
20
25
30
33
36
38
39
39
38
36
34
29
9
7
14
20
25
30
33
36
38
39
39
38
37
34
30
10
7
14
20
25
30
33
36
38
.3.9
39
39
37
34
30
11
7
14
20
25
29
33
36
38
39
39
39
37
35
31
12
7
14
20
25
29
33
36
38
39
40
39
38
35
31
13
7
14
19
25
29
33
36
38
39
40
39
38
35
31
14
7
13
19
25
29
33
36
38
39
40
40
38
36
32
15
7
13
19
24
29
33
36
38
39
40
40
39
36
32,
16
7
13
19
24
29
33
36
38
40
40
40
39
37
32
17
7
13
19
24
29
33
36
38
40
41
40
39
37
33
18
7
13
19
24
29
33
36
38
40
41
41
40
38
34
19
7
13
19
24
29
33
36
38
40
41
41
40
38
34
20
7
13
19
24
29
33
36
38
40
41
41
41
39
35
21
6
13
19
24
29
33
36
39
40
-42
42
41
39
36
22
6
13
19
24
29
33
36
39
41
42
42
42
40
36
23
6
13
18
24
29
33
36
39
41
42
43
42
40
37
24
6
13
18
24
29
33
36
39
41
42
43
43
41
38
25
6
13
18
24
29
33
36
39
41
43
43
43
42
38
26
6
13
18
24
29
33
36
39
42
43
44
44
42
39
27
6
13
18
24
29
33
36
39
42
43
44
44
43
40
28
6
12
18
24
29
33
37
40
42
44
45
45
43
41
29
6
12
18
24
29
33
37
40
42
44
45
45
44
41
f beginning
For < greatest phase
ending
Z of }) 's centre fromN. towards E. =
Z of 3) 's centre from Vertex towards E
f (-O-^-Mn
.=](-0-M [-
I ( ') + "a M'J
(20)
These angles relate to the natural appearance or direct images of the bodies. For
the same angles, as they will appear through an inverting telescope, + 180 must be
applied : this may be simply done by using (180 t) instead of ( i).
116 On Eclipses.
To find the time when the apparent conjunction takes place, let t denote the interval,
in units of an hour, to be applied to the time of the true conjunction, and h the com-
mon hour angle of the bodies at the true conjunction. Then the position of the Sun
not being supposed to be influenced by parallax, the common apparent hour angle of
the bodies, at the time of the apparent conjunction, will be h' = h-\- 15 . t ; and
therefore, at this time,
= l t A a = (P 1 ^^] sin (A + 15 .
so that the condition for apparent conjunction, viz. ' rr A = 0, gives
M (V ~ N U n '( / *- M5 - o = ( 21 )
\ cosD j
for the determination of the interval t which from this equation will be best found,
perhaps, by the usual method of double position. We only want, however, an approxi-
mate value, and may therefore avoid much unnecessary labour in estimating this time.
Thus, at the time of true conjunction, the same approximate formulae may be adopted
as used at page 109, viz :
_.. cos/ .
A P' sin h
cosD
.. / dh , A cos /
A != F -r- sin 1" ) cos h
\ dt J cos D
in which applies to the Moon. It is evident then, as the true positions of the
bodies have no difference of right ascension, that A is the apparent difference of
right ascension ; and consequently, as the relative apparent motion in right ascension
is ! Ai or l P'( sin I'M ^ - cos/t, the correction t to be applied to
\ dt J cos D
the time of true conjunction to get that of the apparent, will be
. cos/
P' -=r- sm h .
. cosD sm/t
D/ / dh . A cos/ cosD / dh . A
P' r- sm 1" cos h oil -=rj r 17 sm * cos h
\dt J cosD P'cos/ \
Make now the following assumptions :
(D + a'corr.) S a cos D'
p = j cos i sii
\ * - - (4)
; corr.) } sin i H a cos D' cos t
AD A a cos D' .
> = r cost smt
k k
Jfr (5)
k k
A q = j- A D sin t -J A a cos D ; cos t
and, observing the above values of x and y, the equations (2), (3), will become
rfrsT+fcsinw (g A?) )
NAUTICAL ALMANAC, 1836. APPENDIX. i
118 On Eclipses.
Let 7, Y'j be determined by the equations
and p, 9, will take the following values
p =: Y CO s (y 4. t )
q r= &y sin (Y' +
It yet remains to determine the values of Ap, A
1 ! COST A J
To simplify the expressions, let
_ [5 -31439] A cosD /
(1 -n)A' cosD
[5 -31439] A [5 -31439] A
c 7 -**. cos D a = -7- sin D
(1 wO A' (1 nC) A'
and
b A ' cos / sin h
-^v~
AD = cA ; sin/ A ; cos /
cos T A
== c A ' sin / a A ' cos / cos h + a A ' tan cos / sin h
These substituted in (5) give
Ap = c cos i sin / cos / 1 a cos i cos h ( a cos t tan b sin t) sin /i 1
A 9 = Arc sin i sin / cos / ! ka sin t cos A (ka sin i tan j- 6 cos t) sin h >
The value of b contains the factor , for which we have
cosD
cos D'
= cos AD (1 + tan D tan AD)
Substitute the first value of tan AD, page 103, and
cosD'
- = cos AD
i
of the fraction,
cos D 1 p sin P {sin / sin D + cos / cos D cos (h) }
Or, putting h instead of (A) in the numerator, which cannot sensibly affect the value
cos D' 1 n
=r- s= cos A D
cosD 1 Wi
On Eclipses. 119
This, supposing cos AD = 1, reduces the values of the constants a, 6, c, to the
following,
~ [5-31439]A j
(l-*i) A' V ..... (9)
c = 6 cos D a = 6 sin D >
If e be a small arc determined by g cose = 6, g sine = a tan, we shall have
a cos i tan -- b sin i == # sin ( t + e) = cj cos (90 + t e)
&a sin t tan - + kb cos L kg cos (t e) = kg sin (90 + t e)
However, as e must always be a very small arc, we may suppose cos e = 1 ; also
= p L 7 sin / + 7 r cos / cos (^'+X) 1
t-(T q) +^smw + L // sin/ 7" cos / cos (f ;/ + X) J
After computing the constants k, p, g, L', L/ 7 , V'j Y'"* b Y rneans of the equations
(1), (7)> ( 8 ) (9)> ( 10 ) (H)j (13), and (14), we shall thus have two numerical
equations for the determination of w and the Greenwich time t of the phase, for any
place whose latitude is / and longitude X. The accuracy of the determination will
principally depend on the proximity of the resulting time t to the assumed time T ;
and therefore the result will be near the truth for all places where the phase will take
place near to this time.
In making these calculations for any particular portion of country, which for the
partial phase will be necessary for both the beginning and ending, it will be best in
the first instance to fix upon a place near the centre and compute the eclipse for that
place, which computation will furnish good mean values for the data D, , 5, ' corr.,
AD, A , t, y A', A, and comp. log (1 nj.
i 2
120
On Eclipses.
= 7" I
= -L" i
(16)
By supposing
7 cos /' = 7 7 I" cos / /7 sr 7"
' sin /' = L 7 " sin /"
the expressions
L 7 sin / + 7 7 cos / cos (Y^ 7 +X)
L 7/ sin / -f 7 /7 cos / cos (Y^' + X)
will take the forms
t! {sin /' sin / + cos I 1 cos / cos (y^ 7 + X) }
/7 {sin I" sin / + cos I" cos / cos (f " + X) }
and, without the factors ', 7/ , will represent the cosines of the distances of the pro-
posed place from two other places whose latitudes are / 7 , l" t and west longitudes
Y*', V"- The former of these two places will be near to the southern pole of the true
relative orbit, and the latter will be near to the orbit itself, and will precede the Moon
by a distance nearly equal to 90.
For purposes, which do not require great minuteness, the preceding equations will
admit of some simplification by neglecting the small angle e. Add the squares of the
equations (13) and (14), observing that c 2 + 2 = fe 8 , and
which give the general relation
L" 8 7 /72
/v rt
By neglecting e, \ ^ 9 + *> cos x ~ sin i, sin x = cos t ; and then
/< = v
(17)
which united with the equations ( 16) give ' = b, %' ^kb, and hence
Or,
C COS t
'l ri b
7' = ' cos l'~b cos /'
= cos D cos t
sin t
= cos D sin i
7" = " cos l"~kb cos /"
,"-m= * is "' x =
COS I COS L
k b cos/ 77 cos/ 77
sin /' = cos D cos i
L 7 = -6 sin/' 7':= 6 cos/'
n' H) = sin D cos i
b
l
sin (V' H) = shit
(20)
7-r- ^ cos D sin t
kb
-^-cos(V" H) = sinDsint l ----- -- (21)
^-BinCV" H) = cost
in which the coefficients c, a, will not be required.
III. TRANSITS OF MERCURY AND VENUS OVER THE Disc OF THE SUN.
These phenomena are, in many respects, analogous to that of an annular eclipse of
the Sun, and admit of a similar calculation; the principal distinction consists in the
negative sign of the relative motion of the Planet in right ascension, which will make
the inclination of the orbit always obtuse, and therefore render some modifications
necessary in the determination of the particular species of the other angles which
enter into the computation. To avoid any confusion that might thus arise, we shall
adopt the Sun as the moveable body, and refer his positions to that of the Planet
which we now suppose to be stationary. Thus,
S = the 0's declination.
D = the planet's declination.
it = the O 's equatorial horizontal parallax.
P = the planet's equatorial horizontal parallax.
= the J s right ascension minus that of the planet.
x = ($' + a'corr.) D.
y = ' COS y.
Xi = the 's motion in declination minus that of the planet.
y l = (0's motion in right ascension minus that of planet) -cos $',
and so we might proceed as with an eclipse of the Sun, only observing that the rela-
tive parallax p (IT P) is a negative quantity, and that the positions of the contacts
on the limb of the Sun, as in the case of an occultation, will be at points opposite
to those which come out in the calculation. However, as the relative parallax
is always very small, the ingress and egress of the planet will be seen at all places on
the earth at nearly the same absolute time ; it will, for this reason, be best to compute
first the circumstances for the centre of the earth, and then to ascertain the small vari-
ations produced by parallax for any assumed place on the surface, which may be readily
deduced from the preceding equations for the reduction of an eclipse of the Sun.
Let w, (), be the values of o>, t, for the centre of the earth, and, by separating the
effects of parallax from the equations (6),
122
On Eclipses.
cos w = p
(0 = (T-?) + ft sin w
A cos w = A p A / ~ A cos { ~ X + w } = sin { i + w } . We shall
therefore have for the time of ingress or egress the following general expression, in
which the terms within the brackets depend on the position of the place of obser-
vation ; also the upper signs apply to the ingress, and the under signs to the egress.
t = T q + k sin w
cos{ i+w} . , / . k cos{ -i+w} , sin{ i + w} . \
sm h cos /
sin w J
kb
sin/ ( sin 5
sm w v sin w
cos/i
Assuming k" = : , this expression will resolve into the following :
psinw
tan i = -
k [3-55630] -
y\
(^ + corr.) D
7 cos"^
cos^
cosw = 7 cos (^ -f
^ = *-/ sin (y + i)
(0= T q + * sinw
(a)
(O
A sin w
L"
= cos{( i) + w}
l^-cosC^" H) = cos {( t) + w} sinS
R,
" p cos / cos W + X) L" /> sin / }
On Eclipses. 123
In these equations
H the O's true hour angle from the meridian of Greenwich, at the time ()
For I exterior 1 contact of limbs, A = { ' + '1
I interior J \ a s]
For contact of centre of planet with O 's limb, A = a
s denoting the true semidiameter of the Planet, and a that of the Sun.
The equations (a), (5), (c), (d), will serve to determine the constants (), 7",
L", ^", for the times of ingress and egress, and then there will result two numerical
equations of the form (?) to reduce the phenomena to any place on the earth's surface.
For the points on the limb of the Sun we shall have
At { in S ress \ angle from N. towards E = |(180-0-w j for ^.
I egress f I (180 0+wj
"~^~ w > for inverted image;
which will be sufficiently accurate for all places on the earth.
The time T may be assumed near to the time of conjunction in longitude, or right
ascension, as it may suit convenience. For Mercury, if very minute accuracy is
wanted, it may be necessary, for more correct values of (0, to assume two times T
near to the times of ingress and egress ; but it is very questionable whether such a
precarious extent of accuracy would sufficiently recompense the time expended on the
calculation.
IV. OCCULT ATIONS OF STARS BY THE MOON.
These may be calculated in the same manner as Eclipses of the Sun, the only differ-
ence in the operation consisting in the star having neither motion, parallax, nor semi-
diameter. But, where great minuteness is not wanted, these particular circumstances
will afford some degree of simplification to the expressions, if that parallax of the Moon
be adopted which would answer to the star as an apparent place, since this parallax,
at the times of immersion and emersion, will then be precisely that of the respective
points of the Moon's limb which come in contact with the star; and thus the augmen-
tation of the Moon's semidiameter will be evaded, so that the true semidiameter may
be employed. For this novel and judicious expedient we are indebted to Carlini. See
Zach's Correspond ance, vol. xviii. page 528.
As in the case of the Sun, let 3 denote the declination, and h the hour angle of the
star ; and let P represent the equatorial horizontal parallax of the Moon. Then, for
the effects of parallax in right ascension and declination, we must substitute S for D',
and h for h 1 in the formulae (2) at page 103, which thus become, disregarding ^ A h,
_ cos /
A = p P sin h
cosD
A D = p P (sin / cos B cos / sin cos A)
As soon as the immersion takes place, these expressions will represent the parallax of
that point of the Moon's limb which is in contact with the star; and, therefore, the
application of this parallax to the centre of the Moon will produce an apparent dis-
tance A ', of the centres, equal to the true semidiameter s of the Moon. Also as the
star, in the course of the occultation, is only affected with its apparent diurnal
motion, the hourly variations of the above values will be
124 On Eclipses.
fdh . ,\ cos /
A , = p P sin 1" ) cos h
\dt ) cos D
A D! = p P ( -T- sin I'M cos / sin 5 sin h
in which is 15 2' 28", the hourly diurnal motion of the Earth, and therefore
gsinl"=[9-419l6].
Assume
, rn cos
U ' = p cos / =r
(1)
which are constant coefficients depending on the latitude of the place ; then
00). p 0(3). p
A = sin h A j = = cos A
cos D cos D
A D = (0 (2J cos 2 C1) sin cos A ) . P A D! = (3) . P sin 3 sin h
If, in the values of A, Ai, we use cos instead of cos D, the values of #, y, a'^y^
page 111, will become
x = (D 3) (0 (2) . P cos S (1) . P sin 3 cos A)
y r a cos 3 (1) . P sin h > XON
arj = D! (3) . P sin 5 sin h
y t = ai COS 5 (3) . P COS h
in which we have disregarded the correction.
With the values of x 9 y, x lt y,, so found, we may then proceed with the equations
(16) and (18), pages 111 and 112, as in the case of a Solar Eclipse.
This method is similar, and, as far as accuracy goes, the same as the recent method
of Professor Bessel, who divides all the quantities by the equatorial horizontal
parallax of the Moon, He assumes
ot. cos B #i cos
0* - *rv *^^
P " ~~p P p
L
P
u = (l) sin h u 1 = (3) cos h
v = (2) cos ^ (1) sin ^ cos h v 1 = (3) sin 5 sin h
so that if we change the signification of the symbols or, y, ari, y t , and suppose them
now to represent the preceding values divided by P, we shall have
x = q v x l z=. q 1 v 1 1 , . s
y = p M y l ^=p l u' )
These values being adopted, in proceeding with the equations (16) and (18) we must
use A ' = -^-, the value of which, according to Burckhardt's Tables de la Lunc,
(Paris, 181.2), page 73, is [9'43537]. Much facility is thus given to the calculation
' sin Z = sin sin w - - - - (8)
because the squares of these three equations added together will give unity on each
side. By these equations we shall hence have
sin D' cos Z = sin D' cos sin w
sin Z cos M sin Z (cos i cos ta 1 -f- sin t sin w')
== (cos a-' sin Z) cos t + (sin w' sin Z) sin t
ES cos i cos w + sin i sin sin w
and consequently
sin / =r cos D' cos i cos w + sin w (sin D' cos + cos D' sin i sin 0)
which now involves only one variable 0. Again, assume two arcs 0, y, which will
fulfil the equations
cos cos y =: sin D' ...... (9)
cos sin y = + cos D 7 sin i - - - (10)
A third equation will follow from these, viz.
sin 6 r cos D' cost ------(11)
because, as before, the squares of these three equations will together make unity. The
value of sin / will now become
sin / cos w sin + sin w cos cos (0 -f V)
The angle + V being the only variable in this expression, it is evident that the
greatest value of / will have + y =r 0, and the least + V = 180. Therefore,
f greatest 1, r , f # + w 1 . r f northern 1 v .,
I least f value of ' = { 6 - w } ' usm w for { southern } hmlt
These would be the extreme latitudes for the appearance of the occultation if the
Earth were a transparent body ; as this, however, is not the case, it will be necessary
that the star should be above the horizon, a condition not included in the preceding
equations. The zenith distance Z must not exceed 90, and therefore cos Z must
necessarily be a positive quantity.
By the equation (6), cos Z must have the same sign as cos0, and this must be the
same as + cos "^ for northern limit, or cos ^ for southern limit, because in the
former case + ^ = 0, and in the latter + V = 180. But, by (9), cos V must
have the same sign as D'. Consequently
For limit > cos Z has the same
sou
t > - D
It is evident, therefore, that the extreme northern limit will have the star below the
horizon and be excluded when D' is negative, and that for the same reason the
southern limit will be excluded when D' is positive. Thus the only admissible
extreme limit will be determined by the equations
(12)
using upper signs when D' is positive, and under signs when D 7 is negative.
The other limit for the actual appearance of the occultation will evidently be one of
On Eclipses.
127
the two places where the other limiting line meets the rising and setting limits, and
will be determined by
cos w t: sin / 2 == cos D' cos -j ( t) + w r - - - (13)
using, as before, upper signs when D 7 is positive, and under signs when D' is
negative.
The equations (11), (12), (13), for convenience in determining the species of the
angles, may be put in the following form,
n-A 7 + n + A 7
cos w
pi P/
sin 9 = cos D' cos t > - - (14)
I, == Wl
sin 4 = + cos D' cos (w a t)
observing that w 1} w 2 , 0, and i, must here take the same sign as D'; also
under } si S ns when
These formulae are applicable to a solar eclipse. For an occultation of a star by
the Moon, P'will be the Moon's horizontal parallax, and A' her semidiameter, which,
as these limits are not wanted very accurately, may be regarded as true quantities ;
also we may neglect u and so take instead of D 7 . Since t= [9'43537] = '2725,
the formulae for an occultation will hence be
tan i =
COS
cos
= + "2725
n = (diff. dec.) cos i
cos w 2 = + + -2725
sin = cos $ cos i
l sin / 2 T cos cos (w 2 t)
(15)
in which we also give to the angles W! , w 2 , i, 0, the same sign as S, and use upper
signs when S is positive, and under signs when 5 is negative. We may also observe,
that,
1 . When 5 is North, / t is the most northern limit ; and when is South, l v is the
most southern limit.
2. When Wi is imaginary, / t will be 90, and of the same name as J. In this case
the occultation will be visible about the pole of the Earth which is presented to the
star ; the visibility will extend beyond the extremity of the disc of the Earth as it
would be seen from the star.
3. When w a is imaginary, / 2 will be the complement of S and of a different name
from . In this case, if we consider the disc of the Earth as seen from the star, the
visibility of the occultation will extend beyond that extremity of the disc which has the
pole on the other side of it.
After an occultation is computed for any particular place, if we deduct the star's
right ascension from the sidereal times of immersion and emersion we shall get the
hour angles of the star, + West, East. By comparing these hour angles with the
semidiurnal arc of the star, we can distinctly ascertain the positions of the star with
respect to the horizon.
128 On Eclipses.
V. ECLIPSES OF THE MOON BY THE EARTH'S SHADOW.
These may be also resolved in the same way as those of the Sun. The absolute
positions of the Moon and Shadow being independent of the position of the spectator
on the Earth, the determination of parallaxes will be here unnecessary, which much
simplifies the calculation of these eclipses. The considerations requisite to be attended
to, by way of distinction, are the following :
Semidiameter of the Shadow -r (P' + IT
A =c(P ?r) m =
s = [9 -43537] P
3. Take out D, S, , D,, for the time T.
h == sidereal time at place minus }) ' s right ascension, to the tenth of a minute,
in arc.
m
n = k cos h
cos D
A = [5 -31439] k sin h [corr. for n\
A ! = Qi n A D! = Q 2 sin /t
Correction for w to be taken from the table on page 107
4,
tan = cos (/t) cot /
tan M = - _____ tan (A)
cos
check - -
G = cos (h) cos /
tan e = tan (0 + D) cos M
B = cos M cos e
sin G
~ 5"
cos (0+D)
M to be in the same semicircle with h
On Eclipses. 131
A sine AD =r [5 '31439] A B [corr. forw,]
s' r= s [corr. for w J
Correction for rii to be taken from the table on page 107
5. D'~D AD
y = ( A a) COS D' y t = ( ai A i) COS D'
or = (D' + a'corr.) S x v = D t A D!
= Wcosj
sin S cos S
_ W cos t [3 -55630]
7. n H
cos w = c =
A cos
= c sn a t == c sn
Time of greatest phase = sum of times of beginning and ending
When n < s 1
132 On Eclipses.
9. FOR A MORE ACCURATE CALCULATION OF THE TlME, &C., OF BEGINNING OF
THE PARTIAL PHASE, assume a convenient time near to the preceding determination.
For this time, take out the quantities D, D M , , oc l9 from the Ephemeris; and pro-
ceed as in Nos. 3, 4, 5, 6, 7, omitting 6, 2) and the times of greatest phase and ending.
Let MI, ( tt>u be the values of the angles in this computation , then, for the posi-
tion of the point of contact on the limb of the Sun,
Angle from iy^ j towards the East = i/Z^Z^M } for direct ima S e<
Angle from towards the East = fiJi; for inverted image.
10. FOR A MORE ACCURATE CALCULATION OF THE TlME, &C., OF ENDING OF
THE PARTIAL PHASE, assume a convenient time near to the first determination. For
this time, take out the values of D, U^ , , t ; and proceed as in Nos. 3,4, 5, 6, 7>
omitting a, t it and the times of beginning and greatest phase.
Let M a , t 2 , w 2 , be the angles in this computation; then, for the position of the point
of contact on the limb of the Sun,
Angle from {y^} towards the East = j [~^ + ^_ M J for direct imag6 '
Angle from j^J towards the East = { [jjfc fc iiwerted i
II. FORMULA FOR REDUCTION TO DIFFERENT PLACES.
11. Instead of Nos. 5, 6, 7> substitute the following :
D 7 = D AD ' = A
y l =(i A0 cosD 7
tant = k =r [3'55630]
2/1 i
(D + 7 corr.) a cos D
p r= 7 cos (V^ + 7 ^ 7 sin (^ + t)
T q = T 7
12. , [5-31439] A
6 = j [corr. for Hj
A
e m minutes = [7-9208] A sin D v = (90 + e
13. H =5 the true Greenwich hour angle of }) at the time T
L 7 L 7/
r= cos D cos i -yy- := cos D sin i
b kb
yl ylt
-7- COS (y'~ H) rr: sill D COS 4 -rT- COS ($" H) = Sill D Sill
o kb
yl yfl
-j- sin (^' H) = cos x -rr- sin (^ 7/ H) = sin x
On Eclipses. 133
14. The constants T 7 , k, p, L 7 , L/ 7 , 7 7 , 7", being so computed, the angle w and the
time t of the phase for any place whose North latitude is / and East longitude X, will
be determined by the two following equations, in which the upper sign relates to the
beginning and the under sign to the ending.
cos a? = p L 7 sin / -r- 7' cos / cos (X -f y 7 )
t = T 7 + k sin w + L" sin / 7 7/ cos / cos (X + y 77 )
The result will be the most accurate when the place is near to that on which the
previous part of the calculation is founded.
III. TRANSIT OF MERCURY OR VENUS OVER THE DISC OF THE SUN.
(Same notation for the Planet as for the Moon.)
1 5. Assume the time T near to the time of conjunction in longitude, or right ascension,
= Sun's right ascension Planet's right ascension in arc
KI =r hourly variation of
D!= Sun's hourly motion in declination minus that of the Planet
For contact of Planet's centre with Sun's limb, A = a
D,
tan i =
k r= [3-55630]
7 cos y = -
cos 3
A cos i
_ cos S
corr.) D
A
cos w = 7 cos (V + q =z ky sin (^ + t)
16. H = the true Greenwich hour angle of at the time T
kb
sinw
L 7/
= cosS cos {( c) + w}
y"
-jji cos (y 77 H) = sinS cos {( i) + w}
^
8 i n(v ,//_H)=sm{(-0 T w
17. Then, for the centre of the earth,
(0 = (T 9) + k sin w
and, for any place whose latitude is / and east longitude X,
*=(*) + {7" p cos / cos (\ + y") L 77 p sin /}
using the upper signs for the ingress, and the under signs for the egress.
The positions of the points of ingress and egress, estimated from the North point of
the Sun's limb towards the East, as the transit would be seen from the centre of the
earth, will be determined in the same manner as for the immersion and emersion of an
Occultation, No. 19, using w for w. These angles may be assumed to be the same for
any place on the surface, the effect of parallax being so very minute.
NAUTICAL ALMANAC 1836. APPENDIX,
134 On Eclipses.
IV. OCCULT ATI ON OF A STAR BY THE MOON.
GENERAL LIMITS OF LATITUDE.
18. (! and D x at true 6 )
D t
tan i = n = (diff. dec.) cos t
! COS
cos Wi = + ~ -2*125 cosw 2 = + ~ + -2725
sin 9 = cos cos t
A = Wj B sin / 2 = + cos 5 cos (w 2 t)
w i> w 2 , t, 0, same sign as
I u PP er 1 signs when is \ P osit ; ve 1
\ under j \ negative J
When w t is impossible, l^ = 90, with the same name as $.
When w 2 is impossible, / a = complement of , with different name from
CALCULATION FOR PARTICULAR PLACE.
19. For the latitude of the place prepare the constants
M
0( 3) = [9 -41916] (U
cot /
which will serve for all Occultations at that place.
For the time of true <^ find
h = sidereal time at place right ascension of Star
and thence determine the time T, as in No. 1. For this time take out the quantities
P, s, D, D!, a, i ; and compute
x = (D S) (0 ( ^.Pcosa (1) . P sin S cos A)
y = cos ^ (1) P sin A
y. = ai COS 2 . P COS A,
With these proceed as in Nos. 6 and 7, using A' = * = [9 "43537] P.
20. For the positions of the points of immersion and emersion on the limb of the
Moon,
At "on'' angle from North towards Jl for direct image.
At {"SSJ 8 ari g lefromNorthtowardsEast = l(Zi)^w} for inverted ima 8 e '
For the same angles from the Vertex we must deduct the parallactic angle for each
time.
2.1. If an accurate calculation is wanted, proceed as with a Solar Eclipse.
On Eclipses. 135
V. ECLIPSE OF THE MOON.
22. Fix on a convenient time near to the time of opposition in longitude, or full moon;
and for this time find P, s, TT, ,
AD t
Partial A 7 - - - - 30 48 '3
Annular A'---- 051 '5
D +
AD +
I)
o / //
19 33 43
35 26
+ 23 49
A oc + 23 15
+ 18 58 17
a' + 34 f + 22 1
' corr. (log +1*53148 \ log + 3 '12090
185829 cos D'+ 9*97574 - - - +9-97574
7 f
\
x -
12 y - + 1 -50/22 y l - + 3 '09664 (1)
138
On Eclipses,
y - + 1 -50722
x - i -07918
4- HO 28
* t (tan S
\cos S 9 '
42804
y l - + 3-09664 (1)
Xi - -f 2 '65514
cot i +0 '44150
54364
i + 19 53 '5 W - + 1 '53554
(S + t) 130 21 '5 - cos 9-81129
n 1 '34683
Partial - - log A ' 3 -26677
w + 9041-1 cos w 8 -08006
cos i +9 '97328
- - + 1 '53554
const. 3 -55630
+ 5 '06512 (2)
H + 1 '96848 (2) (1)
- 8 '08006
-221 2'6
39 40 '4
c 3 -88842
sin a +9 '81732
c 3 '88842
sin b 9 '80510
PARTIAL.
-(S+0
a
b
ANNULAR
Assumed time
Beginning - -
Longitude - -
Beginning - -
n - -
Annular - - log A'
+ 11533''-9 cos vo
130 21*5 c
3 70574
-f 3 -69352
h m s
1 24 39
3 13 -
h m s
t a + 1 22 18
3 13
1 48 21
12 44
Greenwich
mean times,
W. 12 44 W.
1 35 37
Edmburgh
mean times.
1 -34683
1 '71181
H + 1 '96848
9 '63502
9 '63502
2 '33346
9 -96047
c 2 -33346
sin b 9-40711
245 55 '4 sin a +
14 47 -6
Assumed time
Beginning
Longitude
. ."' Beginning
2 -29393
+ 1 -74057
h m s h m s
3 17 t z + 55
3 13 3 13
3 9 43
12 44
Ending 3 13 55^'
( mean times.
W. 12 44 W.
2 56 59
Ending 3 1 111 Edinbur g h
& 1 mean times.
POSITIONS OF CONTACTS FOR DIRECT IMAGE.
(-0 -19'9
w +907
Partial contact at { J^J** ; ] _' * } f rora North towards j ^f
(-0 - 19-9
u 4- 115 '6
Annular contact at ( be g hinin S
{ ending -
from North towards
West.
East.
On Eclipses.
139
For the same angles from Vertex we must estimate them towards the East, and
deduct the angle M, thus
Beginning 135 '5
M H-31'9
Ending + 95 '7
M +31-9
167'4 towards West.
63 '8 towards East.
COMPUTATION FOR l h 48 m , FOR AN ACCURATE DETERMINATION OF PARTIAL
BEGINNING.
/ //
D + 19 19 35-9 a +
D! + 9 26
Edinburgh Sid. Time at Greenwich
f ill nm
O 1 II
18 57 39-3
h m
Mean Noon - - 3 20
1
15 23 '2
27 38
14-4
7-9
Sidereal Equivalei
m _ - _ _ 7 '94712
cos D - - 9 '97481
k - - - - 7 -97231
cos h - - + 9*95707
it for 1 I
48
[ 48
}'s R.A.
{time
arc
Const. - -
5 8
_______ 3 28
32 '2
18 '2
14 -0
i on;O
3' '5
2-1764
~T &<*
5-31439
7 'Q7231
sin h - - -
corr. for n -
/ J i "0 1
370
n - - -
Qi
log - -
A !- -
h
(h) - :
e - - -
D
+ D -
7 '92938
47172
_l_ 2 'Ql 730 O
\ A - - -
- cos - - - -
cot/ - - - -
tane - - - -
sin B - - "-'*-'
- cos - - - -
~r * J L i u *
- + 2 -6466
- + 13' 46"'6 flog -
IAD,
+ 1 -8033
- + 7' 23"
/
- + 25 3'5
- + 6-9
- + 25 10 -4
/
- + 31 36 '7
- + 19 19-6
- + 50 56 -3
+ 1' 4"
+ 9 -95666
+ 9-75001
+ 9 -83256 cos /
+ 9 '78922 G
+ 970667
+ 9*78665
+ 971946
+ 9 -79945 B
tan (A) - - -
o / [tan MI - - -
+ 9 '92001 - - check
+ 9 -67209
+ 9-59210
- + 9 '92002
LCOS M 1
tan(0 + D)
- +
\)
yoyi i -
09068
COSf - - -
T
+
J
9
yuyn
81754
Q
05Q7Q
B - - - -
4.
()
78665
idii -
cos
sin - - - -
- +
- +
9
( J
81754
87733
1 Q*7 1 1
const. - -
5
5
s
31439
10104
1Q71 1
.
iy / 1 1
*-Jl L L
07444
corr for n \
518
flog - - -
3
30333
IAD
+
33
' 30"-6
140
On Eclipses.
log s -
2 -94904
518
{loo- -
0x400
-pv
log -
? /
i c n *n
AD! - -
s
#! - - -
D - -
AD -
/
- +19 19
- + 33
//
35
30
'9
6
A 7 - -
a
A a
30 49 '9
15 23 -2 a, +
+ 13 46'6 A! +
27 38
7 23
' corr.
- + 18 46
- + 18 57
5
2
39
3
'2
3
| '
\log -
cosD'
29 9 '8 +
3 '24299 log - +
+ 9 -97627 +
20 15
3 -08458
9 -97627
1 4
8 22
- - 11 31 '8
S ---- 112 39' '81-
y - 3-21926 y t - + 3 '06085 (1)
x - 2 '83998 x l - + 2 70070
(tan S + '37928 cot t t + '36015
] - -T -
Ism S 9-96510 cos tl +9*96215
--- +
- 3
fi
3 -06085
A' 3-26715
90
+
t - 1
13 34 '5
k +
3 -72475
6 + -24953
x - i
13 32 '2
D +
' corr.
6+3 '97428
o / //
19 19 35 -9 1
2'2 J
cos D' +
2
9
96530
97627
& +
18 57 39'3
A 7
7
sin ^
2
94157
. . A/
3
"12018
_
i//
ooO oo/.o
J
t an y
9
82139
T
t +
23 34 *5
i
cos y +
9
92092
A /
V + l
9 57*7
A'
A'
i
3
19920
-26715
Q321 1
cos(V+0
+ 9*99341
sin
f' + O ~
9
23802
{***" J |**&
+ 9 '92552
A;
""*'"
3
72475
p
+ '84240
--
2
'89488
01
A +
Loner.
25 3'-5
3 10 *9 W.
*
+
1
13 5
48
HP
i . i
o
1 K.
cosD + 9*97481
cost + 9*96215
6 - - + -24953
cosx
tan
sin
L' - - + -18649
sin D + 9*51977
cos t +9 -96215
+ 9*48192
9 -60134
-11942
9-90109
r' H 5246 / -8{ t ?
l*i
H + 28 14 '4 +9 70025
6 - - -24953
y' . . 24 32 '4
- + 9 '94978
cosD + 9*97481
sin i +9 -60200
kb + 3-97428
H + 28 14-4
-j- 110
1-5
L" + 3 -55109
sin D + 9 '51977
sin t +9 -60200
sn
sn
+ 9*12177
+ 9 -96228
+ -84051
+ 9-99552
+ 9-96676
kb - - 3 '97428
7" + 3-94104
142 On Eclipses.
We have hence, for the Greenwich time t of beginning, at any place whose latitude
is /, + North, South, and longitude X, + East, West, the two following
equations, which may be safely depended on for any place in Scotland or the North of
England.
cosw = 0-84240 --[0-18649] sin / + [9'94978] cos/ cos (X 24 32''4)
f2 h l m 5 s [372475] sin w+ [3-55109] sin/ [3-94104] cos/ cos (X+ 110 l'-5)
Contact on 0's limb, w + 23 34' '5 from the North towards the West.
As a check on this calculation take the assumed radical place, Edinburgh, and
/=+5546 / '9, X = 3 10' '9, giving u = 89 6' '9 and* l h 45 m 24% which per-
fectly coincide with the results of the original calculation.
Similar calculations for the ending of the Eclipse give the equations,
coso; = 0-93848 [0-20291] sin /-f [9-88677] cos/ cos (\+ 2/ 6'7)
f == l h 38 m 33"+ [3-66890] sin w + [3-35544] sin/ [3'90073] cos / cos (X+ 153 3'-8)
Contact on 0's limb, w 16 56' -2 from the North towards the East.
Also by calculating with T 2=: 3 h 1 3 m for the annular phase there will result
cos ct' = 29-66600 [175159] sin /+[ 1-46950] cos/ cos (X + 1 42'-4)
*=l h 43 m 7 8 + [2-14475] sin u> + [3-45484] sin/ [3-92550] cos /cos (X+ 131 55'-9)
Contact on 0's limb, 19 53' -5 + o> from the North towards the East,
the upper sign appertaining to the beginning and the under sign to the ending. If
cos tv > l the place will be without the limits, and the eclipse will not be annular.
By taking /= +55 46''9, X= 3 10' '9, the results will exactly correspond with
the special calculation.
Note. The expression of cos u> for the annular phase, as the appearance of this
phase is comprised within narrow limits on the surface of the Earth, will afford a very
convenient and simple determination of the places which range in those limits as well
as those which range in the central line ; and we may expect very accurate results
throughout the portion of country originally taken into consideration. Thus for the
Southern limit we must obviously have cos w = + 1 , for the Central line cos w =r 0,
and for the Northern limit cos u> = 1 ; and hence the following conditions :
{+ 1 ) f southern limit.
> for < central eclipse.
1 J ( northern limit.
By making the assumptions
n 1 cos N' = 7' cos (X+VO I f \ +
n'sinN'^L 7 j "
they will give
f p -f 1 1 [ southern limit 1
n 1 cos (N' + /) = < p > for < central eclipse ?----(*)
( p 1 J ( northern limit J
If we therefore take any meridian whose East longitude is X, these two equations
(r), (s) will serve to determine the extreme latitudes /, on this meridian, between
which the eclipse will be annular as well as that where it will be central.
For the preceding eclipse, these equations will be
n 1 cos N' = [1 -46950] cos (X+ 1 42' '4)
n' sinN 7 = [1 '75159]
{ [1 -45737] ] f southern limit.
[1 -47226] V for < central eclipse.
[1 '48665] J ( northern limit.
On Eclipses.
143
If we take, for example, the meridian of Edinburgh, and use X 3 10''9, there
will result,
o /
Extreme Southern Point of annular appearance, N. 54 19 *7
Point of Central appearance, N. 55 20 '4
Extreme Northern Point of annular appearance, N. 56 21 '7
which are geocentric latitudes.
III. CALCULATION OF THE TRANSIT OF MERCURY,
Nov. 7, 1835.
The conjunction in right ascension takes place about 7 h 38 m ; take therefore
T r= 7 h 40 m , an d we readily find from the ephemeris the following data,
5 16 15' 58" '2
O / //
D 16 22 4 "2
D! 2 32 '6
s 4'8
* + 10 '95
v + 5 32'7
a 16 10 '4
TT 8-66
P 12 -66
With these quantities the calculation, for external contact of limbs, is as follows :
P
a 16 10-4
s 4'8
A 16 15 '2
//
12-66
8-66
4 -00 - - - - -60206
A 2 -98909
+ 1 '03941
b + 7 "61297
i + 2 '52205
cos
+ 9 '98226 ...... +9 '98226
O f //
$16 15 58 '2
corr.
D 16 22
cos + 1 -02167
2)
of
COB) + 2-50431
4-2 acos$+ 1 -02167
D, 2- 32-6 - - 2-18355
+ 2 '5.0431
6 6 -0 - - - - + 2 -56348 i
t 25 32 '3
' t f tan^ + 8 '45819
23 53-6
-0+9-96109 sin
cosw +9 -53566
w + 6955'-4
cosf + 9*99982
+ 2 -5636T
A 2 -98909
, f tan 9 -67924
e qo .q 1
d \cos + 9-95535
A 2 '98909
const. 3 -55630
sinw +9 -97278
k + 3 -99643
7+9 ^7457
k + 3 -99643
9*60749
3 -17849
q O h 25 m 8 8 '3
T + 7_40
T + 8 5 8-3
+ 6 -50074
k + 3 -99643
b + 7-61297
kb + 1 '60940
sin w + 9 ^7278
k" + 1 -63662
cos S + 9 '98226
k" cosS + 1 '61888
sinw +3-96921 - 2 35 15
Mean time of j 1 ^ 688
egress
10 40 23 -Q } for the centre of the Earth -
144
On Eclipses.
CONSTANTS FOR REDUCTION OF INGRESS.
Equa. -f
| time -f
(arc +
- * +
\v
I W
5 29 52 7
16 10 -0
5 46 2*7
8630 / '7
25 32 '3
69 55 '4
44 23 'I
Y<" H 105 58 '3
*" 19 27 -6
- - - - cos + 9 '85410 - - - - sin 9 '84477
sin 5 9 "44733
9 '30143 - 9 '30143
s$ + 1 '61888
L" + 1 -47298
f tan +
{ sin
9 '98290
+ 9 '86187
k" + 1 -63662
7" + 1 -49841)"
CONSTANTS FOR REDUCTION OF EGRESS.
k m s
10 40 23 '9
Equa. + 16 9 -2
H in
[time + 10 56 33 '1
(arc + 164 8'-3
*
"{ 1
95 27 -7
88 28 -0
252 36-3
107 23 '7
- - - COS
sin o
8
-9
'97854
44733
_ - _ _
sin
+
9
99802
+ 8
42587
. . . .
-"V
+
8
42587
{
tan
f
1
57215
\
sin
+
9
'99984
k" cos $
+ 1
61888
k"
-**
+
9
i
'99818
63662
L" -59742
y" + 1 -63480
The former part of the calculation repeated for the times 5 h 30 m and 10 h 40 ra we
shall find more accurate times of ingress and egress, for the centre of the earth, to be
5 h 29 m 56 s and 10 b 40 m 31% which however still cannot be depended on within a few
seconds. More reliance can be placed in the amount of reduction for parallax. The
times reduced for any place whose North latitude is /, and East longitude X, viz. :
Ingress, Nov. 7 d 5 h 29 m 56 8 + [T4730] p sin/ [1'4985] /> cos /cos (X 19 28 7 )
Egress, - 10 40 31 + [G'5974] p sin/ + [1'6348] /> cos/ cos (X 107 24 ; )
will indicate, with considerable accuracy, the difference between the times at any two
places.
The positions of the contacts on the Sun's limb, for an inverted image, will be
; [ Jf ' } from the North towards the {
Contact at
On Eclipses.
145
IV. OCCULTATION OF A STAR.
On January 7, 1836, the star t Leonis, whose right ascension is 10 h 23 m 26*'4 and
declination N. 14 58' 39", will be Occulted by the Moon.
LIMITS OF LATITUDE.
At the time of true 6 in right ascension, viz. 12 h 12 m I) 8 , we have the following
data,
O I II I II
D + 15 33 2 D! 11 47
$ + 14 58 39 ! + 30 41
P + 56 4
D 5 + 34 23
with which we proceed thus :
D! 11 47'- - 2 '84942
cri + 30 41 - - +3 -26505
9 -58437
S + 14 59' cos + 9 '98498
(tan 9 -59939
i 21 41 < -,
Icos + 9*96813
cliff. dec.+ 34' 23" h 3 '31450
n
"T 5 "
const.
O /
WL -f 147 24 - nat. cos
5699
2725
8424
2974
n + 3 '28263
P+56' 4"- -+
_
" + -5699 - - + 9 -75577
iv a + 107 18 - nat. cos
L + 21 41 - log. cos -f 9'968T~(1)
+ 86 37' - log. cos + 8 -8833 (2)
log. cos 5 + 9 '9850 (3)
Q + 63 51 - log. cos + 9-9531 (l) + (3)
/t + 83 33
4 14
/ 2 4 14 log. sin / 2 + 8 -8683 (2)+ (3)
The star may therefore be occulted between the parallels of latitude N. 83 33' and
S.4 14'. The parallel of Greenwich is within these limits; and if the hour angle of
the star be computed roughly for the meridian of Greenwich, the star will be found to
be considerably elevated above the horizon. A special calculation for the Observatory
of Greenwich will consequently serve as an example of the circumstances for a par-
ticular place.
CALCULATION FOR GREENWICH OBSERVATORY.
Constants (1) , 0< f >, 0< 8) .
P ----
cos/
9-99913
9-79610
(l) +9 '79523 +9 '79523
cot/ + 9-90381 Const. - - 9 '41916
0W 1- 9-89142 (3) -f 9*21439
These will be constant for all Occupations at Greenwich.
h in s
Sidereal time at mean noon 19 4 22 '4
Star's right ascension - - - 10 23 26 '4
h at mean noon ----- 15 19 4 '0 --------
Mean time of true 6 - - - -f 12 12 T - - - - ;: - <
+ 2
4 -0
Acceleration
h at true 6
h m
15 19
n 6
1 49-4
35
Acceleration - -f
time 4 11 14 -6
arc - - 62 48'-7
-I
With this and i = 30'- 7 we find, by the table )
at page 129,T = Il h 6 m . J
h at mean noon is put down negatively, in order to have more readily the other values
of h less than 12 1 ' or 180.
146
P 56' 4" -
On J
- + 3 -52686 +
- + 9 '89142 cosh +
Eclipses.
3-52686
9 '65983 sill h
+ 3*52686
-9-94915
cos a - -
+ 3 -41828 +
- + 9 "98499 sin a +
3 -18669
9*41236 sinS
3 -47601
+ 9*41236
r + 3-40327 +
2 -59905
2-88837
I + 42' 11" ?+
9 '79523 ^
+ 9 "21439
D-S -
+ 48 +
+ 38 3
- + 47 22
+ Q JO
2 -39428 I
*i - -
2' 7"
11 42
9 35
y *y
f
_ __ 33/ 54"
fi -
+ 30' 44"
t
cos S - -
P sin h - -
0w ...
3 '30835
- + 9 '98499
t 3 -29334
1 32' 45"
- 3 -47601 P -
- + 9 "79523 Const.
cos a -
;:;: I
3-52686 P ( f s/l
+ 3 -26576
+ 9 "98499
+ 3-25075
+ 29' 41"
* ' t
f 3 -27124 A'-
2 -96223
+ 2 "40108
t 31' 7"
t
+ 4' 12"
{ ;;;.
X - - - -
tan S - -
1' 38"
1 "99123
- + 2-74741
9 -24382 S
956'-6 cotl "
+ 25 29
+ 3 '18441
-2-75967
o -42474
cosS - -
- + 9 "99343 i
20 36 -6 cos i -
+ 9"97128
W - - -
4- 2 'T'i^QS -
-4- 2 "*7 5 3Q8
cos ( S +
0+ 9*93508 (S + i) +
30 33 -2
3 -55630
n - - -
A'
COS la - -
- + 2 -68906
2 -96223
- + 9 '72683 - co - - +
H - -
57 47 *0 - cos w -
>*7 1 3 *8 r>
+ 6-28156
+ 3 -09715
+ 9*72683
+ o > *7nqp
sin a - -
- 9 '66045 b - - +
88 20 '2 sin b -
+ 9 '99982
3 -03077
nh i *?m.Q
/
+ 3 -37014
+ nh OQ m ' 1
u I/ y
1 1 fi
u oy i
HA
1 48 ' 1
H4 *i *1 TVfpfin tinif^
Acceleration
S. T. mean :
1 -8
noon 19 4 *4
Acceleration -^
S. T. mean noon
2 *0
19 4 -4
C KA .0
6^1 '5 - Sid times
Star's R. A.
- 10 23 *4
Star's R. A. -
10 23 "4
f Im. h - -
4 29 *1 = 67
[Em. h == - - -
- 3 31 "9 = 53
IParallactic
Z - 39' 7
\Parallactic Z
- 36'9
( t \
i on A
( i }
, pp. ./
V */
\ i)
J- 5*7 -8
On Eclipses.
14'
These angles are for the inverted image; and, being estimated towards the East,
the negative values must be considered as towards the West. The decimation of the
Star gives for the latitude of Greenwich a semidiurnal arc of 7 h 23 m ; as this exceeds
the value of h both at Immersion and Emersion, the Immersion and Emersion will
both occur above the horizon.
V. CALCULATION OF THE ECLIPSE OF THE MOON,
April 30, 1836.
The Opposition or Full Moon takes place at 19 b 58 m . For the computation assume
the time 20 h O m .
19 h 20 h 21 h
hms hms hms
}) 5 sR.A. 14 32 51'35 - - 14 35 11 '19 - - 14 37 31 '43
0'sR.A. + 12 h - 14 33 52 '38 - - 14 34 1 '91 - - 14 34 1 1 '45
(time 1 1 '03
ami
9 '28
1 space
a = +
15'
' 19
+ 3 19*98
+ 50' 0"
15 15 -, ,/,
'9 + 32 ll <*' = + 32/ 38 "
U
19'
})'sDec. 14 5
a cor. - -
0'sDec. - - + 15 6 35
1 T~l6
II
191
of
+ 61' 16"
x = + 47 21
+ 33 29
20
h
21 h
O
j
//
o
/
n
14
19
58
!;
14
34
32
1
+
15
7
20
+
15
8
6
+
47
21
+
33
29
/ ei
Ji
s - 13' 54
"
a +3-01662 a t + 3 -29181 P60'19"
cos D +9 -98627 +9'98627.
y l+ rS^ P 3 -55859
#, 2-92117
y +3-00289
x +3 -45347
S +19 30'7
tan S + 9 -54942 cot i -0 '33691
cos S + 9 '97431 cos i +9 -96163
9 -99929
3 -55788
23 43 '9 W +3 -47916 +3-47916 ve, it will be readily inferred that the elements of the orbit
can at every time be expressed by means of these co-ordinates and velocities at that
time. Thus, by proper treatment, the equations of motion may be made to express
the differential coefficients of the elements of the ellipse in which (as above) the
planet's place and motion near that time are to be determined by the usual formulae.
This gives only the momentary changes of the elements : a process equivalent to
integration will then give us the total change that has taken place between any one
time and any other time.
(4) Let IA be the Sun's mass^ m l9 m 2 , rag, &c. the masses of the disturbing planets
(the masses being represented by the number of units of velocity which their action
at the unit of distance would give in the unit of time), and for brevity let the letter
m be used for the disturbed planet; also let x, x lt x z , &c., y, y l9 y a , &c., 2, z l9 z a ,
&c., be the co-ordinates of the disturbed and disturbing planets at the time t\ x being
measured from the Sun towards the first point of Aries (supposed invariable), ?/
being measured towards the first point of Cancer (the plane of the ecliptic being
supposed invariable), and z perpendicular to the plane of the ecliptic: let r, r l9 r 2 ,
&c. be the true radii vectores : also let a, e, ts, z, v, be the mean distance, excentricity,
longitude of perihelion, inclination, and longitude of node, of m at that time ; n the
mean angular motion (in the instantaneous ellipse) in one unit of time, measured in
parts of radius : and let e be the epoch of mean longitude of m, or the angle that
must be added to nt, to ^ form in the instantaneous ellipse what in an invariable
ellipse is called the mean longitude ; so that the place of m may be calculated at the
time tj by supposing it to move in the ellipse whose elements are a, ' + (y-yi) f + (z-*i)'}* r ' 8 >
for all the different disturbing planets, we have
d*x ux
w=-"^- flA -
Similarly, putting B for the sum of such quantities as
y-y _
and C for the sum of such quantities as
Wi f _ * ** __ j. il
ft l{^-^) 2 + (y-yO 2 -f (*-*i)*}* ri3
for all the different disturbing planets v we have
These are our fundamental equations.
(6) To discover the best method of combining these, we must, in conformity with
the considerations of (3), express some of the elements in elliptic motion by means of
the co-ordinates and velocities of the planet. Now in elliptic motion = --
(velocity) 2 2 1 f/rfa?Y /c/y\ 8
- -
the equations of (5) the variation of --- 1 12? J "'"("T// ~'"(l7)l
have the variation of -- Also putting h for twice the area passed over in the unit
a
of time by the radius vector, in undisturbed elliptic motion, the projections of this
double area on the planes of #y, yz, and xz, are h cos i, h sin t sin r y h sin i cos v 9
12
152 On the Calculation of Perturbations.
dii dx dz d\i dz dx
but these projections are respectively x ~- 7-7, V -; -- * ~r-> x , -- z -r-i
J dt J dt ' y dt dt ' dt dt '
therefore inferring from the equations of (5) the variations of these latter quantities
we shall obtain the variations of A cos i y h sin i sin v, and h sin i cos v. From these
we shall obtain the variations of i 9 v, and h: and since h = ^{/Ka(l e 2 )}, and
the variation of a is already found, the variation of e will be found. The mean
distance and excentricity being known, the place of perihelion is easily found, as there
is but one place of perihelion which can give the proper values of v and -7 : and in
like manner the epoch is found. This is the general outline of the method which we
shall follow.
