DEFINITIVE DETERMINATION OF COMET 1887‘IV.M

PRESENTED TO THE FACULTY OF THE UNIVERSITY OF. VIRGINIA ON” APPLYING FOR

THE DEGREE OR DOCTOR OF SCIENCE,

MY

FRANK MULLER,

0F VIRGINIA. ‘

‘ .I 225k“

, , C» 32¢":(Inn.3 '2‘.

FROM THE

.ASTRONOMIICAL JOURNAL.

Nos. 174—5.

DEFINITIVE DETERMINATION OF THE ORBIT OF COMET 1887 IV,

BY FRANK MULLER.

This comet was discovered by Mr. E. E. BARNARI) at the

observatory of Vanderbilt University, Nashville, Tenn., on

May 12. It was then 0’.5 in diameter, with a stellar nucleus

of the 11—12 magnitude; about perihelion it was of the 9—10

magnitude, with a tail 2’ long; soon after it became diffuse

and elongated in the direction north and south, and was last

seen by Mr. BARNARD 011 August 11, when its theoretical

relative brightness was 0.3.

1. Preliminary elements, perturbations, and ephemeris. —-

Several sets of elements were published, not resting however

upon sufficient data to indicate any deviation from a para-

bolic path. Finally, Mr. CHANDLER (Astronomical Journal,

No. 160) noted that the observations could not be satisfied

by parabolic elements. From normal places for May 14,

; June 12, and July 12, he computed the following elements,

: definitive :

2 preparation of the normal places.

which represent the observations closely, and are practically

T : June 16.66108 Greenwich M.T.

a) : 15° 8’ 3".72

gg : 24') 13 1.6 .8 1887.0

t: 17 32 53 .4

logq : 0.1441634

log a = 2.5009248

e : 0.9956014

Upon these elements I have based the ephemeris for the

That the differences be-

. tween the ephemeris and Observation might contain only the

j applied to the ephemeris.

errors of observation, the perturbations were computed and

All the planets from filercury/ to

' Saturn were considered; the disturbances were small, and

' were chiefly caused by Jupiter, and by the Earth, with which

" apparition.

the comet was nearly in conjunction during the whole

The perturbations of the rectangular ecliptic

coordinates were computed for every eight days, and the

‘ corresponding changes in right-ascension and declination

.3 were as follows:'. o

; 1887 Gr. M.T. An; 13/ .42 .1"; Alt/3

May 10.0 —242 —73 — 7 —6.44 +3.49

18.0 167 49 6 5.17 1.98

26.0 —106 ——31 — 5 —3.72 +0.81

1887 Gr. MT. 15 11,, .12 Am a;

ll //

June 3.0 — 59 —17 — 3 ——2.27 +0.14

11.0 26 s 2 1.06 —0.08

10.0 7 2 1 0.27 0.05

27.0 0 0 0 0.00 0.00

July {.0 6 2 1 0.25 0.07

13.0 23 9 2 0.01. 0.27

21.0 .37 20 5 1.96 0.56

29.0 90 _ 36 0 3.17 0.89 ’

Aug. 6.0 153 ‘ 57 15 4.48 1.21

1.1.0 —216 ———84 —21 ——.).78 .—1.52

To avoid introducing any irregularity in the interpolation

of the ephemeris, Au and .16 were separately obtained for

every day. The masses and places of the disturbing planets

were taken from the Berlin Jahrbuch.

The reduction to apparent place was computed, by means

of the independent star-numbers ol’ the American Nautical

Almanac, for every four days. By interpolation this cor-

rection also was Obtained for every day.

These corrections being applied, the following is the result-

ing ephemeris for Greenwich mean noon and midnight of

each day :

1887 a 8 logA

h m s o 1 H

May 12.0 15 9 40.94 —30 56 47.5 9.6845

12.5 10 27.95 30 41 52.6

13.0 11 15.34 30 26 4.0.8 9.6790

13.5 12 3.12 30 11 11.9

14.0 12 51.30 29 55 26.1 9.6737

14.5 13 39.86 29 39 23.4

15.0 14 28.82 29 23 3.7 9.6685

15.5 15 18.18 29 6 27.3

16.0 16 7.94 28 49 34.2 9.6634

16.5 16 58.09 28 32 24.4

17.0 17 48.64 28 14 58.0 9.6585

17.5 18 39.58 27 57 15.3

18.0 19 30.93 27 39 16.4 9.6537

18.5 20 22.67 27 21 1.3

19.0 21 14.82 27 2 30.3 9.6491

19