(7) From the equations of (5) we obtain
dx d*x dy d*y dz dV\ 2/* / dx dy
~ '*~' *
d f 2
or, as the quantity under the differential sign on the first side is = ,
1 da dx dy dz
a* dt " dt dt dt
da ( dx dy dz
whence = 2 a I A - \- B -- + C r-
from which the variation of the semi-major axis of the orbit is found.
(8) Now the longitude of m from the node is 6 v (0 being measured as
mentioned in (4) ), and the co-ordinates of m parallel to the line of nodes, perpen-
dicular to the line of nodes in the plane of the ecliptic, and perpendicular to the
ecliptic, are therefore r cos (0 >/), r sin (0 ") cos i, and r sin (e v) sin i.
From these we readily obtain
t T = r {cos (0 r) cos v sin (0 y) ccs i sin r}
y = r {sin (0 >') cos i cos v + cos (0 v) sin v}
z = r sin (0 v) sin i
and "hence
On the Calculation of Perturbations.
153
dx dr x dO
~-j~ =: -T- -- H r -7-
( }
s sin (0 F) cos v cos (0 >') cos z sin y >
du dr y de ( \
~ = -=- + r -7- \ cos (01') cos z cos v sin (0 1>) sin y f
a/ at r at \
dz dr
d9
cos e -
These expressions, it is to be observed, are formed on the supposition that the
elements are invariable; which is correct, because the motion in the actual part of the
real orbit is the same as it would be in the instantaneous ellipse of that instant, sup-
posing that ellipse to remain unvaried.
(9) Substituting these in the expression for -^-,
at
da 2 a 2 dr
Af sin (0 y) cos y -f- cos (0 v) cos i sin y J
B ( sin (0 v) sin v cos (0 v) cos i cos v
C cos(0 v) sin/
which for brevity we shall write
~ ' ~di T ~dt
The calculation of A' involves no difficulty ; that of B' is rendered very easy by the
use of two subsidiary constant angles, "^ and x where tan ^ = tan v cos i t and
tan x = cot y cos i ; whence
sin v .
in (0 v + V ) +B sin (0 v v) C snucos (0- y).
(10) Since ?- 2 ~=k-
we have r 2 r=
( 1 ^ a )\ and J p, =rn a 1 ,
e 2 )
or r-r =
wa a
r
Also
therefore , -^ = e sin (0 cr)
acr
dO
na s 6
154 On the Calculation of Perturbations.
Substituting these values in the expression of (9),
da na 3 e sin (0 si) B
- '<
(11) From the equations of (5) we find
, /x cZ 2 y d 2 # d ( dy dx\
or, by (6), since * _ - y _ = _ (* ^ - y _), &c.
(
-7- (A sin i sin v) = /* (B* Cy)
-7 (^ sin i cos v) =r /* (A^; Ca;)
.... d> dtanv
(12) _ = co S V-^ r -
9 d h sin i sin
= cos V -7-
dt h sin z cos v
, a . a . \ h sin cos v -7- (/i sin i sin v) 7t sin i sin y -y- (h sin z cos ^) I
= -T~: < cos v r- (h sin i sin y) sin v -7- (& sin .1 cos i/) >
A sin z 1 d< di v ' J
. A ( z sin v) + B (z cos v) + C (x sin y y cos v) 1
h sin z v J
If we substitute for x and y the values in (8), we find
x sin v y cos v ~ r sin (0 v) cos z z cot z
Therefore -j- = , . 2 . ^ A sin v sin i + B cos y sin i C cos i !
cfa A sm 5 ^ [
Let A sin v sin i 4- B cos v sin C cos a = C'
On the Calculation of Perturbations* 155
Then, since h = V/* J{ a ( l e *)}> and n = V/* a ""^
/* an
and s= 77V
fi/ an
therefore
dt ^(1 e 2 )
. ,.
2
di 1 . 2 . d tan 2 *
(13) -j- r coti cos-*
/ * s -
2 dt
1 . d (h sin z sin v) 2 + (A sin i cos v) a
- cot i cos s z -r- -^ - - - -- ^ --
2 dt (Acosz) 2
. . f A sin i sin v A sin i cos v
C 8 '
A-CO.H-
(/i sin i sin v) 2 + (A sin i cos
-y- 1 cos i sin y (Ex Cy) + cos i cos v ( A Co;) sin i (Ay B# ) j-
= -^- 1 A (z cos z cos y y sin *) + B (z cos i sin v. + x sin ')
h ( ^
C cos i (# cos v + y sin y) >
Oh substituting the values of x, y, and z, from (8) this is changed to
I A( z cot(e >) sin v J + B (z cot (0 v) cosy J
C z cot (0 r) cotzl
V(l e 2 ) sin
an
r-r- zcot (0 y) C'
r cos (8 v) C'
rf*
= t \ (h cos z) 8 + (h sin i sin v) 2 -j- (/i sin * cos y) 2 V
2 A ct< I
= -~ | A cos i (Ay Bo;) + /t sin i sin v (BsCy) + h sint cos v (Az Ca;)|
(y cos * + z si n * cos v) + B (z sin i sin v x cos z)
)
C (a; sin i cos y + y sin sin y) f
156 On the Calculation of Perturbations.
Substituting the values of #, y, and z, from (8)
1 dh anr i . ( . \
T ' ~dt ~ J(le*) t V m ^ C S * + COS ^ e ~~'') cos * sin v )
+ B f cos (0 i/) cos i cos j/ + sin (0 t>) sin r J C sin i cos (9 y) !
" "
And as h *Jp J{a (1 , sin (6 tu)
or -/77 57- rB' A 7 > -
VO e*) VO e ) r
B/ c
r 1 e
whence
7 O x r% v
na
r
and . = -
(15) Now ^ g ? is found by differentiating
logr=rloga + log(l e 2 ) log {1+e cos (0
which represents the correct value of log r, because by (2) and (4) the same expres-
sions are to be taken to represent the place of m (using the elements of the instan-
taneous ellipse), as those which are employed in undisturbed elliptic motion (using
the elements of the permanent ellipse). Still it is to be borne in mind that the
elements vary from one instant to another; and therefore their variation must be
taken into account in forming - Thus we have for -- -^ - the rigorous
expression
d (log a) ^{log(l e 8 )} tljlogQ + e cos(0 CT))| de_
~dT~ ~~di~ de ' dt
d{log(l-f e cos (e CT))} dcr d [log(l-fe cos (0 CT) )} dQ
dns '~dT~ dO "cU*
This expression, it is evident, has been obtained merely by considering that the place
of m is always represented truly by the elliptic formulae applied to the variable
elements. But by (2) the motion of m is also to be represented truly by the elliptic
formulae for motion applied to the variable elements : and therefore generally the first
differential coefficient with respect to t, of r or 9 or of any function of r or 0, must be
On the Calculation of Perturbations.
represented truly by the elliptic formulae. Now the elliptic formula for - - is
dd
where r is the same as in the former expression. Making the two expressions
equal,
d (log q) rf
d* de dt
d {log (l-fe cos (6 CT))} dor __
rfd "rfT : *
The reasoning of this article is general for any expression in terms of the co-ordi-
nates of the place of m ; and its general result may be stated thus : The differential
coefficient of any function of the co-ordinates (including polar co-ordinates) of m
taking those parts only which depend on the elements, is equal to zero. This would
not be true if the motion of m entered into the function ; for then the differential
coefficient of the function would involve second differential coefficients of the co-ordi-
nates, which, as in (5), have not the same form for undisturbed and for disturbed
motion.
(16) The last equation is
1 da 2e de cos(0 or) de e sin (0 TO)
a dt l e*dt I + e cos (0 TO) dt 1 + e cos (0 TO) c/J
whence
<&
S"
=
djxr _ 1 + e cos (9 TO) / 1 da 2e de\ cos (0 CT)
~dT "~ e sin (0 -or) \ * ' 57 "" I e* ' ~di) ~~ e sin (Q CT)
1+ecos (9 CT) San , na?*J(\ e 2 ) . ,cos (0 CT)
' c sin (e - J)~ ' V ( J - g2 ) ~~ ~~
n c8 . Acos(e^) B-
/ sin (0 CT) r
The factor of B ; in this expression
= 1FJ<^1^{^^^
COS CT
cos (e CT) { 1 -f 2ecos (0 w) + e 2 cos 2 (0 CT) 1
158 On the Calculation of Perturbations*
g
at at
or
(24) The element n (the mean motion) depends entirely upon a, whose variation
has already been found : it may, however, be useful to give its variation separately.
Since n z a 3 ^^, we have
. dn z da ... dn 3 n da
2 no? -r- + 3 TiV - = 0, from which - ------
dt dt dt 2 a dt
(25) Collecting, for convenience of reference, all the expressions that we have
obtained, we have
164 On the Calculation of Perturbations.
n*a*e sin (9 CT)
4- 3tt 2 ay(l-e 2 ) - + ___ r 8n-t 2+e cos -
2VO O cos 2 -
efcr naVO e 2 ) cos(0
" : "~ ""
f g / (1 _, 8) r sin (0-o) {2+
It will easily be seen that, if we assume the third-differences to be sensible, their
effects during one interval will destroy each other in the same manner as those of
the first-differences: the only effects omitted are, therefore, those of the fourth-
differences, which (except the intervals are extravagantly large) may be neglected.
If, however, it is wished to take them into account, the expression is =
the sum of all the calculated quantities [a]
+ - the sum of all the corresponding second-differences
, the sum of all the corresponding fourth-differences.
SECTION IIL
PRACTICAL RULES FOR CALCULATION.
(33) Divide the time through which the variations of the elements are to be
calculated into equal intervals. Experience alone can teach the calculator what will
be the most advantageous length of the intervals : it will depend greatly upon the
positions of the disturbing planets, especially Jupiter ; but it is probable that, when
Jupiter is nearest, intervals of 10 days each would not be found too long, and that
at other times intervals of 20 days each might be safely used. It is desirable to
retain the intervals of the same length through the whole of the time, even though the
calculations at some parts should be made independently for only each alternate
interval, and the others should be filled up by interpolation.
(34) All the calculations which follow are to be made for the middle day of each
interval. Thus, suppose the intervals were of 10 days each, and we wished to
calculate the variations of elements in the 400 days between September 17, 1834, and
October 22, 1835, the calculations must be made for 1834, September 22, October 2,
October 12, &c. In the following rules we shall express the order of the calculation
by the letter p : so that for September 22 (in this instance) p= 1 ; for October 2,
p = 2 ; for October 1 2, p = 3, &c.
m2
168 On the Calculation of Perturbations.
(35) It is supposed that we know tlie planet's mean longitude at the beginning of
the time, e ; its mean distance from the Sun, a ; the number of seconds in its mean
sidereal motion during one interval, N; its longitude of perihelion, w; its excen-
tricity, e; the longitude of its ascending node, v, and the inclination of its orbit to
the ecliptic, i: all for the beginning of the time. The mean longitude and the
longitude of perihelion are supposed to be measured from the first point of Aries on
the plane of the ecliptic, to the node, and then upon the plane of the orbit. With
these elements, the planet's true longitude (measured as the others are measured),
its radius vector r, and its co-ordinates, x 9 y, z, (of which x is drawn from the centre
of the Sun towards the first point of Aries, y towards the first point of Cancer, and z
perpendicular to the plane of the ecliptic towards the north) must be calculated
approximately for the middle day of every interval. It is probable that an accuracy
of 1' in true longitude and of the whole radius vector will be sufficient: but
5000
experience will be the best guide on this point.
(36) The proportions -, -, &c., of the masses of the various disturbing planets
to the Sun's mass are supposed to be known. The radii vectores r l9 r 2 , &c., the
co-ordinates x l9 y l9 2 15 x*, y^ 2 2 , &c., of the various disturbing planets, and A,, X 2 , &c.,
their distances from the disturbed planet, must be calculated for every middle day.
(37) The next step will be to calculate, for every middle day the following
quantities :
(38) Find the angles ^ and y, where tan y = tan v cos z, and tan % = cot v cos i
(y<- and x will therefore be constants) : and calculate for every middle day the follow-
ing expressions :
B' = A j - sin (0 v+V) + B -^- sin (0 v x) C sin i cos (0 v)
C 7 = A sin v sin i + B cos v sin z C cos L
(39) Find the angle 0, such that sin =s e (0 is therefore constant), and calculate
for each middle day the following expressions (where p 9 as before mentioned, is the
ordinal number of the interval)
On the Calculation of Perturbations. 169
[ a ] - f'g N sin \" a 3 tan 0V m ^"" g) A 7 + (* N sin I 77 a 4 COS0")
(\ * - //a rt \ / \ TJ/
i Sill \y TJT ) I a q _* i **
3 N sin I" a tan A 7 [ 3 N 8 sin 1 7/ a cos I
/ 7* \ / 7*
/ N /
(3N^inl 77 anan0)( F -|) S ^^ + ^
+ (N^cos0tan|)^^A 7
,^J_ / tan|\
- { ( 3 N 2 sin 1 7/ a 3 cos ] + I 2 Na -r )r sin (0 w)
I V / r \ cos /
+ f Na tan tan J r sin (0 w) cos (0 17)| B'
, a i
2 COS COS
= Nrf cot
&m<> cos
\ sin (e _ w) B ,
?>/
> si" ( 9 ~ ro ) cos (9 ro) B' + - - z C'
[ ] = - fN sin 1" a* cos ^ ^ ( -^=^ A' + ( N sin 1" a ^)1'
\ / r \ sm0/ r
^Nsin l // acot0 > )rB /
cos sin 2 1
The quantities within the large parentheses are constant.
(40) Collect the whole series of calculated quantities [a], take the first-differences
and second-differences, and supply by estimation a second-difference preceding the first
and one following the last; and with these form a first-difference preceding the first
[a] and one following the last [a]. Then the sum of all the quantities [a] with
of the excess of the last first-difference over the first first-difference will be the
24
whole variation of a, in parts of the earth's mean distance from the Sun. Similar
Operations performed with respect to all the quantities [N], [e], [BT], [v], and p] will
give the whole variations of N, e, or, r, and i 9 in seconds of space; and similar
170 On tlie Calculation of Perturbations.
operations performed on all the quantities [e] will give the whole variation of e in
parts of unity. These whole variations we shall denote by the prefix .
(41) Now, the planet's place at and near the end of the time is to be calculated as
if it were moving, undisturbed, in an elliptic orbit, whose mean distance == a + 5 a,
excentricity = e + Se, longitude of perihelion == CT + &CT, longitude of node = v + Jf,
and inclination = i -f 2 : and its mean longitude in this orbit is to be calculated as
if, at the beginning of the time, its mean longitude had been e + Se, and as if, from
the beginning of the time, its mean sidereal motion had been N + N in every
interval.
(42) If the planet's place is to be calculated for a considerable time before and
after the day to which we have corrected the elements, it will not, perhaps, be
sufficiently accurate to use one set of elements (though this is sufficient for the
ordinary ephemeris for the opposition of a small planet). In that case it will only
be necessary to terminate the summation of the quantities [a], &c., at two or three
different days, and to use the elements, thus corrected to two or three different days,
for the calculation of places for times near to those days.
(43) If the change in the elements through the whole period appears to be great,
the only method of making the calculation accurate will be, to sum the variations for
a short time (as perhaps one half or one third of the whole period) and, correcting the
elements, to use these corrected elements for the calculation of the co-ordinates and
other quantities which are to be used in the calculation of the variations for the next
part of the period. It is probable that this process will seldom be found necessary,
except when the planet near its aphelion is acted on by powerful disturbing forces; a
circumstance which occurs sometimes in the perturbations of ENCKE'S Comet.
(44) It is only necessary to add, that the formulae above suppose the longitudes to
be measured from an invariable line on an invariable ecliptic. To take account of the
alteration in the ecliptic and the first point of Aries, the longitude of the node must be
increased by
general precession during the interval diminution of obliquity during
_the interval X sin v X cott;
the longitudes of the planet and its perihelion from the node must be increased by
diminution of obliquity x sin v X cosec i :
the complete mean longitude therefore of the planet, and longitude of its perihelion,
must be increased by
general precession + diminution of obliquity X tan x siri v.
And the inclination must be increased by
diminution of obliquity X cos v.
On the Calculation of Perturbations. 171
The true longitude, calculated with the elements -thus further corrected and the
mean sidereal motion, for any considerable interval before and after the day for which
the elements are computed, must also be affected with precession proportional to that
interval. . If, however, in the process of calculating true longitudes, the motion of
precession be added to the mean sidereal motion, and if the same motion of precession
be applied to the longitudes of the node and perihelion (neglecting for short times the
effect of change of obliquity) it will not be necessary to take account of precession
afterwards.
G. B. AIRY.
OBSERVATORY, CAMBRIDGE,
Dec. 3, 1834.
172
ON THE
DETERMINATION OF THE LONGITUDE
FROM AN
OBSERVED SOLAR ECLIPSE OR OCCULTATION.
BY W. S. B. WOOLHOUSE,
HEAD ASSISTANT ON THE NAUTICAL ALMANAC ESTABLISHMENT.
AN accurate observation of a Solar Eclipse, or Occultation of a Star by the Moon,
furnishes a favourable opportunity for the calculation of the longitude. This cal-
culation may be effected by various methods, most of which are well known to
astronomers : amongst the most simple and practically useful may be noticed the
method of the late Dr. Young, and the improvements on the same by Mr. Thomas
Henderson, now Astronomer Royal of Scotland, (see Nautical Almanacs from 1827
to 1833, inclusive); also two methods by Mr. Edward Riddle, and another by
Mr. Thomas Maclear, published in the Memoirs of the Royal Astronomical Society,
vol. iv. pages 305 and 531. To obviate the difficulties in the way of the calcu-
lation, these methods, as, indeed, all others that have come within my observation,
suppose in the first instance that the estimated Greenwich time will suffice to take
out the Moon's declination accurately from the ephemeris; or that the motion of
the Moon in her orbit is uniform throughout a wide interval of time ; consequently,
when a good result is to be obtained, and the error of the estimated longitude is
considerable, the computer is generally obliged to repeat the calculation with more
accurate data, deduced for the Greenwich time according to the calculated, instead of
the assumed, longitude. A method well adapted to computation, and, in all cases,
free from inaccuracy or roughness of approximation has long been wanted. The
following brief discussion of the problem is submitted by way of continuation of
my paper on Eclipses, which forms the Appendix to the Nautical Almanac for
1836, and has for its object an easy, practical, and, at the same time, a correct solu-
tion. It is proposed, also, to supersede the necessity of having recourse, in these
calculations, to the elements usually printed in Occultation lists, the use of which
materially augments the chances of inaccuracy; and furthermore, to reduce the
processes of calculation, for an Occultation, to plain and simple rules for the use
of those who may be unaccustomed to analytical formulae.
In the case of an Eclipse of the Sun, the apparent time of observation being
converted into arc, at the rate of 1 5 for an hour, will show the true hour-angle of
the Sun's centre at that instant; and as the declination of the Sun is never subject
to a very rapid daily variation, it may be taken out from the Ephemeris with tolerable
accuracy by the approximate Greenwich time, deduced from an estimated longitude or
a rough longitude by account.
On the Determination of Longitude,
173
In the annexed figure, let Z represent the position of the zenith,
S the Sun and c'c'c' his limb. To illustrate the principle and
simplify the reasoning that enters into the present investigation,
it will be convenient to imagine, merely by way of convention,
the limb c'c'c' of the Sun to be an apparent one as affected by
a parallax equal to the relative parallax of the two bodies : on
this supposition, let s be the true place and c c c the true appear-
ance of the limb as it would be seen from the centre of the Earth.
Then, by the theory of the effects of parallax, the true semidia-
meter Sc' of the Sun will represent the fictitious semidiameter
sc as augmented by the parallax ; and if any point c be taken in the fictitious limb
cccit will be transferred to a corresponding point c 1 on the true limb of the Sun ;
consequently, the true limb of the Moon M being brought in contact with this disc,
the parallax will exactly reduce her apparent limb to a contact with the Sun's true
limb c'c'c'. Moreover, as the hour-angle and declination of the Sun S are known at the
time of observation, and as this position is now viewed as an apparent one, the effects
of the parallax or, in other words, the calculation of the relative right ascension and
declination of s and the diminished semidiameter s c follows directly from the equations
(2) of my paper on Eclipses, page 103. The problem is thus reduced to the deter-
mination of the corresponding Greenwich time when the true disc of the Moon comes
in contact with the given disc ccc placed at a given relative right ascension and declin-
ation; and every consideration relating to parallax is hence eliminated from the inquiry.
Assume,
at the right ascension
h the hour-angle
the declination
or the semidiameter
the right ascension
S the decimation
) cos M
A 3 = p P cos M cos e
= a + A a
$=$+ A
cos (0
tan (A)
174
On the Determination of Longitude
in which M may be regarded as the parallactic angle, and e the altitude of the Sun,
the latter of which will be wanted to take out the diminution of the Sim's semidiameter
with the table on page 1)5, to get
,
cos (D)
or, as the small arcs may be assumed proportional to their sines,
cos (D)
The time of the Moon's passing over this angle, or the time elapsed in passing from
M to m is therefore -- -. - x ' , which deducted from t 0) there results, for the
A! COS (D)
instant of contact, or observation,
The corresponding Greenwich time =
f A . (Q)~( A ) AsinCv + VO
A t A a cos (D)
The longitude from Greenwich is hence determined by taking the difference between
the Greenwich time and that of the observation, previously making them both apparent
or both mean by the application of the equation of time if necessary ; and it will be
when the Greenwich time is the
For an Occultation of a Star by the Moon the calculation will, in some respects, be
slightly abridged. The characters AI, D 1? must then represent the absolute motions of
the Moon in right ascension and declination ; the semidiameter k 's declination and the log. cosecant of the reduced latitude. To the same log. add
the log. cosecant of the >fc's declination the log. secant of the reduced latitude and the
log. secant of the hour-angle corrected. These sums will be the prop. logs, of two
arcs.
The former arc to have the same name as the latitude.
The latter to have
178 On the Determination of Longitude
The sum of these two arcs, having regard to their names, will give the correction
to be applied to the >K J s declination to get the declination corrected.
To the sum (S^ add the constant log. 1*1761 and the log. cosine of the >fc's de-
clination corrected ; the sum will be the prop. log. of an arc in time, to be
subtracted from } the *' s ri S ht ascension, when it is j^f } of the meridian,
to get the >K's right ascension corrected.
In the hourly ephemeris of the Moon, fix on a convenient time at which her right
ascension is near to that of the star corrected ; and, for this time, take out the
right ascension, the declination, and their hourly variations.
Subtract the common log. of the difference between the corrected right ascension of
the star and the right ascension of the Moon from the common log. of the hourly
motion in right ascension; to the remainder add the constant log. 0'477l; to the
same remainder add the prop. log. of the hourly motion in declination. The former
sum will be the prop. log. of a time to be
subtrlcted from } the assumed time when *' s R ' A ' is { ST* } *<"> J >S R ' A '
to get the time corrected;
The latter will be the prop. log. of a correction of the }) 's declination, to be applied
with-
To the common log. of the hourly motion in right ascension, add the log. cosine of
the J) J s corrected declination ; and to the sum (S 2 ) add the prop. log. of the hourly
motion in declination and the constant log. 7' 1427- The result will be the log co-
tangent of the 1st orbital inclination* and must take
is { Sout } f > .,
To the prop. log. of the difference between the star's declination corrected and
the Moon's declination corrected add the constant log. 9 '43 54 and the log. secant
of the preceding orbital inclination ; and from the sum deduct the prop. log. of the
horizontal parallax. The remainder will be the log. secant of the 2nd orbital
inclinationf which must have the name
-VT' [
when the observation is an { immersion
[ emersion
Add together the two orbital inclinations, having proper regard to their names ; and
to the log. cosecant of this sum add the preceding sum (S 2 ), the prop. log. of the
horizontal parallax and the constant log. 8 '1844. The sum will be the prop. log.
of a correction to be applied to the time corrected to get the mean time at Green-
wich : it must be
, f ] r when the sum of the orbital inclinations is < '
* With the parallel of declination. t With the Moon's limb.
from an Observed Occultation, fyc. 179
By applying the equation of time from p. IT. of the ephemeris there will result the
Greenwich apparent time, and the difference between it and the apparent time of
observation will show the longitude of the place from Greenwich ; it will be
' > when the Greenwich time is -j j= r ^ e r I than the observed.
EXAMPLES.
I. SOLAR ECLIPSE.
For a Solar Eclipse take the example directly calculated in the Appendix to 1836,
page 139:
Suppose the beginning of the Solar Eclipse on May 15, 1836, to be observed to
take place at l h 36 m 35 8< 6P.M., apparent time, in latitude 55 57' 20" N., and
longitude about 12 m W.
Here we have
h m
Observed apparent time - 1 36 "6
Longitude- r 12'0 h = + l h 36 m .35-6
Greenwich apparent time 148 '6 = + ' 24 8'- 9
Equation of time - - - - 3 *9
Greenwich mean time - - 1 44 '7
We hence take from the Ephemeris, * = 3 h 29 m 19% S = + 18 57''6,
= 9 "99902
P r 3'51267 cos (A) + 9*96060 - - +9 '96000
p - - 9*99902 cot/- +9*83256 cos/ +9 "75001
F- - 3-51169 e+3i 50 7 7 tane- +9*793i6 G- +9*71061
cos/ - 9 -75001 $+18 57 -6
61183 + ^+50 48'3 cos - +9 '80069 B - +9 '78899
p - +2-87353 (1) +9-92162 check +9 -92162
cosD 9*97484 tan (A) + 9 '64936
+ 2 -89869 tanM +9*57098
8 7 >9 const. 7 '92082 cogM + ^ -97180 - - +9 '97180
6 -6 - - +0-81951 tan (0+S)+0 -08861 cos +9*81719
(A) + 24 2^3 + 4858 /< 3.tan- +0-G6041 B- +9 '78899
s +TT^6 p '- +3 ' 51l6i)
+ 33-3 '- - - - AS - +33 7 18"-4 - - +3 '30068
S +!9 30-9 cos - 9 '97430 (2) ff 15/ //>g
+ 2-89923 (1) (2) dim. - 1 1 -6' P- - 3 '51380
const. 8-82391 , . Q0 .o-. const 9 "43537
ff Q - 1 O O O O " -
{log.- - +1-72314 * H.49'6 2-94917
A + O h O m 52 8 '86 A 30 27 '9
3 29 19
oc n 3 30 12 ^.
180
On the Determination of Longitude
By inspecting the hourly ephemeris of the Moon's right ascension on May 15th
with = 3 h 30 m 12", the most eligible time to assume is evidently (i) = 3 h O m s ; at
this time we have (A) = 3 h 30 m 42 8 '84, (AO r= 2 m O s< 68, (D) = + 19 31' 34"'0,
(DO = + 9' 55"'2, () = 3 h 29 m 31-57, (0 = + 9 S '89, (5) = + 18 58' 2l"'4,
(5i) = + 34"-8 : with these we proceed as follows :
(A,) -
- 2 0-68
9'89
(S } -
- + 9 55 -2
- + 34-8
()- -
A +
- i 5079
h m s
3 29 31 -57
52 '86
AS +
- + 9 20 '4
o / //
18 58 21 -4
33 18 -4
GO' -
3 30 24 -43
Oo) +
19 31 39 '8
(A) - -
3 30 42 -84
f(b)
\lntr
-(A) -
18 -41
_ i o.f)^o^
'
log.
A,- -
2 -04450
D
+ 2 '74850 (1)
m - -
const. -
9 -22055
3 -55630
/log- -
9 -22055
1 -96905
log- -
2 77685
(
I 7 33"-l
O h 9 58 8 '2
(D) +
19 31 34 -0
(0
300
DO +
19 30 '9
'o +
2- i\ 1 O
OU 1 o
(^o) +
19 31 39-8
k
1 38'9
cos (D) - -
const. - -
n - -
9 '97428
2 -04450
1 -17609
3-19487 (2)
19 41 '2
tan^ 9 '55363 (1) (2)
COS rj - - +9 '9/384
k 1 '99520
A - -
1 ^6904
3 -26196
V - - + 92 55 '2
rj y- 112 36 -4 -
corr. l h 4 m 38 ? '5
T
cos^- 8 -70708
- - sin 9 '96528
A - -
const. - -
3 -26196
3 -55630
6-78354 (3)
3-58867 (3) (2)
corr.
45 23 '3
3 56 -0
1 49 19 '3
1 36 35 -6
Greenwich mean time
Equation of time
Greenwich apparent time
Observed
Longitude -
12 43 7 W. of Greenwich.
from an Observed Occupation, fyc.
181
II. OCCULTATION OF A STAR.
Suppose, at Bedford, on January 7? 1836, in latitude 52 8' 28" N., the Immersion
of i Leonis to be observed at 10 h 39 m 22 s * 4 P.M., apparent time, and the estimated
longitude to be about O h l m W. Required the longitude ?
Apparent time (observation) - 10 39
Longitude
1 W.
Apparent time (Greenwich) - 10 40
Equation of time - - - - -
Mean time (Greenwich) - -
- 10 47
Latitude N. 52 8 28
Reduc. - - - 10 57
N. 51 57 31
Reduced or geocentric latitude.
For Jan. 7, at 10 h 47 m , we find, from the Ephemeris, O's R.A. .= 19 h 12 m 40',
D 's dec. = N. 15 50', and }) 's equ. hor. par. = 56' 1"*9.
O's R.A. 19 12 4Q 8
App. time - - 10 39 22
R. A. meridian ------ 5 52 2
10 23 26
's hour angle E
I in ti
\ in a
time -
arc -
corr n . - -
*'s hour angle E. corr d . - -
4 31 24
""67 51'
_J7
67 34
P. L. }) 's hor. par. - '5068
corr. for lat. - - - - 9
P. L. corr d . hor. par. - '5077
sec. red. lat. - - - - '2103
cosec. hour angle - - '0333
sum (SO 7513
cos. }) 's dec. - - - 9 "9832
const.log. 0*3010
P. L. corr n . - - - - i -0355
P. L. corr d hor. par. - 0*5077 p$a- rf w - '5077
sec. *'s dec. - - - - 0*0150 - - - cosec. --.--- 0*5876
cosec. red. lat. - - - - 0*1037 - - - sec. - '2103
N. 42' 33" *0 P.L. 0^6264 " sec ' Cotr * hour gle OJ4184
S. 3 23 '9 P. L. ----- 1 7240
corr 11 . - - -
*'s dec. - -
N. 39 9*1
N.14 58 38 *8
*'sdec. corr d . N.15 37 47 '9
corr 11 .- - - - -
*'sR.A. - -
10 23 26 *39
*'s R. A. corr d . 10 21 13*83
sum (SO ---- 0*7513
const, log. - - - 1 "1761
cos. - - - --- 9 '9836
--- 1 '9110
On referring with the *'s corrected R.A. to the hourly ephemeris of the Moon, it
will evidently be most convenient to take out the data at ll b : for this time we have
3) 's R. A. = 10 h 20 m " 58" -47, hourly motion D 's R.A. = 2 m 2 s '9, }) 's dec. =
N. 15 47' 1'1"*0, hourly motion D 's dec. = S. II 7 41"*5.
NAUTICAL ALMANAC, 1837. APPENDIX. n
182 On the Determination of Longitude
h m s
*'s corr d . R. A. - 10 21 13 '83
J)'s R. A. - - - - 10 20 58 -47
fdiff. ...... 15 '36
\common log. ..... 1 "1864
com. log. h. m. }) 's R. A. - 2 '0896
Remainder - - '9032 - - ........... '9032
const.log. ------ 0-4771 P. L. h. m. D 's dec. - - - 1 '18/4
corr n . --- 729'9 P.L. 1 '3803 corr". - - S. l' 2/' '7 P.L.2 '0906
Time assumed 11 00 - })'sdec. - -N.15 47 11 '0
Timecorr d . - 11 7 29 -9 }> 'sdec.corr d .N. 15 45 43 '3
o / //
com.log. h m. })'sR.A. - - - - 2 '0896 * 's corr d . dec. - - - N. 15 37 47 '9
cos. 3>'s corr d . dec. ------ 9 '9834 J, 's - - - N. 15 45 43 '3
sum (S 2 ) - -. --- - ---- 2-0730 (Jiff. (# S. of }>) - - 7 55 '4
P.L. h.m. }'sdec. ...... 1 '1874 < pl , -
const. log. ---------- 7-1427 lKJj ; .
_ L const, log. ----- 9 '4354
1st Orb. incl. - - N. 2134 / cot. OM031- -sec. ,,,,-,-- 0*0315
2ndOrb.incl.- - S. 6l 9
sum S. 39 35
corr". 19 44 '5 P.L. -9599
Time corr d . 11 7 29 '9
Green w h mean time - - 10 47 45 -4
Equation of time- - - 6 31 '0
Green w h app. time - - 10 41 14 *4
Observed - 10 39 22 '4
Longitude 152'OW.
P. L. }) 's hor. par. - -
'8232
'5068
'3164
0-1957
2 -0730
-5068
8 '1844
nm f^> ~\
P. L. }) 's hor. par. - -
P.S. The principle of reversing the effect of the relative horizontal parallax on the
position of the Sun, instead of using the actual effect on the position of the Moon, may
be advantageously employed in the direct calculation of an Eclipse for a particular
place. It will only be necessary to use the parallaxes for the Sun viewed as an
apparent position, and to diminish the semidiameter by the amount derived from the
table on page 175. Thus, it appears, at the beginning of the Eclipse, for instance,
that the contact may be mathematically tested in two ways. First, we may apply the
actual effects of the parallax to the true position of the Moon, then augment her semi-
diameter, and thus establish a contact of the limbs. But, if we reverse the operation,
and consider the Sun to be an apparent body under the influence of the relative paral-
lax, then clearing it from this supposed influence by reversing the parallax, and
from an Observed Occidtation, fyc. 183
diminishing the semidiameter, a contact will similarly be established with the true
limb of the Moon ; and this principle, in its application to solar eclipses, possesses an
advantage similar to that derived in the case of an occultation, by considering the Star
as an apparent place. (See Appendix to Nautical Almanac for 1836, page 123*.)
The formulae, Nos. 2, 3, 4, and 5, pages 130 and 131 may, according to this method,
be supplied by the following :
P' = p (P *) ro = P' cos /
Q t = [9 '4180] Q 2 = [9 '4180] m sin
s- [9 -43537] P
3. * = --
cosD
A h in minutes = [7 '92082] k sin h
(A) = h A h
tan 6 == cos (A) cot / G r= cos (A) cos /
tanM= ^ --tan (A) tans = tan (6> + S) cos M
check - - -
B =: cos M cos e
sin 9 G
cos (6 -f 5) B
er = ff diminution for e
i f phas*e, A f ^^ \
annular j r [ s
4. * = 5j A = *oin*
A t = Qi A cos /i A &i = Q a sin (A)
5. ^o=5 + A^ '= A
y = ( A a) cosD yi=(i A a cos D
a: r= (D + ' corr.) 5 ^ = D! A 5,
* This was inadvertently ascribed to Carlini j Professor Henderson, by whom a paper has
appeared upon this very point, in the Quarterly Journal for 1828, page 411, informs me that the
method has been long in practice, and that it was employed at an early period by Dr. Maskelyne.
185
ON THE ELEMENTS OF THE ORBIT
OF
HALLEY'S COMET,
AT ITS APPEARANCE IN THE YEARS 1835 AND 1836.
BY LIEUT. W. S. STRATFORD, R.N.,
Supcriuteudcnt of the Nautical Almanac.
THE object of the present paper is to afford the most accurate means of determining
the Elements of the Orbit of Halley's Comet, at the instant of its Perihelion Passage
in 1835, from all the Observations of that Body; and to explain in detail the various
operations which have been performed at the Nautical Almanac Office for its
accomplishment.
It was originally intended to trace the Comet's history from the period of its return
in 1759, but this has been rendered unnecessary by the masterly address of Mr.
Airy, the Astronomer Royal, to the Fellows of the Royal Astronomical Society, at their
annual general meeting in 1837, on tne occasion of presenting the gold medal of the
society to Professor Rosenberger, " for his elaborate calculations relating to the return
of Halley's Comet." It is impossible to mention Mr. Airy without, at the same time,
acknowledging the cordial co-operation which the Author of this paper has expe-
rienced from that gentleman, not only in the particular instance of the cometary
discussions, but at all times, and on all occasions, in which matters connected with
the perfection of the Nautical Almanac and the interests of science have been con-
cerned.
In the SUPPLEMENT to the NAUTICAL ALMANAC for the year 1833, with the view of
attracting the early attention of astronomers to the subject, there was first given an
Ephemeris of the Comet, from Aug. 3, 1835, to Feb. 11, 1836, founded upon the fol-
lowing elements of its orbit, given by M. de Pontecoulant in the Conn, des Terns
pour rAn 1833, page 112.
Passage of the perihelion, 1835, Nov. 7 '2, Paris mean astronomical time.
o / //
Place of the perihelion on the orbit - - - - 304 31 43
Longitude of the ascending node -----55300
Inclination of the orbit - 17 44 24
Ratio of the excentricity to the semi-axis major - 0*9675212
Semi-axis major 17 '98705
Motion retrograde.
This Ephemeris was reprinted in the NAUTICAL ALMANAC for 1835 ; and as the
principal difference of M. de Pontecoulant's elements from those of M. Damoiseau
related to the time of passage of the perihelion, a double Ephemeris was added, for
the purpose of affording means of an early rectification of this element. The double
NAUTICAL ALMANAC, 1839. APPENDIX. O
186 On the Elements of the Orbit of Halleys Comet,
Ephemeris contained, for each 8th day, from Aug. 7, 1835, to Feb. 7, 1836, the Right
Ascension and Declination, each to minutes, of the Comet, on two suppositions of the
arrival at the perihelion, viz., Nov. 3'2 and Nov. 11'2, 1835.
In the same work were also given the co-ordinates of the Sun and Comet, together
with a plan of the Heavens, showing, from three different sets of elements, the paths
of the Comet amongst the fixed stars, and the relative position of the Comet in each
on certain days, so as to indicate the direction in which the Comet should be sought
for, with the greatest probability of its rediscovery.
It appears that the Comet was first seen at Rome by M. Dumouchel, Director of
the Observatory of the Roman College, at O h 20 m , sidereal time at the place, on
August 5, 1835, in Right Ascension 5 h 26 m , and Declination +22 27'. (Ast.
Nach., No. 288.) It was observed generally in Europe after the 20th of August.
From a comparison of observations made at the latter end of August with the
double Ephemeris, it was estimated that the Comet would arrive at its perihelion about
8 '5 days later than the time stated by M. de Ponte'coulant.
With a view to a nearer approximation to this element, another double Ephemeris
was published on September 30, 1835, containing, for the month of October, 1835, the
places of the Comet, on the supposition of the perihelion passage occurring respec-
tively on Nov. 15 '1935 and Nov. 16 '1935, astronomical mean time at Greenwich.
Additional observations indicated that Nov. 16 '1935 might be adopted for the
time of passage, without much liability to error. With this time, and the other ele-
ments of Pontecoulant unchanged, an Ephemeris was computed from Aug. 20*5 to
to Sept. 30*5, 1835, which, united with the October Ephemeris computed previously
from the same elements, embraced the period between Aug. 20*5 and Oct. 31'5,
1835. With this and six other Ephemerides computed in a similar manner from
elements in which a small variation was given to each in succession, whilst the
other five remained constant, a general Ephemeris was formed for the same period,
in which the Right Ascension and Declination consisted each of one known and six
unknown quantities.
Having collected 56 Right Ascensions and 56 Declinations from roughly-reduced
observations, made between Aug. 20 and Oct. 19, 1835, with these and the corre-
sponding Right Ascensions and Declinations interpolated from the last-mentioned
Ephemeris, there were formed 112 equations of condition, from which were deduced
corrections for the assumed elements.
From these approximate elements an Ephemeris was immediately published for the
month of November ; but there being no doubt of some error having crept into the
calculations, a revision of the whole was entered upon, and the following results ulti-
mately obtained.
Perihelion passage, 1835, Nov. 1 5/93546, Mean Astronomical Time at Greenwich.
Semi-axis major - - - . -^ ... -^ ;r .... 18*0779386
Ratio of the excentricity to the semi-axis major '9675509
o / //
Inclination of the orbit - - - - - - - 17 45 56 '7
Longitude of the ascending node 55 8 21 *2) From Mean Equinox
Longitude of the perihelion on the orbit - - 30432 9 '2j of Nov. 15, 1835.
at it # appearance in the Years 1835 4" 1836.
187
With these results an Ephemeris for the month of December was prepared, and pub-
lished on December 1, 1835.
It now remained to ascertain, by a rough comparison with observations, whether
these elements were sufficiently approximate for the calculation of perturbations and
their own final rectification, and for this purpose the following comparisons were
made.
Date.
Right Ascension.
Declination.
1835.
Aug. 20 '5091
Observed.
Computed.
O-C.
Observed.
Computed.
O-C.
h m s
5 40 52
h m s
5 40 54
2
+ 23 30 -0
1
+ 23 29 '4
i
+ -6
Sept. 2 '6526
5 52 10
5 52 12
2
25 10-4
25 10 'I
+ 0*3
Oct. S'3122
8 36 49
8 36 49
+ 57 53'7
+ 57 54 '6
-0-9
Nov. 8 '2233
17 15
1? 14 59
+ 1
12 51 7
12 51 '2
0-5
The observation of August 20 was made at Dorpat by Professor Struve ; that of
September 2, at Hayes, by the Rev. T. J. Hussey; and those of October 8 and No-
vember 8, with the meridian instruments at Cambridge, by Professor Airy.
The results of these comparisons showed that the elements represented the orbit
with sufficient accuracy for the purposes in view, and the calculations were imme-
diately commenced. The first part of the Series containing the Apparent Right
Ascension and Declination and the Logarithm of the true distance of the Comet from
the Earth, between August 1'5, 1835 and March 3T5, 1836, was published on
December 30, 1835, with the view of affording early facilities for the reduction of the
observations of the Comet throughout the whole period of its probable visibility. It is
here reprinted in a different form, in Table X, the Right Ascensions being expressed
in arc instead of time.
The various calculations relating to the Ephemeris of the Comet were performed
agreeably to the method described by Mr. Woolhouse, in the Appendix to the Nau-
tical Almanac for 1835, and those relating to the Perturbations by Professor Airy's
method, in the Appendix to the Nautical Almanac for 1837, using in all cases the
data from the Nautical Almanac.
In order to prevent confusion, it has been deemed expedient to alter the notation
occasionally; and, for facility of reference, it is here collected and arranged in order of
the letters of the small italic, the small roman, the large roman, and the greek
alphabets.
NOTATION.
a The semi-axis major of the Comet's orbit, at mean noon of July 30, 1 835.
[a] The variation of a during one of the equal intervals into which the
whole period, through which the variations of the elements are cal-
culated, is divided.
o 2
188 On the Elements of the Orbit of Halley's Comet,
la The whole variation of a in any given number of intervals, from July
30-0, 1835.
e
sin 1"
' I
2 >
c. J
c 2 > Constants used in the calculation of the Variations.
&c.
e The ratio of the excentricity to the semi-axis major of the Comet's orbit.
\e\ The variation of e during one interval.
5 r- )
^sinz cos( = dimin. of obliq. x sin v tan i
%i = dimin. of obliq. X cos v
Iv = dimin. of obliq. X sin v cotan i
the amount of diminution being reckoned from July 30, assuming O r/< 457 as the
annual diminution.
The results are inserted in Table IX. They represent for any date in the Table the
196 On the Elements of the Orbit of Halleijs Comet,
total amount of alteration which each element of the orbit of July 30*0 has under-
gone by the action of all the disturbing Planets.
Having obtained these values for every fourth day, commencing with July 30, 1835,
the elements of the perturbed orbits were obtained for each fourth day, by the suc-
cessive addition of each , e, &c., to the original elements ; the mean longitude on
July 30 having been taken
c rr CT -j- nt
t being the interval in days between July 30 '0 and the assumed time of passing the
perihelion, viz., Nov. 15*93546.
It now only remained to compute the Right Ascension and Declination of the Comet
from each set of perturbed elements, and by a comparison of the results with those in
the original Ephemeris derived from the imperturbed elements of July 30'0, to ascer-
tain the alterations produced by the disturbing Planets on the Right Ascension and
Declination of the Comet.
The value of e for the orbit of each date was assumed to be the mean longitude in
that orbit on July 30, arid was in each case reduced to the date with the mean
motion belonging to the orbit, as determined from its semi-axis major.
With the longitudes v and cr, reduced to the true equinox of each date, by applying
precession and nutation, the apparent Right Ascension and Declination of the Comet
were determined from each set of perturbed elements for every fourth day, from
July 30, 1835. These calculations were conducted in a manner precisely similar
to those for the original Ephemeris.
Subtracting the and I of the original Ephemeris from the and I derived from
the disturbed elements, the effect of perturbation upon the Right Ascension and De-
clination for each fourth day, from July 30, was obtained, and thence, by interpo-
lation, the daily effect.
These perturbations are inserted in Table X. They are to be applied with the proper
sign to the Right Ascension and Declination of the Comet in the same Table, to fur-
nish the apparent Right Ascension and Declination, such as should be exhibited by
observation, on the presumption that the elements of July 30, 1835, are the true
elements of the Comet's orbit at that period.
CORRECTION OF THE ASSUMED ELEMENTS.
Let it be now supposed that the true elements of the orbit on July 30 *0, 1835, were
T + O d '02 P
a + '01 Q
e + O'OOOl R
CT+ 1' S
v + 1'U
t + 1' V
T, a, &c., being the numerical values of the assumed elements on July 30, (page
186), and 0'02 P, 0*01 Q, &c., the corrections due to those elements.
at its appearance in the Years 1835 $ 1836. 197
If with the elements T-fO'02, o, e t or, v, i y a right ascension ^ and decimation
1 be computed for the time t in a way similar in every respect to that described in
pages 191 and 192, it is plain that t and $ ^ are the variations of right
ascension and declination produced by the given variation (0 '02) ofT alone. Now
if the elements have been obtained sufficiently near in the first instance to justify
the presumption that the variations of the right ascension and declination will be
proportional simply to the variation of the element which produces them, then for the
true variation of T, viz. 0*02 P, the variation of right ascension will be ( i) P,
or reduced to the arc of a great circle, ( cos 5 . P, and of declination (S S,) P.
In the same manner the elements T, a+0'01, e, nj t v, i will furnish a right ascen-
sion or 2 and declination . 2 ; and the variations 2 , $ 2 will be dependent upon
the given variation of a alone. The variations produced by the true variation of a,
viz. O'Ol Q, will therefore be ( a 2 ) cos $ . Q in right ascension, and ( 2 ) . Q in
declination.
By thus varying each of the other elements in succession by a given minute quan-
tity, a knowledge is obtained of its separate influence in altering the right ascension and
decimation, and hence the influence of the assumed unknown total variations, viz.
For variation of e = '0001 R,the variation in J R * A< ~ (""*) cos 5 * R
Dec. - O J.) . R
. = (-*,) cos
Dec. = -W .S
_ u
Dec, = (* - W . U
R. A. = (a ff 6 ) cos 5 . V
Dec. = ($ - J.) . V
Having the variation of and 5 consequent upon a variation of each element singly,
now suppose all the elements to vary together, the total variation of right ascension
and declination of the Comet will be
+ A! P+ A 2 Q+A 3 R+A 4 S+A 5 U+A 6 V = (') cos $ = E
+ A' t P+ A', Q+ A 7 8 R+ A ; 4 S+ A '3 U+ A' 6 V = (S 7 -S) = E'
where P, Q, R, S, U, V are the unknown quantities to be determined.
The variations of right ascension and declination determined
from given minute variation of elements, as before explained.
x , A 2 , &c. 1
\ , A ' 2 , &c. J
' , y An observed right ascension and declination at the time for which
and S have been computed.
On these principles have the equations of condition in Table XI. been formed.
Six different Ephemerides have been computed from six different sets of elements ;
and subtracting the resulting Right Ascension and Declination of each from the
Right Ascension and Declination in Table X, derived from the original elements,
the differences (those of R. A. being first multiplied by cos S,) form the coefficients
of P. Q, R, &c. ; indicating, for any given time within the limits of the table, the
numerical amount of variation caused by a minute variation of each element.
The mode of using the table is as follows : Having a reduced observation of Right
198 On the Elements of the Orbit ofHalleij* Cornet, #c.
Ascension and Declination at a given mean time, find, by interpolating Table XI,
the coefficients of P, Q, R, &c., for that time : find also, by interpolating Table X,
the Right Ascension and Declination , including the perturbations, for the same
instant : subtract the interpolated from the observed Right Ascension, and multiply
the remainder by cos 5, the product is to be substituted for E, the right hand term
in the equations of condition dependent upon Right Ascension.
Subtract the interpolated from the observed Declination (North Declination
being +, and South ), and the remainder is to be substituted for E', the right
hand terni in the equations of condition dependent upon Declination.
The unknown quantities being 6, require absolutely only 6 equations for their
complete determination, but from the uncertainty attached to observations of Comets,
it is desirable to procure as many as possible, and form similar equations of condition,
the resolution of any number of which may be effected by the metfiod of least squares.
Each observation of Right Ascension and Declination furnishes, conjointly with
the Ephemeris, a value of E and E'; and all the equations combined as before men-
tioned, and resolved, will furnish the value of the unknown quantities P, Q, R, &c.,
and hence the corrections to be applied to the assumed orbit of July 30, to obtain the
true orbit of the Comet on that day.
When from a complete discussion of all the observations deserving of confidence,
the assumed orbit of July 30, with the position of the perihelion therein, and the
time of perihelion passage, shall have been corrected, the total variation of each of
the elements for each fourth day, from July 30, given in Table IX., furnishes a ready
means of reducing the orbit and the position of the perihelion to the instant of the
Comet's passage by that point.
W. S. STRATFORD.
TABLES.
200 On the Elements of the Orbit ofHalleijs Comet,
TABLE I.
Containing, for each Mean Midnight at Greenwich (Astronomical time) from
August 1, 1835, to March 31, 1836, the Comet's Excentric Anomaly O), Loga-
rithm of the Radius Vector (r), and True Heliocentric co-ordinates (x, y, z,);
x, being measured on a line passing through the true Vernal Equinoctial point
of the date ; y, on a line in a plane parallel to that of the Equator, and perpen-
dicular to the direction of x; and z, perpendicular to the plane of the Equator,
towards the North.
Date.
u
Log. of r
x
y
z
1835.
O 1 II
Aug. 1 '5
23 13 56'58
0-3021016
+ o -9934592
+ -6196067
+ 0-6400825
2'5
23 6 58 *85
0-2990714
o -9942075
6032837
-6364889
3'5
22 59 58 '31
o -2960188
'9949189
5869050
-6328720
4'5
22 52 54 '63
o -2929266
o -9955925
5704610
o -6292300
5'5
22 45 47 '96
-2898106
o -9962267
5539570
-6255636
6-5
22 38 38 '33
-2866660
o -9968209
5373970
0-6218734
7'5
22 31 25-49
-2834906
o -9973750
5207700
'6181573
8'5
22 24 9 -45
'2802842
o -997*869
5040/90
'6144149
9'5
22 16 50 -08
o -2770462
'9983565
4873210
-6106458
10 '5
22 9 27 -50
o -2737772
'9987832
4705020
'6068508
11 '5
22 2 1 '60
o -2704/60
0-9991612
4536180
o -6030291
12'5
21 54 32 -11
o -2671406
'9995002
4366610
o -5991783
13-5
21 46 59 -20
o -2637726
'9997897
4196410
o -5953005
14'5
21 39 22 '77
o -2603706
1 -0000309
4025540
o -5913941
15'5
21 31 42-81
o -2569340
1 -0002-227
3853990
o -5374591
16-5
21 23 59 '20
-2534626
1 -0003650
3681770
-5834953
17'5
21 16 11 -61
-2499532
1 -0004558
3508750
o -5794993
18'5
21 8 20 -35
-2464082
1 -0004936
3335080
o -5754738
19'5
21 25-36
-2428272
1 -0004786
3160740
o -5714190
20-5
20 52 26-12
-2392058
1 -0004070
2980070
o -5673298
21 -5
20 44 23-12
-2355478
1 ^002793
2809760
0'5632107
22'5
20 36 15 '98
Q'2318502
1 -0000940
2633190
-5590588
23 '5
20 28 4 '45
-2281108
-9998470
2455800
-5548718
24*5
20 19 48 -87
-2243320
'9995412
227/720
-5506526
25 '5
20 11 28-62
-2205090
0-9991710
2098/60
o -5463960
26'5
20 3 4-17
o -2166456
o -9987370
1919HO
-5421066
27-5
19 54 35-12
0-2127380
o -9982370
1738640
-5377804
28 '5
19 46 l -47
'2087864
o -9976680
1557380
0-5334177
29'5
19 37 23 '12
o -2047902
'9970305
1375330
-5290182
30 '5
19 28 39 -96
0-2007480
o -9963210
1192460
-5245807
31 '5
19 19 51 '84
o -1966086
4-0 -9955375
+ 1 -1008750
+ '5201040
at its appearance in the Years 1835 < 1836.
201
TABLE I. continued.
Date.
u
Log. of r
X
y
z
1835.
Aug. 3 1 '5
Sep. 1 '5
2 '5
3'5
4'5
o / //
19 19 51 '84
19 10 58 73
19 2 0-47
18 52 56 '97
18 43 47-97
o -1966586
o -1925220
0-1883362
-1841010
-1798138
+ o -9955375
'9946785
o -9937410
o -9927245
0-9916247
+ 1 -1008750
1 '0824220
I '0638820
1 '0452580
1 -0265400
-f -5201040
-5155884
0-5110315
-5064341
o -5017930
5'5
6-5
7'5
8 '5
9'5
18 34 33 '77
18 25 13-86
18 15 48 '02
18 6 16-51
17 56 39'H
o -1754778
o -1710878
o -1666426
-1621440
o -1575906
0-9904415
0-9891705
o -9878092
'9863554
'9848072
1 '0077480
o -9888602
o -9698734
o -9508074
o -9316510
o -4971120
-4923859
-4876133
-4827964
o -4779343
10 '5
11 '5
12'5
13 *5
14-5
17 46 55 -37
17 37 5 -46
17 27 9'13
17 17 6-38
17 6 56-59
'1529786
0-1483094
-1435812
'1387936
-1339424
'9831605
0*9814.125
o -9795602
o -9776010
o -9755304
o -9123946
-8930468
'8736058
0-8540718
'8344322
o -4730224
o -4680629
'4630536
o -4579945
-4528812
15 '5
16'5
17-5
18 '5
19'5
16 56 40 '13
16 46 16-54
16 35 45 79
16 25 7'51
16 14 21 '67
o -1290300
-1240534
0-1190126
0-1139044
o -1087290
'9733450
o -9710422
'9686180
o -9660667
o -9633349
0-8147010
7948690
o -7749395
o -7549054
o -7347707
0-4477163
o -4424965
o -4372221
0-4318895
o -4264990
20 *5
21 '5
22 '5
23-5
24-5
16 3 28-09
15 52 26-47
15 41 16-77
15 29 58 -38
15 18 31 -63
-1034856
o -0981718
o -0927882
-0873298
0-0818006
o -9605694
o -9576147
0-9545167
o -9512687
o -9473680
0*7145355
o -6941950
o -6737546
0-6531994
-6325483
-4210495
'4155386
o -4099664
-4043276
o -3986265
25 '5
26*5
27-5
28 '5
29*5
15 6 55 '91
14 55 11 -00
14 43 17-03
14 31 13 '11
14 18 59 '48
o -0761956
'0705142
-0647582
'0589216
o -0530076
o -9443070
'9405810
'9366846
o -9326099
'9283524
-611/866
o -5909151
o -5699451
-5488564
o -5276640
-3928567
o -3870176
-3811104
0-3751282
o -3690735
30-5
Oct. 1 '5
2'5
3'5
4 '5
14 6 35 *82
13 54 1 -63
13 41 16-79
13 28 21 -03
13 15 13 -82
o -0470144
'0409398
'0347846
-0285478
o -0222278
o -9239040
0'9192580
0'9144076
o -9093442
o -9040590
-5063664
-4849564
-4634403
'4418178
-4200841
-3629438
o -3567352
'3504477
o -3440792
o -3376257
5 '5
6-5
7*5
8 '5
9'5
13 1 55 -17
12 48 24 '48
12 34 41 -59
12 20 46 '04
12 6 37'48
-0158266
o -0093420
o -0027756
9-9961268
9 -9893966
'8985456
o -8927934
o -8867936
-8805362
'8/40114
-3982486
-3763035
o -3542575
-3321065
-3098547
0-3310880
0-3244616
-3177458
0-3109374
-3040343
10'5
11 52 15 -64
9 '9825862
4-0 '8672078
+ -2875047
+ 0-2970345
NAUTICAL ALMANAC, 183U. APPENDIX.
202
On the Elements of the Orbit of Hattey's Comet,
TABLE L continued.
Date.
u
Log. of r
X
y
z
1835.
Oct. 10 '5
11 '5
12 "5
13'5
14*5
o 1 II
11 52 15 -64
11 37 40 '05
11 22 50 '50
11 7 46 '26
10 52 27 '30
9 '9825862
9 -97569/0
9-9687316
9 -9616898
9 -9545774
+o -8672078
-8601150 :
-8527220
-8450138
-8369808
-fO '2875047
o -2650579
'2425212
-2198898
0-1971773
+ -29/0345
-2899355
"2827358
o -2754305
-2680199
15 *5
16'5
17-5
18*5
19^
10 36 53 '09
10 21 3 '22
10 4 57 '41
9 48 35 -00
9 31 55 '85
9 ^473970
9 '9401526
9 '9328506
9 -9254944
9 '9180934
,
-8286088 '1743849
-8198842 -1515173
'8107940 -1285853
-8013202 -1055881
-7914509 '0825416
'2605010
"2528708
'2451286
o -2372695
o -2292937
20'5
21 '5
22'5
23'5
24-5
9 14 59 '54
8 57 45 '46
8 40 13 -51
8 22 23 '23
8 4 14 '06
9 -9106550
9 '90318/2
9 -8957622
9 -8882118
9 -8807306
o 7811695
7704582
o 7593024
o 7476850
o -7355907
o -0594537
-0363288
+ -0131855
o -0099666
'0331082
o -2211993
-2129819
'2046422
o -1961779
'18/5894
25 '5
26-5
27-5
28 '5
29'5
7 45 46 '69
7 26 59 '92
7 7 53 '64
6 48 27 '88
6 28 42 '66
9 -8732712
9 '8658540
9 -8584958
9 '8512188
9 '8440474
o 7229972
o 7098927
o -6962557
o -6820712
o -6673242
o -0562329
'0/93132
o -1023373
'1252805
-1481181
o -1788723
o -1700299
o -1610593
0-1519619
o -1427398
30-5
31 '5
Nov. 1 '5
2'5
3'5
6 8 37 '86
5 48 13 '56
5 27 30 '00
5 6 27 '36
4456 '08
9 '8370056
9 '8301216
9 '8234252
9'8l69476
9'8107226
o -6519966
o -6360744
o -6195440
o -6023926
-5846093
o -1708276
-1933811
o -2157490
o -2379015
-2598051
o -1333933
-1239252
-1143388
o -1046375
'0948263
4-5
5'5
6-5
7-5
8'5
4 23 26 '85
4 1 30 '13
3 39 16 '85
3 16 47 '94
2 54 4 -57
9 '8047858
9 7991722
9 7939190
9 -7890630
9 '7846406
-5661888
o -5471216
o -5274071
o -5070447
-4860391
-2814225
o -3027204
'3236594
'3442008
'3643039
-0849124
o -0749012
-0648014
o -0546219
o -0443735
9'5
10 '5
11 "5
12-5
13'5
2 31 7-96
2 7 59 '57
1 44 40 '93
i 21 13 77
o 57 39 '86
9 -7806866
9 7772344
9-7743138
9-7719514
9 7701694
o -4643965
'4421295
-4192531
o -3957876
o -3717559
'3839309
-4030404
-4215950
o -4395563
o -4568896
-0340668
o -0237145
'0133293
+o -0029255
-0 -0074825
14*5
15 '5
16'5
17*5
18 '5
34 1 -10
10 19 '56
13 22 '99
37 4 '34
1 42 '58
9 '7689852
97684100
9 7^84498
9 '7691038
9 7703662
o -3471868
'3221121
o -2960645
-2705833
o -2442074
o -4735619
, -48^3424
-5048074
o -5193319
0-5331004
o -0178797
'0282503
-0385803
-0488533
'0590554
19-5
1 24 15 70
9 7722240
+ o 2174783
o -5460981
o -0691720
at its appearance in the Years 1835 & 1836,
203
TABLE I. continued.
Date.
u
Log. of r
x
y
z
1835.
Nov. 1 9 '5
20-5
21 '5
22 '5
23-5
o / //
1 24 15 70
1 47 41 '88
2 10 59 '30
2 34 6 '25
2 57 1 '23
9 7722240
9 7746596
9 7776506
9 7811692
9 7851854
+o -2174783
o -1904396
-1631328
-1356024
o -1078894
-5460981
'5583170
o -5697534
-5804060
-5902805
o -0691720
o -0791901
o -0890973
-0988819
-1085341
24 '5
25'5
26-5
27-5
28'5
3 19 42 '85
3 42 9 78
4 4 20 '93
4 26 15 '50
4 47 52 '40
9 7896660
9 7945750
9 7998768
9 '8055344
9*8115104
-0800351
-0520802
-f -0240623
o -0039370
-0320279
-5993855
0-6077318
-6153356
-6222141
-6283870
-1180451
o -1274063
0-1366118
o -1456570
'1545361
29-5
30 '5
Dec. 1 '5
2 '5
3'5
5 9 11 '21
5 30 11 '45
5 50 52 '58
6 11 14 '39
6 31 16 '68
9 -8177696
9 '8242774
9 '8309996
9 '8379058
9 '8449658
o -0600360
-0879834
-1158424
0-1435917
0-1712102
-6338764
-6337059
-6428983
-6464802
o -6494756
o -1632477
o -1717897
0-1801603
-1883598
-1963884
4 '5
5-5
6'5
7'5
8'5
6 50 59 '34
7 10 22 -67
7 29 26 '30
7 48 10 '63
8 6 35 '94
9-8521522
9 -8594414
9 -8668076
9 -8742314
9-8816940
o -1986796
0-2259913
'2531207
-2800633
'3068094
o -6519104
-6538097
-6551979
o -6560999
'6565386
-2042471
0-2119395
-2194648
o -2268274
-2340303
9'5
10-5
11 *5
12'5
13 '5
8 24 42 '38
8 42 30 -21
8 59 59 '82
9 17 U -55
9 34 5 77
9 -8891782
9 -8966684
9*9041516
9 -9116158
9-9190510
-3333506
o -3596787
o -3857907
'4116820
o -4373501
o -6565376
-6561188
-6553030
-6541116
-6525640
o -2410765
o -2479692
o -2547123
o -2613094
o -2677644
14-5
15 '5
16'5
17-5
18 '5
9 50 42 '67
10 7 2 '88
10 23 6 '65
10 38 54 '54
10 54 2671
9 -9264462
9 -9337952
9 -9410906
9 '9483274
9 '9554990
-4627882
'4880003
-5129837
o -5377420
o -5622698
o -6506786
o -6434727
o -6459639
-6431681
'6401000
o -2740799
'2802610
-2863110
o -2922343
'2980335
19-5
20 '5
21 '5
22'5
23*5
11 9 43 79
11 24 46 '01
11 39 33 75
11 54 7 '53
12 8 27 70
9 -9626026
9 -9696340
9 '9765896
9 -9834690
9 -9902692
o -5865737
o -6106513
-6345044
'6581395
'6815563
o -6367742
-6332049
o -6294039
-6253846
o -6211567
-3037131
o -3092764
o -3147259
-3200668
-3253015
24 '5
25 '5
26'5
27 '5
28 '5
12 22 34 -49
12 36 28 '40
12 50 9 76
13 3 38'95
13 16 56 '11
9 -9969886
o -0036268
0-0101834
o -0166578
-0230484
o 7047555
o 7277430
o -7505212
o 7730940
o 7954590
o -6167336
-6121230
o -6073370
-6023821
o -5972692
-3304329
-3354646
-3403998
-3452415
o -3499912
29'5
13 30 1 79
o -0293576
o -8176264
-5920045
-3546530
204
On the Elements of the Orbit of Halley's Comet,
TABLE I. continued.
Date.
u
Log. of r
X
y
z
1835.
Dec. 29 -5
30 '5
31 '5
1836.
Jan. 1 *5
2 '5
o / //
13 30 1 79
13 42 56 '14
13 55 39-57
14 8 12 -27
14 20 34 76
o -0293576
'0355838
-041 >286
-0477918
o -0537754
o -8176264
-8395930
o -8613670
-8829480
'9043446
o -5920045
'5865984
-5810555
-5753848
'5695897
-3546530
o -3592290
o -3637218
-3681338
o -3724678
3'5
4 '5
5*5
6-5
!*. 7 ' 5
14 32 47 '08
14 44 49 '48
14 56 42 '60
15 8 26 '05
15 20 71
o -0596792
'0655036
o -0712524
o -0769220
-0825180
'9255554
'9465822
o -9674333
'9881102
1 -0086172
o -5636797
-5576599
-5515331
o -5453091
'5389883
o -3767258
'3809095
'3850229
'3890650
o -3930407
8'5
9'5
10 '5
11 '5
12'5
15 31 26 -37
15 42 43 '65
15 53 52-37
16 4 52 '97
16 15 45 '44
-0880382
-09348/0
o -0988622
o -1041670
o -1094006
1 -0289507
1 -0491251
1 -0691320
1 -0889800
1 -1086656
-5325780
'5260810
o -5195026
-5128475
-5061187
o -3969498
o -4007962
o -4045793
-4083024
o -4119655
13-5
14-5
15-5
16*5
17-5
16 26 30 '28
16 37 7 '59
16 47 37 -41
16 58 '05
17 8 15 70
o -1145670
o -1196668
o -1246992
o -1296674
o -1345720
1 -1282024
1 -1475900
1 -1668241
1 -1859141
1 -2048622
o -4993195
'4924528
-4855227
-4785318
-4714824
o -4155723
o -4191239
o -4226202
o -4260641
-4294566
18 '5
19-5
20 '5
21 '5
22-5
17 18 24-46
17 28 26 '49
17 38 21 '85
17 48 10 '98
17 57 53-91
-1394142
'1441950
'1489146
o -1535766
0-1581812
1 -2236687
1 -2423366
1 -2608652
1 -2792653
1 -2975366
o -4643735
o -4572223
-4500181
o -4427634
'435 1630
o -4327990
o -4360926
o -4393375
o -4425371
o -4456916
23 '5
24-5
25'5
26-5
27-5
18 7 30 -59
18 17 1 '38
18 26 26 -37
18 35 45 '55
18 44 59 -12
o -1627276
o -1672194
o -1716568
o -1760400
-1803700
1 -3156733
1 -3336871
1 '3515778
1 -3693435
1 '3869864
-4281209
-4207380
-4133141
'4058548
'3983598
'4488008
'4518678
-4548926
o -4578756
'4608183
28'5
29-5
30-5
31 '5
Feb. 1 '5
18 54 7-29
19 3 10 -13
19 12 7 73
19 21 '19
19 29 47 70
'1846496
-1888784
-1930580
-1971884
o -2012720
1 -4045150
1 -4219270
4392260
4564090
4731870
-3908298
'3832684
o -3756758
'3680542
'3604040
o -4637230
'4665887
o -4694171
-47*22084
o -4749648
2'5
3'5
4'5
5*5
6'5
19 38 30 '30
19 47 8 -07
19 55 41 '13
20 4 9 '53
20 12 33 -37
'2053086
o -2092994
-2132450
o -2171468
0-2210042
4904530
5073140
5240680
5407210
5572660
'3527285
-3450256
o -3373003
-3295531
-3217829
o -4776857
'4803725
'4830257
-4856463
'4882339
7 * 5
20 20 52 '99
-2248214
-1 -5737210
-3139910
o -4907921
at its appearance in the Years 1835 $ 1836.
205
TABLE I. continued.
Date.
u
Log. of r
X
y
z
1836.
Feb. 7*5
8'5
9'5
10-5
11*5
/ //
20 20 52 '99
20 29 7 '99
20 37 18 *83
20 45 25 '58
20 53 28 '15
-2248214
-2285944
o -2323276
o -2360212
o -2396750
-5737210
5900680
6063210
6224810
6385450
-3139910
-3061834
'2983541
'2905048
-2826437
o -4907921
o -4933176
-4958136
'4982807
-5007183
12'5
13 '5
14'5
15 '5
16'5
21 1 26'81
21 9 21 '45
21 17 12-17
21 24 59-18
21 32 42 '41
-2432908
'2468684
-2504082
o -2539120
o -2573796
6545200
6704020
6861920
7018960
7175110
o -2747628
o -2668677
-2589599
'2510363
-2431004
0-5031283
-5055107
-5078655
o -5101936
-5124952
17-5
18 '5
19'5
20*5
21 '5
21 40 21 *83
21 47 57 '92
21 55 30 '28
22 2 59 '18
22 10 24 74
-2608110
o -2642096
o -2675730
o -2709030
o -2742006
7330360
7484860
'7638460
7791240
7943220
'2351553
o -2271943
o -2192253
-2112468
-2032570
o -5147712
o -5170232
o -5192495
'5214518
-5236307
22 '5
23 '5
24 '5
25-5
26-5
22 17 46 '98
22 25 5 '86
22 32 21 -41
22 39 33 '93
22 46 43 -20
o -2774660
o -2806996
-2839010
o -2870734
o -2902144
8094430
8244830
8394410
8543290
8691360
o -1952574
-1872502
o -1792374
o -1712136
-1631855
o -5257862
-5279182
-5300280
-5321165
-5341824
27-5
28-5
29*5
Mar. 1 *5
2 '5
22 53 49 '38
23 52 -75
23 7 52 '89
23 14 50 '21
23 21 44 -56
-2933258
o -2964100
o -2994636
o -3024902
'3054882
'8838690
8985370
'9131230
9276420
9420870
'1551516
o -1471069
'1390603
o -1310079
0-1229515
o -5362271
'5382521
-5402550
'5422388
'5442020
3'5
4'5
5 *5
6-5
7'5
23 28 36 -25
23 35 25 -06
23 42 11 -02
23 48 54 -47
23 55 35 -07
'3084602
-3114052
0*3143228
o -3172162
'3200826
9564670
9707780
'9850160
9991950
2 '0133000
-1148859
-1068224
o -0937545
o -0906788
'0826053
-5461461
-5480715
-5499772
-5518650
o -5537340
8'5
9'5
10 '5
11 '5
12-5
24 2 13 '04
24 8 48 "56
24 15 21 -51
24 21 51 -81
24 28 19 74
-3229238
-3257418
-3285348
-3313032
-3340486
2 -0273360
2 '0413220
2 '0552370
2 '0690850
2 '0828730
o -0745275
-0664464
-0583611
o -0502796
o -0421943
-5555845
o -5574191
o -5592349
-5610338
'5628160
13'5
14-5
15-5
16 '5
17-5
24 34 45 -27
24 41 8 '35
24 47 29-21
24 53 47 -56
25 3 '63
o -3367712
o -3394706
0-3421488
'3448034
o -3474364
2 -0966030
2-1102700
2 -1238840
2 -1374320
2-1509200
'0341058
-0260183
o -0179263
o -0098370
o -0017496
'5645820
-5663312
-5680652
o -5697827
-5714849
18'5
25 6 17-54
'3500484
2-1643540
+ 0-0063418
o -5731712
206
On the Elements of the Orbit of Halley s Comet,
TABLE I. continued.
Date.
u
Log. of r
X
y
z
1836.
Q 1 II
Mar. 18 '5
25 6 17-54
'3500484
2 -1643540
+ 0-0063418
0-5731712
19'5
25 12 29 '15
o -3526392
2-1777300
-0144311
'5748430
, 20 -5
25 18 38 '55
'3552088
2-1910480
-0225203
o -5764996
21 '5
25 24 45 '69
0-3577576
2 -2043100
o -0306062
-5781412
22'5
25 30 50 '83
-3602870
2-2175190
-0386944
o -5797692
23'5
25 36 53 '88
o -3627968
2 -2306750
'0467840
'5813829
24-5
25 42 54 '71
'3652858
2 -2437730
-0548675
0-5829816
25 -5
25 48 53 '50
o -3677556
2 -2568170
o -0629496
-5845670
26 '5
25 54 50 '43
o -3702076
2-2698160
'0710352
-5861395
27-5
26 45 '23
o -3726398
2 -2827560
0-0791168
o '5876977
28 *5
26 6 38 '03
o -3750534
2 ^956460
-0871938
-5892438
29'5
26 12 28 '86
'3774486
2 '3084840
o -0952698
o -5907748
30-5
26 18 IS'Ol
o -3798274
2 -3212790
0-1033490
o -5922947
31 '5
26 24 4 '99
-3821934
2 -3340520
+ -1114423
'5938052
at its appearance in the Yearn 1835 4" 1836.
207
TABLE II.
Containing, for each Mean Midnight at Greenwich (Astronomical time) from August 1,
1835, to March 31, 1836, the Sun's True Geocentric co-ordinates (X, Y, Z,);
X, being measured on a line passing through the True Vernal Equinoctial point of
the date ; Y, on a line in the plane of the Equator, and perpendicular to the direc-
tion of X ; and Z, perpendicular to the plane of the Equator, towards the North.
Date.
X
Y
Z
1835.
Aug. 1 '5
2 '5
3 '5
4 *5
5 "5
0-6377144
o -6507172
'6635348
o -6761640
'6886010
+o -7239212
0-7139433
o 7037629
'6933840
0-6828106
+ 0-3141959
0-3098651
o -3054467
o -3009422
o -2963532
6-5
7'5
8'5
9*5
10 '5
'7008426
7128858
o -7247270
o -7363624
7477904
o -6720445
-6610891
o -6499469
'6386212
0'6271152
0-2916807
'2869258
-2820903
o -2771750
0-2721814
11 '5
12-5
13 '5
14 '5
15'5
o -7590069
o -7700086
o -7807926
07913554
0-8016942
'6154303
0-6035713
'5915402
0-5793399
o -5669735
o -2671100
o -2619630
o -2567409
-2514459
o -2460787
16-5
17-5
18'5
19'5
20'5
0-8118056
-8216860
0'8313320
0-8407414
-8499097
'5544446
'5417562
0-5289113
0-5159127
o -5027664
-2406408
-2351336
-2295588
o -2239171
-2182115
21 '5
22 '5
23 '5
24-5
25 '5
-8588338
0-8675108
o -8759393
-8841147
-8920348
-4894753
o -4760423
o -4624726
'4487694
o -4349377
0-2124431
-2066131
o -2007236
o -1947763
-1887731
26'5
27-5
28 '5
29-5
30 '5
o -8996971
o -9070998
;9142407
0-9211173
o -9277274
0-4209812
o -4069035
o -3927093
-3784028
o -3639887
-1827156
0-1766055
0-1704449
-1642355
0-1579793
31 '5
o -9340702
+ 0-3494706
+ 0-1516782
208
0)i the Elements of the Orbit ofHalleys Comet,
TABLE II. continued.
Date.
X
Y
Z
1835.
Aug. 31 *5
Sep. 1 '5
2-5
3'5
4'5
o -9340702
'9401430
o -9459447
'9314/35
o -9567233
+ 0-3494706
-3348529
'3201410
o -3053374
o -2904466
+ 0-1516782
0-1453338
'1389485
0-1325236
o -1260608
5 '5
6-5
7-5
8'5
9'5
0'9617067
'9664088
'9708324
o -9749755
o -9788376
o -2754731
-2604215
-2452950
o -2300975
0-2148339
o -1195620
o -1130293
0-1064639
o -0998680
-0932431
10*5
11 '5
12-5
13-5
*4 '5
0-9824170
0-9857127
'9887233
0-9914467
'9938820
o -1995069
0-1841211
o -1686809
-1531899
0-1376537
o -0865909
0-0799130
0-0732117
-0664882
o -0597451
15 '5
16'5
17-5
18'5
19'5
o -9960284
'9978838
o -9994476
1 '0007185
1 -0016956
0-1220750
o -1064592
0-0908113
o -0751365
-0594383
-0529836
'0462060
-0394144
'0326112
o -0257979
20-5
21 '5
22-5
23'5
24'5
1 -0023781
1 -0027662
1 -0028587
1 -0026553
1 -0021567
o -0437224
o -0279936
+ 0-0122569
'0034839
'0192232
0-0189768
-0121501
+ -0053200
0-0015119
'0083432
25'5
26'5
27-5
28'5
29'5
1 -0013618
1 -0002713
0-9988861
o -9972056
0-9952314
o -0349567
o -0506792
'0663847
o -0820709
o -0977301
0-0151719
o -0219959
0-0288126
o -0356207
-0424172
30 '5
Oct. 1 '5
2'5
3'5
4'5
o -9929635
0-9904031
o -9875505
o -9344073
o -9809734
0-1133602
0-1289545
'1445099
o -1600214
0-1754841
o -0492011
o -0559695
o -0627209
o -0694533
o -0761646
5 '5
6-5
7*5
8 '5
9*5
o -9772506
0-9732391
o -9689400
'9643546
'9594834
'1908938
o -2062490
-2215423
o -2367707
0-2519284
-0828530
0-0895174
o -0961551
o -1027644
-1093433
10 '5
o -9543277
0-2670131
-1158903
at its appearance in the Years 1835 $ 1836.
209
TABLE II. continued.
Date.
X
Y
z
1835.
Oct. 10 '5
11 '5
12*5
13-5
14-5
o -9543277
0'9488880
0*9431659
0-9371623
'9308786
0-2670131
0*2820198
o -2969437
-3117803
'3265255
0-1158903
'1224035
0'1288809
0-1353204
o -1417202
15*5
16'5
17-5
18 '5
19'5
0-9243160
0'9174755
0-9103593
'9029685
o -8953061
0-3411742
0-3557214
o -3701632
-3844938
o -3987093
'1480782
'1543921
0-1606602
0-1668801
o -1730500
20 -5
21 '5
22 '5
23 '5
24'5
o -8873732
0-8791725
o -8707056
0-8619757
-8529850
0-4128050
o -4267767
o -4406199
-4543289
o -4679004
0-1791678
'1852318
0-1912399
0-1971897
-2030800
25 '5
26'5
27-5
28 -5
29'5
'8437366
-8342334
-8244780
0-8144746
-8042254
'4813287
o -4946128
o -5077449
o -5207219
-5335404
-2089085
0-2146737
o -2203735
-2260058
o -2315694
30 '5
31 '5
Nov. 1 '5
2 "5
3'5
o -7937343
'7830035
o -7720366
o -7608375
'7494084
o -5461961
-5586856
o -5710053
-5831521
-5951215
o -2370624
'2424831
0-2478301
0-2531021
'2582969
4'5
5'5
6-5
7*5
8'5
0-7377541
o -7258759
07137773
o -7014629
-6889342
o -6069100
0-6185154
o -6299339
o -6411615
0-6521953
'2634133
-2684502
o -2734059
-2782789
-2830678
9'5
10 '5
11 '5
12'5
13'5
0-6761963
-6632517
-6501038
o -6367569
0-6232138
'6630313
-6/36663
o -6840969
o -6943192
o -7043302
-2877708
-2923865
0'2969137
'3013504
o -3056957
14'5
15'5
16 '5
17-5
18'5
o -6094795
o -5955570
-5814503
0-5671633
o -5527009
0-7141259
o -7237032
'7330599
0-7421914
0-7510947
o -3099473
0*3141040
-3181650
-3221281
0-3259921
19*5
'5380680
o -7597662
o -3297556
210
On the Elements of the Orbit ofHalleys Comet,
TABLE II. continued.
Date.
X
Y
Z
1835.
Nov. 19 '5
20 -5
21 '5
22 '5
23'5
'5380680
o -5232679
'5083063
-4931886
o -4779192
o -759/662
'7682038
o -7764042
'7843640
'7920813
o -3297556
0-3334175
o -3369766
0-3404311
o -3437806
24-5
25'5
26-5
27-5
28'5
'4625022
-4469443
'4312499
-4154234
o -3994705
7995539
'8067783
0*8137524
o -8204754
o -8269446
o -3470239
-3501597
'3531868
o -3561047
'3589124
29*5
30 '5
Dec. 1 -5
2 '5
3 -5
'3833962
o -3672041
-3509002
'3344888
0-3179752
-8331580
6 -8391143
0-8448118
'8502491
'8554240
o -3616093
o -3641943
o -3666671
o -3690266
0-3712725
4 '5
5'5
6-5
7-5
8'5
-3013635
-2846594
0-2678671
o -2509927
'2340384
0-8603357
-8649822
o -8693622
o -8734741
0-8773168
-3734042
-3754208
0-3773217
0-3/91064
-3807744
9'5
10 '5
11 -5
12'5
13 '5
0-2170117
o -1999168
0-1827584
o -1655416
0-1482738
-8808888
-8841886
-8872146
'8899654
'8924400
-3823249
o -3837572
'3850708
-3862649
o -3873390
14'5
15 '5
l6'5
17-5
18 '5
o -1309573
o -1135992
o -0962042
0-078778!
0-0613261
o -8946376
o -8965570
0-8981975
o -8995575
o -9006370
'3882928
'3891256
'3898372
o -3904274
-3908958
19*5
20 '5
21 -5
22 '5
23 '5
0'0438550
o -0263694
-0088756
+ '0086187
0*0261103
G'9014350
0-9019513
'9021856
0-9021381
0-9018086
'3912420
0'3914662
-3915681
o -3915477
0-3914051
24*5
25 '5
26-5
27 '5
28 '5
o -0435920
-0610586
-0785044
o -0959239
0-1133131
0-9011976
'9003058
0-8991338
o -8976816
o -8959504
-3911402
0-3907531
o -3902444
0-3896141
-3888627
29'5
+ -1306663
-8939408
'3879904
at its appearance in the Years 1835 8f 1836.
TABLE II. continued.
Date.
X
Y
Z
1835.
Dec. 29 '5
30 '5
31 '5
1836.
Jan. 1 *5
2'5
+ '1306663
o -1479764
'1652420
'1824538
o -1996086
'8939408
o -8916536
o -8890896
'8862483
'8831344
-3879904
-3869975
'3858847
'3846513
'3832999
3 '5
4-5
5'5
6-5
7-5
o -2167012
o -2337263
o -2506797
o -2675558
-2843501
o -8797475
-8760878
0-8721565
o -8679550
-8634839
-3818298
0*3802417
'3785354
o -3767122
0-3747721
8'5
9-5
10-5
11 '5
12-5
0-3010571
0-3176716
'3341886
'3506041
0-3669118
'8587450
o -8537391
-8484678
0-8429315
0-8371324
o -3727154
-3705428
-3682549
'3658521
-3633350
13*5
14'5
15 *5
16'5
17*5
-3831065
0-3991837
-4151367
-4309614
0-4466519
-8310725
0-8247514
-8181733
0-8113392
0-8042515
o -3607043
o -3579612
0-3551061
0-3521401
o -3490642
18 '5
19'5
20 '5
21 '5
22-5
-4622034
o -4776097
'4928658
o -5079681
o -5229109
o -7969121
'7893248
o -7814911
0-7734146
o -7650978
-3458791
-3425862
0-3391866
-3356814
o -3320718
23 '5
24 '5
25'5
26 '5
27-5
o -5376908
-5523017
o -5667400
0-5810008
-5950812
o -7565431
o -7477544
'7387340
o -7294867
o -7200125
-3283588
'3245443
o -3206292
-3166155
-3125034
28-5
29'5
30 '5
31 '5
Feb. 1 '5
o -6089770
'6226830
o -6361972
'6495148
o -6626323
o -7103177
'7004046
o -6902761
o -6799352
'6693855
-3082956
-3039931
o -2995972
o -2951092
-2905308
2'5
3'5
4 '5
5'5
6'5
o -6755458
-6882509
o -7007469
0-7130268
o -7250890
-6586305
o -6476734
o -6365163
0-6251638
0-6136183
-2858629
-2811075
o -2762656
-2713381
o -2663271
7-5
+ 07369287
-6018841
'2612341
212
On the Elements of the Orbit ofHalkifs Comet,
TABLE II. continued.
Date.
X
Y
Z
1836.
Feb. 7 '5
8-5
9'5
10-5
11-5
+ 0*7369287
'7485434
'7599287
0*7710818
0-7819977
0*6018841
-5899637
-5778589
o -5655759
0-5531174
0-2612341
o -2560599
'2508063
o -2454750
o -2400677
12 '5
13*5
14-5
15-5
16 '5
o -7926742
0-8031068
0-8132926
'8232278
0-8329091
-5404874
o -5276899
o -5147293
o -5016092
-4883346
*234586l
'2290318
-2234068
o -2177124
o -2119510
17-5
18 '5
19'5
20-5
21 '5
'8423340
-8514993
-8604026
o -8690409
o -8774127
o -4749097
0-4613391
o -4476267
-4337777
0-4197970
o -2061244
-2002344
-1942829
0-1882717
-1822038
22 '5
23-5
24 ! 5
25 *5
26'5
0*8855150
-8933456
o -9009036
0-9081862
0-9151928
-4056887
0-3914576
0-3771077
o -3626450
-3480724
0-1760802
o -1699033
0-1636751
o -1573979
0-1510/29
27'5
28 '5
29'5
Mar. 1 '5
2'5
0-9219204
'9283680
0-9345352
0-9404191
o -9460190
o -3333960
0-3186195
o -3037472
-2887843
o -2737349
-144/036
0-1382896
-1318348
-1253406
0-1188088
3'5
4 '5
5 '5
6-5
7'5
G'9513338
o -9563625
o -9611028
0*9655541
o -9697146
-2586029
o -2433919
'2281080
0-212/539
o -1973345
0-1122411
o -1056393
o -0990056
0-0923411
-0856490
8*5
9'5
10 '5
11 '5
12 '5
'9735837
o -9771600
'9804420
'9834286
0'986ll85
0*1818541
o -1663172
o -1507290
'1350940
0*1194l68
-0789302
o -0721868
ff -0654212
'0586352
0-0518310
13-5
14 '5
15 *5
16 '5
17-5
0*9885118
o -9906069
G'9924031
o -9939002
'9950982
0-1037026
-0879563
-0721828
-0563861
'0405740
0-0450106
-0381762
0-0313301
-0244738
o -0176109
18'5
4-0*995996l
o -0247498
o -0107423
at its appearance in the Years 1835 <$" 1836.
213
TABLE II. continued.
Date.
1836.
Mar. 18 '5
19-5
20-5
21 '5
22 '5
23'5
24 '5
25-5
26'5
27-5
28 '5
29 '5
30-5
31'5
o -9965947
o -9968940
o -9968942
o -9965953
o -9959996
o -9951064
o -9939175
'9924335
'9906551
'9885840
'9862208
'9835663
0-9806227
o -0247498
o -0089170
+ 0-0069177
o -0227492
-0385728
-0543835
0-0701763
'0859477
0'10l6914
0*1174037
0-1330796
'1487153
0-1643064
+ 0-1798484
-0107423
'0038708
+ -0030020
'0098734
0-0167414
-0236038
-0304584
-0373036
0-0441370
o -0509567
o -0577607
o -0645172
-0713143
+ 0-0780601
214
the Elements of the Orbit of Halleys Comet,
TABLE III.
Containing, for Mean Noon at Greenwich, the Heliocentric co-ordinates (a,-, ar n &c.,
2/> 2/n &c., s, z &c.) of the Comet and the disturbing Planets; x, x l9 &c., beijig
measured on a line passing through the Mean Vernal Equinox of January 1, 1835;
2/> yn &c -> perpendicular to .T, #,, &c., in the plane of the Ecliptic (supposed
invariable); and z, 2 1} &c., perpendicular to the plane of the Ecliptic, towards the
North.
HELIOCENTRIC CO-ORDINATES OK
Date.
THE COMET.
MERCURY.
1835.
X
y
z
#1
2/i
*j
Aug. 1
+ 0-99318
+ 1 '74856
o -05926
+ o -35692
-06352
0-03711
5
o -99605
1 -68279
'04645
-33668
+ -05450
-02532
9
'99827
1 '61585
'03360
o -27709
o -16571
o -01059
13
o -99977
1 -54778
o -02073
o -17940
-25364
+ -00555
17
1 -00052
1 '47856
o -00786
+ 0-05418
-30220
'02080
21
1 '00046
1 -40815
+ 0*00503
o -07972
o -30270
o -03275
25
o -99949
1 -33638
o -01792
'20210
'25798
-03984
29
'99752
1 -26322
o -03079
-29857
0'17949
o -04176
Sep. 2
o -99433
1 -18872
o -04361
"36284
+ -08114
o -03913
6
-98992
1 -11270
'05638
o -39447
-02482
o -03297
10
'98408
1 -03500
o -06908
o -39625
o -12912
'02429
14
o -9/670
o -95572
-08165
o -37227
-22519
'01400
18
o -96745
o -87462
o -09409
o -32692
'30845
+o -00292
22
o -95618
0'79l66
-10632
0-26451
o -37570
-00834
26
'94252
o -70681
-11828
o -18919
-42466
o -01919
30
o -92621
o -61993
0-12991
o -10497
o -45370
0-02914
Oct. 4
o -90680
o -53095
0-14109
0-01587
o -46161
o -03773
8
-88377
o -43990
o -15172
+o -07396
o -44764
'04453
12
'85647
o -34673
0-16162
-16005
'41133
o -04910
16
'82431
0-25157
o -17060
o -23743
-35283
'05103
20
o -78639
o -15474
0-17837
-30045
o -27322
o -04987
24
0'74170
+ -05655
-18461
-34263
o -17469
'04529
28
-68924
0-04218
-18890
o -35691
o -06262
o -03704
Nov. 1
o -62786
0-14032
o -19075
-33638
+ -05541
02520
5
o -55672
-23622
0-18961
o -27643
0-16651
0-01046
9
o -47529
o -32763
-18496
-17853
-25419
+o -00567
13
'38379
-41220
0-17640
+ -05316
-30238
o -02091
17
-28356
-48727
'16380
-08074
-30252
-03283
21
0'17673
-55104
o -14740
o -20296
-25748
o -03987
25
+ -06600
o -60262
o -12774
o -29920
0-17877
o -04176
29
-04613
-64230
+ 0-10554
-36320
+ -08033
+o -03910
at its appearance in the Years 1835 fy 1836.
215
TABLE III. continued.
HELIOCENTRIC CO-ORDINATES OF
Date.
THE COMET.
MERCURY.
1835.
X
y
z
*i
2/1
2l
Nov. 29
'04613
-64230
+ -10554
-36320
+ -08033
+o -03910
Dec. 3
o -15751
o -67104
-08152
o -39459
'02564
-03290
7
o -26672
o -69032
-05635
o -39616
0-12991
-02421
11
-37289
o -70155
-03050
0-37199
-22590
o -01392
15
o -47555
0'706l7
+o -00437
'32650
'30902
+ 0-00282
19
'57458
o -70539
-02181
0-26397
0-37615
-00842
23
o -66999
o -70013
o -04786
-18856
-42497
o -01927
27
o -76196
o -69123
o -07366
-10429
-45385
o -02920
31
-85065
o -67934
o -09915
-01516
o -46162
o -03778
1836.
Jan. 4
o -93620
-66500
0-12427
+ o -07465
o -44744
o -04457
8
1 '01892
-64859
o -14902
o -16070
o -41096
'04913
12
1 -09896
'63046
0-17338
o -23799
-35231
-05102
16
1 -17652
o -61089
o -19735
-30087
o -27253
-04986
20
1 -25176
o -59015
o -22092
'34285
o -17408
'04524
24
1 -32483
-56837
0-24412
-35689
o -06171
o -03696
28
1 -39589
o -54573
o -26696
o -33606
+ -05632
-02510
Feb. 1
1 '46508
-52241
-28942
'27585
0-16731
-01034
5
1 -53250
-49838
-31154
0-17764
o -25472
+ -00580
9
1 '59834
'47385
-33334
+ -05213
'30257
-02102
13
1 -66253
-44885
-35480
-08175
'30232
o -03290
17
1 -72536
-42347
o -37597
-20380
o -25699
o -03990
21
1 '78682
-397/6
-39683
-29982
o -17807
o -04176
25
l -84697
o -37174
o -41740
'36356
+ o -07954
o -03906
09
1 -90594
o -34557
o -43770
o -39471
o -02646
-03285
Mar. 4
1 -96372
0-31911
o -45774
o -39606
-13068
-02413
8
2 '02041
o -29241
o -47753
0-37173
'22658
'01383
12
2 -07601
-26564
o -49705
-32608
o -30960
+ o -00275
16
2-13074
-23875
-51636
-26344
-37658
-00850
20
2 -18443
o -21175
-53544
-J8794
-42526
o -01934
24
2 -23723
-18459
-55428
0-10363
o -45398
o -02927
28
2 -28929
o -15747
o -57293
-01448
o -46159
-03785
Apr. 1
2 '34045
o -13017
o -59137
4-0 -07533
o -44726
'04462
216 On the Elements of the Orbit of Halle ys Comet,
TABLE III. continued.
Date.
HELIOCENTRIC CO-ORDINATES OF
VENUS.
THE EARTH.
1835.
X 2
y*
**
r 3
y*
*3
Aug. 1
5
9
13
17
+ '04533
'03582
-11652
o -19574
o -27247
+o -71820
71834
o 70932
o -69129
o -66447
+ -00829
o -01293
0-01742
-02168
-02567
+o -63109
-68234
73053
o 77537
o -81672
'79458
o 75020
-/0243
'65149
-59758
*
21
25
29
Sep. 2
6
o -34572
-41456
-47812
o -53559
'58622
o -62917
-58584
-53504
o -47742
0-41371
o -02934
o -03263
-03551
o -03793
'03986
-85436
-88806
o -91768
'94303
'96405
o -54094
-48178
-42041
-35/12
o -29220
,np r!.^
10
14
18
22
26
-62938
-66452
o -69120
070910
0'71802
o -34472
o -27134
o -19451
-11521
+ -03446
'04128
-04218
'04255
-04239
0-04167
'98064
o -99270
1 -00010
1 -00283
1 -00082
o -22593
0-15862
o -09056
o -02206
+ 0-04657
-
30
Oct. 4
8
12
16
o -71733
'70856
o -69034
'66342
0-62817
o -04673
o -12734
-20634
-282/5
o -35560
'04043
-03868
-03644
o -03373
-03061
o -99112
o -98272
o -96670
o -94609
o -92098
0-11497
-18278
0-24971
'31548
o -379/6
~ *
20
24
28
Nov. 1
5
'58505
-53461
o -47750
'41443
0'34621
-42398
-48705
-54405
o -594-28
0-63714
o -02710
-02325
o -01912
o -01476
'01020
-89140
-85755
o -81956
o 77764
073194
-44223
-50259
-56050
0-61569
o -66787
> -*>;* 1
9
13
17
21
25
o -27370
o '19779
o -11944
o -03962
+ -04068
o -67208
o -69869
071667
'72583
o 72606
-00551
+o -00075
-00402
'00873
'01333
o -68270
0-63010
o -57443
-51591
-45486
o -71682
o 76229
-80406
'84186
o -87553
i "
29
Dec. 3
7
11
15
-12048
-19880
o -27469
o -34722
'41549
o 71735
-69983
o -67373
o -63937
-59/18
0-01777
-02199
-02594
-02958
'03285
'39158
-32638
o -19150
'90483
o -92967
'96554
r ~*" ""Hi
19
+o 4/867
o -54768
o -03572
+ 0-05274
+ '98229
*. ^ .
at its appearance in the Years 1835 $ 1836.
217
TABLE III. continued.
Date.
HELIOCENTRIC CO-ORDINATES OF
VENUS.
THE EARTH.
1835.
^ 2
2/2
*2
#3
2/3
*3
Dec. 1 ( J
+ o --P7867
o -54768
o -03572
+ o -05274
+ '98229
_
23
o -53598
o -49147
-03814
-
-
^Vfi
27
-58672
o -42923
-04011
o -08707
'97942
-
31
o -63027
o -36173
'04159
-
-
J
1836.
Jan. 4
o -66610
-28980
'04257
-22505
0-95711
- -
8
o -69376
-21432
-04300
i,"
^ L'-'j. '
_ _
12
o -71290
o -13621
-04291
-35860
0*91584
_
16
o 72326
0-05641
-04229
-
- ,
.
20
o -72470
+ 0-02408
'04115
0-48511
0-85631
-
28
70084
-18316
o -03739
o -60191
o -779/6
-
Feb. 5
'64230
-33320
o -03176
o 70677
-68788
_ _
13
o -55190
o -46670
o -02457
079781
0-58241
_
21
-43399
o -57690
0-01613
'87315
-46544
_
29
o -29435
-65822
o -00690
'93141
-33944
_
Mar. 8
+ -13990
o -70643
+o -00268
o -97165
0-20691
' "ri ' W *
16
o -02161
071899
'01214
o -99316
4-0 -07032
_ _
24
-18203
o -69510
o -02097
o -99561
o -06767
-
Apr. 1
0-33317
+o -63592
+ '02873
0-97911
'20428
- -
*''-* S t\ : :i?;;;V& [-^i8$ff b* CS
Date.
MARS.
JUPITER.
1835.
#4
y4
m
*"\
#5
2/3
2 5
Aug. 5
-54181
-48934
+o -02664
+ '15810
+ 5 -13050
'02150
13
49982
'58556
-02356
-
-
-'*'.
21
45104
0-67914
'02037
4-0-03617
5 '13854
0-01868
29
39553
o 76959
o -01706
_
-
- o \
Sep. 6
33355
-85641
o -01369
-08574
5 -14375
o -01601
14
26523
0-93918
o -01027
.
_
_ _
22
19078
1 -01744
o -00679
o -20774
5 -14600
-01318
30
11045
1 -09064
+ -00331
" : -
vl" SS^
-
Oct. 8
1 '02452
1 -15838
'00022
o -32959
5-14553
'01050
16
'93338
1 -22017
o -00375
-
-
. - . - ;
24
o -83739
1 -27562
o -00728
-45124
+ 5 '14209
o -00766
NAUTICAL ALMANAC, 1839. APPENDIX.
218
On the Elements of the Orbit of Halley's Comet,
TABLE III. continued.
Date.
HELIOCENTRIC CO-ORDINATES OF
MARS.
JUPITER.
1835.
X 4
y*
*4
*5
y*
2 5
Oct. 24
o -83739
1 -27562
o -00728
'45124
+ 5-14209
o -00766
Nov. 1
o -73690
1 -32422
0-01076
. - ^.*. .
-
-
5
-68512
34586
'01246
i*.
_
-
9
0-63241
36565
'01414
o -57261
5 -13594
o -00496
13
0-57881
38354
'01582
- -
-
-
17
'52441
39949
'01748
o -63317
5-13181
-00353
21
o -46928
41346
o -01912
-
-
-
25
0-41348
45241
o -02072
o -69365
5 12696
-00211
29
o -35706
43526
'02230
T ;.;'
_
-
Dec. 3
0-30011
44301
'02384
o -75409
5-12135
o -00075
7
o -24274
44867
'02536
_ _
_
-
11
-18500
45218
'02683
'81442
5 -11505
+o -00060
15
o -12697
45347
'02824
.
-
-
19
o -06375
45255
o -02961
-
't ! <
-
23
'01043
44941
o -03095
-
t' s
-
27
+o -04790
44401
-03224
o -93457
5 -10047
o -00347
31
0-10618
43632
-03348
-
-
-
1836.
Jan. 4
-16428
42636
o -03467
-
-
*'?/
8
-22210
41414
'03580
_
_
-
12
o -27955
39965
'03688
1 -05436
5 -08323
o -00619
20
-39305
36386
'03886
_ _
_
-
28
-50394
31899
'04055
1 47355
5 -06302
-00892
Feb. 5
0-61145
1 '26532
'04199
*V f
-
-
13
o -71474
1 -20298
0-04316
1 '29205
5 -04011
0-01181
21
-81319
1 -13235
-04401
-
-
- -
29
o -90590
1 -05385
'04452
1 '41000
5 -01453
0-01455
Mar. 8
o -99222
o -96792
o -04477
_
-
-
16
1 -07152
'87512
'04469
1 -52700
4 '98632
o -01729
24
1 -14319
o -77609
'04429
-
-
l -
Apr. 1
+ 1 -20670
o -67151
-04356
1 -64327
+ 4-95541
+ '02005
'?* >' . ' "^ A '
at its appearance in the Years 1835 Sf 1836.
219
TABLE III. continued.
Date.
HELIOCENTRIC CO-ORDINATES OF
SATURN.
THE GEORGIAN.
1835.
X 6
2/6
2<>
x 7
y?
2 7
Aug. 5
-8 -89774
3 -88947
+ -42209
+ 17 -0573
10 -4973
o -26210
21
8 '86645
3 -97109
o -42219
17 '0899
10 -4472
'26231
Sep. 6
8 '83446
4 -05257
'42224
17 -1214
10 -3966
-26251
22
8 '80195
4 -13352
-42226
17*1538
10 -3452
o -26272
Oct. 8
8 76859
4 -21415
-42227
17-1854
10 -2941
o -26292
24
8 73474
4 -29447
-42222
17-2171
10 -2431
-26313
Nov. 1
8 71738
4 -33460
"42218
>
! -
9
8 -69981
4 -37462
-42213
17 -2484
10-1920
-26334
17
8 '68210
4 '41440
o -42206
U V
-
! ;
25
8 '66423
4 -45410
o -42199
17-2795
10*1405
'26354
Dec. 1 1
8 -62800
4 -53367
-42180
17 -3101
10 -0890
o -26374
27
8 -59112
4 -61233
o -42167
17-3412
10 -0370
o -26395
1836.
Jan. 12
8 '55342
4 -69105
-42142
17 -3724
9 '986o
o -26416
28
8 -51530
4 76947
'42115
17-4028
9 *9344
o -26436
Feb. 13
8 -47618
4-84719
'42086
17 -4333
9 '8828
o -26457
29
8 -43665
4 -92436
'42057
17 -4639
9 *8308
o -26478
Mar. 16
8 -39615
5 -00138
'42018
17 -4936
9 7787
'26488
Apr. 1
8 '35526
-5 -07797
+ -41981
4.17-5235
9 7268
-26509
i* ,
q 2
220
On the Elements of the Orbit of Halley's Comet,
TABLE IV.
Containing, for Greenwich Mean Noon of each fourth day, from Aug. 1, 1835, to
April 1, 1836, the united Effects (A) of the attractions of the disturbing Planets
upon the Comet in the direction of the co-ordinate x, expressed in 10,000,000,000th
parts of an unit, and distinguishing the separate Effect of each Planet.
Date.
Mercury.
Venus.
Earth.
Mars.
Jupiter.
Saturn.
Georgian
A.
1835.
Aug. 1
+ 36848
+ 11611
+ 17620
1116
+ 196320
8558
+ 97
+ 252822
5
42145
6705
19032
1104
190031
8398
90
248501
9
41413
+ 1537
20367
1091
183562
8232
83
23/639
13
30642
3781
2l6l6
1077
176930
8061
76
216345
17
+ 10536
9111
22773
1061
170155
7883
69
185478
21
11180
14318
23831
1042
163284
7700
62
152937
25
26320
19277
24780
1021
156341
7511
55
127047
29
32794
23886
25610
998
149332
7314
48
109998
Sep. 2
33032
28053
26301
972
142265
7109
41
99441
6
30023
31713
26831
943
135164
6894
33
92455
10
25728
34806
27158
911
128052
6671
25
87H9
14
21124
37288
27207
876
120921
643/
17
82420
18
16592
39123
26818
837
113761
6190
9
77846
22
12213
40286
2o62/
795
106585
5930
+ 1
72989
26
7923
40762
22652
748
99401
5657
~ 7
66956
30
3584
40530
+ 14783
696
92202
5367
15
56793
Oct. 4
+ 996
39585
10274
639
84976
5056
23
+ 30395
8
6069
37916
110218
575
77723
4723
30
69670
12
11984
35512
356598
503
70435
4368
37
314599
16
19294
32346
185594
421
63116
3986
43
139980
20
29065
28372
41067
327
55773
3571
50
+ 11451
24
43494
23494
2076
219
48367
3118
58
62896
28
61638
17522
+ 9898
- 93
40863
2624
72
92088
Nov. 1
62110
10080
14024
+ 57
33326
2086
88
97263
5
49829
- 397
15260
239
25834
1500
101
89164
9
34105
+ 13233
15201
459
18374
866
108
80398
13
+ 12161
34958
14462
724
10939
188
102
72954
17
10130
75992
13318
1026
+ 3631
+ 517
90
84264
21
25236
163607
11907
1329
3453
1239
75
149318
25
31375
+ 95881
10307
1526
10273
1962
61
+ 67967
29
31111
379845
8563
1404
16801
2668
52
415174
Dec. 3
27607
159455
6708
+ 721
23033
3354
44
199356
7
23508
64310
4761
- 487
28985
4011
36
108554
11
22283
24629
2742
1721
34681
4635
28
75965
15
27377
3809
+ 662
2466
40156
5217
19
67948
19
29934
+ 9163
1466
2652
45427
+ 5760
10
64566
at its appearance in the Years 1835 $ 1836.
221
TABLE IV. continued.
Date.
Mercury.
Venus.
Earth.
Mars.
Jupiter.
Saturn.
Georgian.
A.
1835.
Dec. 19
29934
+ 9163
- 1466
2652
45427
+ 5760
10
64566
23
23121
18202
3630
2486
50516
6266
1
55286
27
14508
24939
5822
2172
55457
6737
+ 8
46275
31
7112
30130
8033
1825
60285
7174
17
39934
1836.
Jan. 4
583
34157
10251
1495
65003
7577
26
35572
8
+ 5756
37230
12468
1198
69614
7948
35
32311
12
12401
39462
14686
935
74141
8288
44
29567
16
19643
40921
16909
702
78607
8598
52
27004
20
27431
41656
19129
495
83013
8879
60
24611
24
3500/
41697
21339
309
87360
9132
68
23104
28
40235
41074
23544
140
91660
9359
76
24600
Feb. 1
39220
39809
25748
+ H
95926
9562
84
32985
5
28021
37932
27954
157
100160
9740
91
52173
9
+ 7558
35473
30160
290
104363
9894
99
81209
13
14278
32469
32359
413
108544
10024
106
112169
17
29373
28959
34531
529
112709
10132
113
136880
21
35787
24988
36635
638
116860
10217
120
153319
25
35984
20606
38612
741
120999
10281
127
163840
29
32924
15869
40369
838
125129
10323
134
171258
Mar. 4
28534
10837
41812
930
129256
10345
141
177349
8
23802
5574
42839
1017
133382
10347
147
182938
12
19136
+ 149
43370
1098
137508
10229
153
188385
16
14651
5367
43386
1174
141637
10293
159
193415
20
10313
10901
42902
1245
145769
10239
165
198236
24
6007
16378
41990
1312
149910
10167
171
202635
28
1576
21724
40748
1374
154070
10078
177
206489
Apr. 1
+ 3162
26863
39257
+ 1430
158248
+ 9972
+ 182
209622
(, ~ ' - - i * .? I
222 On the Elements of the Orlit of Hatteys Comet,
TABLE V.
Containing, for Greenwich Mean Noon of each fourth day, from August 1, 1835, to
April 1, 1836, the united Effects (B) of the attractions of the disturbing Planets
upon the Comet in the direction of the co-ordinate y, expressed in 10,000,000,000th
parts of an unit, and distinguishing the separate Effect of each Planet.
Date.
Mercury.
Venus.
Earth.
Mars.
Jupiter.
Saturn.
Georgian
B.
1835.
Aug. 1
5216
+ 57349
17205
- 171
406607
1181
+ 100
372931
5
+ 8207
56610
15626
214
374253
1251
101
326426
9
25951
55158
13876
257
343563
1320
102
277805
13
43508
53098
11937
299
314472
1389
103
231388
17
52882
50506
9784
341
286878
1458
103
194970
21
49407
47440
7386
383
260718
1526
104
173062
25
36660
43938
4688
424
235887
1594
104
161891
29
21620
40039
1616
464
212329
1662
104
154308
Sep. 2
+ 8683
35776
+ 1945
504
189992
1729
104
145717
6
988
31188
6180
543
168775
1794
103
134629
10
7785
26320
11395
582
148579
1858
103
120986
14
12466
21226
18102
617
129376
1921
102
104950
18
15689
15966
27247
651
111145
1981
101
86152
<22
17922
10602
40670
684
93796
2038
100
63068
26
19462
+ 5207
62251
714
77242
2094
99
31955
30
20468
150
101095
741
61494
2147
97
+ 16192
Oct. 4
20975
5395
179393
765
46574
2194
94
103584
8
20893
10450
322598
785
32398
2235
91
255928
12
19957
15237
+ 142675
799
18893
2269
87
+ 85607
16
17648
19669
270096
804
6133
2294
82
316562
20
13162
23651
169564
800
+ 5794
2308
77
203614
24
6516
27069
88291
784
16937
2306
72
107957
28
4848
29767
47178
747
27331
2283
70
57422
Nov. 1
- 6950
31491
24781
684
36851
2234
69
29220
5
+ 11045
31769
11301
581
45367
2154
66
+ 10673
9
32602
29539
2483
415
52783
2034
6l
50975
13
44395
21919
+ 3658
154
59022
1868
47
83181
17
42131
+ 1103
8144
+ 256
64012
1652
29
114023
21
29800
87166
11528
886
67726
1386
+ 11
195731
25
14477
407564
14142
1803
70263
1074
- 6
507169
29
+ 395
+ 128198
16181
2975
71763
720
19
218773
Dec. 3
11754
31315
17773
4115
72380
330
29
50840
7
23274
46031
18996
4662
72280
+ 90
38
26685
11
35285
45337
19911
4269
71616
531
47
15658
15
42472
42287
20547
3216
70524
986
56
10458
19
36628
38508
+ 20932
+ 2034
+ 69034
+ 1454
- 64
+ 18164
at its appearance in the Years 1835 fy 1836.
223
TABLE V. continued.
Date.
Mercury.
Venus.
Earth.
Mars.
Jupiter.
Saturn.
Georgian.
B.
1835.
Dec. 19
36628
38598
+ 20932
+ 2034
+ 69034
+ 1454
- 64
+ 18164
23
28757
34561
21077
1034
67152
1936
72
27809
27
25462
30235
20999
+ 288
65009
2426
79
32946
31
24555
25650
20706
244
62721
2919
86
35811
1836.
Jan. 4
24285
20837
20205
619
60267
3416
93
38054
8
23845
15830
19502
884
57617
3917
99
40378
12
22682
10679
18605
1073
54823
4418
105
43307
16
20136
5431
17524
1208
51927
4919
110
47485
20
15257
140
16268
1305
48925
5419
115
53795
24
6764
+ 5132
14854
1372
45807
5918
120
63455
28
+ 6499
10328
13298
1418
42593
6416
124
77592
Feb. 1
24053
15388
11624
1447
39302
6911
128
95703
5
41413
20250
9862
1462
35930
7403
132
113264
9
50646
24855
8052
1466
32473
7891
136
1*22315
13
47144
29143
6249
1459
28938
8375
140
118250
17
34506
33058
4513
1444
25330
8854
144
104673
21
19663
36547
2916
1422
21650
9327
147
88534
25
+ 6965
39566
1520
1394
17896
9795
150
74198
29
2462
42071
+ 373
1358
14072
10256
153
62799
Mar. 4
9054
44027
506
1317
10180
10710
157
53883
8
13600
45405
1156
1271
6220
HI 56
160
46594
12
16773
46185
1643
1220
+ 2193
11595
163
40174
16
19047
46354
2078
1165
1904
12025
166
34019
20
20714
45911
2582
1106
6074
12446
169
27712
24
21939
44862
3215
1043
10307
12859
171
21046
28
22788
43225
4039
977
14604
13263
173
13907
Apr. 1
23220
+ 41025
5116
908
18965
+ 13657
-175
+ 6298
-'- "VI. LJi'S + ;U& + *\-''ilIj ~"~ i'vU ~-j Ci
-'.: J;c,-i Jk *U'-.4 tc8 -" ' -
224
On the Elements of the Orbit of Halley's Comet,
TABLE VI
Containing, for Greenwich Mean Noon of each fourth day, from August 1, 1835, to
April 1, 1836, the united Effects (C) of the attractions of the disturbing Planets
upon the Comet in the direction of the co-ordinate z, expressed in 10,000,000,000th
parts of an unit, and distinguishing the separate Effect of each Planet.
Date.
Mercury.
Venus.
Earth.
Mars.
Jupiter.
Saturn.
Georgian.
C.
1835.
Aug. 1
3800
- 57
- 98
+ 15
- 9965
+ 383
4
13526
5
3143
+ 342
89
16
6842
400
3
9319
9
- 1577
740
74
16
4037
417
2
- 4517
13
+ 877
1123
54
16
1535
435
1
+ 861
17
3484
1482
24
16
+ 679
453
6090
21
5149
1810
+ 19
16
2639
472
4 1
10106
25
5441
2104
84
17
4378
491
2
12517
29
4761
2361
181
18
5906
509
3
13739
Sep. 2
3701
2578
332
18
7231
527
4
14391
6
2629
2758
571
19
8379
545
5
14906
10
1686
2899
966
20
9377
563
6
15517
14
896
3004
1649
21
10228
581
7
16386
18
+ 242
3073
2902
22
10933
599
8
17779
22
307
3109
5394
24
11512
616
q
U
20357
26
776
3113
10918
26
11982
633
10
25906
30
1182
3092
25176
28
12341
649
11
40115
Oct. 4
1525
3051
70609
32
12585
664
12
85428
8
1783
2998
251704
37
12726
677
13
266372
12
1871
2944
691560
43
12776
689
14
706155
16
1550
2907
373844
50
12726
699
15
388691
20
113
2913
113050
59
12569
704
16
129198
24
+ 4445
3006
42511
72
12306
704
16
63060
28
13336
3266
19803
89
11938
699
16
49147
Nov. 1
12114
3847
10738
113
11460
6s6
16
38974
5
5105
5084
6427
147
10869
663
16
28311
9
3783
7806
4127
192
10159
629
17
26713
13
5027
14496
2785
255
9329
579
16
32487
17
6143
34615
1945
342
8392
514
15
51966
21
6192
119356
1387
457
7364
431
13
135200
25
5417
518654
998
595
6264
333
11
532272
29
4348
285750
711
721
5111
221
Q
/
296871
Dec. 3
3333
48633
489
763
3920
+ 97
i
57242
7
2471
10795
311
656
2706
38
4
16905
11
1591
+ 2015
158
426
1477
180
1
+ 6489
15
+ 254
84
+ 22
182
+ 240
328
+ 1
- 470
19
114C
2018
105
+ 4
1007
481
1
4748
at its appearance in the Years 1835 fy 1836.
225
TABLE VI. continued.
Date.
Mercury.
Venus.
Earth.
Mars.
Jupiter.
Saturn.
Georgian.
c.
1835.
Dec. 19
1140
2018
105
+ *
1007
481
1
4748
23
1768
2594
229
101
2270
639
3
7604
27
2057
2913
354
154
3543
800
5
9826
31
2341
3100
485
176
4824
963
7
11896
1836.
Jan. 4
2676
3208
626
183
6115
1127
9
13944
8
3051
326l
781
180
7422
1293
11
15999
12
3442
3270
955
175
8744
1460
12
18058
16
3803
3238
1153
168
10083
1628
13
20086
20
4039
3178
1381
162
11439
1796
15
22010
24
3966
3087
1645
156
12816
1964
16
23650
28
3295
2957
1951
150
14215
2132
18
24718
Feb. 1
1720
2804
2305
144
15637
2300
19
24929
5
+ 725
2625
2715
139
17083
2467
21
24325
9
3287
2423
3182
135
18556
2634
22
23665
13
4868
2201
3707
132
20057
2800
23
24052
17
5056
1961
4280
129
21588
2965
24
25891
21
4279
1709
4882
126
23151
3129
26
28744
25
3146
1448
5483
123
24747
3292
27
31974
29
2040
1183
6034
121
26377
3453
28
35156
Mar. 4
1105
918
6488
118
28043
3612
29
38103
8
+ 351
657
6792
116
29747
3770
30
40761
12
260
406
690/
114
31490
3926
31
43134
16
775
- 169
6825
112
33275
4079
32
45267
20
1233
+ 48
6558
110
35104
4230
33
47220
24
1661
242
6144
108
36977
4379
34
49061
28
2080
410
5634
106
38896
4525
34
50865
Apr. 1
2500
+ 549
5076
105
40863
4669
K i 35
52699
"T '
226
On the Elements of the Orbit of Halley's Comet,
TABLE VII.
Containing, for Greenwich Mean Noon of each fourth day, from August 1, 1835, to
April 1, 1836, the Values of A', B', C', expressed in 10,000,000,000th parts of
an unit.
Date.
A'
B'
C'
Date.
A'
B'
C'
1835.
1835.
Aug. 1
400192
+ 379032
+ 141231
Dec. 7
+ 11485
+ 101222
f 47929
5
301357
358155
128032
11
17539
69220
27928
9
211518
329654
112257
15
24926
59426
18386
13
141858
292643
93706
17
102748
252128
74645
19
24389
59864
14810
23
17936
58438
11451
21
90639
217317
58853
27
13211
55640
7974
25
89146
193228
48127
31
10821
53820
4916
29
84787
177804
41372
1836.
Sep. 2
73714
166523
36611
Jan. 4
9730
53215
2265
6
57437
156071
32437
10
38417
144762
28140
8
9118
53611
+ 103
14
- 18463
131718
23336
12
8320
55031
2240
18
+ 1635
116138
17587
16
6727
57842
4084
22
22026
96485
+ 9889
20
3923
62825
5414
26
43585
69390
2335
24
+ 317
71329
5668
30
67851
+ 26861
26810
28
1407
84977
+ 3844
Oct. 4
94611
50570
91816
Feb. 1
+ 5544
104188
1213
8
+ 91421
182278
315760
5
31087
125261
9655
12
125628
1588
766170
9
79727
139831
19132
16
128719
+ 356292
349965
13
141938
141288
25805
20
+ 544
226084
84652
17
201580
131390
27870
24
52187
119597
25473
21
250151
116780
26453
28
75176
54565
13733
25
288374
102408
23510
Nov. 1
72601
+ 8246
7670
29
320097
90076
20348
5
52487
44330
6502
Mar. 4
348516
79732
17511
9
+ 26453
87584
14205
8
375450
70725
15107
13
557
112055
27186
12
401858
62406
13092
17
23152
137530
48286
16
427372
54192
11245
21
61543
229148
125518
20
452446
45735
9491
25
233150
262744
578345
24
476649
36849
7676
29
- 90035
+ 329951
424816
28
499672
27466
5678
Dec. 3
+ 1952
181003
113288
Apr. 1
+ 520954
+ 17609
3389
7
+ 11485
+ 101222
- 47929
-
at its appearance in the Years 1835 $ 1836.
227
TABLE VIII.
Containing the Variations of the Elements of the Comet's Orbit for each interval
of four days, between the Noon of July 30, 1835, and the Noon of April 3, 1836,
Greenwich Mean Time; the tabular date being the middle of each interval.
* # * The figure in a parenthesis, at the head of a column, indicates the number of
cyphers to be prefixed to all the values in that column.
Date.
M
M
M
M
M
M
1835.
0'(2)
0'(5)
//
//
//
a
Aug. 1
16336
26591
+o -11876
o -37360
1 -60324
0-91704
5
14500
23503
'08438
-32982
1 -47658
o -86797
9
12507
20180
'05353
-28125
1 -32495
-80042
13
10549
16974
o -02757
o -22796
1 '15008
o -70786
17
08993
14511
+ '00832
o -17603
o -97468
o -59770
21
08075
13152
-00420
'13429
0-83170
o -49304
25
07645
12589
-01224
o -10604
o -73334
'41255
29
07377
12253
-01807
-08781
o -66679
o -35717
Sep. 2
07047
11770
o -02266
o -07467
o -61397
-31813
6
06578
11022
-02595
o -06337
-56339
-28726
10
05958
09999
-02758
-05248
-51025
o -26016
14
05188
08707
o -02704
-04139
_0 '45250
o -23475
18
04243
07101
'02348
o -02952
-38839
o -21037
22
03028
05018
o -01492
-01563
0-31387
-18683
26
01341
02101
+o -00392
+ -00345
o -21975
-16348
30
+ 01292
+ 02486
0"'04942
o -03677
-08391
'13809
Oct. 4
05904
10548
0-18382
0-11572
+ -14815
o -09948
8
+ 12478
+ 22003
o -679/8
-36132
-53807
+ 0-01473
12
05187
09472
1 -75710
'78360
+ 0*13846
'35552
16
25208
44683
o -34717
0-31333
-82022
o -23627
20
13279
23452
'21425
-06448
'54292
+ -01356
24
05189
09083
o -06673
-01583
'32026
'10245
28
00389
00545
-03681
-00652
-21084
o -19390
Nov. 1
+ 02334
+ 04282
o -02076
-00247
o -13961
-23658
5
05154
09287
o -01749
o -00107
o -05269
o -26766
9
07611
13666 .
o -03728
+ "00005
+ 0-01254
-30519
13
09168
16452
-06805
-00426
-04645
-33634
17
11410
20483
'11223
-01516
-04872
o -42751
21
19649
35284
-26254
-05820
-02358
079611
25
+ 27384
+ 49289
1 -04830
-34892
o -29122
0-85178
29
21037
37484
+o '63619
'31053
+o -69732
+ 1-57194
228
On the Elements of the Orbit of Halleys Comet,
TABLE VIII. continued.
Date.
[]
M
M
M
W
[]
1835.
0'(2)
"(5)
//
//
//
ii
Nov. 29
21037
37484
+o -63619
-31053
+o -69732
+ 1 ^7194
Dec. 3
12683
22616
'13105
-09588
-33595
-86342
7
07103
12645
'03832
'04551
-20249
'49779
11
04997
08886
o -01209
o -02910
o -14724
o -35496
15
04475
07954
+ '00114
-02068
-13443
-32039
19
04221
07474
-00458
o -01775
o -15017
-32548
23
03644
06411
o -00778
-01448
o -16087
o -31196
27
03121
05455
'00833
'01056
0-16190
o -29140
31
02779
04823
o -00692
o -00678
0-16363
o -27892
1836.
Jan. 4
02567
04425
-00399
-00323
o -16760
o -27406
8
02430
04162
+ -00022
+ -00015
0-17396
o -27479
12
02341
03976
-00551
-00339
-18317
'28028
16
02286
03845
-01144
o -00633
o -19665
0-29149
20
02277
03784
o -01697
-00858
o -21716
-31165
24
02366
03877
o -01963
o -00916
o -24971
o -34794
28
02661
04309
+ 0-01456
+ '00633
0-30101
-41156
Feb. 1
03318
05359
-00498
-00203
o -37505
o -51292
5
04418
07193
-04268
0-01639
o -46219
-64562
9
05766
09518
o -09049
o -03292
-53426
o -77337
13
06940
11615
o -12991
o -04496
-56422
-85163
17
07661
12974
-14868
0-04911
-55240
o -86922
21
07978
13636
-14895
o -04711
o -51949
-84892
25
08070
13894
"13924
-04227
-48343
-81621
29
08077
13986
-12638
o -03691
"45205
'78468
Mar. 4
08065
14032
o -11374
-03203
o -42626
o 75796
8
08053
14070
-10236
-02784
'40421
o -73495
12
08044
14113
o -09234
-02430
'38386
o -71378
16
08018
14129
-08239
-02101
-36283
o -69122
20
07978
14123
o -07211
'01784
'34001
o -66621
24
07914
14083
o -06037
-01451
'31449
o -63751
28
07822
14001
o -04616
o -01078
-28596
-60464
Apr. 1
07698
13869
'02844
o -00647
+ -25435
+o -56728
at its appearance in the Years 1835 fy 1836.
229
TABLE IX.
Containing the total amount of Variation of each of the Elements of the Comet's
Orbit, on every fourth day, commencing at the Noon of July 30, 1835, and ending
at Noon of April 3, 1836, Greenwich Mean Time.
*** The figures in the parentheses indicate the number of cyphers between the
decimal point and the first significant figure.
Date.
$a
Se
Zv
ft
TJ
Se
1835.
ti
//
n
ii
July 30
'0000000
o -oooooooo
'00
'00
'00
o -oo
Aug. 3
'(2) 16329
'(5) 266
+ 0'13
0-37
1 -60
o -92
7
30822
501
'23
0-70
3 '08
i 79
11
43330
703
0'30
0-98
4 -40
2 -58
15
53896
'(5) 873
'34
1 '20
5 '56
3 -29
19
62915
'(4)1018
0'36
1-37
6 '53
3-89
23
71010
1150
o -37
1 '51
7-37
4-39
27
78662
1276
o -37
1-61
8 '10
4 -80
31
86036
1398
'36
170
877
5 -16
Sep. 4
93077
1516
0'35
1*77
9*38
5 -48
8
'(2) 99649
1626
0'34
1 '83
9'95
577
12
'(1)105601
1726
'32
1 '88
10 -46
6-03
16
110782
1813
'31
1-92
10 -91
6-26
20
115014
1884
'30
1-94
11 -30
6-47
24
118022
1934
"30
1'95
n -6l
6-66
28
119324
1954
'32
1 '95
11 '83
6 -82
Oct. 2
H7949
1928
'38
1-91
11 '91
6-96
6
111963
1820
o -59
178
11 76
7-06
10
100495
1619
1 -31
1 '41
11 '25
7-04
14
105780
1715
3 '00
0-66
11 '14
670
18
129657
2138
3'87
'33
11 '91
6-47
22
143096
2376
4 -12
"26
12-46
6 -45
26
148422
2469
4 '20
"24
1278
6-56
30
148898
2476
4 '25
'23
12-99
675
Nov. 3
146560
2433
4'29
'22
13'13
6-98
7
141421
2340
4'32
'22
13 -19
7-25
11
133848
2204
4-37
'22
13 -18
7-56
15
124652
2039
4 '45
'22
13 -13
7*90
19
o -(1)112992
1830
4 '58
'23
13-09
8 -34
23
'(2) 93364
1477
4 '88
'30
13'05
9'12
27
0'(2) 68320
'(4)1027
+ 5'39
0'63
1275
-9'87
230
On the Elements of the Orbit of Halleys Comet,
TABLE IX continued.
Date.
Sa
I 1 }
h
SCT
*e
1835.
Nov. 27
'(2) 68320
'(4)1027
+ 5 // '89
o"-63
12 75
II
-9-87
Dec. 1
86991
1359
6 '54
o -93
12 -09
8 '43
5
0'(2) 99790
1587
670
1 '03
11 75
7-55
9
'(1)107038
1716
6-75
1-07
11 '54
7 -05
13
112101
1806
678
1 '10
11 '39
6-69
17
116587
1886
679
1 -12
11 '26
6-37
21
120795
1961
6 '80
1 '13
11 '11
6-04
25
124441
2025
6'80
1 '15
10-95
573
29
127570
2079
6'81
1 '15
1079
5 '44
1836.
Jan. 2
130354
2128
6'81
1-16
10 -62
5 '16
6
132924
2172
6'82
1-16
10 '46
4'89
10
135356
2214
6-84
1-16
10 '28
4 '6l
14
137698
2253
6'85
1 '15
10-10
4 '33
18
139986
2292
6-88
1 -14
9*91
4 -04
22
142267
2330
6'91
1 -13
9-69
3 73
26
144642
2369
6'94
1 '12
9 '44
3 '38
30
147318
2412
6-97
1 -11
9'14
2-97
Feb. 3
150654
2466
6 -98
1 -11
8 76
2-46
7
155082
2538
6-94
1 -12
8 '30
1 -81
11
160841
2633
6-87
1 -15
777
1 '04
15
167762
2749
675
1-19
7-21
0'19
19
175406
2879
6 -61
1 -24
6-66
+o -67
23
183375
3015
6-48
1 '28
6'14
1 -52
27
191441
3154
6'35
1 '32
5 '65
2 '34
Mar. 2
199517
3293
6 '24
1 '36
5'20
3-12
6
207582
3434
6-14
1 -38
478
3'88
10
215635
3574
6-05
1 -41
4-37
4 '61
14
223678
3716
5 -97
1 -43
3'99
5-33
18
231695
3857
5'90
1 -45
3 '63
6'02
22
239672
3998
5 '84
1 '46
3-29
6'68
26
247585
4139
579
1 '48
2*98
7-32
30
255406
4279
5'76
1 '48
2-69
7-93
Apr. 3
'(1)263103
'(4)4418
+ 574
-1-49
2-44
+ 8'49
at its appearance in the Years 1835 fy 1836.
231
TABLE X.
Containing, The Apparent Right Ascension and Declination, and the Logarithm of
the True Distance from the Earth, of HALLEY'S Comet, from August 1-5, 1835,
to March 31*5, 1836, Mean Time at Greenwich, deduced from approximate Ele-
ments of its orbit, on the supposition that those Elements continued invariable
during the interval : and the Perturbations in Right Ascension and Declination
produced by the disturbing Planets, on the assumption that the approximate
Elements represent the actual orbit in which the Comet was moving at Mean Noon
at Greenwich on July 30, 1835.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1835.
Aug. 1 '5
2 '5
3'5
'
1 //
81 21 56'0
81 33 55 '2
81 45 55 '5
n
*2
0-3
0'3
o / //
+ 21 55 40 '8
21 59 49-1
22 4 1 -0
//
o-o
o-o
o-o
'40743
-40252
o -39750
4'5 81 57 56 '7
5'5 82 9 58 '8
6 '5 82 22 2 *1
0-3
0'3
0'4
22 8 16 '8
22 12 36 '7
22 17 1 -0
0*0
o-o
o-o
o -39239
0-38717
0-38186
7'5
8'5
9'5
82 34 6 *3
82 46 1 1 '8
82 58 18'4
0-4
0-2
O'O
22 21 29 '9
22 26 3 7
22 30 42'6
o-o
o -o
O'O
o -37643
o -37090
-36525
10 '5
11 '5
12'5
83 10 26-3
83 22 35 "7
83 34 46 '4
+ 0-1
0*2
0-2
22 35 27-1
22 40 17 '3
22 45 13 '8
o -o
o-o
o -o
o -35949
-35361
o -34760
13-5
14'5
15 '5
83 46 58 '7
83 59 12 '7
84 1 1 28 '6
+ 0-3
0*3
0'4
22 50 l6'5
22 55 26'3
23 43 '3
+ 0-1
0*1
O'l
0-34147
-33521
-32881
16-5
17-5
18-5
84 23 46-3
84 36 6'0
84 48 277
+ 0-4
0-5
0-6
23 6 8 'I
23 11 41 '3
23 17 23-0
+ 0'1
0*2
0*2
-32228
0-31560
-30878
19'5
20-5
21 '5
85 51 '8
85 13 18 '4
85 25 47 '6
+ 0-6
0-6
0-6
23 23 14 -0
23 29 15 -0
23 35 26 '4
0-3
0-2
+ 0-1
0*30180
o -29467
o -23737
22 '5 85 38 19 '4
23 '5 85 50 54 '7
24 -5 36 3 33 '4
+ 0-6
0-7
0-7
23 41 48 '9
23 48 23 '6
23 55 10*5
+ 0-0
o-i
o-i
o -27991
o -27226
-26444
25 '5 86 16 15 7
26 '5 86 29 2 '3
27 '5 86 41 53 '4
+ 0-7
0-7
0-8
24 211-2
24 9 26 '2
24 16 56 -8
o-o
+ 0-1
o-i
! ~*
-25643
-24823
-23982
28-5
29*5
30 '5
86 54 49 '8
87 7 51 '9
87 21 0'4
+ 0-8
0-9
0-9
24 24 43 '6
24 32 48 -1
24 41 11 '5
o-i
0*2
0-3
0-23120
o -22237
0-21330
31*5
87 34 l6'l
+ I'D
+ 24 49 55 -3
+ 0'3
'20400
On the
of lh<> O-rltit of //"//V//.V Cornel,
TABLK X. continued.
Da*.
Apparent
Right A0cunvion
IVrfm-
I'ilhOIIM.
A|.|,;.r-nf
Declination.
Pertwr-
bationi.
I.M^. -f TII... |)nt.
In, in Ih-- K.iilh.
1835.
/ //
jj
/ n
Aug.31 '5
87 :',! 16-1
+ '0
+ 24 49 55'3
+ 0-3
--0100
Sep. 1 '5
87 47 397
o
24 59 0'9
o -:i
'19445
2*5
88 1 12-1
1
25 8 29*9
0'3
o -1816:*
3 '5
88 14 54 '8
+ '1
25 18 24 '3
+ 0'3
'17455
4'5
88 28 48 7
'2
25 28 46 '2
0'3
-16418
88 42 54 '9
2
25 39 37 '0
'2
0-15351
6 :,
88 57 15*4
+ '2
25 51 O'l
+ O'l
'14253
7 '5
89 11 51*6
2
26 2 .17 '9
o -o
0-13121
8 '5
89 26 45 '6
3
26 15 33 '4
O'l
o -i 195:.
9 '5
89 41 59'3
+ '3
26 28 49 '6
O'O
'10752
10 T,
89 57 35 '4
4
26 42 50 '8
+ '1
o -09509
11 '5
90 13 36'7
5
26 57 41 '0
-2
'08226
1 2 Ti
90 30 6 '5
+ '6
27 13 24 '6
+ 0'4
o '06900
i :i :,
90 47 8 '2
7
27 30 7 '2
o -:t
o (:,., -,
14 '5
91 4 46 '6
8
27 47 55 '1
0'2
o -oi ior,
Iff 'ff
91 23 6'7
+ 1'9
28 6 54 7
+ O'l
o -026:50
16 :,
91 42 13 '9
2-0
28 27 14-3
o -o
'() I 1 00
17 '5
92 2 15 '0
2*1
28 49 2 '6
- O'l
9 -99509
18 '5
92 23 18 '9
+ 2 '3
29 12 30 '3
O'l
9 '97855
19'5
92 45 34 '8
2 '4
29 37 49 '2
0-2
9 '96m
20 '5
93 <) 14 '2
2 '6
30 5 13 '1
0'2
9-9n:ii
21 * 'i
93 34 31 '6
+ 2'8
30 34 59*0
0'2
9 '92456
Ol J
94 1 43-5
2-9
31 7 25 '5
o -:t
9 -90-192
23'5
94 31 11'8
31 42 56'7
0'4
9 '88433
24 '5
95 3 21 7
+ 8 '3
32 21 58 '2
0-4
9 '86273
25 '5
1)5 38 46 '9
:i -6
33 5 4 -0
0'6
9-84000
26'5
96 18 8 7
3 '8
33 52 53 '4
0*8
9 '8 1606
27 :,
97 2 20 '2
+ 4'1
34 46 13 '2
1 '1
1) 79078
28 '5
97 52 32-8
4 ' 1
35 46 4 '5
1 '4
9 -76403
29-5
98 50 19-0
4'4
36 53 37 '6
1'S
9 73566
: \'cur.\ 1835 M l 6
106 5 .V; I
4- 4-5
4 -8
5'1
4:t 16 33-2
43 48 15 -8
44 22 14 '6
- 3'7
3-7
3'8
9 -60269
:r .->!):* 18
9 '58352
4 '25
4 '5
7I
106 45 '7
107 26 58'9
108 11 42 '3
+ 5-4
6-7
44 57 2 -8
45 33 14 -0
46 10 51 -2
-3'9
4-1
4'3
9 '*7I70
9 -,6373
9'fj loo
5*0
:> -2.-,
5'5
108 59 39 '4
109 51 10-2
no 46 39-7
+ 7*4
7 '9
8*4
46 49 58 '1
47 30 37 -7
48 12 53 -4
4'5
4-7
4-9
9*54331
9 '53288
9 '52229
5 75
6'0
6 '125
111 46 34 '7
112 51 26 '8
113 25 54'8
4- 8*9
9'3
9 '5
48 56 47 '6
49 42 22 '2
50 5 47 '5
S'l
5'4
5*5
9'51156
9'5<<
9 '495 19
6
6 -375
6 -5
114 1 50-6
114 39 20
115 18 27*7
4- 9 '6
\n
9'8
50 29 37'9
50 53 53 '8
51 18 34 7
- 5*7
5 -8
6 '0
9'4i**7
9*494!fl
9 '47854
6 ', ' ,
6 ?l
6-875
115 59 20*1
116 42 4'2
117 26 47 o
1- '.> '7
')<>
9'5
51 43 40 '5
52 !> 10'8
52 35 4 '9
- 6'2
fr-4
6-6
9 '47293
9 -46729
9-4616:5
7'0
7*125
7*w
118 13 35'4
119 2 37 '8
119 54 2-8
9'4
9 '2
8 -9
53 1 22'4
53 28 2'3
53 55 3 '6
- 6-8
7-0
7 '3
9 '45595
9 '45024
91I44H
7 '375
7 '5
7 '625
120 48 O'O
121 44 38*8
122 44 9 '8
+ 8'5
8'1
7'6
54 22 24 '8
51 50 4'6
55 18 1 'O
-7-5
7-8
8'1
9 '438/8
'.) '4 1
9 '42728
7'7>
7-875
8-0
46 44 '3
124 52 34 -5
126 1 52'6
-H7'i
fr'J
5-9
55 46 1 1 '<>
56 14 r*'8
56 43 4 '6
8'5
*<9
'42152
9'4U7I
'J''(>
J') ',
11 -1
9 *38/08
9 '38 Ml
9 '3757*
8'875
1
136 3 47 '3
O'l
4-60 20 '8
-11 '3
9 -37019
NAUTICAL AMUHAC 1839,
234
On the Elements of the Orlit of Halley's Comet,
TABLE X.-- continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1835.
Oct. 8 '875
9*0
9'125
o / //
136 3 47 '3
137 49 29 "2
139 40 51'9
//
o-i
1 '2
2'4
1 II
+ 60 20 '8
60 26 57 -i
60 52 45 'I
ii
11 '3
11 -5
11-7
9-37019
9 '36465
9-35916
9*25
9'375
9'5
141 38 8'3
143 41 29-6
145 51 4-5
- 3-7
5 '2
6-8
61 17 34-1
6l 41 12'6
62 3 27-9
11 '8
11 '9
12 -0
9 -35375
9'34841
9*34316
9'625
9-75
9-875
148 6 58 '2
150 29 12-1
152 57 41-7
- 8-7
10-9
13'2
62 24 6 '6
62 42 54 '3
62 59 36-0
12-0
ll'9
11 '8
9 '33800
9 '33295
9*32801
10 '0
10-125
10'25
155 32 17 '6
158 12 42'3
160 58 32 *1
15-5
17-8
19-9
63 13 55 '9
63 25 38 '5
63 34 27 '9
-11-7
11 '5
11-2
9 '32320
9*31853
9-31401
10-375
10-5
10 '625
163 49 14 '9
166 44 11 '9
169 42 36 -5
21-8
23 '5
24'9
63 40 9*0
63 42 26 '9
63 41 8 -4
1 1 -0
10-7
10-2
9 -30964
9 '30545
9 -30144
10-75
10 '875
11 -0
172 43 34 '9
175 46 9-8
178 49 19*3
26-1
27 '2
28-1
63 36 1 '6
63 26 57 *3
63 13 48 -0
-9'7
9'2
8-6
9 -29764
9 -29404
9 -29066
11-125
11 -25
11 -375
181 52 2 -0
184 53 16'5
187 52 4 '2
-28 '9
29-5
29'9
62 56 29 -0
62 34 58 '9
62 9 18 '8
8-1
7-6
7'2
9-28751
9 -28460
9 -28196
11 '5
11 '625
11 75
190 47 30 '9
193 38 49 'I
196 25 18*4
30-2
30 '2
30'0
6l 39 32 '4
6l 5 46-0
60 28 8 "4
-6-8
6-5
6'2
9 -27957
9 -27746
9 -27562
11 '875
12 '0
12-125
199 6 26 -3
201 41 47-1
204 11 3 '3
-297
29-3
28'8
59 46 50 '7
59 2 5-4
58 14 6'3
-6-0
5-7
5'5
9 -27407
9 -27283
9 ,-2/188
12-25
12 -375
12-5
206 34 4 -0
208 50 44'7
211 1 5-8
28 -3
277
27-1
57 23 8 '4
56 29 27 '3
55 33 19 'I
5-3
5'1
5'0
9-27124
9 -27090
9 -27087
12 '625
12-75
12 -875
213 5 12-0
215 3 10 '7
216 55 13 '2
-26-6
26'2
25 '8
54 34 59 '9
53 34 45 '8
52 32 53 -0
4-8
4'7
4-6
9-27116
9-27175
9 -27264
13'0
13*125
13 -25
218 41 32 '0
220 22 21*3
221 57 56-6
25'4
24 '9
24'5
51 29 36*9
50 25 12-1
49 19 53 -0
4'5
4'4
4 '3
9 '27384
9 -27533
9-27712
13-375
223 28 33 '2
24 -0
+ 48 13 52 '9
4'2
9-27919
at its appearance in the Years 1835 < 1836.
235
TABLE X. continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1835.
Oct.13'375
13 '5
13 '625
f //
223 28 33 '2
224 54 26 '9
226 15 52 -2
//
24-0
23-6
23 -1
1 //
+ 48 13 52'9
47 7 24 -8
46 40 '4
/;
4-2
4 -1
4-0
9*27919
9 -28154
9-28415
13 75
13 '875
14'0
227 33 5 -1
228 46 19-8
229 55 50 '7
22 *7
22 '2
21 -8
44 53 50 '9
43 47 6 '3
42 40 35 '8
-3-9
3-8
3-7
9 -28702
9 -29014
9 '29349
14 '125
14 '25
14-375
231 1 50 '6
232 4 33 -0
233 4 9 '5
21 -4
21 -0
20-6
41 34 27-4
40 28 49 "2
39 23 48 -2
3 '5
3 -3
3-1
9 '29706
9 '30085
9 '30484
14-5
14 '625
1475
234 051-6
234 54 49 "6
235 46 13-5
20 '2
20 -0
19 '8
38 19 30 -2
37 16 o-i
36 13 23 -0
-2-9
2-6
2-2
9 '30902
9 -31337
9*31789
14 -875
15 -0
15-125
236 35 12 -6
237 21 56'0
238 6 31 -1
19 '6
19'4
19 '2
35 11 42'9
34 11 3-3
33 11 26-5
1 -8
1 -3
0-9
9 '32257
9 '32738
9 '33232
15 -25
15 -375
15 -5
238 49 6 -0
239 29 48 -0
240 8 43 -4
-19*0
18 -8
187
32 12 55-3
31 15 31 -5
30 19 16-5
0-4
+ 0-1
0-6
9 '33738
9 '34255
9 '34782
15 '625
15 -75
15 -875
240 45 58 -1
241 21 38 -2
241 55 49 'I
18-5
18-3
18 -1
29 24 10 '7
28 30 15-4
27 37 31 -1
-f- i-o
1 '4
1'9
9 '35318
9-35861
9-36411
16-0
16-25
16-5
242 28 35 '2
243 30 12 '2
244 27 4 -3
-17-8
17-3
16-7
26 45 57 -8
25 6 22 -0
23 31 24-8
+ 2-3
3-0
37
9 '36967
9 '38095
9 '39239
1675
17*0
17-25
245 19 39 -3
246 8 24 '0
246 53 40 -1
16-2
15 -6
15-1
22 59-3
20 34 57 -2
19 13 7 '9
+ 4'2
4-6
4-9
9 -40392
9*41551
9 '42712
17-5
1775
18 -0
247 35 48 '7
248 15 6 -0
248 51 48 -0
14-5
14 -0
13 -4
17 55 21 -4
16 41 25 -1
15 31 8 7
+ 5-2
5-3
5'4
9*43871
9 -45026
9 -46174
18-25
18 '5
18-75
249 26 7 -8
249 58 18 7
250 28 30 -2
12-9
12 -4
12 -0
14 24 20-1
13 20 48 '6
12 20 22 '9
+ 5-4
5-3
5 'I
9-47313
9 '48442
9 '49559
19*0
19 "25
19-5
250 56 53 -1
251 23 34-9
251 48 44 '8
-11-6
11 -3
11 -0
11 22 52-5
10 28 7'3
9 35 57 -6
+ 4-9
4-6
4-3
9 -50663
9-51753
9 '52828
1975
252 12 28 -5
-10-7
+ 8 46 14'2
+ 4'0
9 '53889
-236
the Elements of the Orbit of Halley's Comet,
TABLE X. continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1835.
Oct. 19 '75
20 -0
20'25
o / //
252 12 28 '5
252 34 53 '3
252 56 4-1
n
-107
10 -4
10-1
/ //
+ 8 46 14-2
7 58 48 '5
7 13 32 -1
n
+ 4-0
37
3'5
9*53889
9 -54935
9 -55964
20 '5
20-75
21 -0
253 16 6-3
253 35 4 7
253 53 3 '9
- 9'9
97
9'5
6 30 17 -8
5 48 57-6
5 9 25 '6
+ 3'2
3-0
27
9 "56979
9 '57977
9 -58960
21 '25
21 '5
21 75
254 10 6'9
254 26 18 -4
254 41 40 -6
-9'3
9-1
8-9
4 31 34'9
3 55 20 '3
3 20 35 '8
+ 2'5
2 -3
2-1
9 '59928
9 -60879
9*61816
22-0
22 -25
22-5
254 56 17-2
255 10 10 -0
255 23 22-9
8-8
87
8-6
2 47 167
2 15 17 -8
1 44 34 '9
+ '8
6
5
9 '62/37
9 -63643
9 '64534
2275
23 -0
23 -25
255 35 56-6
255 47 54 -4
255 59 17 -4
8-5
8-4
8-3
1 15 3'3
46 39 '5
-r- 19 19 '6
+ '3
1
o
9-65411
9 -66274
9-67122
23-5
23*75
24-0
256 10 8 '6
256 20 28 -6
256 30 197
8-2
8-1
8'1
6 59-5
32 21 -2
56 48 '6
+ 0-9
0-8
07
9 '67957
9 -68778
9-69587
24-25
24-5
24 75
256 39 42 -5
256 48 39 -4
256 57 10 -0
8-0
7'9
7-8
1 20 24 '5
1 43 11'4
2 512*3
+ 0-6
0-5
'4
9'70382
971165
971935
25-0
25 -25
25'5
257 5 17-1
257 13 -5
257 20 22 '6
~ 77
7-6
7-5
2 26 29 -0
2 47 3'9
3 6 59 -0
+ 0-3
0-3
0-2
9 -72694
973441
974176
25 -75
26-0
26-25
257 27 23 -2
257 34 4 '2
257 40 25 -1
-7'4
7-4
7-3
3 26 15 '9
3 44 567
433'1
+ 0-1
'1
o-o
9 -74900
9-75613
976315
26-5
2675
27-0
257 46 28 '2
257 52 12 7
257 57 40 *5
- 7-2
7-1
7-0
4 20 36 -6
4 37 39 -3
4 54 11-9
o-o
O'O
o-o
9 77007
9 77688
9 78360
27-25
27-5
2775
258 2 51 -1
258 7 46 -6
258 12 25 '9
- 7-0
6-9
6-8
5 10 16-0
5 25 52 '9
5 41 37
o
o-
o
979021
9 -79673
9 -80316
28-0
28-25
28 '5
258 16 51 -0
258 21 '9
258 24 58-4
- 67
6-6
6-6
5 55 49 '5
6 10 11 '7
6 24 ll'O
o-
o
0-2
9 '80949
9 -81 .-,73
9 -82189
29-5
258 38 35 7
-6-3
7 16 38 '9
0-2
9 -84567
at its appearance in the Years 1835 4' 1836.
237
TABLE X. continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1835.
Oct. 29 -5
30-5
31 '5
o / //
258 38 35 7
258 49 3 '3
258 56 42 '3
//
-6-3
6-1
5'9
/ II
7 16 38 '9
8 4 10-8
8 47 30 7
//
0-2
0*3
0'3
9 '84567
9-86818
9 '88953
Nov. 1 '5
2-5
3 '5
259 1 49'3
259 4 38'4
259 5 21 '5
- 57
5 '5
5 '4
9 27 15 -o
10 3 53'6
10 37 51 '2
0-4
0'4
0'5
9 -90979
9 -92905
9 '94737
4 '5
5 '5
6-5
259 48'8
259 1 9 '3
258 56 31 '2
5-4
5'4
5'4
11 9 28'9
11 39 4'4
12 6 52'4
0*5
0'6
07
9 4 96482
9*98144
9 '99729
7'5
8 '5
9 '5
258 50 21 -2
258 42 46 '2
258 33 52 '0
5 *5
57
5 '8
12 33 5 '8
12 57 55 '4
13 21 30 '5
-8
0-9
i -o
'01240
-02681
'04056
10 '5
11 '5
12-5
258 23 44 "J
258 12 30 -0
258 12 *9
-6-0
6-0
5 '8
13 43 59'2
14 5 28 *2
14 26 4-1
1 -1
1 '2
1 'I
0-05367
o -06617
o -07309
13 '5
14-5
15'5
257 46 59 '2
257 32 53 *8
257 18 2'2
5'5
5'1
5 *1
14 45 51 '6
15 4 55 '4
15 23 19 '3
1 'I
1 -0
1 -0
'08945
o -10027
o -11057
16'5
17-5
18 '5
19*5
20 '5
21 '5
257 2 29*4
256 46 20 '4
256 29 39 '8
256 12 32'3
255 55 2 '8
255 37 15 '2
5*4
5 '8
6'3
- 67
6-8
6-8
15 41 7'0
15 58 21 '5
16 15 5 '2
16 31 207
16 47 9*9
17 2 34'9
1 'I
I '2
1 -3
1 '4
1 -4
1 '4
-12037
o -12968
0*13853
o -14692
'15489
o -16243
22 '5
23'5
24 '5
255 19 13'5
255 1 1 '4
254 42 42 '5
-6-8
6-9
7-2
17 17 37'2
17 32 18'4
17 46 397
1 '5
1 '5
1-6
0-16957
o -17633
0'18271
25'5
26-5
27-5
254 24 19'8
254 5 56-4
253 47 34 '3
-7'6
7*9
8*2
18 42'5
18 14 27 7
18 27 56-6
- l'7
1 '8
1*9
-188/3
0-19441
-19975
28 '5
29 '5
30-5
253 29 16 '6
253 11 4'6
252 52 59 '9
8'4
8 '5
8'5
18 41 lO'O
18 54 9'1
19 6 54 '6
- 1'9
2 -0
2'0
-20478
0-20951
0-21394
Dec. 1 '5
2'5
3 '5
252 35 4'1
252 17 18 *5
251 59 43 7
8-5
8 '5
8*5
19 19 27*2
19 31 47-9
19 43 57 -2
2-0
2-0
2-1
0-21809
'22198
0-22561
4'5
251 42 20 '6
-8-6
-19 55 56-1
2-1
'22899
238
On the Elements of the Orbit of Hollers Comet,
TABLE X. continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1835.
o ; //
ii
o / //
//
Dec. 4 '5
251 42 20 '6
-8-6
19 55 56*1
2'1
'22899
5'5
251 25 8 *5
8-6
20 7 45 '2
2 'I
'23213
6'5
251 8 9 '5
87
20 19 24 '8
2'1
-23505
7-5
250 51 22'4
-8-9
20 30 55 '9
2-0
o -23775
8'5
250 34 47*1
9-2
20 42 19-1
2*0
'24023
9*5
250 18 23 7
9'5
20 53 34 7
2'0
-24252
10'5
250 2 11-8
-97
21 4 43 '4
2-1
-24461
11 '5
249 46 10 '5
97
21 15 45 '8
2 '3
-24652
12-5
249 30 19 -5
9'6
21 26 42 *2
2'4
'24824
13'5
249 14 37-6
- 9'6
21 37 33 -0
-2-6
-24979
14'5
248 59 5 -0
9'6
21 48 18 '9
27
0-25116
15'5
248 43 40 -2
9'6
21 59 O'l
2'8
-25238
l6'5
248 28 22 '4
-97
22 9 37-2
-2-9
-25344
17-5
248 13 9*9
9 '9
22 20 10 -6
2'9
-25434
18 '5
247 58 3 -0
10-2
22 30 40 '6
2'9
-25510
19-5
247 42 59 '5
-10*5
22 41 7 '5
3-0
o -25571
20 -5
247 27 58 '8
10'8
22 51 3i 7
3 -1
-25618
21 '5
247 12 59 "8
11 -0
23 1 53 '6
3 *2
'25651
22-5
246 58 0'8
11 -1
23 12 13'2
3'2
o -25672
23'5
246 43 -3
11'2
23 22 31 '5
3 '3
'25679
24'5
246 27 58 -1
11 '3
23 32 48 '1
3'4
o -25674
25*5
246 12 52'2
11 '4
23 43 3 '5
3'5
-25656
26*5
245 57 41 -8
11-6
23 53 17 '9
37
o -25627
27-5
245 42 24 "9
11 -8
24 331'8
37
-25586
28 '5
245 27 1 '4
12-0
24 13 45 '0
3'8
-25534
29*5
245 11 28-8
12 -2
24 23 58 'I
3-9
o -25470
30 '5
244 55 46 '8
12 '3
24 34 1 1 "2
3-9
o -25396
31 '5
244 39 53 '4
12-4
24 44 24 '5
4'0
0-25311
1836.
Jan. 1 '5
244 23 47 '3
12-5
24 54 38 *2
4-1
-25215
2'5
244 7 27 -0
12-6
25 4 52 '4
4'2
0-25109
3'5
243 50 51 *3
12 '8
25 15 7'6
4'2
o -24994
4'5
243 33 59 '2
13 -0
25 25 23 *3
4'3
-24868
5'5
243 16 47 '6
13'2
25 35 40 -2
4 '4
o -24733
6-5
242 59 17-4
13-5
25 45 58 '2
4'5
-24589
7-5
242 41 24-9
13'8
25 56 17-5
4 '5
0-24435
8 '5
242 23 9 7
14*1
26 6 38 '2
4-6
o -24272
9'5
242 4 29 -1
14 *3
26 17 0*2
- 47
0-24100
at its appearance in the Years 1835 $ 1836,
239
TABLE X. continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1836.
Jan. 9 '5
10 '5
11 '5
o / //
242 4 29 -1
241 45 22 -6
241 25 48'0
//
14-3
14 '3
14*3
/ //
26 17 -2
26 27 23 '6
26 37 48 '5
//
-47
4 '8
4'8
0-24100
0'23919
o -23730
12*5
13'5
14-5
241 5 43 '8
240 45 7 '5
240 23 57 -2
14-4
14*4
14-5
26 48 14 '8
26 58 42 '6
27 9 117
~4'9
5-0
5*1
-23532
-23326
0-23112
15 '5
l6'5
17-5
240 211 '8
239 39 48 '7
239 16 45 7
14-5
14-6
147
27 19 42'1
27 30 13-8
27 40 46 '5
5-2
5'3
5*4
-22890
o -22661
-22424
18 '5
19*5
20-5
238 53 1 'I
238 28 33 '1
238 3 19*4
14-8
15 -1
15 '4
27 51 20 -0
28 1 54 '1
28 12 28 '6
5-5
5-6
57
o -22179
o -21927
o -21669
21 -5
22'5
23-5
237 37 17'0
237 10 24 'I
236 42 39 7
-157
16'1
16 '5
28 23 3-1
28 33 37'3
28 44 10 *8
5'8
5*9
5'9
-21404
0-21132
-20854
24'5
25-5
26 '5
236 14 '3
235 44 23 '3
235 13 477
16'9
17'3
17-8
28 54 43 'I
29 5 13'5
29 15 42-1
-6-0
6-1
6-2
o -20570
-20281
-19986
27-5
28 -5
29*5
234 42 10-8
234 9 297
233 35 42 "6
18-1
18 '5
18 '9
29 26 7 -6
29 36 29 7
29 46 47 '2
-6-2
6-3
6-4
o -19687
-19382
0-19073
30-5
31 '5
Feb. 1 '5
233 46 '5
232 24 39 '9
231 47 19-2
-19'3
19 '5
197
29 56 59-9
so 7 6-4
30 17 5*6
-6-5
6-5
6*6
0'18760
0-18443
0-18124
2'5
3 '5
4'5
231 8 42 '8
230 28 48 '8
229 47 34 '2
-19'9
20 'I
20*6
30 26 56 '6
30 36 38 '2
30 46 8-9
-6-6
6-6
67
0-17801
0-17475
o -17148
5 *5
6'5
7'5
229 4 567
228 20 54 '5
227 35 23 '9
21 '4
22 '4
237
30 55 27-1
31 4 32 -0
31 13 21 -0
-6-8
6-9
6-9
o -16819
-16489
0-16158
8'5
9'5
10 '5
226 48 25 '4
225 59 54 '8
225 9 50 '6
24 '6
25'3
25 '8
31 21 52'8
31 30 5 '5
31 37 56'9
-6-9
7-0
7-0
'15828
-15498
0-15170
11 '5
12 '5
13 '5
224 18 12 '2
223 24 56 '9
222 30 4 '4
26'2
26-4
267
31 45 24'5
31 52 26-5
31 59 0'4
- 7-0
6-9
6-9
0*14844
0*14520
'14200
14'5
221 33 34 '5
-27 -2
32 53-3
-6-9
0-13884
240
On the Elements of tlie Orbit of Halky's Comet,
TABLE X. continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist
from the Earth.
1836.
Feb. 14 '5
15 '5
16'5
/ .'/
221 33 34 '5
220 35 25 -0
219 35 37 'I
11
o*: '2
277
28 '5
o / //
32 5 3'3
32 10 32 '8
32 15 26-0
//
-6-9
6-8
67
'13884
o -13573
-13268
17'5
18'5
19-5
218 34 11 '3
217 31 6-0
216 26 24-8
29*3
30 "2
31 '2
32 19 40-2
32 23 12 -1
32 25 59 '2
-6-6
6-5
6-3
0-12971
'12680
0-12399
20 *5
21 '5
22'5
215 20 8 '6
214 12 19'2
213 2 59 -1
32*2
33-2
34'2
32 27 58-4
32 29 6 7
32 29 21'0
-6'2
6-0
5 '8
o -12127
0-11866
o -11617
23-5
24 *5
25 -5
211 52 12-3
210 40 3 '2
.209 26 34 '3
35'2
36-4
37 '5
32 28 38 '5
32 26 57'0
32 24 13 -0
- 5'6
5 '2
4 *8
0-11380
0-11157
'10948
26'5
27-5
28 '5
208 11 52 '7
206 56 2 '9
205 39 9*3
38'6
39-6
40 '2
32 20 24 '5
32 15 29'2
32 9 24 '3
4 '4
4 -0
3-6
0-10755
0-10579
-10420
29-5
Mar. 1 '5
2 *5
204 21 21 '8
203 2 45 '6
201 43 28 '8
40-6
41 '0
41 '3
32 2 9'2
31 53 41 '6
31 44 07
3'3
2 '9
2-6
0-10279
0-10158
'10058
3'5
4 '5
5'5
200 23 37-5
199 3 22'5
197 42 51 -0
41 -5
41 '6
42-0
31 33 57
31 20 55 '9
31 7 31 '6
2 '2
1'9
1 '5
o -09978
o -09919
'09884
6'5
7'5
8*5
196 22 10 '7
195 1 32 '4
193 41 3'3
42-5
43-0
43-6
30 52 53 '8
30 37 27
30 19 59-4
1 'I
-0-6
o-o
0-09871
-09881
0-09915
9'5
10 '5
11 '5
192 20 51 '3
191 i 6-0
189 41 57'2
44-2
44'8
45'4
30 1 45'6
29 42 24 '4
29 21 58 '3
+ 0-6
1 '2
1 '8
o -09974
0-10058
'10165
12-5
13'5
14'5
188 23 30 '6
187 5 54'9
185 49 17-9
-45'7
46-0
46-1
29 o 29'6
28 38 2 -0
28 14 39'4
+ 2'3
2'9
3'5
0*10298
-10455
'10637
15-5
l6'5
17-5
184 33 45 -0
183 19 24-5
182 6 21 '9
-45-9
45'4
44 '8
27 50 25'0
27 25 24 -0
26 59 40 '6
-r- 3-8
4-0
4'2
0-10843
0-11073
o -11327
18-5
19'5
20 '5
180 54 41 -4
179 44 28 '5
178 35 47 *1
44-4
44 '3
44-2
26 33 18'5
26 6 22 '9
25 38 58 '3
+ 4'5
4'9
5'4
o -11603
0-11903
-12223
21 *5
177 28 41 '6
44 '3
25 11 9'4
+ 6-0
-12565
at its appearance in the Years 1835 <$* 1836.
241
TABLE X. continued.
Date.
Apparent
Right Ascension.
Pertur-
bations.
Apparent
Declination.
Pertur-
bations.
Log. of True Dist.
from the Earth.
1836.
Mar.21 '5
22'5
23 *5
24 '5
25*5
26*5
27-3
28 '5
29 '5
30 '5
31 '5
177 28 41 '6
176 23 13-5
175 19 26-0
174 17 21 '8
173 17 I'l
172 18 24 '9
171 21 34-4
170 26 29 *4
169 33 9*3
168 41 33 -0
167 51 38 '9
n
44-3
44 '4
44 '2
44 -0
43 '6
43'3
-43 '0
427
42-4
-42 '2
-42*2
o / //
25 11 9'4
24 43 '7
24 14 36'6
23 46 1 '5
23 17 19'6
22 48 34 7
22 19 50 '8
21 51 11 '6
21 22 39 '8
20 54 18 '5
20 26 9 '9
ii
6-0
6-3
6-3
6-3
6-4
6-5
6-9
7'4
7'8
'12565
o -12927
-13309
0-13709
0-14127
0-14561
0-15011
0'15475
0-15953
0-16445
o -16949
242
On the Elements of the Orbit of Halley's Comet,
TABLE XL
Containing 730 Equations of Condition for correcting the assumed Elements
Orbit of HALLEY'S Comet, on July 30, 1835.
of the
Date.
Equations of Condition
dependent upon Right Ascensions.
1835.
//
II
n
u
II
II
Aug. i -5
+ 5'1
P
55*8
Q +
301 -0
R +
42-6 S
+ i-o
U -
i-o V
= E
2*5
+ 5-1
P
56-1
Q +
302-4
R +
42'6 S
+ I'l
u
0-9 V
= E
3*5
+ 5-0
P
56-5
Q +
303-8
R +
42*6 S
+ I'l
u
0-9 V
Tjl
4-5
+ 4*9
P
- 56*7
Q +
305*1
R +
42-7 S
+ 1-2
u -
0-8 V
t= E
5-5
+ 4'8
P
56-9
Q +
306*4
R +
42 '7 S
+ 1-1
u -
0-8 V
^ E
6-5
+ 4-7
P
- 57-2
Q +
307-9
R +
42-8 S
+ I'l
u
0-8 V
= E
7*5
+ 4-7
P
57*3
Q +
309*5
R +
42-8 S
+ 1-4
u
0-6 V
= E
8*5
+ 4-5
P
57*9
Q +
311-0
R +
42-9 S
+ 1-2
u
0*6 V
= E
9'5
+ 4 '6
P
58 -0
Q +
312-7
R +
43-0 S
+ 1-4
u
0-6 V
= E
10*5
+ 4-4
P
58 '2
Q +
314-4
R +
43-0 S
+ 1*4
u
0*5 V
= E
11*5
+ 4-2
P
58'5
Q +
315-9
R +
43-0 S
+ 1-3
u -
0-6 V
= E
12 '5
+ 4-2
P
53-8
Q +
317 '8
R +
43*1 S
+ 1*5
u
0*5 V
=5 E
13-5.
+ 4-0
P
59-2
Q +
319 *7
R +
43-1 S
+ 1*4
u
0*4 V
= E
14*5
+ 3-9
P
59*5
Q +
321-5
R +
43-2 S
+ 1*5
u
0-4 V
= E
15 '5
+ 3-7
P
59*8
Q +
323 '3
R +
43-4 S
+ 1-5
ti-
0-3 V
= E
16*5
+ 3*6
P
60-2
Q +
325-3
R +
43*4 S
+ 1-6
ll
0*2 V
= E
17*5
+ 3'3
P
60-6
Q +
327-3
R +
43-4 S
+ 1-7
u
0-3 V
T?
18*5
+ 3-2
P
60-9
Q +
329*6
R +
43-5 S
+ 1-6
u -
o-i V
= E
19*5
-f 3-0
P
61-4
Q +
331 '7
R +
43-6 S
+ 1-5
u
*
= E
20-5
+ 2-8
P
6l'8
Q +
333 '8
R +
43-7 S
+ 1-6
u +
o-i V
= E
21*5
+ 2-6
P
62 -3
Q +
336*1
R +
43*7 S
+ 1'6
u +
o-i V
= E
22-5
+ 2*4
P
62-6
Q +
338 -5
R +
43-8 S
+ 1-7
u +
0-2 V
= E
23-5
+ 2'3
P
63-1
Q +
340 '9
R +
44*0 S
+ 1-5
u +
0-4 V
= E
24*5
+ 1 '9
P
63*4
Q +
343 -5
R +
44-0 S
+ 1'7
u +
0-4 V
= E
25-5
-f 1-6
P
64*0
Q +
346 -1
R +
44-2 S
+ 1-9
u +
0-5 V
= E
26-5
+ 1 -5
P
64-5
Q +
348 -8
R +
44-3 S
+ 1-9
u +
0-5 V
= E
27-5
+ 1-4
P
64-9
Q +
351*8
R +
44-6 S
+ 2'0
u +
0-7 V
=r E
28-5
+ 0-8
P
65-5
Q +
354-5
R +
44-5 S
+ 2-1
u +
0*8 V
= E
29-5
+ 0-6
P
66*0
Q +
357-6
R +
44-6 S
+ 2-1
u +
i-o V
= E
30*5
+ 0*2
P
66-7
Q +
360-7
R +
44-7 S
+ 2-1
u +
i-o V
= E
31-5
0-3
P
67-3
Q +
363*9
R +
44 '9 S
+ 2'2
u +
1-1 V
= E
at its appearance in the Years 1835 $ 1836.
243
TABLE XL continued.
Containing 730 Equations of Condition for correcting the assumed Elements
Orbit of HALLE Y'S Comet, on July 30, 1835.
of the
Date.
Equations of Condition dependent
upon Declinations.
1835.
w
II
//
/t
/;
II
Aug. i
5
4'2
p +
16-2 Q
- 88 '9
R
11-5 S
+ 14 '5
U -
4-3 V
= E 7
2
5
4-3
p +
16-4 Q
90-3
R
11-5 S
+ 14-4
U
4-2 V
= E 7
3
5
4-5
p +
16-8 Q
91 '8
R
n-6 S
+ 14-5
Ti-
3-9 V
= E 7
4
5
4-5
p +
17*0 Q
93-3
R
11-8 S
+ 14 '6
ll
3-7 V
= E 7
5
5
4-6
p +
17-3 Q
94-6
R
11-8 S
+ 14-8
IT -
3-5 V
= E 7
6
5
4-8
p +
17'5 Q
96 -4
R -
12 '1 S
+ 14-7
U -
3-3 V
= E 7
7
*5
5-0
p +
17 '9 Q
98-1
R
12-1 S
+ 14*8
U
3-1 V
= E'
8
5
5'0
p +
18-3 Q
99-6
R
12-6 S
+ 15'0
U -
2-9 V
rr E 7
9
5
4 '9
p +
18-5 Q
101 -4
R
12-5 S
+ 15 -1
U -
2-6 V
== E 7
10
5
5*4
p +
18-8 Q
103*2
R
12-8 S
+ 15-0
u
2-4 V
== E 7
11
5
5-4
p +
19 '2 Q
104-9
R-
12-8 S
+ 15-2
Ti-
2-2 V
= E 7
12
5
5-6
p +
19-4 Q
107*0
R
13-0 S
+ 15-2
ll -
2~-0 V
= E 7
13-5
5-5
p +
19'9 Q
103 '9
R-
13-1 S
+ 15J4
U -
1-6 V
= E 7
14
5
5-6
p +
20-2 Q
110-9
R
13*4 S
+ 15*5
U
1-3 V
= E 7
15
5
5-7
p +
20-6 Q
113-0
R
13-7 S
+ 15-6
U -
1-1 V
= E 7
16
5
6-0
p +
21-1 Q
115-2
R
13-9 S
+ 15-6
U -
0-9 V
= E 7
17
5
6-0
p +
21 -4 Q
117-6
R-
13 '8 S
+ 15-8
U
0-6 V
= E 7
18
5
6-2
p +
21-8 Q
120*2
R
14-1 S
+ 16-0
U
0-3 V
== E 7
19
5
6-3
p +
22-4 Q
122 -2
R -
14 : 4 S
+ 16-0
U
*
= E 7
20
5
6-8
p +
22-7 Q
124 *9
R-
14-8 S
+ 16-1
u +
0-3 V
= E 7
21
5
6-8
p +
23-2 Q
127-7
R
15-0 S
+ 16-2
u +
0-6 V
== E'
22
*5
6-9
p +
23-8 Q
130 -2
R-
14-9 S
+ 16-5
u +
0-9 V
= E 7
23
5
7-5
p +
24-1 Q
133-5
R
15-7 S
+ 16-3
u +
i-o V
= E 7
24
5
- 7'3
p +
24-8 Q
136-2
R
15-7 S
+ 16-7
u +
1-6 V
= E/
25
5
7*7
p +
25'3 Q
139*5
R
16-1 S
+ 16-9
u +
2-0 V
= E 7
26
5
Q O
O 1
p +
25-9 Q
142'7
R
16-7 S
+ 17-0
u +
2-3 V
= E 7
27
5
8-4
p +
26-3 Q
146-5
R-
16-8 S
+ 17-0
u +
2-7 V
= E 7
28
5
8-6
p +
27*2 Q
149-9
R -
17-2 S
+ 17-4
u +
3-1 V
= E 7
29
5
8-9
p +
27-8 Q
153-6
R -
17-0 S
+ 17-5
u +
3'6 V
= E-'
30
5
9'0
p +
f\
157-8
R
17-5 S
+ 17-6
u +
4-0 V
= E 7
31
5
p +
29 *4
162-1
R
18-0 S
+ 17-9
u +
4-5 V
= E 7
244
On the Elements of the Orlit of Halky's Comet,
TABLE XL continued.
Date.
Equations of Condition dependent upon Right Ascensions.
1835.
Aug. 31 B 5
Sep. i-5
2-5
// // // // // //
-3 P 67 '3 Q 4- 363 '9 R + 44 '9 S + 2 '2 U 4- 1 '1 V = E
'5 P 67 '8 Q + 367 '2 R 4- 45 -1 S 4- 2 *4 U + 1 '3 V =r E
o -9 P 6s -6 Q 4- 370 -7 R 4- 45 -2 S 4- 2 -3 U 4- i '4 V = E
3*5
4'5
5-5
1 -3 P -- 69 -1 Q 4- 374 -5 R 4- 45 -4 S 4- 2 -4 U 4- 1 '5 V =r E
1 -8 P 69 '8 Q + 378 -2 R + 45 '5 S 4- 2 -5 U 4- 1 '6 V = E
2 '2 P 70 '5 Q 4- 382 -1 R 4- 45 '7 S 4- 2 -6 U 4- 1 '8 V = E
6'5
7-5
8'5
2 '9 P 71 -3 Q 4- 386 '2 R + 45 -8 S 4- 2 -6 U 4- 2 '1 V = E
3 -4 P 72 -1 Q + 390 -5 R + 45 '9 S 4- 2 *7 U 4- 2 '2 V = E
4 -i P 73 -o Q 4- 394 -7 R + 46 -o S 4- 2 -7 U + 2 -3 V = E
9'5
10-5
11 '5
4'8 P 73-8 Q 4- 399'3 R 4- 46-3 S 4- 3-0 U 4- 2 "6 V = E
5 '5 P 74 '7 Q + 404 -1 R 4- 46 '4 S + 3 '0 U + 2 '8 V = E
6 '4 P 75 '6 Q 4- 409 -3 R 4- 46 -8 S + 3 -1 U + 2 -9 V ==: E
12 '5
13-5
14*5
7'3 P - 76-5 Q 4- 414-3 R 4- 46-8 S 4- 3-2 U 4- 3 '2 V = E
8'2 P 77-3 Q + 419-8 R 4- 47'2 S 4- 3'4 U + 3 '5 V = E
9 '4 P 78 -5 Q 4- 425 -4 R + 47 '3 S 4- 3 '4 U 4- 3 "] V = E
15*5
16-5
17-5
10 '5 P 79 '7 Q + 431 -0 R + 47 '5 S + 3 '5 U + 4 -0 V = E
11 -9 P so -8 Q + 437 -o R + 47 '7 S + 3 -7 U -f 4 -4 V = E
13 '3 P 81 '7 Q + 443 '8 R + 48 -0 S + 3 "9 U + 4 "] V =r E
18*5
19*5
20 -5
15'0 P 83-1 Q + 450'4 R + 48-1 S + 3*9 U + 5 '1 V = E
16 '8 P 84 '5 Q + 457 -0 R + 48 '2 S + 4 -3 U + 5 '4 V = E
18 '8 P 85 7 Q + 463 -9 R + 48 -5 S + 4 '4 U + 5 '9 V = E
21 '5
22 '5
23 '5
21 'I P 87 '2 Q + 471 '3 R + 48 '6 S + 4 '6 U + 6 '4 V = E
23 '6 P 89 'I Q + 478 '6 R + 48 '8 S + 4 '9 U + 7 -0 V r= E
26 -6 P 90 -o Q + 485 -9 R + 48 -8 S + 5 'i U + 7 -6 V = E
24 '5
25 '5
26 '5
29 9 P 90 -9 Q 4- 493 -6 R 4- 48 -9 S + 5 -4 U + 8 -2 V = E
33 '9 P 92 '5 Q 4- 500 '3 R 4- 48 -8 S + 5 '9 U + 8 '8 V = E
38 '3 P 94 '3 Q 4- 506 '6 R 4- 48 '6 S 4- 6 -1 U 4- 9 '8 V = E
27*5
28'5
29*5
43-6 P 94-7 Q 4- 512-6 R 4- 48-1 S 4- 6-6 U + 10-7 V = E
49 '8 P 96 -0 Q 4- 516 -4 R 4- 48 -0 S 4- 7 -1 U 4- 11 -8 V =r E
57 -4 P 96 -6 Q + 518 -o R 4 46 -7 S + 7 -7 U + 12 -9 V = E
30'5
Oct. 1 '5
1-75
66'3 P 96-5 Q 4- 516-3 R 4- 45-1 S 4- 8'5 U 4- 14-2 V =3 E
77-6 P 95-3 Q + 506-8 R 4- 42-8 S 4- 9'2 U 4- 15-9 V = E
80-6 P 94-7 Q + 504-1 R + 42-3 S 4- 9 '7 U 4- 16-4 V = E
2-0
2 '25
2-5
84 -o P 94 -i Q 4- 499 '6 R + 41 -4 S + 9 '7 U 4- 16 -9 V = E
87-5 P 93-3 Q 4- 495-0 R 4- 40-6 S +10-0 U 4- i7'4 V = E
91 -5 P 92 -3 Q 4- 489 -s R 4- 39 -6 S + 10 -i U 4- is -o V = E
2'75
95 '2 P 91 -0 Q 4- 483 -2 R 4 38 -8 S -J- 10 '5 U 4- 18 '6 V = E
at its appearance In the Years 1835 8; 1836.
245
TABLE XL-
-continued.
Date.
Equations of Condition
dependent
upon Declinations.
1835.
II
II
a
//
//
II
Aug. 31 '5
9
"0 P
+ 29-4
Q-
162-1 R
18 -0
S
+ 17-9
u +
4-5 V
= E'
Sep. 1-5
10 -o P
+ 30-2
Q-
166-2 R
18 '2
S
+ 18 -0
u +
4-9 V
= E'
2-5
10
1 P
+ 31 '2
Q-
171 -2 R
18-8
S
+ 18-3
u +
5-4 V
= E 7
3 '5
10
5 P
+ 32-0
Q-
176-1 R
19 '3
S
+ 18-6
u +
6-0 V
= E'
4-5
11
2 P
+ 32 '8
Q-
181-6 R
19-8
S
+ 18-7
u +
6-4 V
= E'
5'5
11
4P
+ 34-0
Q-
187-0 R
20-2
S
+ 19-1
u +
7-1 V
= E 7
6-5
11
8 P
+ 35'2
Q-
193 -o R
20-8
S
+ 19*3
u +
7'7 V
= E'
7'5
12
4 P
+ 36'1
Q-
199*4 R
21 '2
S
+ 19-7
u +
8-4 V
r= E'
8 '5
12 -9 P
+ 37-3
Q-
206 -2 R
22-1
S
+ 19'8
u +
8-9 V
= E'
9 '5
13
5 P
+ 38 '8
Q-
213-6 R
22*5
S
+ 20'1
u +
9 '7V
= E'
10 '5
14
2 P
+ 40*1
Q-
221 -3 R
23-3
S
+ 20 '4
u +
10-2 V
= E'
11 '5
14
9P
+ 41 '4
Q-
229-9 R
23-8
S
+ 20-8
u +
11-1 V
= E'
12 '5
15
6P
+ 43'2
Q-
238 '9 R
24-8
S
+ 21 -1
u +
12-0 V
= E'
13-5
16
sP
+ 45 -1
Q-
248'7 R
25-7
S
+ 21-6
u +
12*8 V
= E'
14 '5
17
3 P
+ 46-8
Q-
259-4 R
26-5
S
+ 22*0
u +
13-8 V
=r E ;
15'5
1 O
1 O
3 P
+ 49-0
Q-
271 -i R
27'5
S
+ 22-3
u +
14-7 V
= E'
16 -5
19
4 P
+ 51 '4
Q-
283-6 R
28 -6
S
+ 22'8
u +
15-8 V
= E'
17'5
20
6P
+ 53'9
Q-
297-5 R
29 '6
S
+ 23-2
u +
16-8 V
s= E'
18 -5
21
9 P
+ 56-7
Q-
313-2 R
31 -2
S
+ 23*8
u +
18-1 V
= E'
19-5
23 '4 P
+ 59-6
Q -
329 *5 R
32 -4
S
+ 24 '4
u +
19'4 V
= E'
20 '5
25
oP
+ 63-1
Q-
348 -3 R
34-1
S
+ 25-1
u +
20-8 V
= E'
21 '5
26
9 P
+ 66-8
Q-
369-3 R
36-0
S
+ 25'6
u +
22-2 V
= E'
22 '5
29
oP
+ 71-4
Q-
392-4 R
38 -1
S
+ 26 '4
u +
23 f 9 V
= E'
23 '5
31
5 P
+ 75-7
Q-
419-0 R
40-4
S
+ 27'0
u +
25 '9 V
= E'
24*5
34
oP
+ 81 '5
Q-
448-7 R
42-8
S
+ 28 *0
u +
27-6 V
= E'
25 '5
37
2 P
+ 87'5
Q-
483-0 R
45 -8
S
+ 29-0
u +
29-6 V
= E'
26-5
40
9 P
+ 94-1
Q-
522-6 R
49 *2
S
+ 29-8
u +
31 -8 V
= E'
27-5
45
o P
+ 103 '2
Q-
566 -9 R
53-1
S
+ 30-9
u +
34-2 V
=r E'
28-5
49
. 9 p
+ 112-3
Q-
621 -o R
- 57-6
S
+ 32 -1
u +
36 '8 V
r= E'
29-5
55
6 P
+ 123-4
Q-
683 -8 R
63*0
S
+ 33-3
u +
39-8 V
= E'
30*5
62
3 P
+ 137-2
Q-
757-2 R
69*4
S
+ 34-7
u +
42 '9 V
= E'
Oct. 15
70
2 P
+ 153'1
Q-
847 '4 R
76-9
S
+ 36-0
u +
46-3 V
=r E'
1-75
~ 72
sP
+ 157-6
Q-
372-2 R
79'1
S
+ 36-6
u +
47-2 V
T?'
J j
2-0
;4
7P
+ 162-3
Q-
898 '2 R
81 -4
S
+ 37*0
u +
48-1 V
= E'
2 -25
77
1 P
+ 167-4
Q-
925-6 R
83-7
S
+ 37'4
u +
49-1 V
= E'
2'5
79
5 P
+ 172-5
Q-
954-5 R
86-1
S
+ 37'5
u +
50'0 V
= E'
2 '/5
82
o P
+ 178-3
Q-
984 '8 R
88 -4
S
+ 38-0
u +
51'0 V
= E'
246
On the Elements of the Orlit of Halleys Comet,
TABLE XL continued.
Date.
Equations of Condition dependent upon Right Ascensions.
1835.
Oct. 2-75
3-0
3'25
if 11 n it n
95'2 P 91 -0 Q + 483-2 R + 38-3 S +10-5 U
99-5 P 89-6 Q + 475-7 R + 37-6 S +10-7 U
104-0 P 37-8 Q + 467-1 R + 37-0 S +11-1 U
+ 18-6 V = E
+ 19-1 V = E
+ 19-7 V == E
3 '5
3-75
4-0
108-8 P 86-0 Q + 456-9 R + 35-0 S +11'4 U
113-8 P 84-0 Q + 4i5'4 R + 33 '6 S + 11 '9 U
119-3 P 81-8 Q + 431'S R + 31 '8 S +12 -2 U
+ 20 -2 V r= E
+ 21-0 V = E
+ 21 -6 V *= E
4-25
4-5
4-75
125-1 P 79-5 Q + 416-4 R + 29-8 S +12-7 U
131-4 P 76-3 Q + 398-7 R + 27-3 S +13-1 U
138-0 P 72-7 Q + 378-8 R + 25-1 S +13 '7 U
+ 22 -2 V = E
+ 22-9 V = E
+ 23-7 V == E
[ vsr 5 *
5-25
5-5
145-2 P 68 '7 Q + 355 '9 R + 22 '3 S + 14 '0 U
152-7 P 63-9 Q + 330-4 R + 19'6 S +14-7 U
161 -4 P 58 -4 Q + 300 '9 R + 15 -9 S + 15 -1 U
+ 24 *6 V = E
+ 25 '5 V = E
+ 26 *4 V = E
5-75
6-0
6*125
170-3 P 52-3 Q + 268-0 R + 12-3 S +15'7 U
180 ; P 45*3 Q + 230-0 R + 7 '8 S + l6 '3 U
185-0 P 41-4 Q + 208-8 R + 5'5 S +l6'6 U
+ 27-4 V =r E
+ 28 -4 V = E
+ 28-9 V = E
6-25
6-375
6-5
190-2 P 37-3 Q + 186-2 R + 3-1 S +17-0 U
195 -7 P 33 -o Q + 161 -9 R + o -5 S .+ 17 -2 U
201-4 P 28-3 Q + 135-9 R 2 '2 S +17 '5 U
+ 29 -4 V = E
+ 29 -9 V = E
+ 30-4 V = E
6 '625
6-75
6-875
207-2 P 23-3 Q + 108-0 R 5-1 S +18-0 U
213-3 P 17-8 Q + 78-1 R 8-2 S +18-5 U
219-8 P 11-9 Q + 46-0 R 11-5 S +18-8 U
+ 31 -o V = E
+ 31-6 V = E
+ 32 -1 V = E
7-0
7-125
7-25
226-5 P 5-7 Q + 11-7 R is-o S +19*1 U
233-2 P + 0-8 Q 24-8 R 18-6 S +19 '6 U
240-3 P + 7'8 Q 63-8 R 22-4 S +20-0 U
+ 32-7 V = E
+ 33 -4 V =: E
+ 34-1 V == E
7-375
7-5
7-625
247-8 P + 15-2 Q 105-7 R 26-6 S +20-4 U
255 -6 P + 23 -3 Q 150 '4 R 31 *0 S +20 '8 U
263 -5 P + 32 -2 Q 198 -0 R 35 -6 S +21 *3 U
+ 34-7 V = E
+ 35 -3 V = E
+ 36-0 V = E
7*875
8 -0
271 -6 P + 41 -8 Q 249 -o R 40 -5 S +21 9 U
280 -3 P + 51 '9 Q 303 "} R 45 '6 S +22 '3 U
289*4 P + 62-7 Q 362-1 R 51-1 S +22-7 U
+ 36 -6 V = E
+ 37 -2 V == E
+ 37-8 V = E
8-125
8 '25
8-375
298 '7 P + 74 -3 Q 424 -4 R 57 '0 S +23 '4 U
308-1 P + 86-7 Q 490-9 R 63-3 S +24-0 U
318-1 P + 99*9 Q 561-6 R 70*1 S +24-4 U
+ 38 '4 V = E
+ 39 -0 V =r E
+ 39-7 V = E
8-5
8-625
8-75
328 -6 P +114-0 Q 636-9 R 77-2 S +24-8 U
339-0 P +129*0 Q 716-9 R 84-6 S +25-5 U
349 -5 P + 144 '9 Q 801 -8 R 92 '4 S + 26 '1 U
+ 40 -3 V = E
+ 40 -9 V = E
+ 41-5 V = E
8 -875
360-3 P +161-8 Q 892-6 R 100-6 S +26-6 U
+ 42 -1 V = E
at its appearance in the Years 1835 fy 1836.
247
TABLE XL continued.
Date.
Equations of Condition
dependent
upon Decimations
1835.
Oct. 2 -75
82
'oP
+ 178-3 Q
II
984 -8
R
//
88 -4 S
//
+ 38 -0
u +
Bl'-O V
E'
3-0
84
7 P
+ 184-2 Q
1016 '9
R
91 -0 S
+ 38-5
u +
51-8 V
E'
3*25
87
5 P
+ 190-6 Q
1050-5
R
93-7 S
+ 38 -8
u +
52-8 V
: '
V
3 *5
90
4 P
+ 197-1 Q
1086-1
R
96-7 S
+ 39-1
u +
53-6 V
E'
3 -75
93
3 P
+ 203-9 Q
1123 -4
R
99'8 S
+ 39-5
u +
54-6 V
E'
4'0
- 96
5P
+ 211 '0 Q
1162-9
R
103-2 S
+ 39-9
u +
55-6 V
;
E'
4*25
99
7P
+ 218 -0 Q
1204 *5
R
106-7 S
+ 40 -4
u +
56-6 V
=
E'
4'5
102
8 P
+ 226-2 Q
1248-2
R
110-6 S
+ 40-8
u +
577 V
E 1
4-75
106
1 P
+ 234-6 Q
1294-0
R
114 -4 S
+ 41-1
u +
58-6 V
r
E'
5*0
109
6 P
+ 243 '2 Q
1341-9
R
118 *4 S
+ 41-5
u +
59-4 V
E'
5'25
112
8 P
+ 252-7 Q
1392 -o
R
122-4 S
+ 41-9
u +
6o-4 V
: '
E'
5'5
116
sP
+ 262-5 Q
1444-8
R
126-8 S
+ 42 -0
u +
61-0 V
*"-*
E'
5'75
119
7 P
+ 272-7 Q
1499 '5
R
131 -3 S
+ 42*4
u +
61-8 V
=
E'
6-0
123
1 P
+ 283 -0 Q
1557*3
R
135-6 S
+ 42-5
u +
62-3 V
r ;
E'
6-125
124
6 P
+ 288-5 Q
1587 '2
R
-137 -9 S
+ 42-7
u +
62-5 V
r~*
E 7
6-25
126
l P
+ 294-0 Q
1617-7
R
140-4 S
+ 42-8
u +
62-7 V
T~*
E 7
6-375
127
4P
+ 299-7 Q
1648 '9
R
142*9 S
+ 42-9
u +
63-0 V
r
E'
6-5
128
7P
+ 305-5 Q
1680 -6
R
145-6 S
+ 42-9
u +
63-2 V
r~r"
E 7
6 -625
130 -0 P
+ 311-4 Q
1712*9
R
148-2 S
+ 42-9
u +
63-4 V
=
E'
6-75
131
2 P
+ 317-4 Q
1745-6
R
150-8 S
+ 42-9
u +
63-6 V
r
E 7
6-875
132
2 P
+ 323-5 Q
1778-6
R
153-3 S
+ 43-0
u +
63-7 V
=r
E'
7-o
133
1 P
+ 329-6 Q
1812 -0
R
155 '9 S
+ 43*0
u +
63-8 V
=
E 7
7'125
133
8 P
+ 335-6 Q
1845-6
R
158-5 S
+ 43-0
u +
63-9 V
r-r
E 7
7-25
134
3P
+ 341-7 Q
1879-5
R
161-1 S
+ 42-9
u +
63-9 V
i**""
E 7
7-375
134
5 P
+ 348'1 Q
1913-6
R
163-5 S
+ 42*8
u +
63-8 V
=
E 7
7-5
134
6 P
+ 354-4 Q
19477
R
165-9 S
+ 42-6
u +
63-6 V
r
E 7
7-625
134
2 P
+ 360-6 Q
1981-5
R
168-3 S
+ 42 '4
u +
63-5 V
_
E 7
7-75
133
5 P
+ 366-8 Q
2015-0
R
170-7 S
+ 42*2
u +
63-3 V
=
E 7
7-875
132
4 P
+ 372-8 Q
2048 -0
R
173-1 S
+ 42-0
u +
62-9 V
r~*
E 7
8-0
^-130
9 P
+ 373-6 Q
2080-2
R
175-4 S
+ 41-7
u +
62-4 V
==
E 7
8*125
128
9P
+ 384-2 Q
2111 '5
R
177-5 s
+ 41'3
u +
62-1 V
=
E 7
8-25
126
5 P
+ 389-6 Q
2141 -4
R
179-4 S
+ 40-8
u +
61-8 V
;
E 7
8-375
123
5 P
+ 394-6 Q
2169 -4
R
181 -0 S
+ 40-4
u +
60-9 V
E 7
8 '5
119
8 P
+ 399-2 Q
2195 -a
R
182*3 S
+ 39'9
u +
59-9 V
~
E 7
8 '625
115
sP
+ 403*3 Q
2218-7
R
183-6 S
+ 39'3
u +
59-0 V
,T
E'
8-75
110
o P
+406-9 Q
2239 -0
R
184-5 S
+ 38*7
u +
58-1 V
:r
E'
8-875
103
7 P
+ 409-9 Q
2255-7
R
184*8 S
+ 38-1
u +
57-1 V
r
E 7
248
On the Elements of the Orbit of Halley s Comet,
TABLE XI
. continued.
Date.
Equations of Condition
dependent
upon Right Ascensions.
1835.
//
II
II
a
II
II
Oct. 8-875
36o -3
P
+ 161-8 Q
892-6
R
100 -6
S
+ 26-6
u +
42-1 V
- "P
9-0
371 -3
P
+ 179-7 Q
937-6
R
109-3
S
+ 27-1
u +
42-6 V
= E
9 '125
382-3
P
+ 198-5 Q
1088 -6
R
118 -4
s
+ 27*7
u +
43-1 V
= E
9*25
393 -3
P
+ 218 '3 Q
1195-0
R
127*9
s
+ 28*3
u +
43-5 V
= E
9-375
404 '3
P
+ 239-3 Q
1306 '7
R
137-6
s
+ 28-9
u +
43 '9 V
s= E
9'5
415 '0
P
+ 261 -o Q
1423*4
R
147-7
s
+ 29-5
u +
44-2 V
= E
9 '625
425-4
P
+ 283-6 Q
1545-2
R
158*1
s
+ 30-0
u +
44-3 V
= E
9-75
435 -4
P
+ 306-8 Q
1671-3
R
168-8
s
+ 30-5
u +
44-3 V
= E
9-875
445 -0
P
+ 330 -8 Q
1801*3
R
179-7
s
+ 30-8
u +
44-1 V
= E
10-0
453 '8
P
+ 355*2 Q
1934*2
R
190-8
s
+ 31 -1
u +
43-8 V
= E
10 '126
461 '7
P
+ 380-1 Q
2069 -3
R
202-0
s
+ 31-3
u +
43-4 V
=r E
10-25
468 -6
P
+ 404-9 Q
2205*3
R
213*1
s
+ 31 -5
u +
42 '8 V
1'.
30 '375
474-5
P
+ 429*7 Q
2341 -2
R
224-1
s
+ 31-4
u +
42*0 V
T?
10*5
479-0
P
+ 454-1 Q
2475*4
R
234 *7
s
+ 31-3
u +
41 -0 V
= E
10-625
481-1
P
+ 473*0 Q
2606*2
R
244 -6
s
+ 31 -0
u +
39-8 V
= E
10-75
481 -4
P
+ 501-0 Q
2732 -4
R
253 -8
s
+ 30*6
u +
38 -4 V
= E
10-876
480 -5
P
+ 522*6 Q
2852*3
R
262 *4
s
+ 80*1
u +
36-9 V
= E
11 4
477*9
P
+ 542-7 Q
2964-9
R
270 -2
s
+ 29'6
u +
35-2 V
= E
11 '125
473 -5
P
+ 561-2 Q
3068 *8
R
277-2
s
+ 29-0
u +
33*4 V
= E
11-25
467*4
P
+ 577-8 Q
3163-4
R
283*4
s
+ 28 -3
u +
31-4 V
= E
11 *375
459-7
P
+ 592*3 Q
3247-4
R
288 *6
s
+ 27*6
u +
29-4 V
= E
11 -5
450 *4
P
+ 604-7Q
3320 -5
R
292 -6
s
+ 26-8
u +
27-3 V
= E
11-625
439*7
P
+ 615-0 Q
3382 -0
R
295 -1
s
+ 25*8
u +
25 -0 V
= E
11 '75
427*9
P
+ 623*2 Q
3432-1
R
296-6
fc
+ 24-4
u +
22 *7 V
= E
11 '875
415-3
P
+ 629-2 Q
3470 -6
R
297-6
s
+ 23-3
u +
20 -3 V
= E
12-0
402 -0
P
+ 633-2 Q
3498-3
R
297-9
s
+ 22-3
u +
18-0 V
s= E
12 -125
387 *7
P
+ 635 *3 Q
3515*4
R
-296-9
s
+ 21 '3
u +
15*7 V
-p
12-25
373-1
P
+ 635-7 Q
3523*0
R
295*2
s
+ 20*2
u +
13-5 V
= E
12*375
358 -3
P
+ 634-5 Q
3521 '7
R
292-8
s
+ 19-1
u +
11*3 V
= E
12-5
343 *2
P
+ 631 -7Q
3512-3
R
289*8
s
+ 18'0
u +
9-1 V
= E
12 -625
327-8
P
+ 627-4Q
3495*6
R
286 -4
s
+ 16-8
u +
7*1 V
= E
12-75
312*3
P
+ 621-8 Q
3472*2
R
282-5
s
+ 15-7
u +
5*2 V
= E
12-875
297 *4
P
+ 615*4 Q
3442-8
R
278 *2
s
+ 14-7
u +
3-4 V
-_ r*
13*0
282-7
P
+ 608-2 Q
3408 -1
R
273-6
s
+ 13-8
u +
1-6 V
V
i *
13*125
268 -2
P
+ 600-5 Q
3369 -1
R
268 *7
s
+ 13-1
u
o-i V
= E
13-25
254-0
P
+ 592-2 Q
3326 -3
R
263*5
s
+ 12*3
u
1*7 V
= E
13-375
240 -4
P
+ 583-3 Q
3280-3
R
238 -0
s
+ 11-3
u -
3 -2 V
= E
at its appearance in the Years 1835 fy 1836.
249
TABLE XL continued.
Date.
Equations of Condition dependent upon Declinations.
1835.
Oct. 8 '8 75
9-0
9*125
// // // // // //
103-7 P +409-9 Q 2255-7 R 184*8 S +38 "1 U + 57 '1 V = E 7
96-4 P +412-1 Q 2268*1 R 184-6 S +37*4 U + 56*1 V = E 7
88 *0 P +413 -3 Q 2275 -3 R 184 *1 S +36 '7 U + 54 '8 V = E 7
9*25
9-375
9'5
78*4 P +413*4 Q 2277-3 R 183'0 S +35*9 U + 53 *5 V = E 7
67-6 P +412*4 Q 2272-8 R -181-1 S +35-0 U + 52*1 V = E 7
55*4 P +410-1 Q 2261*1 R 178*4 S +34-0 U + 50 '7 V = E'
9*625
9*75
9*875
41*8 P +406*2 Q 2241*3 R 175*0 S +33*1 U + 49 *1 V = E'
26*7 P +400*6 Q 2212*5 R 170*8 S +32*1 U + 47 *4 V = E 7
10*3 P +393*3 Q 2173-9 R 165*8 S +30-7 U + 45-8 V = E'
10*0
10*125
10 '25
+ 7*7 P +384-1 Q 2124*9 R 159*8 S +29*2 U + 44 *1 V = E'
+ 27*2 P +372*7 Q 2064*9 R 152*8 S +28*0 U + 42 *4 V = E'
+ 47*8 P +359-3 Q 1993-6 R 144*9 S +26*8 U + 40*7 V = E 7
10-375
10*5
10 '625
+ 69*6 P +343*7 Q 1910*7 R 136-1 S +25*3 U + 39*o V = E'
+ 92*5 P +326-0 Q 1816-3 R 126*2 S +23*8 U + 37 *4 V = E 7
+ 116*7 P + 306-1 Q 1710*2 R 115*1 S +22-6 U + 36*1 V = E 7
10-75
10-875
11-0
+ 141-5 P +284-3 Q 1593*5 R 103 '2 S +21*4 U + 34 *9 V = E 7
+ 166*6 P +260*6 Q 1466-9 R 90*7 S +20-2 U + 33*9 V = E 7
+ 191*8 P +235-4 Q 1331*7 R 77-5 S +19'0 U + 33*1 V r= E'
Il f 125
11-25
11 '375
+ 217*1 P +208-9 Q 1189-0 R 63*7 S + 18-2 U + 32*7 V = E 7
+ 242*0 P +181*4 Q 1040-8 R 49 *6 S +17*4 U + 32 '5 V = E 7
+ 265-9 P +153-1 Q 888-8 R 35*6 S +l6'5 U + 32 *5 V = E 7
11*5
11-625
11-75
+ 288*9 P +124*6 Q 734-8 R 21-5 S +15*6 U + 32 *7 V = E 7
+ 310*9 P + 96*2 Q 580-3 R 7*5 S +15-3 U + 33-5 V = E 7
+ 331*6 P + 68*1 Q 427-0 R + 6*4 S +15-0 U + 34*5 V = E 7
11*875
12*0
12*125
+ 350*6 P + 40-8 Q 277*0 R + 20*0 S +14*9 U + 35*7 V = E 7
+ 368-0 P + 14*4 Q 131*3 R + 32*4 S +14-8 U + 37 *2 V = E 7
+ 383-6 P ll'O Q + 8*4 R + 44*3 S +14 '8 U + 38 *8 V = E 7
12-25
12*378
12*5
+ 397-7 P 35*0 Q + 141*5 R + 55*6 S +15 '0 U + 40-8 V = E 7
+ 409-5 P 57*2 Q + 267'0 R + 66-2 S +15*4 U + 43*2 V = E 7
+ 419*7 P ~ 77-6 Q + 384*5 R + 76*2 S +16-0 U + 45*8 V = E 7
12 *625
12-75
12*875
+ 429 *0 P 96 -0 Q + 493 *5 R + 85 -3 S + 16 *7 U + 48 *2 V = E 7
+ 436-8 P 112*7 Q + 593*7 R + 93*4 S +17*4 U + 51-0 V = E 7
+ 442*4 P 127*8 Q + 684-8 R + 100 *7 S + 18 *1 U + 53 '8 V = E 7
13*0
13*125
13-25
+ 446-4 P 141-2 Q + 767-2 R +106*9 S +18-8 U + 56 -6 V = E 7
+ 449-1 P 153-1 Q + 840*8 R +112*4 S +19*6 U + 59 *6 V = E 7
+ 450*5 P 163-5 Q + 906-0 R +117*2 S +20-5 U + 62*5 V = E 7
13-375
+ 450'7 P 172*3 Q + 962-9 R +121*4 S +21*4 U + 65-3 "V = E'
NAUTICAL ALMANAC, 1839, APPENDIX.
250
On the Elements of the Orbit of Halley's Comet,
II
TABLE XL continued.
Date.
Equations of Condition dependent upon Right Ascensions.
1835.
Oct. 13 -37
13-5
13-62
" // // // // //
240-4 P +583-3 Q 3280 '3 R 258 '0 S +11-3 U 3 '2 V = E
227-2 P 4-574-2 Q 3231-5 R 252 '4 S +10 '3 U 4 -6 V = E
214-2 P +565-2 Q 3180-3 R 246 '8 S + 9 '6 U 5 -ft V = E
13*75
13-87
14'0
201 -6 P +556 -1 Q 3127 -2 R 241 -1 S + 8 -9 U 7 ! V = E
189-6 P +546-7 Q 3072-4 R 235-4 S + 8'2U 8 '1 V = E
178-0 P +537-0 Q 3016-5 R 229-7 S + 7-6 U 9 -o. V = E
14-12
14-25
14-37
166-8 P +527-1 Q 2959-8 R 224-0 S + 7-0 U 10-0 V = E
156 -2 P +517 -2 Q 2902 -7 R 218 '3 S + 6 '5 U 11 '0 V = E
146 -2 P +507 -3 Q 2845 5 R 212 '6 S + 5 -8 U 11 '7 V = E
14*5
14-625
14-75
136-6 P +497*3 Q 2788-2 R 207-0 S + 5-2 U 12-3 V = E
127-3 P +487-2 Q 2730-8 R 201-4 S + 4-7 U 13-0 V = E
118-3 P +477-0 Q 2673-6 R 195-9 S + 4-3 U 13-6 V = E
14-875
15-0
15125
109-8 P +466-6 Q 2616-8 R 190-6 S + 3-9 U 14 -i V = E
101-7 P +456-3 Q 2560-6 R 185-5 S + 3 '6 U 14 -6 V = E
93-9 P +446-2 Q 2505-1 R 180 '4 S + 3 *2 U 15M V = E
15 '25
15 '375
15*5
> 86t-6 P +436-3 Q 2450 '5 R 175 '3 S + 2 '9 U 15 '6 V = E
79. '8 P +426-7 Q 2396-8 R 170-4 S + 2-4 U 16-0 V = E
73'3 P +417-3 Q 2344*1 R 165 '6 S + 2 '0 U 16 -3 V = E
15 '625
15 '75
15-875
67-0 P +408-1 Q 2292 *3 R 160 '9 S + 1-6 U 16-7 V = E
60-9 P +399-1 Q 2241-4 R 156-2 S + 1 '3 U 17 '0 V == E
55-3 P +390-4 Q 2191-7 R 151-8 S + 1 '0 U 17 '3 V = E
16 o
16-25
16-5
49 '9 P +381 -9 Q 2142 '9 R 147 '5 S + -8 U 17 *5 V = E
39-8 P +365-3 Q 2048-3 R 139 '0 S + '3 U 17 '9 V = E
30-8 P +349-6 Q 1958-7 R 131-4 S '2 U 18 '3 V = E
16-75
17-0
17-25
22-3 P +334-5 Q 1873 '1 R 123 '8 S '5 U 18 5 V = E
15-2 P +320-0 Q 1791-9 R 117-0 S 0-8 U 18-5 V = E
8 -2 P +306 -3 Q 1714 -4 R 110 '5 S 1 '0 U 18 '6 V = E
17-5
17-75
19-0
2-6 P +293 -3 Q 1641-7 R 104-3 S 1-3 U 18-6 V = E
+ 3-0 P +280-9 Q 1572-6 R 98-3 S 1-4 U 18-6 V = E
+ 7 -9 P +269 -4 Q 1507 -4 R 92 '8 S i -6 U is -4 V = E
18 '25
18 -5
18-75
+ 12 -7 P +258 -4 Q 1445 '2 R 87 *4 S 1 "J U 18 '4 V = E
+ 16 -2 P +248 -1 Q 1387 -0 R 82 -9 S 1 -8 U 18 -2 V = E
+ 20 "3 P +238 -3 Q 1331 '9 R 78 "1 S 1 '9 U 18 '1 V = E
19-0
19'25
19-5
+ 23 -3 P +228 -8 Q 1280 -0 R ?4 -1 S 1 '8 U 18 '0 V = E
+ 26-6 P +220-1 Q 1230-3 R 69-8 S 2 '0 U 17 '9 V = E
+ 29 -i P +211 -9 Q 1183 -7 R 66 >4 S 2 -4 U 17 '6 V = E
19 75
g
+ 31-8 P +204-0 Q 1139-1 R 62-5 S 2'2 U 17'6 V = E
^^ ." '' r^-"**". - ' ' " " "-' . 'J-tJ--^-. . . ^ S.
at Us appearance in the Years 1835 fy 1836.
251
TABLE XL-
-continued.
Date.
Equations of Condition dependent
upon Declinations.
1835.
a
n
n
Oct. 13 *375
+ 450-7
p
172-3 Q
+ 962*9
R
+ 121M S
+ 21 -4
U
+ 65," 3
V =
E 1
13-5
+ 449*6
p
130-0 Q
+ 1012*0
R
+ 124-9 S
+ 22 -3
U
+ 68*1
V =
E 1
13*625
+ 447-6
p
187-3 Q
+ 1053*6
R
+ 127-2 S
+ 23*2
u
+ 70-9
V =r
E'
13*75
+ 444-5
p
193-7 Q
+ 1088 *0
R
+ 129-8 S
+ 24*0
u
+ 73 '6
V =
E 1
13*873
+ 440-3
p
198-8 Q
+ 1115 *6
R
+ 131-1 S
+ 24*6
u
+ 76-1
V S3
E'
14*0
+ 435-5
p
202-8 Q
+ 1187*0
R
+ 131 '9 S
+ 25-2
u
+ 78-5
v =
E f
14123
+ 430-2
p
205-8 Q
+ 115* -1
R
+ 132-5 S
+ 26-1
u
+ 80-9
v =
E f
14-25
+ 424 -2
p
207 -9 Q
+ 1163 9
R
+ 132 -6 S
+ 27-0
u
+ 83-2
V =
E 7
14*375
+ 417-3
p
209*4 Q
+ 1169*6
R
+ 132-1 S
+ 27*5
u
+ 85/3
V =
E 7
14-5
+ 410*0
p
210-2 Q
+ 1171-0
R
+ 131-2 S
+ 28*0
u
+ 87*3
V =
E'
14 -623
+ 402 -8
p
209-8 Q
+ 1168-6
R
+ 130-2 S
+ 28-7
u
+ 89*2
V =
E 7
14-75
+ 395-4
p
208 -6 Q
+ 1162*6
R
+ 123-9,S
+ 29-3
u
+ 91-0
V ss
E 7
14*875
+ 337-6
p
206 -6 Q
+ 1153 -a
R
+ 127-4 S
+ 29-7
u
+ 92*5
V =8
E 7
15-0
+ 379*
p
203 -9 Q
+ 1141 -4
R
+ 125-6 S
+ 30*1
u
+ 93*9
V r=
E 7
15-125
+ 371 '6
p
201 -0 Q
+ 1127-4
R
+ 123-8 S
+ 30*6
u
+ 95-3
V =5
E 7
15-25
+ 363-4
p
*-i97'9 Q
+ 1111-5
R
+ 121-8 S
+ 31*1
u
+ 96*5
V rs
E 7
15-375
+ 354-9
p
194-5 Q
A, 1093-5
R
+ 119-4 S
+ 31 *3
u
+ 97*5
V =
E 7
15-5
+ 346-5
p
190-9 Q
t 1074*1
R
+ 116-9 S
+ 31-4
u
+ 98*3
V =
E 7
15*625
+ 338-2
p
187 -o Q
+ 1054 *0
R
+ 114-6 S
+ 31*7
u
+ 99*1
v =
E 7
15*75
+ 330-0
p
189*9 Q
+ 1032 -9
R
+ 112-3 S
+ 31*9
u
+ 99*8
V =
E 7
15*875
+ 321 -6
p
173-8 Q
+ 1010*9.
R
+ 109-6 S
+ 31-9
u
+ 100*3
V =
E 1
16 '0
+ 313-2
p
-174-6Q
+ 988-0
R
+ 106-9 S
+ 31*9
u
+ 100*7
V S3,
E 7
16-25
+ 297-3
p
165 '9 Q
+ 941 *0
R
+ 101-8 S
+ 32-1
u
+ 101*5
V =
E 7
16*5
+ 281*7
p
157-3 Q
+ 892-8
R
+ 96 -6 S
+ 31 -8
u
+ 101*7
V
E 7
16*75
+ 266-9
p
149*0 Q
+ 843*8
R
+ 91 -6 S
+ 31-9
u
+ 101*8
V S3.
E 7
17'0
+ 252-5
p
s141 *0 Q
+ 795-4
R
+ 86-5 S
+ 31-5
u
+ 101 -5
V =
E 7
17-25
+ 239-4
p
132-9 Q
+ 748*8
R
+ 81 -9 S
+ 31-5
u
+ 101-2
V S3,
E'
17*5
+ 226-3
p
125*4 Q
+ 703-6
R
+ 77'2 S
+ 30*6
u
+ 100*4
V S3
E 7
17-75
+ 214-4
p
117-6 Q
+ 661 -0
R
+ 73 -o S
+ 30-4
u
+ 99-7
V =
E 7
18-0
+ 202*8
p
110-1 Q
+ 620-1
R
+ 68 -6 S
+ 29-7
u
+ 98-7
V S3.
E 7
18 25
+ 192*3
p
102*9 Q
+ 581-7
R
+ 64 -8 3
+ 29-2
u
+ 97-6
Vra
E 7
18-5
+ 182*2
p
9* "3 Q
+ 345-0
R
+ 61 -0 S
+ 28*4
u
+ 96*4
V =
E 7
18-75
+ 172-9
p
89-9 Q
+ 510-4
R
+ 57-6 S
+ 27*9
u
+ 95-3
V *==.
E 7
19-0
+ 164 -2
p
84-0 Q
+ 477-8
R
+ 34-3 S
+ 27-4
u
+ 94-2
V =
E 7
19-25
+ 155-9
p
78 -6 Q
+ 446*8
R
+ si -a S
+ 26*7
u
+ 92*8
V =:
E 7
19-5
+ 148-1
p
73-7 Q
+ 417-7
R
+ 48-2 S
+ 26*1
u
+ 91-5
V S3
E 7
19-/5
+ 141*
69*1 Q
+ 390*6
ft
+ 45 '6 S
+ 25*5
u
+ 90-1
V =
E 7
6
p
-
ft 9
252
On the Elements of the Orlit of Halley's Comet,
TABLE
XI
. continued.
Date.
Equations of Condition
dependent upon Right Ascensions.
1835.
Oct. 19-75
20-0
20-25
//
+ 31 -8
+ 33 -8
+ 35*9
P
P
P
If
+ 204 -0
+ 196-3
+ 189-3
Q 1139-1
Q 1097 -5
Q 1057-5
R
R
R
n
62-5 S
59-5 S
56-2 S
II
2 *2
2-6
2-6
U
U
U
II
17-6 V
17-6 V
17-4 V
= E
= E
= E
20'5
+ 37 '9
P
+ 182-6
Q 1019-4
R-
53*2 S
2 *4
U -
17-0 V
= E
20-75
+ 39-8
P
+ 176-2
Q-
983*4
R
50-1 S
2-3
U
16-6 V
= E
21 *0
+ 41 *2
P
+ 170-1
Q-
949-8
R-
47-5 S
2-4
U -
16-5 V
= E
21*25
+ 42-8
P
+ 164-3
Q-
916-9
R
44*7 S
2-1
U -
16-3 V
= E
21*5
+ 44 *0
P
+ 158-9
Q-
886*4
R-
42*4 S
2-4
U -
16-0 V
= E
21 *75
+ 45*5
P
+ 153-6
Q-
856*9
R
39*8 S
2-3
U -
15-9 V
= E
22*0
+ 46-4
P
+ 149-0
Q-
829*0
R
37-9 S
,2-5
U
15-7 V
= E
22*25
+ 47*9
P
+ 144*4
Q-
802*0
R
35-6 S
2*1
U
15-5 V
= E
22*5
+ 48*3
P
+ 139*9
Q-
777-0
R-
34*0 3
2 *4
U
15-3 V
= E
22*75
+ 49-6
&
+ 135-5
Q~
752*8
R-
31*7 S
2*2
U
15-1 V
= E
23*0
+ 50*3
P
+ 131-4
Q-
729-9
R
30-2 S
2*2
U
14*8 V
=2 E
23'25
+ 61 *2
P
+ 127*2
Q-
707-8
R
28-2 S
2*0
U -
14-6 V
= E
23-5
+ 51 *6
P
+ 123 -4
Q-
687 o
R
26*8 S
2*1
U -
14-4 V
= E
23*75
+ 52*6
P
+ 120*2
Q-
666-7
R
24 '9 S
1*9
U -
14-1 V
= E
24*0
+ 53*0
P
+ 116*6
Q-
647-6
R
23*6 S
2*1
U
14-0 V
= E
24*25
+ 53*8
P
+ 113-2
Q-
629-0
R-
22*0 S
I *8
U
14-0 V
= E
24*5
+ 54*1
P
+ 110-2
Q-
611-8
R
20*9 S
2*2
U
13-9 V
= E
24*75
+ 54*9
P
+ 107-1
Q-
594*7
R-
19-2 S
1-9
U -
13-6 V
= E
25*0
+ 55 M
P
+ 104*2
Q-
578*8
R
18*3 S
2 *2
U -
13-5 V
= E
25*25
+ 55-7
P
+ 101. *5
Q-
562*8
R
16*8 S
1 -8
U
13-3 V
= E
25-5
+ 56 -0
P
+ 98*8
Q-
548*3
15*9 S
2-1
U
13-1 V
= E
25*75
+ 56*6
P
+ 96-0
Q-
533 -8
R
14*4 S
1 -8
U -
13-0 V
= E
26 *o
+ 66-7
P
+ 93-4
Q-
520*4
R
13*6 S
2-1
U
12-9 V
= E
26-25
+ 57*4
P
+ 90-9
Q-
507*1
R
12-3 S
1-7
U -
12-4 V
r= E
26-5
+ 57*3
P
+ 88*5
Q-
494-8
R-
11-8 S
2-0
U
12-6 V
= E
26-75
+ 58*0
P
+ 86*3
Q-
482-3
R
10-5 S
1-7
U
12-4 V
= E
27-0
+ 58*1
P
+ 84*3
Q-
470-8
R-
9*8 S
1-9
u-
12 -3 V
= E
27*25
+ 58*7
P
+ 82*4
Q-
459-0
R
8*6 S
1-5
u
11-9 V
= E
27-5
+ 58-5
P
+ 80*4
Q-
448-6
R
8*1 S
1-9
U -
12-0 V
= E
27*75
+ 59-2
P
+ 78-7
Q-
437-7
R
6*9 S
1-6
U
11-5 V
= E
' 28-0
+ 59-1
P
+ 77-1
Q-
427*9
R
6-4 S
1-8
U
n-6 V
=r E
28 '25
+ 59-8
P
+ 75-6
Q-
417-7
R
5-28
1 -2
u-
11-2 V
== E
28*5
+ 59-4
P
+ 73-6
Q-
409-0
R
5*0 S
1*9
u
11*4 V
s E
29*5
i==
+ 60 -4
.M
P
MB
+ 67-5
'
Q-
375-4
R-
2-28
1*6
U -
10'8 V
= E
at its appearance in the Years 1835 fy 1836.
253
TABLE XI. continued.
Date.
Equations of Condition dependent upon Declinations.
1835.
Oct. 19'75
20 '0
20 '25
20 '5
20 -75
21-0
21 *25
21 '5
21 '75
22 '0
22-25
22 '5
22 '75
23'0
23-25
23'5
23-75
24-0
24-25
24-5
24 '75
25-0
25 -25
25'5
25-75
26 -0
26 '25
26-5
26-75
27 -o
27-25
27-5
27-75
28 -0
28 '25
28 '5
29 5
// // // // // //
+ 141-6 P 69-1 Q + 390-6 R +45-6 S +25-5 U + 90 *i V
+ 134-0 P 64-7 Q + 364 '7 R +43-0 S +24 '7 U + 88 '8 V.
+ 127 -8 P 60 '7 Q + 340 -8 R + 40 '8 S + 24 '2 U + 87 '4 V
+ 121-8 P
+ 116-3 P
+ 110-9 P
+ 106-3 P
+ 101-5 P
+ 97 '4 P
+ 93 -1 P
+ 89 -3 P
+ 85 -6 P
+ 82 -3 P
+ 79 -0 P
+ 76 "2 P
+ 73 '2 P
+ 70 '5 P
+ 67 -6 P
+ 65 -6 P
+ 63 -3 P
+ 61-1 P
+ 59 -0 P
+ 57-1 P
+ 55 -3 P
+ 53 -6 P
+ 51-9 P
+ 50 -3 P
+ 48 -6 P
+ 47 -4 P
+ 46-1 P
+ 44 -8 P
+ 43 -5 P
+ 42 -3 P
+ 41-1 P
+ 40-1 P
+ 39 -0 P
E'
E'
E 7
E'
E'
E'
E'
E'
E'
E'
E'
E'
E'
E'
E'
E'
E 7
E'
E'
57-0 Q + 318*0 R +38*5 S +23*3 U + 86 -0 V =
53-3 Q + 297-1 R + 36-5 S +22-8 U + 84 '6 V =
49-7 Q + 277-4 R +34-3 S +21-9 U + 83*8 V =
46-2 Q +259-4 R + 32*8 S +21*5 U + 82 -1 V =
43-0 Q + 242*3 R + 30-9 S +20-8 U + 80 '8 V =
39-8 Q + 226*3 R +29 '5 S +20-4 U + 79-6 V =
37-0 Q + 211*4 R + 27*8 S +19-7 U + 78 *3 V =
34-3 Q + 197-6 R + 26-6 S +19-2 U + 77-3 V =
31-9 Q +184-3 R + 25*1 S +18-6 U + 75 '9 V =
29 -9 Q + 172 -o R + 23 *9 S +18 -i U + 74 '7 V =
27-9 Q + 160-3 R +22-8 S +17-5 U + 73-6 V =
26-0 Q + 149-5 R +.21-8 S +17-1 U + 72-6 V =
24*3 Q + 139*2 R +.20-7 S +16-5 U + 71'5 V =
22-7 Q + 129-5 R + 19 '7 S +16-0 U + 70-3 V =
21 -2 Q + 120 -3 R + 18 -8 S + 15 -5 U + 69 '3 V =
19-2 Q + 111-8 R + 18-1 S +15-2 U + 68-3 V =
17-9 Q + 103-7 R + 17-2 S +14-6 U + 67-3 V =
16-6 Q + 96-3 R + 16-5 S +14-3 U + 66-4 V =
15-3 Q + 89'3 R + 15-7 S +13-8 U + 65 *4 V =
14-1 Q + 82-8 R + 15-1 S +13-4 U + 64 '3 V =
is-i Q + 76'7 R + 14-6 S +13-0 U + 63-7 V =
12-2 Q + 70-7 R + H-o S +12-6 U + 62-7 V =
11 -4 Q + 65 -2 R + 13 -4 S + 12 -2 U + 6l '9 V =
10-6 Q + 59-8 R + 12-9 S +11-8 U + 61-0 V =
10 -0 Q + 54 -7 R + 12 -2 S + 11 -3 U + 60 -0 V . ==
9 -0 Q + 50 -3 R + 12 -0 S + 11 -2 U + 59 -5 V =
8 -1 Q + 45 '9 R + 11 -6 S + 10 -8 U + 58 '7 V =
7 -5 Q + 4i -8 R + 11 -i S + 10 -4 U + 57 -9 V =
6-7 Q + 37-9 R + 10-7 S +10-1 U + 57-2 V =
6 -i Q + 34 -3 R + 10 -4 S + 9 '7 U + 56 -4 V =
5-4Q+ 30-7 R + 9-9 S + 9'3 U + 55-7 V =
4 -7 Q + 27 -6 R + 9 '7 S + 9 -2 U + 55 -o V =
r- 4 -1 Q + 24 -3 R + 9 '3 S + 8 '7 U + 53 -8 V rr
+ 35 -4 P 2 -1 Q + 13 -4 R + 8 -2 S + 7 '6 U + 51 -7 V = E A
E'
E f
E'
E'
E'
E'
E 1
E'
E'
E'
E'
E 1
E'
E'
E'
254
On the Elements of the Orbit of Halley's Comet,
TABLE XI
. continued.
Date.
Equations of
Condition dependent upon Right Ascensions.
1835.
w
n
n
II
II
Oct. 29 -5
4- 60-4
P 4-
67-5 Q
375 -4
R
2-28
-6
U
10-8 V
" E
30 '5
4- 61-2
P4-
62-2 Q
345*9
R
+
0-3 S
-4
U -
10-4 V
sr E
31-5
4- 6i -9
P 4-
57-4 Q
320*7
R
+
2-4 S
-4
U
9-9 V
= E
Nov. 1-5
4- 6e-5
P 4-
53 -8 Q
298*8
R
4-
4-3 S
-3
U -
9-5 V
17
2-5
+ 63-0
P -f
50-2 Q
- 279*9
R
4-
6'0 S
-o
u
9'2 V
= E
3-5
4- 68-7
P 4-
47 '5 Q
- 263-3
R
+
7-6 S
-0
u
8-7 V
E
4-5
4- 64-2
P4-
44-8 Q
249*8
R
+
8*9 S
1 *0
u
8-2 V
E
5-5
4- 64-7
P +
42-8 Q
2S/-8
R
+
C
0-8
u
7-7 V
=z E
6-5
4- 64-9
P 4-
41-0 Q
228*2
R
4-
11-1 S
0-8
u
7'5 V
= E
7'5
4- 65-6
P 4-
39'5 Q
220*1
R
4-
12-3 S
0-8
u-
7-3 V
= E
8-5
4- 66-1
P -f
33-2 Q
2] 3 -5
R
4-
13-3 S
0-7
u
6-9 V
= E
9'5
+ 66-5
P 4-
37-5 Q
208 *5
R
4-
14-1 S
0-6
u
6-7 V
= E
10-5
4- 66-8
P 4-
36 -7 Q
204 *9
R
4-
15*0 S
0-6
u
6-3 V
= E
11-5
4- 67-0
P4-
36-1 Q
202 -2
R
4-
15-6 S
0-4
u
6-1 V
= E
12-5
4- 67*7
P 4-
35-8 Q
200*6
R
4-
16-8 S
0-5
u -
5-9 V
as E
13-5
+ 67-7
P4-
35-8 Q
* 200-0
R
+
17-0 S
o-i
u
5-5 V
= E
14-5
+ 68-0
P 4-
36-1 Q
200-3
R
4-
17-6 S
0-4
u
5-4 V
s= E
15-5
4- 68-1
P +
36-0 Q
i 201 -2
R
4-
18*3 S
0-2
u -
4-9 V
= E
16-5
4- 68-0
P 4-
36-1 Q
203*1
R
+
18-7 S
0-3
u -
4-9 V
= E
17*5
4- 68-1
P +
36 -7 Q
205 -3
R
4-
19 '0 S
0-2
u
4-5 V
r E
18*5
4- 68-1
P -f
37-2 Q
208 -1
R
4-
19'9 S
O'l
u-
4-3 V
!T E
19-5
4- 68*0
P 4-
37'7 Q
211 -2
R
4-
20-2 S
*
__
4-0 V
"P
20-5
4- 67-9
P 4-
38-2 Q
214-9
R
4-
21*1 S
o-i
u-
3-8 V
= E
21'5
4- 67'6
P 4-
38 '8 Q
218 -9
R
4-
20*9 S
o-i
u-
3-7 V
s E
22-5
4- 67-3
P 4-
39 '3 Q.
222 '9
R
+
21-2 S
0-2
u
3-3 V
= E
23-5
+ 67-0
P 4-
40-3 Q
226 '9
R
4-
21-7 S
*
3'2 V
= E
24-5
4- 66-7
P +
41-2 Q
831/3
R
4-
22-0 S
4-0-2
u -
3-1 V
= E
25-5
4- 66-4
P4-
42 '0 Q
235-7
R
+
22-3 S
4- o-i
u -
2-9 V
= E
26-5
4- 65-8
P 4-
42-6 Q
240-2
R
4-
22-6 S
*
2-8 V
= E
27-5
4- 65-4
P 4-
43-4 Q
244 *1
R
+
22*9 S
*
2*6 V
= E
28-5
4- 64-9
P 4-
44 X) Q
- 248 '7
R
4-
23-2 S
*
_
2-4 V
= E
29-5
4- 64-4
P 4-
44 -7 Q
252-7
R
4-
23-5 S
+ o-i
u -
2-2 V
rr E
30-5
+ 63-9
P +
45-4 Q
36 r 6-9
*
+
23-8 S
4-0-1
u
2-1 V
= E
Dec. 1-5
4- 63-5
P +
46-3 Q
261 -o
R
4-
24-3 S
4- 0-2
u -
1-8 V
= E
2*5
4- 62 -9
P 4-
47-1 Q
264-7
R
4-
24-3 S
4- o-i
u -
1-7 V
= E
3-5
4- 62-2
P 4-
47 '7 Q
268-6
R
4-
24-6 S
*
.
1-6 V
= E
4-i
+ 61-7
P 4-
48-4 Q
272-7
R
4-
24-8 S
4-0-2
u -
1-4 V
s= E
at Its appearance in the Years 1835 fy 1836.
255
TABLE
XI. continued.
Date.
Equations of Condition
dependent
upon Decimations.
1835.
u
fl
a
a
Oct. 29 -5
+
35-4 P
2-1 Q 4-
13 -4 R
4-
8-2 S
4* 7 *6
u +
5i-7 V
=r EJ
30-5
4-
32-3 P
o-s Q 4-
4-6 R
4-
7-4 S
4-6-6
U 4-
49-3 V
C"""t Ft
31-5
+
29-6 P
4-
0-5 Q
2-6 R
4-
6-7 S
4-5-6
U +
47-1 V
s= E'
Nov. i *5
4-
27'4 P
4-
1-6 Q
8-4 R
+
6-1 S
4- 4-7
U +
44-9 V
= E'
2-5
4-
25-5 P
4-
2-5 Q
13-1 R
4-
5-7 S
4- 4-1
U 4-
43-1 V
rr E'
3 '5
4-
23 '7 P
4-
3-0 Q
17-1 R
+
5-2 S
4- 3-i
U 4-
41-2 V
C5 E'
4-5
4-
22-4 P
4-
3-7 Q
20'4 R
+
4-9 S
4-2-5
U 4-
39-6 V
cs E ;
5-5
21-2 P
4-
4-2 Q
23-1 R
4-
4-78
4- 1 *9
U +
38 '0 V
ns E'
6-5
+
19 '7 P
4-6Q -
25-6 R
4-
4-6 S
4- 1-2
U 4-
36-4 V
E'
7-5
+
19-1 P
4-
5-0 Q
27-6 R
4-
4-3 S
4- 0-7
U 4-
35-0 V
= E'
8-5
4.
18-3 P
4-
5-3 Q
29'4 R
4-
4-2 S
4- o-i
U 4-
33 '6 V
=5 E'
9'5
+
17'5 P
4-
5'5 Q
31-1 R
4-
4-0 S
0-4
U4-
32'3 V
S3 E'
10-5
+
16-8 P
4-
5-8Q-
32 -6 R
4-
3-9 S
0-9
U 4-
31 -0 V
5= E'
11 -5
4-
16-0 P
4*
5-8 Q^
34-1 R
4-
3-7 S
1 -5
U 4-
29-6 V
S3 E'
12 '5
+
15 '6 P
4-
6-2 Q .
35'0 R
+
3 -a s
1-7
U 4-
28-6 V
=3 E'
13-5
+
15-1 P
4-
6-4 Q
36-3 R
4-
3-8 S
1-8
U 4-
27-5 V
S3 E'
14-5
4-
14-6 P
4-
6-8 Q
37-3 R
+
3-8 S
2'5
U 4-
26 -4 V
S3 E'
15 '5
14-2 P
4-
6 -8 Q
3d "2 R
+
3-8 S
2-9
U 4-
25 -3 V
S3 E'
16 '5
4-
13 -9 P
4-
7-1 Q -
39-3 R
4-
3-9 S
3-1
U4-
24-3 V
=3 E ;
17'5
4-
13-6 P
4-
7 ; 3Q
40-0 R
4-
3-9 S
3-4
U 4-
23-3 V
S3 E'
18 '5
+
13 "2 P
4-
7'4 Q-
41'2 R
4-
3* S
3-8
U 4-
22'3 V
S3 E'
19'5
+
12 -a P
4-
7-5 Q
42-2 R
4-
3-8 S
4-1
U 4-
21-3 V
23 E'
20-5
4-
12 '4 P
4-
7'7 Q~
43-2 R
4-
3 -a S
4*4
u +
20 % V
S3 E'
81 -5-
-f
1'2 ! P
4-
7-8 Q
44-1 R
4-
3-) S
47
U 4-
19-3 V
S3 E'
22 '5
4-
11 "9 P
4-
7'7 Q-
44-7 R
+
3-8 S
4-6
U 4-
18 'i V
=s E'
23-5
4.
11 -6 P
4-
8-2 Q
45-7 R
4-
3-9 S
5-1
U 4-
17 * V
S3 E'
24'5
4-
11-3 P
4-
8-3 Q
46-6 R
4-
3-3 S
5-4
U 4-
!* V
S3 E'
25-5
+
11-0 P
4-
8'5 Q
47-4 R
4-
3-9 S
5-6
U 4-
157V
= E'
26-5
_l_
10-7 P
4-
8-4 Q
48-3 R
4-
3-98
5-8
U 4-
14-8 V
5= E'
27'5
4-
10 -5 P
4-
8-7 Q
48-9 R
4-
3-98
-6-1
U +
14'* V
= E ;
28-5
+
10-5 P
4-
8-8 Q
49-7 R
4-
4-0 S
6-2
U 4-
13 -1 V
= E'
29-5
+
10-0 P
4-
8-9 Q
50'3 R
4-
4-0 S
6-4
U 4-
12-3 V
S3 E'
30-5
+
9-8 P
4-
9-1 Q -
51 -0 R
+
4-1 S
6-6
U 4-
u* V
S3 E 7
Dec. i -5
4-
9'6 P
4-
9-2 Q -
51 -8 R
4-
4-2 S
6-8
U 4-
10-7 V
S3 E'
2-5
4-
9'4P
4-
9'3 Q -
52-4 R
4-
4-1 S
7'0
U 4-
9-9 V
S3 E 7
3-5
9-1 P
4-
9>4Q-
53*1 R
4-
4-1 S
7 '2
U4-
9-0 V
= E'
4-5
4-
8-9 P
4-
9'5 Q-
53-8 R
4-
4-2 S
7-2
U 4-
8-3 V
= E/
256
On the Elements of the Orbit of Halleys Comet,
TABLE
XI
. continued.
Date.
Equations of Condition
dependent upon Right Ascensions.
1835.
//
II
N
It
a
Dec. 4-5
+ 61-7
P +
48-4
Q-
272-7
R +
24-8 S
4-0-2
U
1-4 V
= E
5-5
4- 6l -2
P +
49-3
Q-
275-4
R +
25-2 S
4-0-2
U
1-1 V
= E
6-5
4- 60-6
P +
49-5
Q-
279-1
R +
25-4 S
4-0-1
U
1-1 V
= E
7-5
+ 59'9
P +
50'1
Q-
282-7
R +
25-6 S
4-0-1
U
i-o V
T?
8-5
+ 59-5
P +
50-8
Q-
285-6
R +
25-9 S
4-0-1
u
0-8 V
= E
9'5
+ 58-9
P +
51 -5
Q-
289-1
R +
26-1 S
4- o-i
u
0-7 V
= E
10'5
4- 58-3
P +
51 '6
Q-
292-1
R +
26-3 S
4-0-1
U ~
0-7 V
= E
11-5
+ 57*9
P +
52-4
Q-
295 -1
R +
26-6 S
4-0-2
U
0-5 V
= E
12 '5
+ 57'5
P +
52-7
Q-
297'7
R +
26-8 S
4-0-2
u-
0-3 V
TJ 1
13*5
+ 56-9
P +
53 '3
Q-
300-6
R 4-
27 '1 S
4- o-i
u
0'3 V
= E
14-5
4- 56-5
P +
54-0
Q-
303-5
R +
27-3 S
4-0*2
u
o-i V
= E
15 '5
+ 55-9
P +
54*4
Q-
306-1
R +
27-5 S
4-0'2
u
*
TT
16-5
+ 55-5
P +
64'7
Q-
308-7
R +
27-8 S
4- o-i
u +
o-i V
= E
17'5
+ 55-1
P +
55 '4
Q-
311 -2
R +
28-1 S
4-0-2
U 4-
0-3 V
= E
18 -5
+ 54-7
P +
56 -0
Q-
313-6
R +
28*3 S
4- o-i
u +
0-3 V
= E
19-5
+ 54-2
P +
56-6
Q-
316-6
R +
28*4 S
*
+
0-4 V
= E
20-5
4- 53-9
P +
56-8
Q-
318 '9
R +
28-8 S
*
_J.
0'5 V
= E
21 -5
+ 53-5
P +
57-0
Q-
321-9
R +
29*0 S
*
+
0-6 V
= E
22-5
+ 52 '9
P +
67'3
Q-
324-2
R +
29-1 S
0-3
u +
0-4 V
~ E
23-5
4- 52-9
P +
58*3
Q-
326-6
R +
29-6 S
*
0-7 V
= E
24-5
4- 52-5
P +
59-0
Q-
328 -9
R +
29'9 S
4- o-i
u +
0-9 V
= E
25-5
+ 52'2
P +
58*9
Q-
331 -5
R +
30-1 S
+ o-i
u +
1-1 V
= E
26-5
+ 51-9
P +
59'7
Q-
334-1
R +
30-5 S
*
+
i-o V
= E
27'5
+ 61-6
P +
60-0
Q-
336 -7
R +
30-7 S
*
1-1 V
= E
28*5
4- 51*3
P +
60-5
Q-
339-2
R +
31*0 S
O'l
u +
1-2 V
= E
29 '5
4- 51 !
P +
61 -o
Q-
341 -8
R +
31-4 S
o-i
U 4-
1-3 V
= E
30-5
4- 50-9
P +
61-5
Q-
345 -2
R +
31-7 S
o-i
u +
1-4 V
= E
31-5
+ 50-7
P +
62-3
Q-
347-6
R +
32 '0 S
*
+
1-5 V
= E
1836.
Jan. i -5
+ 50'4
P +
62-0
Q-
351 -2
R +
32'3 S
o-i
U 4-
5 V
= E
2-5
4- 50-3
P +
62-6
Q-
353-5
R +
32-6 S
0-2
U +
6 V
= E
3-5
4- 50-1
P +
63-6
Q-
356 -0
R +
33'0 S
0-2
U +
7V
= E
4-5
4- 49-9
P +
63-7
Q-
359 '5
R4-
33-3 S
0-2
U +
8 V
=r E
5-5
+ 49-8
P +
65-0
Q-
362 -o
R +
33-7 S
0-2
u +
8V
= E
6-5
4- 49'7
P +
65-2
Q-
365-8
R +
34-1 S
o-i
U +
2-0 V
= E
7 '5
+ 49-5
P +
65-6
Q-
368 '6
R +
34-5 S
0-3
U 4-
2-0 V
= E
8 -5
4- 49'4
P +
65-7
Q-
372-4
R +
34-9 S
0'3
U +
1-9 V
= E
9'5
+ 49-3
P +
67-1
Q-
375 -1
R +
35-3 S
0-3
U 4-
2-1 V
= E
at its appearance in the Years 1835 # 1836.
257
TABLE XL continued.
Date.
Equations of Condition dependent
upon Declinations.
1835.
Dec. 4 -5
4
8-9P 4
9'5 Q
53'8
R 4
4-28
7-2
U 4
81 V
E 7
5-5
4
8'8 P 4
9'8 Q
53 '5
R4
43 S
7'3
U 4
7-5 V
:
E 7
6-5
4
8'5 P 4
9'8 Q
55 '0
R 4
4'3 S
7'6
U 4
6-7 V
=
E 7
7-5
4
8'4 P 4-
9'9 Q
55-5
R4-
4'4 S
7*7
U 4
6-0 V
E 7
8 '5
4
8'2 P 4
10 o Q
56 !
R 4
4'5 S
7'9
U 4
5-3 V
r-
E 7
9 '5
4
8-1 P 4
10'2 Q
- 56-7
R4
4*4 8
8'0
U 4
4-5 V
!"**
E 7
10 '5
4
7-8 P 4
10*1 Q
57 '7
Ri
T
4 '5 S
8 '2
U +
3 7 V
-
E'
11 *5
4
7'8 P 4
10 '4 Q
57'8
R 4
4-6 S
8 '3
U 4
3-1 V
=
E 7
12-5
4
7'6 P -f
10-5 Q
58 '4
R 4
4 7 S
8*4
U 4
2-4 V
rs
E 7
13-5
4
7'4 P +
10-6 Q
59 '1
R 4
4-7 S
8'6
U +
1-6 V
=
E 7
14*5
4
7'3 P 4
10-7 Q
~ 59'7
R 4
4-78
8-7
U 4
0-9 V
:
E'
15'5
4
7-i P 4
io-7 Q
60 '2
R4
4- 9 S
8 '8
U +
o-i V
f
E 7
16-5
4
7-o P 4
10-9 Q
60 '7
R 4
5'1 S
9*0
U -
0-6 V
=
E 7
17-5
4
6 8 P 4
11-0 Q
61 5
R4
5'0 S
9'1
U
1-3 V
r"
E ;
18'5
4
6 -8 P 4
11-2 Q
62*0
R 4
5'1 S
9'2
U
2-0 V
~
E 7
19'5
4
6-6 P 4
11'3 Q
62*6
R4
6'2 S
9*3
U
2-6 V
.
E 7
20'5
4.
6-5 P 4
11-4 Q
63-2
R 4
5'2 S
9'5
U
3'3 V
=
E 7
21-5
4-
6-4 P 4
11-5 Q
64 '0
R 4
5'4 S
9'5
U
4-0 V
=
E 7
22 '5
+
6-0 P 4
11 '4 Q
64-8
R 4
5-28
*r-iO*0
U
5-0 V
...
E 7
23'5
4
6-1 P 4
n-7 Q
65'3
R 4-
5-4 S
9'9
U
5'4 V
~
E 7
24-5
+
6-1 P 4
12-1 Q
65'8
R4
5'6S
10-1
U
6-1 V
^*
E 7
25*5
4
6-0 P 4
12'0 Q
66*4
R4-
5-78
10 '0
U
6-8 V
=
E 7
26 '5
4
5'8 P 4
12-1 Q
67-2
R 4
5- 9 S
10-2
U
7-5 V
~
E 7
27-5
4
s-7 P 4
12-3 Q
67-6
R 4-
6-0 S
10-3
U
8-1 V
T***
E 7
28 '5
4
5-sP 4-
12-3 Q
68'5
R 4
6-0 S
10'5
U -
9-0 V
~
E 7
29'5
4
5-4 P 4-
12'4 Q
69-2
R 4
6-28
10 '6
U
9 . 7 v
-s
E 7
30-5
4
5-4 P 4
12-6 Q
69-9
R4-
6-3 S
10 -7
U
10-3 V
=
E 7
31 -5
4
5-0 P 4
12-7 Q
70'6
R 4
6-38
10 '8
U -
11-0 V
.
E 7
1836.
Jan. i '5
4
6'2 P 4
13-0 Q
71-1
R +
6*5 S
10-9
U
11-7 V
r=
E 7
2 '5
4
5-0 P 4
12-9 Q
72-0
R 4
6-5 S
11 -o
U
12-5 V
r~*
E 7
3-5
4
5-1 P 4
13-3 Q
72-4
R 4-
6-9 S
11 -0
U -
13-0 V
E 7
4-5
4
4-9 P 4
13-2 Q
73 '3
R 4
6-98
11 '2
U
13 -9 V
E 7
5'5
4
4-7 P 4
13-4 Q
73-9
R 4
7-0 S
11 '5
U
14-6 V
=
E 7
6 '5
+
4-6 P 4
13-5 Q
74-7
R4
7-1 S
11-6
U
15-4 V
-
E 7
7 '5
4
4 -5 P 4
13-6 Q
75'4
R 4
7-28
11 '8
U
16-2 V
=r
E 7
8 '5
4
4-1 P 4
13-7 Q
76-0
R4
7-48
11 '8
U -
16-8 V
~~
E 7
9 "5
+
4'4 P 4
14 '0 Q
76-6
R 4
7-sS
12'0
U
17-5 V
=
E'
258
On the Elements of the OMt of Hdleys Comet,
TABLE XL continued.
Date.
Equations of Condition dependent upon Right Ascensions.
1836.
//
II
j
n
II
II
Jan. 9 '5
+ 49-3
P
+ 67-1
Q
375-1
R +
35-3 S
0-3
U +
2-1 V
== E
10-5
+ 49-2
P
+ 67-9
Q
378 -7
R +
35 '7 S
0'4
U +
2-1 V
= E
11 '5.
+ 49-1
P
+ 68-7
Q
382-3
R +
36-2 S
0-4
u +
2-1 V
E
12-5
+ 49-0
P
+ 68-7
Q
686-8
R +
36-8 S
0-4
u +
2-2 V
= E
13-5
+ 49 -o
P
+ 69-3
Q
390*7
R +
37-3 S
0-4
u +
2-3 V
= E
14-5
+ 49-0
P
+ 70-3
Q
394*8
R +
37-78
0-5
u +
2-5 V
= E
15-5
+ 49-o
P
+ 70-7
Q
399'1
R +
38-2 S
0-4
u +
2-4 V
= E
16-5
+ 48-8
P
+ 71 -2
Q
403 -3
R +
38-8 S
0-4
u +
2-3 V
= E
17 '5
+ 48-3
P
+ 72-9
Q
407 -6
R +
39-3 S
0-5
u +
2-5 V
2= E
18-5
+ 4-9
P
+ 73*7
Q
411-3
R +
40-1 S
0-4
u +
2-6 V
= E
19-5
+ 48 -8
P
+ 74-2
Q
4l6'7
R +
40-4 S
0-7
u +
2-6 V
= E
20-5
+ 48 -8
P
+ 74-6
Q
421 "9
R +
40-1 S
0-6
u +
2-5 V
= E
21 -5
+ 48'9
P
+ 75-7
Q
426-9
R +
41-8 S
0*7
u +
2-6 V
ZZ2 E
22'5
+ 48-8
P
+ 76-6
Q
431 -9
R +
42-3 S
0-8
u +
2-5 V
= E
23-5
+ 48*7
P
+ 77-7
Q
437-5
R +
43-1 S
0-5
u +
2-6 V
s= E
24-5
+ 48 -8
P
+ 79-0
Q
443*0
R +
43-7 S
1*1
u +
2-4 V
3= E
25*5
+ 49-0
P
+ 80*4
Q
448 *3
R +
44-7 S
0-9
u +
2-7 V
= E
26-5
+ 49-0
P
+ 81 -2
Q
453-8
R +
45*4 S
1-1
u +
2-6 V
= E
27-5
+ 49 '6
P
+ 82-0
Q
460-4
R +
46-2 S
l -o
u +
2-6 V
E
28-5
+ 49-0
P
+ 83-1
Q
466 '9
R +
47 -o S
1-0
u +
2-5 V
r= E
2$ -5
+ 49*0
P
+ 83-9
Q
473*2
R +
47 -8 S
1-2
u +
2-4 V
=s E
30-5
+ 49'1
P
+ 85-4
Q
'480 -3
R +
48 *'9 S
1-4
u +
2-5 V
ss E
31-5
+ 49-3
P
+ 86-4
Q
487-4
R +
49-7 S
1-3
u +
2-4 V
= E
1*5
+ 49-3
P
+ 87-6
Q
493-3
R +
50-9 S
1-3
u +
2-5 V
s E
2-5
+ 49-2
P
+ 89-3
Q
501-2
R +
51*6 S
1 "7
u +
2-2 V
= E
3-5
+ 49'3
P
+ 91 -1
Q
507 '3
R +
52*7 S
1 -5
u +
2-2 V
zzz E
4-5
+ 49'8
P
+ 91-5
Q
516-2
R +
53-7 S
I'/
u +
2-1 V
z E
5-5
+ 49-3
P
+ 93-7
Q
523 -3
R +
54-8 S
2-1
u +
2-0 V
= E
6-5
+ 49-4
P
+ 94-6
Q
531 -5
R +
55-9 S
2-0
u +
1-8 V
=5 E
7*5
+ 49-3
P
+ 95-9
Q
539-1
R +
67 '-0 S
- 2-2
u +
1-5 V
= E
8-5
+ 49-4
P
+ 97-6
Q
548-9
R +
58-3 S
2-2
u +
1-5 V
= E
9*5
+ 49'4
P
+ 99'3
Q
556-4
R +
59-5 S
2-5
u +
1-2 V
= E
10-5
+ 49'3
P
+ 101 *0
Q
564-8
R +
61-0 S
2-4
u +
1-2 V
= E
11-5
+ 49-3
P
+ 101 -7
Q
574 -o
R +
62-1 S
2-6
u +
0-9 V
= E
12-5
+ 49 '3
P
+ 104-1
Q
582-4
R +
63-4 S
2-8
u +
0-7 V
= E
13-5
+ 49-2
P
+ 105-9
Q
591*0
R +
65-0 S
2-9
u +
0-5 V
= E
14-5
+ 49'0
P
+ 107-1
Q
600-8
R +
66-0 S
3-1
u +
0-3 V
=s E
at its appearance in the Years 1835 Sf 1836.
259
TABLE
XI
. continued.
Date.
Equations of Condition dependent upon Declinations.
1836.
ii
II
it
a
ii
H
Jan. 9*5
4-
4'4 P
4 14-0
Q-
76-6
R +
7-5 S
12 -0 U
17'5
V -
E ;
10-5
+
4-2 P
4 14'0
Q-
77'4
R +
7-6 S
12-2 U
18 -3
V =:
I*
11'5
4-
4-1 P
+ 14'1
Q-
78'1
R4-
7'7 S
12-3 U
19-1
V =2
E'
12-5
+
3'9 P
4 14'2
Q~
78-8
R4-
7'9S
12-4 U
19'9
V =2
E 1
13'5
+
3-9 P
4 14-4
Q-
79'4
R 4
8-1 S
12-5 U
20-6
V =
E'
14-5
4-
3-7 P
4 14-5
ft..f
80-0
R 4
8-1 S
-12 -9 U
21 '2
V S3
E'
15-5
+
3-5 P
+ 14 '6
Q~
80-6
R4-
8-2 S
12-9 U
22 '2
V =
E'
16 '5
+
3-4 P
+ 14-7
Q-
81 '2
R 4
8'4 S
18'0 U
23 '0
V =
E'
17'5
+
2-8 P
4 14*9
Q-
81-7
R +
8'8 S
13-1 U
23-8
y .
E'
18 '5
4-
3-1 P
4 15-0
Q~
82-2
R 4-
8-8 S
13 '3 U
24-6
V S=5
E'
19'5
*
3-0 P
4 15 'I
Q-
82-7
R +
8-8 S
13-5 U
25'4
V =
E ;
20-5
4-
2-9 P
4 15'2
Q~
83'1
R +
9-0 S
13-8 U
26-3
V =
E'
21*5
4-
2-7 P
4 15-2
Q-
83 '4
R4-
9-1 S
13-7 U
27*1
V as
E'
22'5
+
2'5 P
4 15*3
Q-
83 '9
R +
9 *2 S
~14'0 U
28 -0
V =
E'
23'5
4-
2-4P
4- 15*4
Q-
84-1
R +
9-5 S
14-1 U
28-8
Y =
E 7
24-5
+
2-1 P
+ 15M
Q-
84'3
R4-
9-5 S
14-2 U
29-7
V s=
E'
25-5
4-
i-7P
4- 15-3
Q~
84'6
R4-
9-4 S
14-6 U
30-8
V =
E'
26-5
+
1-7 P
4 15 '5
Q~
84 4
R4-
9'7 S
-14-5 U
31 '5
V =
E'
27-5
4-
1-6P
4 15-6
Q~
84 '4
R +
9-8 S
^14 -7 U
32'4
V =
E ;
28 '5
4
1-3 P
4- 15'5
Q-
84*2
R 4-
9'9 S
14-8 U
33'3
V r=
E'
29'5
4-
0-9 P
4-15*4
Q~
84'1
R +
9-9 S
15'0 U
34-4
V SB
E 7
30 '5
+
0-8 P
+ 15-5
Q~
83-6
R +
10-2 S
15-1 U
35 -2
V =
E'
31 -5
+
0-6 P
4- 15-5
Q-
83-0
R +
10-3 S
15-3 U
36-0
V =
E'
Feb. 1-5
4-
0-3 P
+ 15-3
Q-
82'5
R 4-
10-2 S
15'5 U
37-2
Va=
E'
2'5
*
4- 15-2
Q~
81-6
R +
10-3 S
-15 -7 U
38 '0
V =
E ;
3 '5
0-3 P
+ 15-1
Q~
80 '6
R4-
10-4 S
15-7 U
39'1
V =
E 7
4-5
0-7 P
+ 14'8
Q-
79'5
R +
10'3 S
16-0 U
40-1
V =
E 7
5-5
1-0 P
4- 14-4
Q-
78-4
R4-
10-1 S
16-5 U
41
Y
E 7
6-5
1-3 P
4- 14'4
Q-
76-6
R 4-
10-3 S
16-3 U
42-1
V =
E 7
7-5
1-7P
4- 141
Q-
75 -o
R4-
10-1 S
16-6 U
43 '2
V =
E 7
8'5
2-1 P
4 13-6
Q-
72-8
R4-
10 -0 S
16-8 U
44 *3
V SS
E 7
9'5
2'4 P
4 13-4
Q-
70-6
R +
10-0 S
16-9 U
45'2
V s=
E-'
10 -a
2-8 P
+ 13 '0
Q~
68-1
R +
10 -o S
16-9 U
46 '3
V E=
E 7
11 '5
__
3-2 P
4 12*4
Q-
65*5
R +
9'6 S
17-3 U
47-5
V =
E 7
12-5
3-6P
4- 11 '8
Q-
62'5
R4-
9-4 S
17-5 U
48-6
V s=
E 7
13'*
4'2 P
4 11 '3
Q-
59*0
R4-
9-3 S
-17 '5 U
49'5
Y =
E 7
14-5
4'5 P
4 10-6
Q~
55'5
R 4-
9-2 S
-17 7 U
50-6
V r=
E'
260
On the Elements of the Orbit of Halley's Comet,
TABLE
XI. continued.
Date.
Equations of Condition dependent
upon
Right Ascensions.
1836.
//
II
n
a
Feb. 14 -5
+ 49-0
P
Q-
600-8 R
+ 66-0
S
3-1
u +
0-3 V
=r E
15-5
+ 49-0
P
+ 109*0
Q-
6os -7 R
+ 67-7
S
3-0
u
*
^ . -pi
16-5
+ 48-7
P
+ 111 -0
Q-
618-2 R
+ 69-2
S
3-5
u
0-5 V
= E
17-5
+ 48 -1
P
+ 112-7
Q-
628-1 R
+ 70-7
S
3-5
u -
0-8 V
= E
18-5
+ 48-4
P
+ 114-2
Q-
636-5 R
+ 72-3
S -
3-7
u
i-o V
= E
19'5
+ 48 -0
P
+ 116-2
Q-
645-1 R
+ 73-6
S
3-9
u
1-5 V
= E
20-5
+ 47-9
P
+ 117'3
Q-
654-5 R
+ 75-2
c
4-0
u -
1-9 V
= E
21'5
P
+ 119'0
Q-
663-1 R
+ 76'8
s
4-3
u -
2-4 V
= E
22-5
+ 47-2
P
+ 120-8
Q-
671 -o R
+ 78-4
s -
4'5
u -
2-8 V
= E
23-5
+ 46-7
P
+ 122-9
Q-
679-2 R
+ 79-7
s
4-7
u
3-3 V
= E
24-5
+ 46-3
P
+ 124-1
Q-
687 '6 R
+ 81-3
s
5 -0
u
3-8 V
= E
25-5
+ 45-8
P
+ 125-3
Q-
694-9 R
+ 82-6
s
5'2
u
4-5 V
= E
26-5
+ 45-2
P
+ 126 -9
Q-
702-4 R
+ 84-2
s -
5*2
u -
4-9 V
= E
27-5
+ 44-7
P
+ 127-8
Q-
709-1 R
+ 85-7
s
6-1
u
5-3 V
= E
28-5
+ 44 '2
P
+ 129-1
Q-
715-3 R
+ 87-4
s
6-2
u
5-9 V
=s E
29-5
+ 43-5
P
+ 129-9
Q-
721-1 R
+ 88-6
s
6-4
u
6-4 V
= E
Mar. 1-5
+ 42 -8
P
+ 131 -2
Q-
726 -s R
+ 89-9
s -
6-6
u -
7-0 V
= E
2-5
+ 42 -0
P
+ 132-1
Q-
731 -5 R
+ 91-2
s
6-3
u
7 -7V
= E
3-5
+ 41-2
P
+ 133-4
Q-
734 -9 R
+ 92-4
s
6-2
u -
8-1 V
= E
4-5
+ 40 -3
P
+ 134-5
Q-
738 -9 R
+ 93-1
s -
6-6
u
9-0 V
=r E
6-5
+ 39-7
P
+ 134-8
Q-
741 -5 R
+ 94-8
s -
6-5
u
9-2 V
= E
6-5
+ 38*6
P
+ 135-5
Q-
742 -9 R
+ 95-4
s
7-0
u -
10-0 V
= E
7-5
+ 37*5
P
+ 135-0
Q-
744 -4 R
+ 96-2
s -
7-1
u -
10-9 V
= E
8-5
+ 36 -7
P
+ 135-1
Q-
745 -3 R
+ 97*2
s
7-2
u
11-4 V
= E
9'5
+ 35-7
P
+ 135-4
Q-
745 -4 R
+ 98-1
s
7-4
u -
12-0 V
= E
10-5
+ 34-7
P
+ 136-5
Q-
743 '9 R
+ 98-8
g
7-4
u
12-4 V
= E
11-5
+ 33-6
P
+ 135 -3
Q-
742-5 R
+ 99-3
s
7-7
u -
13-2 V
T?
12-5
+ 32-5
P
+ 135-0
Q-
740-1 R
+ 99*7
s
7-8
u
13-8 V
~ E
13-5
+ 31-5
P
+ 134-7
Q-
737 -o R
+ 100-1
s
8-1
u
14-2 V
=r E
14-5
+ 30-5
P
+ 133-8
Q-
733-2 R
+ 100-2
g
7-9
u -
14-8 V
= E
15-5
+ 29*4
P
+ 133-6
Q-
728-5 R
+ 100-5 S
7-1
u
15-3 V
=E
16 '5
+ 28-3
P
+ 132-8
Q-
723-1 R
+ 100-6
g
7'9
u
15-6 V
= E
17-5
+ 27-3
P
+ 131 -4
Q-
717-8 R
+ 100-5
s
8 -0
u -
16-1 V
= E
18 -5
+ 26-2
P
+ 130-8
Q -
711 -2 R
+ 100-5
s
8-0
u -
16-7 V
= E
19-5
+ 25 -2
P
+ 129-7
Q-
704-9 R
+ 100-2
s
8-2
u
17-1 V
= E
20-5
+ 24-2
P
+ 128-5
Q-
697-8 R
+ 100 -2
s
8 -4
u -
17 -4V
== E
21 '5
+ 23-2
P
+ 126-9
Q-
690 -6 R
+ 99'8
s -
8-1
u
17-9 V
=r E
at its appearance in the Years 1835 < 1836.
261
TABLE XI
. ' continued.
Date.
Equations of Condition dependent
upon Declinations.
1836.
~"(i
i
II
n
u
w
II
Feb. 14-5
4
5P +
10-6
Q-
55*5
R +
9-28
17-7
U
50-6 V
= E ;
15-5
4
9P +
10-1
Q-
51'4
R +
8-6 S
17-8
U -
51-7 V
= E'
16-5
5
3 P +
9-3
Q-
46-6
R +
8-28
18-0
U
52-8 V
= E 1
17-5
6
oP +
8 -4
Q-
42 -2
R +
7-98
18-1
u -
53-7 V
= E ;
18-5
6
3P +
7-5
Q-
37-5
R +
7-sS
17-8
u
54-9 V
=r E 1
19-5
6
8 P +
6-5
Q-
32-1
R +
6-9 S
18-3
u
56-1 V
= E'
20*5
"""* 7
3 P +
5-6
Q-
26-1
R +
6-48
18 '5
u -
57-2 V
r=E'
21 -5
7
8P +
4-5
Q-
19'9
R +
5*8 S
18-6
u
68-2 V
= E'
22-5
8
4P +
3-3
Q-
13-6
R +
5-1 S
18-8
u
59-3 V
= E'
23*5
"" "" 8
9P +
1 -8
Q~
7-0
R +
4-2 S
18-9
u
60-6 V
= E'
24'5
9 -2 P +
0-9
Q +
0-6
R +
3-58
19 '0
u
61-3 V
= E'
25'5
- 9
8 P-
0-4
Q +
8*1
R-f
2*8 8
18 -9
u -
62-2 V
= E'
26-5
10
3 P
1-9
Q +
15-9
R +
1-8 S
19-1
u -
63-4 V
= E 7
27 -5
10
8 P
3-3
Q +
23 '9
R +
0-98
18-4
u
64-4 V
=: E 7
28-5
11
3P-
4-8
Q +
32-1
R
*
18-6
u -
65-4 V
= E 7
29-5
11
6 P
6-2
Q +
40-7
R
1-1 S
18-5
u
66-2 V
B E 7
Mar. i -5
12
1 P-
7-8
Q +
49-6
R-
2-1 S
18 -6
u
67-2 V
T?/
JCt
2-5
12
4 P
9-3
Q +
58-6
R-
3-4 S
19 '2
u -
68-0 V
T?/
- rj
3-5
12
9P-
10-9
Q +
67-4
Xv ~~~
4-3 S
^19-2
u
68-7 V
B E 7
4-5
13
2 P
12 '8
Q +
76-3
R-
5-7 S
19*4
u
69 -7V
= E'
5-5
14 -0 P
14-9
Q +
85 -0
R-
7-3 S
19-7
u -
70-9 V
B E 7
6-5
13
7P-
16-1
Q +
94-2
R
7'9 S
19-2
u -
71-2 V
= E ;
7'5
14
o P-
17-5
Q +
102-8
R
9-1 S
19 '3
u -
71-9 V
B E'
8-5
14
2 P
18'8
Q +
112-0
R-
10-2 S
18 '9
u
72-2 V
= E 7
9-5
14
4 P
20-8
Q +
120-2
R
11 '8 S
18 -2
u
73-1 V
B E 7
10-5
14
5P
22-6
Q +
128-4
R-
12-8 S
19-0
u
73-6 V
= E 7
11-5
14
5 P
23'6
Q +
136-3
R-
13-9 S
18-7
u
74 -i V
B E 7
12-5
14
6 P
25-0
Q +
143-8
R
15-1 S
18 '9
u -
74-6 V
B E 7
13 '5
14
8 P
26-6
Q +
150*7
R
16-3 S
18 -8
u
75-1 V
= E 7
14*5
14
5P -
27-5
Q +
157-8
R-
17-3 S
18-6
u -
75-2 V
= E 7
15-5
14
5 P
29-0
Q +
163-8
R
18-4 S
18-5
u
75-7 V
B E 7
16-5
14
3P-
30-1
Q +
169-7
R
19'5 S
18-4
u -
76-0 V
B E'
17-5
14
1 P
30-8
Q +
175-2
R
20-3 S
18 -3
u
76-0 V
B E 7
18 -5
14
o P
32-1
Q +
179-9
R
21-3 S
18-1
u
76-1 V
= E 7
19-5
13
8 P
33-0
Q +
184-3
R
22*2 S
18-0
u -
76-4 V
= E 7
20-5
13v6 P
33-5
Q +
188-3
R-
23 '0 S
17-8
u -
76-5 V
B E 7
21 -5
13
3P -
34-2
Q +
191 -6
R-
23-9 S
17-8
u -
76-6 V
= E 7
262
On the Elements of the Orbit of Halletfs Comet,
TABLE ILL continued.
Date.
Equations of Condition dependent upon Right Ascensions.
1836.
Mar. 21 -5
23 '5
23'5
24-5
25*5
26 5
27-5
28 -5
29 '5
30 '5
31 '5
+ 23-2 P +126-9 Q 690-6 R + 99-8 S 8-1 U 17-9 V = E
+ 22'3 P +126-1 Q 682-8 R + 99 '3 S 8 '2 U 18 "2 V = E
+ 21 -3 P +124 '7 Q 674 '4 R + 93 '7 S 8 -0 U 18 '4 V == E
+ 20 '3 P + 123 -0 Q 666 '2 R +
+ 19'6 P +121-5 Q 657-4 R +
+ 18'5 P +120-1 Q - 648-4 R +
98 '3 S 8 ! U 18 -8 V = E
97-9 S 7'8 U 18-8 V = E
97-1 S 8-1 U 19-4 V = E
+ 17-8 P +118-2 Q 639-6 R + 96-9 S 7-9 U 19*4 V = E
+ 16*8 P +116-9 Q 630-2 R + 95-7 S 7'9 U ~ 19'7 V = E
+ 16-0 P +114-9 Q 621-4 R + 94-8 S 7-9 U 19-9 V = E
+ 15 ! P +113-9 Q 611 -6 R + 94-1 S 7 '9 U 20-3 V r= E
+ 14-6 P +114 o Q 600-5 R + 93-3 S 7-9 U 20*2 V = E
at Its appearance in the Years 1835 < 1836.
263
TABLE XL continued.
Date.
Equations of Condition dependent upon Declinations.
1836.
Mar. 21 -5
22 *5
23-5
24 '5
25 '5
26-5
27*5
28
29
30 '5
31 -5
13 -3 P 34 *2 Q + 191 -6 R
13-0 P - 34'9 Q + 194-6 R
12'6 P 35-3 Q + 196-9 R
12-3 P 35'7 Q + 199-0 R
12 -0 P 36 -1 Q -f 200 -8 R
11 -6 P 36 -4 Q + 201 -5 R
11 '3 P 36 '4 Q + 202 '2 R
11-0 P 36-6 Q + 202-5 R
10-6 P 36-5 Q + 202-5 R
10-3 P 36-3 Q + 201-7 R
9 -3 P 37 -9 Q + 199 '9 R
23 '9 S 17 '8 U
24-3 S 17-6 U
24'9 S 17-4 U
25-6 S 17-3 U
26 '2 S 17 -0 U
26 -6 S 16 -9 U
26 '9 S 16 '7 U
27-1 S 16-1 U
27-6 S 16-4 U
27'8 S l6 -2 U
28 -o S 16 -1 U
76 -6 V = E'
76 -5 V = E'
76 -4 V = E'
76-5 V = E'
76 -3 V = E'
76 -2 V = E'
76 -o V = E'
75-7 V = E'
75 -6 V = E'
75 -3 V = E'
74-9V = E'
265
ON THE PERTURBATIONS OF URANUS.
BY J. C. ADAMS, ESQ., M.A.,
FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE ;
FELLOW OF THE BOYAL ASTRONOMICAL SOCIETY; AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.
(Read before the Royal Astronomical Society, November 13, 1846.)
1. THE irregularities in the motions of Uranus have for a long time engaged the
attention of Astronomers. When the path of the planet became approximately known,
it was found that, previously to its discovery by Sir W. Herschel in 1781, it had
several times been observed as a fixed star by Flamsteed, Bradley, Mayer, and
Lemonnier. Although these observations are doubtless very far inferior in accuracy
to the modern ones, they must be considered valuable, in consequence of the great
extension which they give to the observed arc of the planet's orbit. Bouvard, how-
ever, to whom we owe the Tables of Uranus at present in use, found that it was
impossible to satisfy these observations, without attributing much larger errors to the
modern observations than they admit of, and consequently founded his Tables exclu-
sively on the latter. But in a very few years sensible errors began again to show
themselves, and though the Tables were formed so recently as 1821, their error at
the present time exceeds two minutes of space, and is still rapidly increasing. There
appeared, therefore, no longer any sufficient reason for rejecting the ancient observa-
tions, especially since, with the exception of Flamsteed's first observation, which is
more than twenty years anterior to any of the others, they are mutually confirmatory
of each other.
2. Now that the discovery of another planet has confirmed in the most brilliant man-
ner the conclusions of analysis, and enabled us with certainty to refer these irregu-
larities to their true cause, it is unnecessary for me to enter at length upon the reasons
which led me to reject the various other hypotheses which had been formed to account
for them. It is sufficient to say, that they all appeared to be very improbable in them-
selves, and incapable of being tested by any exact calculation. Some had even
supposed that at the great distance of Uranus from the Sun, the law of attraction
becomes different from that of the inverse square of the distance. But the law of
gravitation was too firmly established for this to be admitted, till every other hypo-
thesis had failed, and I felt convinced that in this, as in every previous instance of
the kind, the discrepancies which had for a time thrown doubts on the truth of the
law, would eventually afford the most striking confirmation of it.
3. My attention was first directed to this subject several years since, by reading
Mr. Airy's valuable Report on the recent progress of Astronomy. I find among my
266 On the Perturbations of Uranus.
papers the following memorandum, dated July 3, 1841: "Formed a design, in
the beginning of this week, of investigating, as soon as possible after taking my
degree, the irregularities in the motion of Uranus which are yet unaccounted for ; in
order to find whether they may be attributed to the action of an undiscovered planet
beyond it, and if possible, thence to determine approximately the elements of its
orbit, &c., which would probably lead to its discovery." Accordingly, in 1843, I
attempted a first solution of the problem, assuming the orbit to be a circle, with a
radius equal to twice the mean distance of Uranus from the Sun. Some assumption
as to the mean distance was clearly necessary in the first instance, and Bode's law
appeared to render it probable that the above would not be far from the truth. This
investigation was founded exclusively on the modern observations, and the errors of
the Tables were taken from those given in the Equations of Condition of Bouvard's
Tables as far as the year 1821, and subsequently from the observations given in the
Astronomische Nachrichten, and from the Cambridge and Greenwich Observations.
The result showed that a good general agreement between theory and observation
might be obtained ; but the larger differences occurring in years where the observa-
tions used were deficient in number, and the Greenwich Planetary Observations being
then in process of reduction, I applied to Mr. Airy, through the kind intervention of
Professor Challis, for the observations of some years in which the agreement appeared
least satisfactory. The Astronomer Royal, in the kindest possible manner, sent me,
in February 1844, the results of all the Greenwich Observations of Uranus.
4. Meanwhile the Royal Academy of Sciences of Gottingen had proposed the Theory
of Uranus as the subject of their mathematical prize, and although the little time
which I could spare from important duties in my college prevented me from attempt-
ing the complete examination of the theory, which a competition for the prize would
have required, yet this fact, together with the possession of such a valuable series of
observations, induced me to undertake a new solution of the problem. I now took
into account the most important terms depending on the first power of the eccentricity
of the disturbing planet, retaining the same assumption as before with respect to the
mean distance. For the modern observations, the errors of the Tables were taken
exclusively from the Greenwich Observations as far as the year 1830, with the excep-
tion of an observation by Bessel, in 1823; and subsequently from the Cambridge
and Greenwich Observations, and those given in various numbers of the Astrono-
mische Nachrichten. The errors of the Tables for the ancient Observations were
taken from those given in the Equations of Condition of Bouvard's Tables. After
obtaining several solutions differing little from each other, by gradually taking into
account more and more terms of the series expressing the Perturbations, I communi-
cated to Professor Challis, in September 1845, the final values which I had obtained
for the mass, heliocentric longitude, and elements of the orbit of the assumed planet.
The same results, slightly corrected, I communicated in the following month to the
Astronomer Royal. The eccentricity coming out much larger than was probable,
and later observations showing that the theory founded on the first hypothesis as to
the mean distance, was still sensibly in error, I afterwards repeated my investigation,
supposing the mean distance to be about g-Vth part less than before. The result,
On the Perturbations of Uranus. 267
which I communicated to Mr. Airy, in the beginning of September of the present
year, appeared more satisfactory than my former one, the eccentricity being smaller,
and the errors of theory, compared with late observations, being less, and led me
to infer that the distance should be still further diminished.
5. In November, 1845, M. Le Verricr presented to the Royal Academy of Sciences
at Paris, a very complete and elaborate investigation of the Theory of Uranus, as dis-
turbed by the action of Jupiter and Saturn, in which he pointed out several small
inequalities which had previously been neglected j and in June, of the present year,
he followed up this investigation by a memoir, in which he attributed the residual
disturbances to the action of another planet at a distance from the Sun equal to
twice that of Uranus, and found a longitude for the new planet agreeing very nearly
with the result which I had obtained on the same hypothesis. On the 3 1st of August
he presented to the Academy a more complete investigation, in which he determined
the mass and the elements of the orbit of the new planet, and also obtained limiting
values of the mean distance and heliocentric longitude. I mention these dates merely
to show that my results were arrived at independently, and previously to the publica-
tion of those of M. Le Verrier, and not with the intention of interfering with his just
claims to the honours of the discovery ; for there is no doubt that his researches were
first published to the world, and led to the actual discovery of the planet by Dr. Galle,
so that the facts stated above cannot detract, in the slightest degree, from the credit
due to M. Le Verrier.
6. In order not to have an inconvenient number of equations of condition, I divided
the modern observations into groups, each including a period of three years, and as
Mr. Airy had shown that the error of the Tabular Radius Vector was sometimes con-
siderable, I either selected those observations which were made near opposition, or
combined the others in such a manner that the result should be nearly free from the
effects of this error. From the observations of each group, the error of the Tables in
heliocentric longitude was found, corresponding to the time of mean opposition in the
middle year of the group. Thus were formed 21 normal errors of the Tables, cor-
responding to as many equidistant periods between 1780 and 1840. The error for
1780 was found by interpolating between the errors of 1781, 1782, and 1783, and
those given by the Ancient Observations of 1769 and 1771, and though not entitled
to the same weight as the others, cannot, I think, be liable to much uncertainty. In
my last calculations, I might have used more recent observations, but in order to
obtain the effect due to the change of mean distance, it was necessary that the in-
vestigation should be founded on the same elements as before, and the later observa-
tions might be used as a test of the theory.
7. In order to satisfy myself that there was no important error in Bouvard's Tables,
I recomputed all the principal inequalities produced by the action of Jupiter and
Saturn, and found no difference of any consequence except in the equation depending
on the mean longitude of Saturn minus twice that of Uranus, the error of which had
been already pointed out by Bessel. The principal equation depending on the action
of Jupiter, also required correction in consequence of the increased value which has
been lately obtained for the mass of that planet. The corrections to be applied to
Bouvard's Tables on these accounts, are the following :
268 On the Perturbations of Uranus.
+ 1 '918 sin {0! 20 2 13 1 '5}
+ 1 '085 sin {0 <& }
0> 0i, 02 being the mean longitudes of Jupiter, Saturn, and Uranus, respectively.
In the Reduction of the Greenwich Observations, the latter correction was already
taken into account. M. Hansen having also found some new inequalities in the
motion of Uranus, depending on the square of the disturbing force, I re-computed
the values of these, following the same method as that given by M. Delaunay in the
Conn.des Temps for 1845, and my results agreed very closely with his, the terms to
be added to the longitude being
H O I
+ 32 -00 sin {302 60! + 20 + 22 18 '8}
8 *35 sin (20260! + 20 +39 10 '5}
1 '49 sin {402 60! + 20 + 34 48 '4}
With respect to the inequalities of higher orders neglected by Bouvard, I considered
that the most important of them would be, either those of long period, or those whose
period was nearly equal to that of Uranus. During three-fourths of a revolution of
the planet, the effects of the former class would be nearly confounded with those
arising from a change in the epoch and mean motion, and those of the latter class
with the effects produced by a constant change in the eccentricity and longitude of the
Perihelion. The position of the planet to be determined would therefore be little
affected by these terms, and the others would probably be much smaller than thos
which would necessarily be neglected in a first approximation to the perturbations
produced by the new planet.
8. Taking into account the several corrections above-mentioned, the residual differ-
ences between the theoretical and observed heliocentric longitudes were the following:
Ancient Observations. Modern Observations.
Year. Observation Theory. Year. Observation Theory.
a H
1690 +61 '2 1780 + 3 '46
1712 +927 1783 + 8*45
1715 +73-8 1786 +12*36
1750 47-6 1739 +19-02
1753 39'5 1792 +18*70
1756 45'7 1795 +21-38
1764 -34'9 1798 +20-95
1769 19*3 1801 +22-21
1771 2'3 1804 +21-16
1807 +22-07
1810 +23-16
1813 +22-00
1816 +22-88
1819 +2069
1822 +20-97
1825 +18-16
1828 +10-82
1831 3*98
1834 20-80
1837 42 -66
1840 66 '64
On the Perturbations of Uranus. 269
9. It is easily seen that the series expressing the correction of the Mean longitude
in terms of the corrections applied to the elements of the orbit, is more convergent
than that which gives the correction of the true longitude, and the same thing is true
for the perturbations of the mean longitude, as compared with those of the true.
The corrections found above were accordingly converted into corrections of mean lon-
gitude by multiplying each of them by the factor -T-, r being the Rad. Vector, and
a and b the semi-axes of the orbit. Hence these latter corrections were found to be
the following :
Ancient Observations. Modern Observations.
Year. Observation Theory. Year. Observation Theory.
16 ( JO +62" '6 1780 +3 '42
1712 +81 '5 1783 + 8 -19
1715 +67'2 1786 + 11'74
1750 51-8 1789 +i7'75
1753 43'2 1792 +17-22
1756 50-1 1795 +19-52
1764 37-8 1798 +19 '06
1769 20 '5 1801 +20-21
1771 2-1 1804 +22 -19
1807 +20-52
1810 +21 '89
1813 +21'I9
1816 +22-50
1819 +20 78
1822 +21-50
1825 +18'97
1828 +11 -50
1831 4 '29
1834 22 '63
1837 46-/0
1840 73 09
These numbers form the basis of the subsequent investigations.
10. Let 5e, 5, 5e, and 3tu denote the corrections to be applied to the Tabular
Elements of Uranus, then the correction of the mean longitude at any time t is
e
= Sf + 2e 8 3CT + t %n {2 cos (ra + f CT) + cos 2 (nt + f. }] ccrz
+ {2 sin (n t + t-rc) + sin 2 (w^ + t-tz) j Ze
If we include the small term 2 c 2 c w in the quantity 8 <-, this correction may be put
under the following form :
S + t S n + cos n t I x l + sin n t $ i/ l + cos 2 n t$ x z + sin 2 n tSt/i
in which expression
;e a = _ e {cos (e cr) dx^ + sin (f re) ci/^
?/ 2 = -e {sin (e-^-cj) 3 # cos (E w)
270 On the Perturbations of Uranus.
11. Also, adopting the notation of Ponte*coulant's "Theorie Analytique," the per-
turbations of mean longitude
= . 2 Fi sin i (nt-n't + -t')
+ m'e 2 GiSin {z (nt n't + e-c') (rc <-f -rr)}
+ m'e 1 S H, sin { i (n t-n't + e- E ')-
Where the accented letters belong to the disturbing planet, t takes all integral
values, positive and negative, except zero, and if we put t (n n 1 ) = z 9 the values of
F Gi and H,- are the following :
3 i n* in* 2n* 2 dA i
n) da
' , 3^ 8 _)
^ 2 -7 2 + 2r(2r ) (0-2 /Of
; 2) 2z 2 n*
c^^h-
(3 (i-\) i (^-IX 1 (f- 1) (2 i-l)n)
ti (^-w) 2 ^ (0 g) + 3 (02 > r Ai - !
12. Now, if we assume -^7 or a = sin 30 = 0-5, the values of the fundamental
db t d*b
quantities 6, a -7- , a 2 --j , will be
log. 6 =0-33170; log. a ^= 9 '53 765 log. a 8 ^ = 9 77848
log. ft, = 9 '74 197 ; lo:r. a ~ = 9 '83868 ; log. 8 ~ = 9 70857
log. 6, =9'32425; log. ~ = 9 '68012 log. a 2 ^;= 9 '87776
log. 6 3 = 8 '9 1670 ; log. a 3 = 9 '463 1 $ log. a 2 = 9 '86253
On the Perturbations of Uranus. 271
Hence the principal inequalities of mean longitude, produced by the action of a
planet whose mass is , that of the Sun being unity, and the eccentricity of whose
orbit is -- will be the following :
36-99' sin \nt-ri t + e e'}
+ 58 '97 m 1 sin 2 {nt n't+e e 1 }
+ 5 '80 m' sin 3 {nt n't + t t?}
+ 2 *06 m' sin [n't+e 1 &}
4 -30 wV sin {n't+e' vs 1 }
+ 31 '25 m' sin {nt- 2n't+e 2 e'
-12-14 m'e' sin {nt-2 n't+e-2 e'
+ 48 '55m' sin {2 n t-3 n 1 1+2 #3
-93-01 wV sin {2 nt 3 ra'*+2 e S
To these may be added the following, which are of two dimensions in terms of the
eccentricities :
+ 0*57' sin 3 {nt-n't+ (. (.'}
-1 '08 m'e'sin. {3 (nt-n 1 t+e-f 1 ) -w+w'}
These expressions may be put under the following form :
hi cos (nn 1 ) < + A 2 cos 2 (?i n f ) t+h a cos 3 ( n fi, f )t
+ ki sin (nri) t+k^ sin 2 (n n') <+A 3 sin 3 ( n n')t
rit-\-q z sin (n 2^') ^ + ^38^ (2n 3n')t
13. Let the time of the mean opposition in 1810 be taken as the epoch from which
t is reckoned ; this date, expressed in decimal parts of a year, will be 1810- 328. Also,
let 3 synodic periods of Uranus, = 3 * 0362 years, be taken for the unit of time ; then
the change of the mean anomaly in an unit of time will be 130 / '5; also w=13 0''6,
? i / =436 / '0 .-. n w'=8 24'-6, n 2ra'=3 48' '6, 2^-3^ = 12 13' '2. Hence
the equations of condition given by the modern observations will be of the form
II 01 01
c SE + $XI cos {13 '5} < + 3a; 2 cos {26 1 '0} t
+ cyi sin {13 '5} *+<5y 2 sin {26 1 '0} t
01 01 01
cos { 8 24 -6} t+ h z cos {16 49 '2} t+h 3 cos {25 13 '8} t
sin { 8 24 '6} t+ k z sin {l6 49 '2} *-f 3 sin {25 13 '8} t
+ Pi cos { 4 36 -0} t+ p 3 cos { 3 48 -6} t+2>* cos { 12 13 '2} t
+ q l sin { 4 36 '0} t+ q z sin { 3 48 '6} t+q 3 sin {12 13 '2} t
in which t assumes all integral values from 10 to +10 in succession, and the
several values of c" are contained in the table given in Article 9.
272 On the Perturbations of Uranus.
14. The final equations for the corrections of the elliptic elements will be found by
multiplying each equation successively by the co-efficients of e, Sn, $Xi and ^y,,
which occur in it, and adding the several results.
Let the equations be treated in a similar manner with reference to the quantities
/*i> ,? h kz> ^3, & P*> P^ fr-
it will be seen that, in consequence of the arrangement which has been given to the
equations of condition, the equations thus formed naturally separate themselves into
two groups, one of which involves only <), Sx 19 Sa? 2 , with the quantities h and p, while
the other involves Sn, ty H o>y 2 > with the quantities k and q.
Also, the co-efficients in these equations are easily calculated by the following for-
mulae, putting t=\Q in their right hand members:
sinm
, 1 , 1 sin m
Zco*m< ^^ +
oo
22 sm
15. By performing the calculations, the equations of the first group are found to be
the following :
(0 15) '48= 21-0000&-+ 6 -0670 a*!- 4'4358d^ 2
+ 13 '6320 h, + '4043 2 - 4 '5608 7i 3
+ 18-6046^+19-3384 jt? 2 + 7-3721 p*
(x) 246-48= 6'06703e+ 8 '2821 ^x,+ 4 '1762 3 a:,
+ 7-40417^+ 8-2523 h*+ 4*6963 h 3
+ 6-5389^+ 6-3978 ^ 2 + 8-1831 p.,
C/O 20974= 13-6320^+ 7 -4041 a*!- 0-23375^
+ 10 -7022 ^ + 4-5356 h 2 0*0018 h. d
+ 12*7013^ +12-9883 p z + 8 '0038 p 3
'M 242 '68= -4043 h + 8 '2523 80:,+ ^ '5650 ^,
+ 4-5356^+10-2960 A 2 + 8-1944 h 3
+ 1-7866^+ 1-3667 jo 2 + 7-6671 p 3
(A a > 86-67=- 4*5608 a + 4-69633^+10-50235^
-00187*^ 8-1944 A,+ 10-7071 h a
3-0812;?, 3-5347 ^8+ 3 '8855 /?,
On the Perturbations of Uranus. 273
# 1 -3 -4948 a# 2
+ 12*9883^ + 1 '3667 7* 2 -3*5347 7* 3
-1-17 '2795 j^ + 17 -9106 jp,+ 7 "5423 p 3
212-56= 7'372lS + 8 '1831 5^ + 3 '40/1 a ar,
+ 8-00387*!+ 7'6671 7* 2 +3-8855 h a
+ 7*6127^ + 7'5423 p 2 +8*2019 p,
16. By means of (<) eliminate It from each of the other equations, and these latter
become
n
(x-) 202 -72 6 -5294 Sa^ + 5 '4577 z 2 +3 *4658 7* t + 8 '1355 7< 2
+ 6-0139 7* 3 +l-l640 ^ + 0-8109^2 + 6-0533^3
(^) 111-41= 3 *4658^ + 2 '6458 S tf a + 1 '8531 7* t + 4 '2731 h*
+ 2*95887i 3 +0-6243 p 1 + 0'4349/? 2 + 3-2183jr? 8
(7O 239 '76= 8 -1355^ + 7 '6504 a# 2 +4'2731 ^+10 '2882 h 2
+ 8 -2822 7i 3 +l '4284 J9 t + '^944 p 2 + 7 '5252 p a
(7* 3 ) 119'57= 6 -0139 &tfi + 9 '5389 a # 2 +2 '9588 h,+ 8 "2822 h*
+ 97166 A3+0'9593 ^ + 0-6652^+ 5 -4866/73
26'50=r 0-8109 ^^ + 0-5900 ^^+0-4349 7*,+ '9944 7* 2
+ 0-6652 /* 3 +0 '1470 ^ + 0-1024^+ 0'7535p 3
189'38= 6'0533Sx! + 4 -9643 Jx 2 +3 '2183 A!+ 7 -5252 A 2
+ 5-4866 ^3+1*0815 ^,+0-7535^+ 5-6139^3
1 7- Again, by means of (a;) eliminate 5 x l from each of the other equations, and we
find
3 -807 = --0 '2512 to 2 +0 '0135 ^0 '0452 7* 2 '2334 h z
+ -0065^-J-O -0045^+0 -0052^3
(7* 2 ) 12 '821= '8502 Sx 2 0*0452 A t 4-0 -15157*2+0 -7890 A 3
'0219^0 -0160 p 2 -0171^3
(A 3 ) 67 '149= 4 -5120 &e 2 -2334 7* t + *7890 7* 2 +4 '1775 h a
-1 128 ^ *0817jtv- -0888 p s
1 -327= '0878 r*+0 -0045 h, *0l60 h z '0817 7* 3
+ '0024^ + 0017# s +0 -0018p 3
1 -448 = ^0 -0955 a# 2 + '0052 h^ *0l7l ^ 2 -0 '0888 h 3
+ 0*0024^ + '0018^+0 -0020^3
18. Similarly, the equations of the second group are found to be
O) 171 -27 = 77 -0000 *n + 9 "3938 ^ 1 '2183 a y %
+ 8-8463 ,+ 7-3034 k t '5927 3
+ 5-7519 ^1+ 4-8755 q a + 9*5583 ^ s
(y) ^166*33= 93'9380^+12*7179Syi+ I'8907ay 2
+ 11-2022 ^+11*0848 k*+ 2*6731 k 3
+ 7-0956 ^+5-9913 ry,+ 1
274 On the Perturbations of Uranus.
_]82-87 = 88 -1630 8n + 11 "2022 S^ '3210 $ y*
+ 10-2978 ^+9*0964 k 2 + 0'406l k 3
+ 6-6370 q,+ 5-6l63 2 +l 1-3346 q 3
89 -07 = 73 -0340 fa + 11 -0848 ?/!+ 4 '8266 Jy 3
+ 9-0964 ^+107040 ^ 2 +5-4376 k.,
+ 5-5855 q,+ 4 '6976 q z + 10 '9375 q. A
+124-80= - 5-9270 J+ 2-6731 ^+10-42535^
+ 0-4061 ,+ 5-4376 2+10-2929 k 3
'2497 q l '2643 # 2 + 2 '1788 ? 3
107*02= 48-7550 8n + 5'9913 J^ 0'66l4$y a
+ 5-6163 A!+ 4 -6976 ^ 2 0-2643 ^ 3
+ 3-6475 ^+3-0894 ^+6-0897 g ?
_175 -89 = 95 '5830 Sn + 12 7441 Sy,+ 1 '3845 Jy 8
+ 11-3346 ^+10-9375 ^ 2 + 2'1788 ^ 3
+ 7'2084 q,+ 6'0897 ^ 2 + 12 7981 q 3
19. By means of (w), eliminate S n from each of the other equations, and we have
(//) 42-61= 1 '2578 fyi+ 3 -3771 &/2+0 '4100 ^ + 2 '1748 k 2
+ 3 -3962 3 + '0785 ^ + '0433 # 2 + 1 '0833 q 3
(k,) 13-90= 0-4100 fy t + 1-0787 ai/ 2 + -1346 ^+07057^ 2
+ 1-0871 3 + 0-0288 ^ + 0*0150 ^+0 -3534^
(A z ) 73-38= 2-1748 fy,+ 5 '9822 S# 2 +0 705? ^ + 3 7767 A 2
+ 5-9998 4+ 0-1298 ^+0-0732^+1-8715^
( 3 ) 111-62= 3-3962 3^+10-3315 tys+l -0871 ^ + 5-9998^
+ 10-2473 ^ 3 + 0-1930 ^ + 0-1110 ^+2 '9145 q 3
(ry 2 ) 1 '42= -0433 fy t + '1100 y a +0 '0150 A x + '0732 k z
+ 0-1110 A 3 + 0-0055 ^ + 0-0023^+0-0375^3
(^ 3 ) 36 72= 1 -0833 3^+ 2 -8969 %/ 2 +0 -3534 ^+ 1 '8715 k a
+ 2 '9145 A 3 + -0684 ^ + -0375 ^+0 *9330 q 3
20. Again, eliminating 5^ by means of (y) we find
(& -009= -0221 3y 2 +0 -0010 ^-0 -0032 ^-0 '0200 ^ 3
+ -0032 ^+0 -0009 <7 2 +0 *0003 q 3
(A 2 ) -0-301 = 0-1430 fy 2 -0 '0032 ^+0-0162 2 +0-1274 3
-0 -0059 ^-0 -0017 &-0 -0016 q 3
(A 3 ) -3 -443 = 1 -2129 fya-0 '0200 ^ + '1274 k*+l '0769 &,
-0 -0189 ^,-0 -0059 ?2-0 -0105 (7 3
-0 -045= -0062 fy 2 +0 -0009 ^-0 -0017 2 -0 '0059 k 3
+ '0028 ?, + -0008 (7s+0 '0002 q s
+o-oi7=-o-on6 ^2+0-0003 ^-0-0016^2-0-0105 k. A
+ -0008 ^ + -0002 ft+0 -0000 jr,
On the Perturbations of Uranus. 275
21. From the equations remaining in the two groups after the elimination of $ , 5 n,
& x n 2y\ it will be easy, when approximate values of the mass and mean longitude of
the disturbing planet have been found, to deduce the final equations for determining
these quantities more accurately by the method of minimum squares.
It maybe observed, however, that the equations in each group are very neaily
identical with each other, aud therefore two final equations may be formed by simply
adding together the several equations of each group, after giving the unknown quan-
tities the same sign in them all. Thus we find
86 '552= -5 7967 3a: 2 +0 '3018 7^-1 '0188 ^ 2 -5 '3704 h 3
+ -1460^ + -1056^+0 '1149 p 3
3 725=-l '3958 fy 2 +0-0254 ^-G'1501 2 -l '2407 k 3
+ '0316 ^+0 '0095 # 2 +0 '0127 q 3
22. If in the expressions before given for^x a and $y z we substitute e = '046679
and e iz~ 50 1 5' "8, we obtain
&a? 2 = -007460 Sa^ + -008974 #!
Sy 2 =-0-008974 Sarj + -007460 iyj
Substituting these values in the equations (x) and (?/) and in those just found, it
may be seen that by adding to the latter equations
-006768 Cr) + -040287 (y)
and '001869 (z) + '008187 (#) respectively,
S x l and ?/, will be eliminated, and we shall obtain the following equations :
(1) 89-641 = 0-3252 ,-0-9637 2 - 5-3297/* 3
+ -0165 h +0 '0876 2 +0 '1368 k 8
+ -1539^+0 -1111 j0 2 +0-1559/? 3
+ '0032 q, +0 -0017 ? 2 +0 '0436 q 9
(2) 3-695=: -0-0065 A, -0-0152^-0-01I2^ 3
-f '0288 Aj '1323 ^ 2 -l '2129^
'0022^ '0015^20 -0113^3
+ '0323?! +0 -0099 ? 2 +0 '02 15 q 3
23. These equations would be sufficient for determining the mass of the disturbing
planet and its longitude at the epoch, if the eccentricity of the orbit were neglected.
We will now proceed to find equations from the Ancient Observations for determining
the eccentricity and longitude of the Perihelion.
The equations of condition given by the Ancient Observations are the following :
62-6= Se- 0-8776 a^ + 0'5402 ^ + 0-87127*, + 0-5 180 2
-39-31 S/i-0-4795 c^ + 0-8415 %/ a -f 0-4909^ + -8554 2
+ 0-0314 7i 3 0-9999 JV- -8640^ 2 -5Q55p a
+ 0-9995 >& 3 + 0-0145 #! 050352 8 --0 -8628^
'27b On the Perturbations of Uranus.
84-5=3 & + 0-4975 ^-0-5050 &TJ + -0288 Ar-0 -9984 A 2
32-30271 0-8675 c^0-8631 B*/. 2 + 09996 A! 4- 0-0573 A,
0-0860 A 3 0-8534 p l 5456^ + -8220^ 3
0-9963 V-0-5213 ft 0-8380 ft 0-5695 ft
67-2= c + -6732 San -0935 &r 2 -11207*, -9749 /*
31 -31 Jw 7394 fyi -9956 fya+0 -9937 A' t '2227 # a
+ 0-3305 A 3 0*8105 j9 t '4912 p 2 +0 '9206p 3
0-9438 3 0-5857 ft '8711 ft "3905 ft
51 '8= Se0-2616 Sav- 0-8631 &K a -9649 A t + '8618 A,
19 '59 Sw + -9652 fy! '5050 Sy 2 '2627 A t + *5073 A 2
'6982 ^30 '0023 J3 t + '2650 /> 2 '5090 p s
0-7159 ^ 3 l-0000 ft -9642 ft-fO '8607 ft
43 -2= Se '4741 Sx t '5505 &r 2 '9154 ^ + '6758 A 2
18-58 Sw+0'8805 5^0 '8348 fy 2 '4025 A t + '7371 k 2
0-3220 /i 3 + 0-0787 ^ + 0-3291^20-6814^3
-0 -9467 3 -9969 ft-0 -9443 ft+0 7319 ft
50-1= Sf 0-6430 &T! 0-173I ^ 2 '8543 ^,+0 '4599 h z
-17 '68 Sw+0 -7659 fy; '9849 ^ 2 '5198 ^ + '88/9 ^ 8
+ 0-0686 Ag+0'1510 /^ + -3848^2 '8085^3
-9976 V-0 '9885 ft -9230 ft+0 -5885 ft
37-8= Se 0-9492 S^ + 0'8021 ^ 2 0-6189 A! 0-2340 7i 2
15-25 S7Z+0-3145 ^ t '5972 5y g *7855 A t + '9722 ^
+ 0-9085 7i 3 +0-3396 j^+0 '5287^0 '9939 ;? 3
0-4179 A 3 -9406 ft -8488 ft+0 -1100 ft
20 -5= h '9985 ^ + '9942 5^20 -4128 h l '6591 //
13-60 2w 0-0538 ^, + 0-1074 57/20-9108^ + 0-7520^
+ 0-9571 ^ 3 + -4607 ^ + 0-6182^20-9711^3
+ 0-2899 ^ 3 0-8875 ft -7860 ft '2385 ft
2 -4= & "9633 S^ + -8560 Sx, -2807 ^0 '8424 h a
12-64 tin '2684 5^ + '51/0 cy 2 '9598 ^ + '5388 k t
+ -7536 7/3+0 -52/9 jt? t + '6670^0 '9023^8
+ '6574 7^ 3 -8493 ft 7451 ft '4310 ft
21. From each of these equations eliminate e, 8n t Sx it and Sy l9 by means of the
equations (*), (/z), (a?), and (?/) before found, and we have the following :
-14-2-0= 1-/265 fce 2 +0 -84127/i+l '9521 7? 2 + 1*3230 A,
1 1 -3691 J?/2+3 -6001 ,2 -8793 #2 10 -9578 & 3
1 -6779^1 -6400 jt? 2 + -2249^8
+ 2 -6815 ft+ 1 '8369 ft+ -2995 ft
On the Perturbations of Uranus. 2/7
105%= 0-4681 2av-073ii hi i -2776;*, 0-06097*,
9-6249 Sy 2 + 3 7087 7^2-1926 V- 9 '5426 3
--1 -7765^ 1 '4924 p z +
+ 1 -6997 ?i + l '1014 8 , q 3 ; it will be proper, therefore, to combine the above equations in such a manner
that these quantities may acquire the largest co-efficients possible. This will be done
by multiplying each equation by a quantity nearly proportional to the co-efficient of
each of the unknown quantities p 3 and q 3 , and adding together the several results.
It was thought unsafe to employ the first of the above equations, since it is derived
from the single observation of Flamsteed, made in 1690, twenty-two years anterior to
any other observation.
278 On the Perturbations of Uranus.
Hence the equation for finding p 3 may be formed by multiplying the above equations,
taken in order, by
-0'8, '6, +1 -0, +1 '0, +0 '9, +0 '6, +0 '4, +0 '3
beginning with the second ; and the equation for q* by multiplying the same equations
by
I'O, 1 -0, 0'5, 04, 0-3, 0-2, 0*1, '1,
Hence we obtain
474"-! = 4 -114&c 2 2 '8 17 /*!+ 7 '837 h t + 4 *528 h s
-20 745 fy a 2 789 7^-6 '551 * 2 20 '666 k,
+ 0-193^ + 0-377^ 1-489^3
1 -660^ 1 -078^ 0'054 ? 3
485 -0 = '446 az 2 3 '308 h, '442 h z + I '629 h s
32 -961 cy 2 + 8 *267 k, 8 *805 2 32 '546 3
4-473^3 -643/? 8 + 0'037^3
2-278^+ 2-086^
26. Eliminate cx z and Sy 2 from these equations by means of (cc) and (y) and
they become
( 3 ) _ 476"-7 = 2 '930 A t + 7 -572 A, + 4 '332 k a
2 751 k, 6 -348 2 20 '350 7< 3
1 -653 and -A- are
m'
a
^=48*' '55 sin (3 y6) 93 '01 e' sin (3 ')
^ = 48 '55 cos (3 ,6) 93 '01 e' cos (3 ft')
where e &' = fi. Equating these to the values given above, we find e' = 3 '2206,
ft' =262 28', and /. ro' = 315 27'. Hence long, of Perihelion in 1846 = 315 5?'.
Lastly, substituting the values just obtained in equation (1), we find m'= "82816.
31. Hence the values of the mass and elements of the orbit of the disturbing
planet, resulting from the first hypothesis as to the mean distance, are the following :
!=;'
Mean Long, of the planet, October 6, 1846, 325 7
Longitude of the Perihelion 31557
Eccentricity of the Orbit - - - - - -O'l6l03
Mass (that of the SUN being 1) - - - - '0001656
These are the results which I communicated to the Astronomer Royal in October,
1845.
I
32. I next entered upon a similar investigation, founded on the assumption that the
mean distance was about T Vth part less than before, so that , or = sin 3 1 = '5 1 5.
The method employed was, in principle, exactly the same as that given before ; but
the numerical calculations were somewhat shortened by a few alterations in the process,
which had been suggested by my previous solution,
On the Perturbations of Uranus. 281
33. Assuming then that = sin 31, the values of the quantities ft, ^~, **~d
will be
log. l> = '33385 ; log. ^ = 9 '57333 ; log. 2 ^ = 9 '8291 1
log. b, = 9 -76106; log. 27 = 9 '86149; log. 2 ^~ = 9 '76573
log. b, = 9 '35361 ; log. a ^ = 9 71359 ; log. of^f = 9 '92466
log. b, = 8 '98918 ; log. * g-jp = 9 -50854; log. 2 = 9 '91563
Hence, by means of the formulae given before, the principal inequalities of the
mean longitude of URANUS, produced by the action of a planet whose mass is
that of the SUN being unity, and the eccentricity of whose orbit is ^-, may be found
to be the following :
n
42 *33 m 1 sin (ntn't+t '}
+ 76 -55 m' sin 2 {nt n't + e e'}
+ 7 -25 m' sin 3 {ntn't +EE'}
+ 2 -34 m 1 sin {n't+e 1 vs }
4 -74 m'e'siu {n't+e. 1 w'}
+ 41 '72m 7 sin {nt 2n't+e 2e'+}
16 '47 w'e'sin {nt 2n't+e 2e'+cj'}
+ 33 -93 m' sin {2nt 3n't+2e 3e'
63 -41 mVsin {2w< 3^+2e
To these we may add the following, which are of two dimensions in terms of the
eccentricities :
u
+ 0'40m ; sin 3 {nt n't+e e?}
74 m' J sin {3 (^ n't+e c') CT+OT'}
34. Now, on our present assumption, ra = 13 0' *6, w'= 4 48' *5, n n f ~8 12' *1,
w-2w'=3 23' -6, 2rc~3ra'=ll 35' 7.
Hence the equations of condition given by the modern observations will be of the
form
II O / O /
c= h +5^008 {13 0'5}*+S# 2 cos{26 1 '0} t
+$# 1 sin {13 0-5}*+fy a sin {26 1 '0} t p
+ ^ cos{ 8 12'1}<+ ^ 2 cos {16 24 -2}*+ a cos{24 36'3}j!
+ * sin { 8 12 -1} t+ k t sin {16 24 '3} t+k a sin {24 36 '3}*
+ p l cos { 4 48 '5} t+ p* cos { 3 23 '6} t+p s cos {11 35 7} t
+ ^sin { 4 48 '5}^+ & sin { 3 23*6}< + ^ 3 sin {11 357}<
u 2
282 On the Perturbations of Uranus.
35. Treating these equations of condition in the same manner as before, the equa
tions in the first group, derived from them, are found to be the following :
(e) 151'48= 21*00003e + 6 *0670 3^ 4 '4358 3x 2
+ 13 '9515 A! + 0-9471 h 2 4 '5965 h*
+ 18-3916^+19-6752 p*+ 8'4184 p s
(x) 246*48= 6'06703e+ 8 '2821 3^ + 4*l7623x 2
+ 7 '3540 7^ + 8-3027 h 2 + 5'096l 7i a
+ 6-5793^ + 6-3319 jt? 2 + 8 -0850 p a
(A,) 207-58= 13 '9515 3e + 7 -3540 3^ *4177 3a? s
+ 10-97357^ + 4'6775 7*2 0-0005 h a
+ 12-8697^+13*4050 jt? 2 + 8-4781 p a
(7* 2 ) 245-17= 0*94713 + 8-3027 ^ t + 7*23625^
+ 4 -6775 A!+ 10 -0259 7* 2 + 8 '3220 7/ 8
+ 2-3661^+ 1'6727 p*+ 7*3073^3
(7i) 103 -48= 4 -5965 3e+ 5 '0961 ^+10 '5558^
- -0005 ^+ 8 -3220 7/ a + 10 '9749 h a
v 3-7316 p z + 3-5852 p a
36. Similarly the equations in the second group, are
(n) 171 -27= 77 '0000 ra+ 9 '3938 3^ 1 '2183 Sy 2
+ 87355 k,+ 7-6213 k z O-OSQO A,
+ 5-9764 ?!+ 4-3875 ^ 2 + 9-6152 ^
(y) -166-33= 93*93803^+12-7179^+ 1-8907^
+ 11-0393 ^+11*3717 ^ 2 + 3-3196 k a
+ 7*3747 ?!+ 5.-3S25 ^+12-6816 ^ 3
(*) 181 -31= 87 -3550 3w + ll *0393 c^ '3758 3# 2
+ 10-0264 ^+ 9'2740 7^ 2 + 0-9476 k a
+ 6*8054 ^+ 4*9866 # 2 +ll-1971 q a
99*51= 76-2130 Src+11 -3717 3^+ 4-4810^ 2
+ 9*2740 #!+ 10 -9740 2 + 5'6294 k a
+ 6-0523 # t + 4-3916 #2+11-0843 # 8
113*14 = 0*5900 $n+ 3-31963^+10*21123^
+ 0*9476 k,+ 5*6294 ^+10*0251 ^
+ 0-1746 ? t + 0-0454 # 2 + 2*4791 # 3
37. The equations (p 2 ), (j^s) of the first group, and (# 2 )> (#3) of the second were not
formed, as our previous solution shewed that when Se, SM, Jaf 1? and Jy n were eliminated,
the co-efficients of the remaining unknown quantities in these equations would be
extremely small. It will be preferable to combine the equations (Ai), (A 2 ), (^ 3 ), and
(*i)> (^z)j (^3) before, instead of after, the elimination of &, 8n, $x lt and &/!, from them.
On the Perturbations of Uranus. 283
If then we change the sign of the third equation in each group, and add it to the
fourth and fifth, we obtain
141 -07= 17 '6009 $ + 6 '0448 S^+18 '2097 &e 2
6'2965 7^4-13-6704 7< 2 +19-2974 h. A
13 '3971 ;?, 15-4639 p*+ 2-4144 p s
194 '94= 11 '7320 Zn+ 3-6520 ?/,+ 15 '0680 ty 8
+ 0-1951 k,+ 7'3294 &+ 14 -7069 3
'5785 q, '5496 q z + 2 '3663 q 3
38. By means of (t) and (n) of Articles 35 and 36, eliminate oe and bi from (x)
and (?/), and also from I he equations just found, and we have
a
(x) 202 72= 6'5294 a^+ 5-4577 fog + 3'3234 h+ 8 '0291 7* 2
+ 6-4240 A 3 + 1-2659 ^ + 0-6477^2+ 5-6529^3
42'6l= 1-2578 3y t -f- 3 '3771 ^+0*3822^+ 2*0739^
+ 3-3916 k B + 0*0836 = 76"'55 sin 26
-i-, = -42 *33 cos e, 2 , = 76 '55 cos 2
771 */*
^ = 7%5 sin 30+0 -007460 ' + '008974 f
m Tfi wi
^= 7'25 cos 30-0 '008974 0+ 0-007460^
)= -20 sin (0-^8)- -074738 \^j cos 20 -- ^ sin 2 0}
Wt (.'W Wi J
-}=- 0-20 cos (0-y6) + -074738 |^5 sin 20 + ^cos 20}
g= 32 '91 sin (20-^) + -259765 jocose -^ sin 0}
^J= 32-91 cos (2 0-y8) + -259765 {^ sin +|/ cos0}
46. Substituting these expressions in the above equations, and putting for ft its
value 50 1 5' '8, we obtain
0= (1 -24872) sin0 + (l '32231) cos0 (1 '48110) sin 20 + (2 '24265) cos 2
(1 '48373) sin 30+ (2 '22809) cos 3 0+(9 '26254) ~,-(.9 '50079) J?
+ (8'44376) |> cos Jv sin 01- (8 '
-(8-17031) j|? cos 20- !' sin 20}-(8 '0686l) {7 sin 20 + ^ cos 2 0}
On the Perturbations of Uranus. 287
0= (I -65190) sin 0+ (2 -06584) cos + (2 "30220) sin 20-(2'60306) cos 20
-(2 -19916) sin 30- (2 -15032) cos 3 (0 *14305) / (9 '60933)
+ (9 '34981) { cos 0- $ sin0}-(9 '31615) {**+ cos 0}
-(8 -85046) j^> cos 2 0-? sin 2 0}-(9 '11828) j|, sin 2 + $ cos 2 0}
0= (1 -91407) sin 0- (2 -55189) cos 0- (2 '62790) sin 20- (2 '64230) cos 2
(2*25331) sin30-(2-34185) cos 3 + (9 '96344) ?+ (0 '56029) $
-(9-83835) {^cos0-$sin0} + (9'64968) ||? sin0 + J*cos 0}
+ (9 -45371) |: cos 20-$ sin 20j + (9 '47306) {^ .in 20 + $ cos 2 0}
where the numbers enclosed within parentheses denote the logarithms of the corres-
ponding co-efficients, as before.
47. From these equations, we find, by the same method as before
= -4655', ^=:138"'92, $=-109'83
Hence, since e = 217 55', e' = 264 50', the mean longitude of the disturbing
planet at the epoch 18 10 '328. The sidereal motion in 36 synodic periods of
URANUS = 57 42', Precession = 30'. .*. Mean Longitude at the time 1846 '762, or
October 6, 1846, = 323 2 ; .
Also, the expressions for --/ and - "J are
> = 33 '93 sin (3 /8)-63 '41 e' sin (3 0-;6')
$ = 33 '93 cos (3 0^-63 '41 e' cos (3 0-0')
wnere ro'rr ft.
Equating these to the values given above, we find e' =2 '4123, /6' = 279 1 1', and
.' . CT' = 298 41'. Hence longitude of the perihelion in 1846 = 299 1 1'.
Lastly, substituting the values just obtained in equation (1) of Article 39, we find
w*'=0 -75017.
48. Hence the values of the mass and elements of the orbit of the disturbing planet,
resulting from the second hypothesis as to the mean distance, are the following:
Mean Longitude of the Planet,0ctober 6, 1846, 323 2
Longitude of the Perihelion ...... 29911
Eccentricity of the Orbit ....... '120615
Mass (that of the SUN being 1) .... '00015003
288 On the Perturbations of Uranus.
49. From the values of m', 0, ~/and -^ found above, the values of the quantities
h t k, p, and q, corresponding to each hypothesis, are immediately determined. Thus
we find
1st Hypothesis. 2nd Hypothesis.
=0-5 = 0-515
A t = 23-98 A l= = 19-07 A t == 23'19 ^= 21-69
A 2 = 47 '58 A 2 = 11 *00 A 2 = 57 '30 k z = 3 '83
A 3 = - 1-93 A 3 = 7 -64 A 3 = 3 -40 k 3 = 5 76
Pl = 9-93 ^1 = - 8-31 ^= 6'52 gi= 7*34
p a 8-54 ^ 2 = 55 '36 ^ 2 = 11 '62 ft =r 54 '39
p 3 = 224-90 5-3= 171-63 j9 3 = 104-21 ^ 3 = 82-39
50. And by substituting these values in the equations (), (w), (z), and (y), we
obtain
1st Hypothesis. 2nd Hypothesis.
a
~rf =
0'5
a
0-515
IT
f
1
49-77 2
= o 702
B = 43 -23
^w = 0-5417
^=130-69 c
Jy l= = 222 '38
^= 1 77
$y l= = 123 '98
; 2 = 1 -02 I
?t/ 2 = 2 -83
^X 2 = 1 *13
gy 8 = -91
and the corresponding corrections of the elliptic elements will be
-=0 '00000999 = ' 0000 77 l
It will be seen that the corrections of the eccentricity and longitude of perihelion
vary very rapidly with a change in the assumed mean distance.
5 1 . If these quantitities be substituted in the expressions before given, we obtain the
following theoretical corrections of the mean longitude, each of these corrections being
divided into two parts, of which the first is due to the changes in the elements of the
orbit of Uranus, and the second to the action of the disturbing planet.
On the Perturbations of Uranus.
289
HYPOTHESIS I.
Ancient Observations.
Modern Observations.
Year.
1712
288 '0 + 365 '8= +77
8
Year.
1780
a it
126-12+129
27=+ 3-15
1715
283
1 + 357
*l = + 74
o
1783
180
28 + 188
70=+ 8
42
1750
+ 210
5260
7= 50
2
1786
227
66 + 240
36= + 12
70
1753
+ 218
1267
0= 48
9
1789
265
70 + 281
63= + 15
9 3
1756
+ 214
0260
0= 46 '0
1792
292
25 + 310
38= + 18
13
1764
+ 154
0186
7= 32
7
1795
305
84 + 325
27= + 19
43
1769
+ 79
6100
7= 21
i
1798
305
67 + 325
72= + 20
05
1771
+ 27
6 41
8= 14
2
1801
291
77 + 312
05 = + 20
28
1804
264
'95 + 285
38= + 20
43
1807
226
78 + 247
51 = + 20
73
1810
-179
43 + 200
76=+21
33
1813
125
59 + 147
72= +22
13
1816
- 68
21+ 91
02= + 22
81
1819
10
40+ 33
18ac+22
78
1822
+ 44
84 23
'64 = + 21
20
1825
+ 94
69- 77
64= + 17
05
1828
+ 136
73-127
48=+ 9
25
1831
+ 168
94172
17=- 3
23
1834
+ 189
85211
04= 21
19
1837
+ 198
51243
59= 45
08
1840
+ 194
54269
36= 74
82
HYPOTHESIS II.
Ancient
Observations*
Modern Observations.
Year.
a n
H
Year.
H
1
,,.
1712
133
7+211
9= + 78
2
1780
133
10 + 135
98=+ 2
88
1/15
-117
7 + 191
5= + 73
8
1783
149
47+157
87=+ 8
40
1750
+ 85
2134
4=-49
2
1786
160
15 + 172
99= + 12
84
1753
+ 73
8122
2 48
4
1789
-164
52+180
64 = + i6
12
1756
+ 59
1 105
2= 46
1
1792
162
30+180
58 = + 18
28
1764
+ 2
7 36
4 = 33
7
1795
153
59+173
07= + 19
48
1769
43
1+ 20
8 = 22
3
1798
138
87 + 158
86= + 19
99
1771
- 6 9
'9+ 54
7=-15
2
1801
118
95 + 139
08 = + 20
13
1804
- 94
'96+115
21 = + 20
25
1807
- 68
25+ 88
85 = + 20
60
1810
40
33+ 61
*6 1 = + 2 1
28
1813
12
72+ 34
91 = + 22
19
1816
+ 13
08+ 9
'88= + 22
96
1819
+ 35
71 12
74= + 22
97
1822
+ 54
04 32
68= + 21
36
1825
+ 67
18- 50
08 = + 17
10
1828
+ 74
52 65
37=+ 9
15
1831
+ 75
74- 79
21 = 3
47
1834
+ 70
85 92
31= 21
46
1837
+ 60
08105
25 = 45
'I?
1840
+ 43
98-118
38 = 74
'40
290 On the Perturbations of Uranus.
52. Comparing these with the corrections of mean longitude derived from observa-
tion, we find the remaining differences to be the following :
Ancient Observations. Modern Observations.
Observation-Theory. Observation Theory.
Year. Hypoth. I. Hypoth. II. Year. Hypoth. I. Hypoth. II
1712 467 4 6-3 1780 40" '27 40" '54
1715 - 6 '8 6'6 1783 0'23 0-21
1750 1-6 2-6 1786 0-96 l-io
1753 457 +5'2 1789 41'82 41-63
1756 4-1 4-0 1792 0-91 1'06
l/6l 5-1 4-1 1795 40-09 40'04
1769 40-6 41-8 1798 0'99 0-93
1771 411*8 412-8 1801 0-04 40-11
1804 41 76 41 -94
1807 0'2i 0-08
1810 40-56 40-61
1813 -94 1 '00
18J6 0-31 0-46
1819 2-00 2-19
1822 40*30 40-14
1825 41 -92 41 '87
1828 42-25 42'35
1831 1 '06 '82
1834 1-44 1-17
1837 1-62 1-53
1840 4173 41-31
The largest difference in the above table, viz. 9 that for 1771> is deduced from a
single observation ; whereas the difference immediately preceding it, which is deduced
from the mean of several, is very small.
53. The results of the two theories agree very closely with each other, and with
observation, till we come to the later years of the series ; and it is to be observed that
the difference between the theories becomes sensible at precisely the point where they
both show symptoms of diverging from the observations, the errors of the second
hypothesis, however, being less than those of the other.
Recent observations show that the errors of the theory soon become very sensible,
though decidedly less for the second hypothesis than for the first. The following are
the differences of mean longitude as deduced from theory and observation, for the
oppositions of 1843, 1844, and 1845 :
Observation Theory.
Year. Hypoth. I. Hypoth. II.
1843 4 7" '11 4- 5*77
1844 4 8 79 4- 7'05
1845 412-40 410 '18
Oil the Perturbations of Uranus. 291
For the observations of the last two years, I am indebted to the kindness of the
Astronomer Royal. The three years nearly agree in showing that the errors of the
first hypothesis are to those of the second in the ratio of 5 to 4, from which I inferred,
in a letter to the Astronomer Royal, dated September 2, 1846, that the assumption of
o,
, = sin 35 = *574, would probably satisfy all the observations very nearly.
54. The results which I have deduced from Professor Challis's observations of the
planet, strongly confirm the inference that the mean distance should be considerably
diminished. It is of course impossible to determine precisely, without actual calcu-
lation, the alteration in longitude which would be produced by such a diminution in
the distance. By comparing the values of 9 given by the two hypotheses, it may be
seen, however, that if we took successively smaller and smaller values for the mean
distance, the values found for the mean longitude in 1810 would probably go on
diminishing, while at the same time the mean motion from 1810 to 1846 would
rapidly increase, so that the corresponding values of the mean longitude at the present
time would probably soon arrive at a minimum, and afterwards begin again to in-
crease. This I believe to be the reason why the longitude found on the supposition
of too large a value for the mean distance agrees so nearly with observation. In
consequence of not making sufficient allowance for the increase in the mean motion,
I hastily inferred, in my letter to the Astronomer Royal mentioned above, that the
effect of a diminution in the mean distance would be to diminish the mean longitude,
55. I have already mentioned that I thought it unsafe to employ Flamsteed's obser-
vation of 1690 in forming the equations of condition, as the interval between it and
all the others is so large. The difference between it and the theory appears to be very
considerable, and greater for the second hypothesis than for the first, the errors being
4-44"*5 and +50' 7 '0 respectively. These errors would probably be increased by
diminishing the mean distance. It would be desirable that Flamsteed's manuscripts
should be examined with reference to this point.
56. The corrections of the Tabular Radius Vector of Uranus may be easily deduced
from those of the mean longitude by means of the following formula:
Ll_ L^Ts r _JL lL*y l ^_ 1 ede _? i *
m f
+ m'e S Di cos {* (nt n't+e ')~ ^t
+ mV2 E; cos {i (nt ra'<+ e t') nt
where & denotes the whole correction of the mean longitude at the, time t t
1 dr 3 e 3
- -j- = e sin \nt-\-e. cr}+ sin 2 [nt-\-e. cr} nearly,
On the Perturbations of Uranus.
f assuming all integral values positive and negative not including zero.
57. By substituting in this formula the values of m', %a, Se, &c., already obtained,
and putting a = 19 '191, we find the following results corresponding to the two assumed
values of the mean distance.
HYPOTHESIS I.
a a dr
+ 0'000069 cos {nt n't+e e'}
+ '000259 cos 2 {nt n't+e E'}
+ '000109 cos 3 {nt n't + e e f }
+ '000016 cos {n't + e' ro}
-000168 cos {nt 2n't+e 2e'+cr}
+ '000078 cos {ntZn'f + EZe'+n'}
'000049 cos {2nt3n't + 2e3e'+ia}
+ '000209 cos {2nt Zn't+Ze
HYPOTHESIS II.
a dr
+ '000073 cos {nt n't+e e'}
+ -000266 cos 2 {nt n't + e e'}
+ -000115 cos 3 {nt?i't + /}
+ '000016 cos {rit+E'rz}
0-000188 cos {nt 2n't + e
+ -000068 cos {nt 2n't + 2e
'000053 cos {2nt 3n't+2e
+ -000165 cos {2nt3n't+2e3e f
58. The values of 3 and - for several late years, are the following
HYPOTHESIS I.
Year.
1834
^s
ii
21-19
dt
ii
20 -93
1840
1846
74-82
148 -65
32-34
-39 '94
HYPOTHESIS II.
// ii
1834 21 -46 20 -85
1840 74 -40 31 '62
1846 145 -91 38 -30
On the Perturbations of Uranus. 293
Hence, by means of the above formulae, we find the corrections of the tabular
radius vector, to be
Year.
1834
1840
1846
59. By far the most important part of these corrections arises from the term
2 T ~di } an( ^ ma y tnere f re ke immediately deduced from a comparison of the ob-
Hypothesis I.
+ -00505
+o -00722
+ '00868
Hypothesis II,
+o -00492
+o -00696
+ -00825
served angular motion of Uranus with that given by the Tables. In fact, the cor-
rections given by this term alone for the epochs above-mentioned, are
Year. Hypothesis I. Hypothesis II.
1834 +0'00447 +0-00445
1840 +0-00694 +0-00678
1846 +0-00853 +0*00818
which, as we see, differ very little from the complete values just found. The correc-
tion for 1834, very nearly agrees with that which Mr. Airy has deduced from obser-
vation in the Astronomische Nachrichten. The corrections for subsequent years
are rather larger than those given by the Greenwich Observations, the results of the
second hypothesis being, as in the case of the longitude, nearer the truth than those
of the first.
60. I made some attempts, by discussing the observations of latitude, to find ap-
proximate values of the longitude of the node and inclination of the orbit of the dis-
turbing planet, but the results were not satisfactory. The perturbations of the latitude
are in fact exceedingly small, and during the comparatively short period of three-
fourths of a revolution, are nearly confounded with the effects of a constant alteration
in the inclination and the position of the node of URANUS, so that very small errors
in the observations may entirely vitiate the result.
61. The perturbations of Saturn produced by the new planet, though small, will
still be sensible, and it would be interesting to inquire whether, if they were taken
into account, the values of the masses of Jupiter and Uranus found from their action
on Saturn would be more consistent with those determined by other means, than they
appear to be at present. The reduction of the Greenwich Planetary Observations
renders such an inquiry comparatively easy, and it is to be hoped that English astro-
nomers will not be the last to avail themselves of the treasures of observation thus laid
open to the world.
St. John's College, Cambridge,
November 12, 1846.
295
ON THE CORRECTION OF A LONGITUDE
DETERMINED APPROXIMATELY BY THE OBSERVATION
OF A LUNAR DISTANCE.
BY THE REV. J. CHALLIS, M.A., F.R.S., F.R.A.S.
PLUMIAN PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF CAMBRIDGE.
IN the usual method of determining the Longitude by the observation of a Lunar
Distance, the measurement of the arc between the Moon and the Sun or Star, is
accompanied by observations of the altitudes of the extreme points of the arc. These
auxiliary observations enable the observer to reduce the measured distance to the dis-
tance as seen from the Earth's centre ; whence, by means of results of theoretical calcula-
tion furnished by the NAUTICAL ALMANAC, the Greenwich Mean Solar Time of obser-
vation, and, by consequence, the Longitude of the place of observation, may be inferred.
The peculiar advantage of the method is, that it does not require the Longitude to
be previously known even approximately. It is not, however, capable of a great
degree of accuracy. The sources of error are, first, the error of observation of the
Lunar Distance, and, to some amount, the errors of the observed altitudes ; next, the
errors in those data of the observer's calculation which relate to the locality where
the observation was taken ; and lastly, the errors of the results of theoretical calcula-
tion in the NAUTICAL ALMANAC. The initial determination of the Longitude by a
Lunar Distance is, therefore, to be regarded only as a first approximation, which
maybe subsequently employed either in deducing from the same observations a more
correct value, or as auxiliary to more accurate methods, such as the observation of
an occultation of a Fixed Star or Planet by the Moon, or of the beginning or end
of a Solar Eclipse, the calculating of which requires an approximate knowledge of
the Longitude.
In order to deduce the Longitude from a Lunar Distance with as much accuracy
as the kind of observation will admit of, it is necessary to go through a calculation
supplementary to that which conducted to the first approximation. An obvious
method of conducting this calculation is, first, to use the observed altitudes merely
for clearing the Lunar Distance of refraction, for which purpose they will in general
be sufficiently accurate ; and then to compute the apparent Lunar Distance for the
place of observation and the approximately known Greenwich Mean Time, in such
manner as to take account of small unknown corrections which the data of the
computation may require, including among them the correction of the assumed
Longitude. An expression will thus be obtained for the true apparent Lunar
Distance, which, being equated to the observed Lunar Distance corrected for refrac-
tion, furnishes a provisional equation of condition between the unknown corrections,
by means of which the correction of the assumed Longitude is determined, when by
any independent means the values of the other corrections have been ascertained.
x2
296 On the Correction of a Longitude determined
If an occultation of a Fixed Star or a Planet by the Moon, or the beginning or
end of a Solar Eclipse, happen to be observed at a place whose Longitude has been
approximately found by a Lunar Distance, this first determination gives the means
of calculating the Occultation or the Eclipse, and thus obtaining a correction of the
approximate Longitude. The calculation maybe made precisely in the same manner
as the above mentioned supplementary calculation of a Lunar Distance, the only
difference being that it admits of some simplification in consequence of the small
distance of the Star, or the Sun, from the Moon's centre. An equation of condition
between the unknown corrections is obtained as in that case, possessing greater value
than the equation of condition for the Lunar Distance, on account of the greater
degree of accuracy which the observation of the Occultation or Eclipse is capable of.
The main object of this communication is to point out an exact and expeditious
method of obtaining the provisional equation just spoken of. The corrections of all
the assumed data will be taken account of (which, as far as I am aware, has not
hitherto been done), and formulae convenient for this purpose, not requiring long or
complicated calculation, will be investigated. An example will be added for the
sake of illustration.
Let it be supposed that the Moon's Geocentric Right Ascension (R'), Geocentric
North Polar Distance (X'), Equatorial Horizontal Parallax (P'), and Geocentric
Semidiameter (S')> at the assumed Greenwich Mean Time, have been obtained by
interpolating to second differences from the NAUTICAL ALMANAC.
Let there be known by observation, the Right Ascension of the zenith, or sidereal
time of observation, at the place () ; and the Lunar Distance (A), that is, the arc
between the Moon's Limb and the Limb of the Sun or a Planet, or between the
Moon's Limb and a Star, which arc is supposed to be cleared of refraction.
Also let I = the assumed Geocentric Co-latitude, that is, the Astronomical Co-
latitude increased by the angle of the vertex.
6' = R' Z, = the Geocentric Hour Angle of the Moon's centre, supposed
to be East.
P = the Horizontal Parallax at the place.
In the annexed figures, A, C, m,
are respectively the positions in
space of the place of the observer,
the Earth's centre, and the Moon's
centre, at the time of observation.
P is the North Pole of the heavens,
Z and M' are the intersections on
the celestial sphere of the prolonga-
tions of C A and C m, and M on the
arc Z M' produced is the intersec-
tion of the sphere by a line through
C parallel to A m.
by the Observation of a Lunar Distance. 297
Let A C = />, the Earth's Equatorial radius being unity, and let C m = r. Then
we have,
r sin P' = 1 r sin P = p
Hence, sin P = p sin P' (1)
Also if the angle z A m = z, and the angle A m C = p, the plane triangle
m A C gives,
sin p p
sin z ~ r
Hence sin p = sin P sin z (2)
By the spherical triangle P M M',
sin P MM'
sin M M' sin P M'
And by the spherical triangle P Z M,
sin ZM sin P Z
"sinZPM = sin P MM'
Hence, by multiplying together the corresponding sides of the two equations,
sinZMsinMPM' _ sin P Z
sin MM' sin ZP M = JnPM"'
Now thearc ZM - z, MM' = p, PM' = X', PZ = /, the angle, Z P M' = 0',
and if = the apparent hour angle Z P M, the angle M P M' = 0'. Substituting
these symbols in the above equation, we have
sin z sin (00') sin /
sin p sin ~~ sin X'
Hence, by substituting for sin p from (2),
sin P sin / .
sin (00') = g^~x/~ sm e (3)
Let G = sin P sin / cosec X' (4)
Then sin (0-0') = G sin = G sin (0-0'+ 0')
Consequently, tan (00') = ^ _ Q. cos Q' ( 5 )
The formulae (4) and (5) serve for calculating 00', and, by consequence, the
apparent hour angle 0. The calculation by the latter formula is facilitated by the
use of a Table of Subtraction-Logarithms.
Formulae for calculating the Moon's apparent North Polar Distance may be
investigated as follows.
Let X = P M, the Moon's apparent North Polar Distance,
and z' = Z M', the Moon's Geocentric Zenith Distance.
Then by the spherical triangle Z P M',
cos X' = cos z' cos / -f sin z' sin / cos P Z M'.
298 On the Correction of a Longitude determined
And by the spherical triangle Z P M,
cos X = cos z cos / -f- sin z sin / cos P Z M.
Hence, dividing these equations respectively, by sin z' and sin 2, and subtracting
the latter from the former,
cos X' cos X
cos /sin (z z'}
sin z sin z'
cos / sin p
sin z sin z'
Substituting now for sin p from (2) and multiplying by sin z f ,
cos X' cos X -^-^ = sin P cos /
sin z' sin X'
But
and
sin0' - sinPZM'
sin z sin X
sin == sin P Z M
sin z' sin 6' sin X'
Consequently, . -- = . z . r-
J ' sin 2 sin 6 sin X
Hence, by substituting in the foregoing equation,
cos X' cot X sin X' sin 0' cosec = sin P cos /.
This equation gives for determining X,
cot X = (cot X' sin P cos I cosec X') sin cosec 0' ..... (7)
To adapt this expression to logarithmic computation, let
tan = sin P cos / cosec X'
so that ^ is a small angle, positive or negative according as / is less or greater than
90, that is, according as the place of observation is North or South of the Equator.
Thus
cos (X' +0) sin
cotX= cos * sin V sine' .............. (9)
The formulae (8) and (9) are applicable for calculating X, at whatever position on
the earth's surface the observation is made.
If S be the Moon's apparent Semidiameter, we have (by the figure),
S Cm sin z
Henceb y( 6) ; S = S'. r .......... (10)
This equation serves for calculating the Moon's apparent Semidiameter, and X
having been previously found.
by tlte Observation of a Lunar Distance. 299
The Moon's apparent Right Ascension (R) is equal to + 0, the hour angle
being supposed to be East.
Let the apparent North Polar Distance of the Sun, or Planet, or Star, be 7, and
the difference between its apparent Right Ascension and that of the Moon be ij.
Then the apparent distance (c) of the Moon's centre from the centre of the Sun or
Planet, or from the Star, is given by the equation,
cos c = cos X cos 7 + si* 1 ^ sin 7 cos t] -- (11)
In the case of an Occultation or Solar Eclipse it will be advisable to make use of a
modification of this formula on account of the small value of c. Since
cos c = 1 2 sin 2 and cos t\ = 1 2 sin 2 ^
it will be seen that
1 2 sin 2 - = cos ( X 7) 2 sin X sin 7 sin 2 ^
= 1-2 sin 2 ^-ZJ - 2 sin X sin 7 sin 8 3
Hence, sin 2 = sin 2 ~2 4. sin X sin 7 sin 2 5
If therefore d = the difference between X and 7, and if c, d, and rj, be each expressed
in seconds of space, by substituting the small arcs for their sines, we have
c 8 = d* + sin X sin 7 if
, / sin X sin 7 ii 8 \
Hence c = d (l + ^J 1 J * (12)
which formula is convenient for calculation if a Table of Addition-Logarithms be at
hand. Otherwise a subsidiary angle \ mav be used such that
tan x = -j * / sin X sin 7
Then, c = d sec x
Now let * be the Tabular apparent Semidiameter of the Sun or Planet, and sup-
pose that the distance between the nearest limbs was measured. Then the computed
value of the Lunar Distance is
c - S - s
Let the true value be,
Then, assuming the observation to be correct, we have the final equation,
Or, if c be the excess of the observed above the computed Lunar Distance, the
equation of condition is,
$c-$S-S* = .............. (13)
300 On the Correction of a Longitude determined
In the observation of an Occupation, or of the beginning or end of a Solar Eclipse,
A = 0, and refraction has no effect. Hence for these cases the equation of
condition is,
The variations <> S and s are the corrections to be added to the values of the
apparent Semidiameters of the Moon and of the Sun or Planet adopted from the
NAUTICAL ALMANAC, in order to reduce them to the true values. The amounts of
these corrections for given distances from the Earth's centre, I shall suppose to be
known by repeated and exact measurements. The remainder of this investigation
will be mainly employed in obtaining an expression for the correction a c, in terms of
small corrections of the elements of the calculation which conducted to the value of c.
For this purpose it will be necessary to obtain expressions for the correction (S 0)
of the Moon's apparent Hour Angle and the correction ( X) of the Moon's apparent
North Polar Distance. Resuming the equation (3), viz.
/n n/\ sin P sin / .
sin (0 ) = sm
sin X'
it will be seen that is a function of 0', X', I and P. Hence to the first order of
small quantities,
*._ ^*0^_i ^ ^ . dQ ,. d ..
where S 0', SX', $ /, and & P, are respectively the corrections of the assumed values of
0', X', /, and P ; and their multipliers are the partial differential coefficients of with
respect to the same quantities. For obtaining these partial differential coefficients it
will be convenient to use, instead of equation (3), its logarithmic equation, viz.
Log sin (0 0') = Log sin Log sin X' + Log sin I -f Log sin P
Hence, differentiating with respect to 0',
= cot (0 - 0') sin cos (0 - 0')
dO' cot (0 - 0') - cot ~ sin0'
( ,
Differentiating with respect to X',
. dO ___ cot X' _ cot X' sin sin (0 0') , ,.
' d X' cot (0 - 0') - cot " sin0'
Differentiating with respect to /, since / and X' are similarly involved in the
logarithmic equation, we have by (16),
dO _ cot / sin sin (0 0')
dl ~ ~
by the Observation of a Lunar Distance. 301
So by differentiating with respect to P,
dO _ cot P sin sin (9 0') , .
d~P ~ "liiTe 7 ""
Again, resuming the equation (7), viz.
cot X = (cot X' sin P cos / cosec X') sin Q cosec 6',
which, since is a function of 0', X', /, and P, shows that X is a function of the same
quantities, we have to the first order of small variations,
To obtain conveniently the partial differential coefficients, the logarithmic equation
of (7) will be made use of, viz.
Log cot X = Log (cos X' - sin P cos /) + Log sin - Log sin 0' - Log sin X'.
Differentiating this equation with respect to 0',
dX de
sec X cosec X , = cot - cot
a a
And by (1 5) -7-7- = sin cosec 0' cos (0 - 0')
(I
Hence 7-7- = sin cosec 0' sin X cos X 3111 (0 0') - - - - (19)
a
Differentiating with respect to X',
d X sin X' d
- sec X cosec X -^ = - _,___ _ cot X +^0-^-,
But ^ W> cos V - sin P cos/ = tan X sin Q cosec Q> cosec X
And by (16), -^7 = cot X' sin cosec 0' sin (0 0')
, i
Substituting these' values it will readily be found that
sin V
Differentiating with respect to /
C S
. d X sin P sin / a
sec X cosec X j-r = -- r/ - = =K -- j + cot -r-r
d I cos X' sm P cos / n d I
But by (7), cos X' - sin P cos / = tan X sin B COS6C B ' COS6C X '
(IB
And by (17), -T-T- = cot / sin cosec 0' sin (0 0')
d X sin P sin sin X .
Hence -77 = - . fl , . ., (sin X sin / + cos X cos / cos 0)- - - (21)
a i sin v sin A/ ,
302 On the Correction of a Longitude determined
Lastly, differentiating with respect to P,
. d X cos / cos P do
sec A, cosec A, 7^5- = -- r-i - : ^ - r 4- cot d
d P cos X' sm P cos / ~
But as before, cos X'-sinPcos/ = tan X sin e cosec e ' cosec X '
And by (18), -- = cot P sin d cosec 0' sin (0 - e')
= cos P sin 2 cosec 0' sin I cosec X' by (3).
d X cos P sin sin X
d P sin 0' sin
Hence -- = " e T sin " x 7 " (sin X cos / - cos X sin / cos 0) (22)
We have next to obtain expressions for the small variations 5 0', S X', J /, and 5 P.
Since by supposition the Hour Angle is East,
8' = R'- I
HenceS0' = SR'-H
Suppose now that the true Longitude of the place of observation is equal to the
assumed Longitude + r, an East Longitude being reckoned negative. Also let the
true Right Ascension of the Zenith be + t. Then the correction of the assumed
Greenwich Mean Time of observation is p (t + r), p. being the factor [9,99881]
which converts an interval of Sidereal Time into an interval of Mean Solar Time.
Hence if t and T be expressed in seconds, and if a be the increment of the Moon's
Geocentric R. A. at the time of observation in one second of Mean Time, the cal-
culated Geocentric R. A. requires the correction + p, a. (t + r) on account of the
error of the assumed Greenwich Mean Time. Let a be expressed in seconds of arc,
and let the correction of the Tabular R. A. of the Moon at the same time be x in
seconds of arc. Then
S R' = p a (t + T) + x
The correction (& ) of the Right Ascension of the Zenith expressed in arc is
15 1. Hence
S0' = n<*(t + T) - 15* + x
So if y6 be the increment of the Moon's N. P. D. at the time of observation in one
second of Mean Time, and y be the correction of the Moon's Tabular N. P. D.,
S V = p fi (t + r) + y
The values of a and y6 are readily obtained after the calculation of the interpolated
values of R' and X'. For by the usual formula of interpolation to second differences,
h h* , h h*
R' = a , + b, |j + c, jp- X' = a 2 + 6 8 jj + c 2 jji
h being the algebraic excess of the time of observation above the nearest Epoch,
and H being the common interval (l h ) between the Epochs, both expressed in
seconds. Hence, R' being in time,
by the Observation of a Lunar Distance. 303
*'*
.
Th H H* 15
rfX' 6. Zc,h
r .
dh
The correction / is the sum of the correction of the assumed astronomical co-
latitude, and the correction of the angle of the vertex. Let us suppose that & / = v
in seconds.
The error of P depends on the error of the Tabular Equatorial Parallax, the
error of the assumed distance of the place of observation from the Earth's centre,
and the error of the assumed Greenwich Mean Time of observation. By varying the
symbols in equation (1) we have,
cos P X P = p sin P' + p cos P' J P',
or, omitting small quantities of an order that may be neglected,
s p = p' $ p + p a P\
*
If the true Equatorial Horizontal Parallax of the Moon for a given distance from
the Earth's centre has been ascertained to be P, (1 -f- 0,001 yw t ), P t being the corres-
ponding value in the NAUTICAL ALMANAC, the true Equatorial Horizontal Parallax
at the time of observation may be taken to be P' (1 -f- 0,001 m^. Also if i be the
increment of the Equatorial Horizontal Parallax at the time of observation in one
second of Mean Time, the correction of P' for error of the assumed Greenwich
Mean Time is -f- p. t (t 4- T). Hence
S P' = 0,001 P' m l + n i (t -f r).
Let the true distance of the place of observation from the Earth's centre be
p (i 4. o,001 z 2 ), so that S p = 0,001 m z p. Then
5 P = 0,001 m t p P' + 0,001 m l p P' + pt p (t + T).
Hence, putting m for m l + m s and supposing that p = I in the third term, which
is very small,
$ P = 0,001 m P + p i (t + T).
If the interpolated value of P' be
. h' h n
and h', H' be expressed in seconds, then with sufficient approximation,
dh'
(25)
For the purpose of facilitating the logarithmic computation by the eight formulae
(15 22), and for the sake of brevity of expression, the following substitutions will
now be made :
304 On the Correction of a Longitude determined
sin 9 L sin X
L = ^Te' Q = TirTAT = Q cos X N = Q sin X
A ^ B - C d& P de
de' d\' Tl = '
A '_ ^_ X TV d ^ P , d\
J j at 15 = -JT/ O = -7-= D
x
1000 dP
By these substitutions the formulae are changed to the following :
A = L cos (0d f ) ............... !_._
B = - L cot X' sin (0-9') ............... (27)
C = L cot Jsin (9-9') ................. / 28 \
D = 0,001 PL cot P sin (9-9') ............. (29)
A' = M sin X' sin (9 9') ............... _ /^QN
B' - N sin X' -f- M cos X' cos (9-9') ........... ( 31 \
C' = N sin P sin I M sin P cos / cos 9 ........ (32)
D' = 0,001 PN cos P cos/- 0,001 P M cos P sin /cose ____ (33)
Most of the logarithms required for computing by these formulae, will have been
already obtained in the calculation of 9 and X.
Now since 5 R = 5 Z, + 5 0, by referring to the values which have been found for
S 0, H, S 0', 5 X', $ /, S P, it will be seen that
SR= Ibt + Adiaty + r) - 15*+ X )
+ B (p ft (t + T ) + y)
which equation takes the form,
So the expression for 5 X is,
5 X = A' (p. a (t + r ) - 15 t -f a:)
(34)
- - - - (35)
which takes the form,
U = o'< + b'r + A'x + E'y + C'v + D'
(36)
and a' = ^ (A' a +B'6) - 15 A' + L' = b > - ISA' ...... (37)
sin c
by the Observation of a 'Lunar Distance. 305
By making all the symbols vary in equation (11) we obtain,
sin c & c = (siii X cos 7 cos X sin 7 cos >)) X
4- (cos X sin 7 sin X cos 7 cos r)) 7
4- sin X sin 7 sin tj j
Hence, as may be readily shewn,
S c= I sin (7 4- X) sin 2 4- sin (X 7) cos 2
4- ( sin (7 4- X) sin 2 sin (X 7) cos 2 V-~
4- sin X sin 7 sin -n ^
sin c
Suppose the correction of the assumed R. A. of the Sun, Planet, or Star, to be e
in seconds of space, and the correction of the assumed N. P. D. to be/. Then
7 = /, and TJ = $ R e, or e S R, according as the Moon is more Eastward or
Westward than the other body. Putting d for the difference between the arcs
X and 7, let the three quantities N t , N 2 , N 3 , be calculated by the formulae
^ sin d 2 T]
sin A sin 7 sin rj
In the case of an Occultation, or Solar Eclipse, on account of the small values of
c, d, and 17, the following formulae may with sufficient accuracy be used instead of
the foregoing.
(41) N 2 = sin (7 + X) .... (42;
N 3 = - sin Xsin7 -- ............ - (43)
c
r" being the number of seconds in an arc equal to radius, and c, d, ij, being all
expressed in seconds of space.
Now let T = N 2 N! and U = N 2 + N 15 the upper or lower sign being taken
according as X is greater or less than 7 ; and let \V = N 3 , the upper or lower sign
being taken according as the Moon's apparent place is more Eastward or Westward
than that of the other body. Then
5 c = W ($ R - e) + T U + U/
The error of the Moon's calculated apparent Semidiameter (S) depends on the
error of the Tabular Geocentric Semidiameter, and the error of the assumed Green-
wich mean time of observation. It has been shewn that
sin 8 sin X
sin 0' sin X"
_
or fc
306 On the Correction of a Longitude determined
Hence 5 S = Q S S' + S' $ Q
If the true Geocentric Semidiameter for a given distance from the Earth's centre be
known from exact measurements to be S t (1 -J- 0,001ra), Sj being the corresponding
value in the NAUTICAL ALMANAC, the true value at the time of observation may be
taken to be S' (1 + 0,001 ri). Also if icbe the increment of the Geocentric Semi-
diameter at the time of observation in one second of mean time, the correction of S'
for error of the assumed Greenwich mean time is + p K (t -f- T). Hence
J S' = 0,001 S'n + n K (t + T)
and Q J S' = 0,001 Sn + JJL K (t + r), since Q = 1 nearly.
*
sin sin X
The equation Q = gin tf sin v gives,
-^ = cot J 8 - cot 0' J 0' + cot X S X - cot X' X'.
By considering only the parts of 5 Q and $ X which depend on 5 Q' and J X', which is
allowable, the following result to the first order of small quantities may be arrived at.
T " = - sin 2 X sin (0 - 0') 5 0' - sin (X - X') J X'
It will in no case be necessary to take account of the term involving 5 X', (which
c
term when multiplied by -77 becomes exceedingly small), and it will suffice to put
-15*forS0'. Thus
S. sin* X sin (0-
Hence, if a = ^-sin 2 X sin (0 - 0') - --- ..... (44)
we have S S = 0,001 S + jne(* + r)+a/
If the interpolated value of S' 'be
h' h*
a * 4 ~H~ + C * H 71 "
and h', H' be expressed in seconds, then very nearly,
dh r ~ H'
We are now prepared to substitute in the equation of condition (13), the values of
S c and S S which it was proposed to investigate.
The substitution gives,
e + $ * = W ($ R - ) + T $ X + U/- o,001 S n - p K (t + r ) - GJ t,
or, putting for S R and 5 X their values obtained above,
by the Observation of a Lunar Distance. 307
+ T (a' t + b f r + A'ar+ B'y + C'v+D'wi)
-(/iic + Gj)<-/iKr-W + U/- 0,001 Sn
Consequently, if the values of [1], [2], [3], [4], [5], [6], be obtained by the
formulae,
[1] = aW + a'T-/iK-t3 [2] = bW + b'T - pK [3] = AW + A'T
[4] - B W + B' T [5] = C W + C' T [6] = D W + D' T
the final equation, completely calculated is,
e + 5 * = [1] t + [2] r + [3] x - We + [4] y + U/+ [5] v + [6] m - 0,001 S n.
For the sake of simplicity the Hour Angle has all along been supposed to be East.
For Hour Angle West, 0' = 2, R', the calculations of and X are the same as for
Hour Angle East, R = Z, 0, and the signs of B, C, D, A', and a, are changed by
the change of sign of and Q'. No other alterations are required for adapting the
formulas to this case.
The foregoing investigation has been conducted so as to take fully into account
any error in the elements of the calculation that can possibly affect the calculated
result of the observation. This degree of exactness may be required in the case of
an Occultation, especially the disappearance or reappearance of a star at the Moon's
dark limb, which is an observation admitting of great precision : but in the ordinary
case of a Lunar Distance much of the calculation may be omitted, as relating to
quantities too minute to be worth taking into account.
To facilitate the use of the method above investigated, I propose now to collect
the formulas of calculation, and arrange them in the order in which they are to be
employed, and at the same time to point out those parts of the calculation which
may be omitted in the case of a Lunar Distance. The formulae, according to their
relation to each other, will be contained in separate articles numbered I, II, III, &c.,
for the sake of future reference.
I. The initial calculations are the interpolations of R', X', P', S', to second differences
from the NAUTICAL ALMANAC. The formulae are,
R' = , + *, A + Cl ^_ X' = 2 + b* A+ cj^
P' = 3 + ^ +Ca ~ S' = , may be taken from Tables of the
values of these quantities calculated for different latitudes on an assumed ellipticity.
See Appendix to the NAUTICAL ALMANAC for 1836, pp. 57 and 58.
sin P = p sin P'
sin P sin /
G -
tan (6 - 6') =
sin A/
G sin 0'
1 - G cos 0'
III. Formulas for calculating the Moon's apparent N. P. D. (X).
sin P cos /
tan = : r-y = G cot /
sin X'
sin
', L cos (X' + 0)
cot X = - } . x ,
cos sin X'
IV. Formulae for calculating the Moon's apparent Semidiameter (S).
L sin X
^"shTT 7 "
S = Q S'
V. Formulae for calculating the apparent Distance between the centres of the two
bodies (c).
The Moon's apparent R. A. (R) is -f- or 0, according as the Hour Angle
is East or West. The difference between R and the apparent R. A. of the other
body is ij.
cos c = cos X cos 7 + sin X sin 7 cos rj.
Transformations of this formula convenient for logarithmic computation are well
known. Tables of Addition and Subtraction -Logarithms might be used to cal-
culate by it in its present form.
In the case of an Occultation, or Solar Eclipse, d being the difference between
X and 7, and c, d, ij, being expressed in seconds of space,
= d (\ + Y sin x sin y\ *
by the Observation of a Lunar Distance. 309
Or, making use of a subsidiary angle,
tan x = -j * / sin X sin 7
c = d sec X-
VI. Formulae for calculating the co-efficients of /, r, a:, y, v, m, in the expression
for $ R, viz.,
It will be advisable to commence this part of the calculation by finding
log sin (0 0') by the formulae
Log q = log tan (0 0')
Log sin (0 e') = log q -f- log cos (0 0'),
Log tan (00') being taken as deduced by the formulae of Art. II., and the
upper or loiver sign being attached according as the hour angle is East or West.
By using the resulting value of log sin (0 0') in all the subsequent calculations, no
farther attention to the double sign is necessary, the same formulae applying whether
the Hour Angle be East or West.
As it is known that the correction & P of the Horizontal Parallax will always be
p
a small quantity, it will be sufficiently accurate to take -77 cot P = 1 in the value of D.
A = L cos (0-0')
B=-Lcot X' sin (0-0')
C = L cot /sin (0-0')
D = 0,001 r" L sin (0- 0') = [2,31443] L sin (0- 0')
= 6-15 (A-l).
Iii the expression for &, P must be in seconds. For a Lunar Distance, the term
containing i is to be omitted, and, generally with sufficient accuracy, A = l, B = 0,
a p. a = b.
VII. Formulae for calculating the coefficients of t, r, x, y, v, m, in the expression
for S X, viz.,
U = a'* + Vr + A'x + B'# + C'v + D'm.
For the reason above given we may put cos P = 1 in the expression for D'.
M = Q cos X
A' = M sin X' sin (0 - 0')
N = Q sin X
B' = N sin X' + M cos X' cos (0 - 0') = n, -f ^
C' = - sin P (N sin / 4- M cos / cos 0) = - sin P (n\ -f ' a )
D' = 0,001 P (N cos / - M sin / cos 0) = 0,001 P (n'\ + " a )
a' = b f - 15 A'.
310 On the Correction of a Longitude determined
P, as before, is to be expressed in seconds. For a Lunar Distance, the term con-
taining i is to be omitted, and, generally with sufficient approximation, A' = 0,
B' = 1, a' = p ft = b'.
VIII. Formulae for calculating T, U, W, and ts.
For a Lunar Distance,
sin d cos 2 -
ft-i. 3 *-
sin c
sin (7 -f- X) sin 2 y
sin c
sin X sin 7 sin rj
sin c
For an Occultation, or Solar Eclipse,
N 2 = j-^ sin (7 + X) = [4,08351] ^-sin (7 + X)
N 3 = sin X sin 7
c
c, d, and ij, being in seconds of space.
T = N 2 -4- N A or N 2 N, \ according as the Moon's Apparent N. P. D. is
U = N 2 N! or N 2 + N, f greater or less than that of the other body.
W = -f N 3 or N 3 , according as the Moon is more Eastward or more Westward
than the other body.
1 e o
GJ = jj- sin 2 X sin (6 - 6') = [5,8617] S sin 8 X sin (8 - 0').
For a Lunar Distance, the calculation of & may be omitted.
IX. Formulae for calculating the co-efficients of t, r, x, y, v, m, in the Final
Equation, viz.,
+ $* = [1] < + [S] r + [3] x - We + [4] y + U/+ [5] v
+ [6] m - 0,001 S n.
[1] = a W + a' T - ^ K - ts
[2] = b W + V T - fjL K
[3] = A W -f A' T
[4] = B W + B' T
[5] = C W + C' T
[6] = D W + D' T
For a Lunar Distance, p K and CD are to be neglected.
In general, e= A + S 4- s c. For the beginning or end of a Solar Eclipse,
e = S + s c ; for the occultation of a Planet, e = S s c, according as the
contact was exterior or interior ; for the occultation of a Star, e = S c.
by the Observation of a Lunar Distance. 311
The correction (S s) of the Tabular Semi-diameter of the Suri or the Planet may
be left out of consideration in an observation of a Lunar Distance, as being small
compared to the probable error of the observation.
The small quantities t, 7f>
Qnrn T off
1 4- Q 7^211
b*
_ . _ _ 9 31046
oum i^og
T C7,/ 0,\. J.
Los 30
+ A "ihA.^
Lo ff 2
i a i 004.
^ H* "
Tncr h
- 1 1 ^Q6
" g H 2
Loa-A -
- 1 1 ^Q6
- ft ft^^l
iJUg /t
"TjOff f _
- Q 7782
Log d - - - -
Sum = Log ^ + 6,3692 Sum = Log^ - - -f 6,1262
312 On the Correction of a Longitude determined
.'. ^j 1 = 4- o"56508 .-. |p = - o",20439
??* = + 0,00023 = + 0,00013
Sum = = -f- 0,56531 Sum = ft = - 0,2042(
A. C. Log H' h 5,3645 A. C. Log H' h 5,3645
Log 6 3 - 1,0917 Log& 4 - 0,5250
Sum = Log i - - 6,4562 Sum = Log K - - 5,8895
It is unnecessary to take out the values of i and K.
2. Calculation of e by the formulae in Article II.
Right Ascension of the Zenith in arc () - - 287 42 49,05
Geocentric R. A. of the Moon in arc (R') - - 336 57 3,45
Difference = the Hour Angle (0') 49 14 14,40
The Hour Angle is East.
o / //
Assumed Astronomical Co-latitude 37 47 8,37
Assumed Angle of the vertex Oil 12,00
Sum = Assumed Geocentric Co-latitude (/) - 37 58 20,37
Assumed Log p - - - -f 9,9990916 A. C. Log G cos 0' \- 2,1579647
Log sin P' - .... 4. 8,2376810
_ZU__ A. C. Log (1 -G cos 6') -f 0,0030292*
Sum = Log sin P- h 8,2367726 Log G + 8,0271706
n 0' -f 9,8793371
9,9986753
A. C. Log sin V ... + 0,0013247
Log sin / ...... + 9,7890733
Log sin P ...... + 8,2367726 o ,
7 ' 9 95369
.-. e - e' = o 27 54,76
Sum = Log G- - - - + 8,0271706 and 0' = 49 14 14j40
Log cose' ..... + 9,8148647
Sum = Log G cos 6' - + 7,8420353
Sum = 9 = 49 42 9,16
Also P = 59' 18",11 = 3558",!!
3. Calculation of X and S by the formulae in Articles III. and IV.
+ 8,0271706 . 04653,14
..... +0,1076227
Sum = Log tan - - + 8,1347933 Sum =
* Taken out of a Table of Subtraction-Logarithms, the argument being the Logarithm im-
mediately above. 9
by the Observation of a Lunar Distance. 313
Log sin -h 9,8823521
A. C. Log sin 0' h 0,1206629 Log sin X 4. 9,9981498
A. C. Log sin X' - - + 0,0013247
Sum = Log L - - - + 0,0030150
LogL 4- 0,0030150
Log sec + 0,0000404 \
A. C. Log sin X'- - - -f- 0,0013247 Sum' = Log Q . . - 4. 0,0024895
Log cos (X' + 0) - - - 8,9617724 Log S' + 2,9874786
Sum = Log cot X - - - 8,9661525 Sum = Log S - - - + 2,9899681
.-. X = 95 17' 5",82 .-. S = 977",17
4. Calculation of c by the approximate formulae in Art. V.
Assumed apparent N. P. D. of Star (7) = 95 1 2,30
X = 95 17 5,82
Difference (d) = 16 3,52 = 963", 52
O I II
Z,= 287 42 49,05
= 49 42 9,16
Hour Angle being East, R = 4- = 337 24 58,21
Assumed apparent R. A. of Star in arc = 337 26 59,55
Difference (r,) = 2 1,34 = 121 ",34
Log d ..... -f 2,9838607 A. C. Log tan 2 x - + 1,8032309
A. C. Logd ---- h 7,0161393 Log sec 2 x ----- h 0,0067789*
Logij ..... 4- 2,0840040
_ _ \ __ Log sec x --- - + 0,0033894
d ...... + 2,9838607
^ ?
Sum = Log - + 9,1001433
2Log^ ----- h 8,2002866 .-. c = 97W
Sum = Log c ---- h 2,9872501
c = 97W
S = 977 > 17
Log sin X - - - + 9,9981498
Log sin 7 ---- -f 9,9983327 .'.S c= +6,10 = 6
Sum = Log tan 2 x + 8,1967691
5. Calculation of A, B, C, D, , , by the formulae in Art. VI.
Hour Angle being East, Log q = -f- Log tan (0 0').
Log q ........ + 7,90954 Log L ........ + 0,00302
Log cos (0 - 0') - '- - - + 9,99999 Log sin (0-0') ---- + 7,90953
- Log cot X' ..... -f. 8,89333
Sum = Log sin (0 - 0') - + 7,90953
Log L 4- 0,00302 Sum = Log B + 6,80588
Log cos (0-0') + 9,99999
Sum = Log A 4- 0,00301
* Taken out of a Table of Addition-Logarithms, the argument being the Logarithm imme-
diately above.
314
On the Correction of a Longitude determined
Log L-f Log sin (0-0') +7,91255 Log ; 001 P ..... +055122
cot/ ....... +0,10762
Sum = Log C + 8,02017
Log L sin (0 - 9') - - - + 7,91255
Constant Log +2,31443
A - C - Log 0,001 P --- + 9,4488
+9,9988
-6,4562
Sum = Log -JLL . , _ 5;9038
Sum = Log D + 0,22698
Log D + o,2270
+ 9,99881 Sum = Lo J ^f~ - ~ 6,1308
Log +9,75229
' P A a =+ 0,56769
Sum = Log p a .... +9,75110 ^ B =- 0,00013
Log A + 0,00301 J- 000 P P * = _ 00014
P
Sum = Log /z A a - - - + 9,75411 Sum = b = + 0,56742
Log f, + 9,99881 also A = + 1.00696
LOST /8 -------- - Q31018 1 (\ ( A i \ '
and - 5 (A - 1) = - 0,03480
Sum = Log fi ft 9,30899
Log B . + 6,80588 S " m = ~ 15 (A - 1) = - 0,10440
and b = + 0,56742
Sum = Log fjL B ft 6,11487
' Sum = a = + 0,46302
The numbers answering to Log B, Log C, and Log D will not be required.
6. Calculation of A', B', C', D', 6', ', by the formula- in Art. VII.
L g Q + 0,00249 Log M _ 8 96680
L cosX - 8,96431 Log cos V ..... 1 8 , ? 89201
Sum = Log M ... -"7,96680 L g COS ^ ~ e ') + 9,99999
Log sin X' + 9,99868
Logsin(e-e')- - - - + 7,90953 Sum = L g *' ' ' ' + 7,85880
Sum = Log A' 6,87501
- + 9,75110
i--- = + 0,99844
- - - = + 0,00722
Sum = Log/zA'a - - - 6,62611
LogQ
Log sin X
+
' = + 1,00566
+ 0,00249
+ 9,99815 LogB'-
Sum = Log N +
L g sin x ' + 9,99868
Sum = Logw,. _f_ 9,99932
+ o,00245
- 9 ' 308 "
Sum =
P/y6 - - - 9,31144
by the Observation of a Lunar Distance.
315
Log N
Log sin I
Sum = Log*',- - -
Log coil
+ 0,00064
+ 9,78907
4- 9,78971
+ 0,10762
Sum = Logw", - - - + 9,89733
LogM 8,96680
Log cos /
Log cos
Sum = Log n\
- Log tan /
Sum = Log n'\
- - + 9,89670
- - + 9,81074
- - - 8,67424
- - - 9,89238
- - + 8,56662
n\ = + 0,61618
' 8 = - 0,04723
n\ 4- n\ = 4- 0,56895
Log (n\ 4- ',) ---- + 9,75507
- Log sin P - - - - -- 8,23677
Sum = Log C' ----- 7,99184
The numbers answering to Log C' and
Log D' will not be required.
w", = + 0,78946
n" z = + 0,03686
+ n "v = + 0,82632
Log (n'\ + ",) - - -
Log 0,001 P - - * - -
Sum = Log D' - - - -
1000 /At
4- 9,91715
+ 0,55122
+ 0,46837
- 5,9038
- 6,3722
IQOOD'/tt
P
- 0,00024
and A' /A a = - 0,00042
B'/A/8 = - 0,20485
Sum = b' = - 0,20551
Also - 10 A' =
- 5 A' =
/. - 15 A' =
and b' =
4- 0,00750
4- 0,00375
4- 0,01125
0,20551
Sum = a' = - 0,19426
7. Calculation of T, U, W, Q by the approximate formulae in Art. VIII.
The Moon has greater N. P. D., and is more Westward, than the Star.
A. C. Logc - 4-7,01275 Log*) ----- h2,08400 Constant Log +5,8617
Logrf- - - - 4-2,98386 A. C. Log c - 4-7,01275 2 Log sin X - 4-9,9963
L g sinX --+9,99815 Log sin(0-0') 4-7,9095
^ . _ ^^ Log S - - - +2,9900
r
l 4-9,
7 = 95 1' 2'
X = 95 17 6
74-X = 190 18 8
Sum = Log N 3 4-9,09323 Sum = Log a 4-6,7575
Log K - - - 5,8895
N 8 = -0,00000
- - - 4-9,9988
2Logrj - - - 4-4,16801
A. C. Logc - -f 7,01275
Constant Log- 4-4,08351
= N 8 4-N,= 4-0,99222
= N 2 -N t = -0,99222
= LogN 2 -4,51674 W = -N 3 = -0,12395
- 5,8883
= -0,00008
= +0,00057
= +0,00049
316 On the Correction of a Longitude determined
8. Calculation of [1], [2], [3], [4], [5], [6] by the formulae in An. IX.
Logo- - - - 4-9,66560 Log b - - - - +9,75390 Log A - - - -f-0,00301
LogW ---- 9,09323 LogW - - - -9,09323 Log W - - - -9,09323
LogaW - - -8,75883 Log b W ---- 8,84713 Log AW --- 9,09624
Loga' ----- 9,28838 Log b' --- -9,31283 Log A' --- -6,87501
LogT- - - - +9,99661 Log T - - - +9,99661 Log T - , - +9,99661
Loga'T- - -- 9,28499 Log b' T --- -9,30944 Log A' T - - -6,87162
'T = -0,19275 b'T= -0,20391
aW = -0,05739 bW = -0,07033 A'T = -0,00074
-(/iK + t3)= -0,00049 -IJLK= +0,00008 AW = -0,12481
Sum = [1] = -0,25063 Sum = [2] = -0,27416 Sum = [3] = -0,12555
LogB ----- h6,80588 Log C ---- - +8,02017 Log D ---- hO,22698
LogW - - - -9,09323 LogW ---- 9,09323 LogW ---- 9,09323
LogBW - - -5,89911 LogCW - - -7,11340 LogDW- - -9,32021
' - - - +0,00245 LogC' --- - -7,99184 Log D' ---- f-0,46837
LogT --- +9,99661 LogT --- +9,99661 LogT - - : +9,99661
LogB'T --- +9,99906 LogC'T ---- 7,98845 LogD'T- - +0,46498
B'T= +o,99784 C'T = -0,00974 D' T = +2,91729
BW = -0,00008 CW= -0,00130 DW= -0,20903
Sum = [4] = +0,99776 Sum = [5] = -0,01104 Sum = [6] = +2,70826
FINAL EQUATION :
+ 6",10 = - 0,2506 t - 0,2742 r - 0,1256 x + 0,1240 e + 0,9978 y - 0,9922/
- 0,0110 v + 2,7083 m - 0,9772 n
To ascertain the correction of the Longitude given by this equation, I shall now
substitute for the other small corrections the most probable values that I have been
able to obtain.
By the result of observations of the Moon on the meridian at Greenwich on the
same day (Sept. 14), x = - 20",70 and y = + 9",19-
Taking the place of K Aquarii from the Twelve-year Greenwich Catalogue,
e = - 0",90 and/= + 2",65.
Adopting + 2",6l for the correction of the Moon's Semi-diameter as determined
by Greenwich Observations (See Introduction to vol. for 1847, p. ci.) the resulting
value of n is 2,773.
Adopting Henderson's correction of Burckhardt's Constant of Parallax, viz.
+ 1",30, (Memoirs of the Royal Astronomical Society, vol. x., p. 294) the value of
w 8 is + 0,380.
by the Observation of a Lunar Distance. 317
The ratio of the Earth's axes assumed by Henderson is that of 299 to 300, while
the ratio employed in the foregoing calculation is that of 297 to 298. The correction
from this to the former ratio gives m l = + 0,014. Hence m, -f m s = -f- 0,394,
which is the value of m.
The alteration of the ellipticity requires the angle of the vertex to be corrected
by 4",47, which is the value of v, the astronomical co-latitude being supposed to
require no correction.
As the observation was considered to be very exact, it will be supposed that t = (X
By substituting these values the equation becomes,
+ 6,'lO = - o'!2742 r 4- 2,600 - o'l!2 -f &170 - 2,629 4- 0,'o49 + l"o67 - 2,707
or, 0",2742 T = + 1",34.
Hence the correction of the Longitude by this observation is -f- 4 9 ,89- I have no
ground for thinking that the Longitude of the Cambridge Observatory requires a
correction to any such amount. The instance is evidently unfavourable to the
determination of the Longitude on account of the small value of the multiplier of r.
Slight errors in the adopted places of the Moon and the Star might account for the
above result without supposing any error in the assumed Longitude. The result,
however, serves to show the advantage of taking account of small corrections which
the data of calculation may require, whether the calculation applies to an Occupation
or a Lunar Distance ; for on the supposition that none of the data except the
assumed Longitude required correction, we should have had,
- 0",2742 T = + 6",10,
or the correction of the Longitude = 22 S ,25.
o
But the assumed Longitude is really not one second in error ; consequently, as the
observation was probably not in error more than a small fraction of a second, nearly
the whole of the above result is attributable to errors in the elements of the calcula-
tion, and will serve to indicate the effect of such errors independently of the degree
of accuracy of which the observation was capable. In fact, as we have seen, by
correcting those errors as nearly as was practicable, the result was reduced to -f- 4 S ,89-
It may be remarked that this instance is confirmatory of the correction applied to
the Moon's Semi-diameter.
CAMBRIDGE OBSERVATORY,
September 10, 1850.
LONDON :
'tinted by WILLIAM CLOWES AND SONS,
Stamford-street.
,
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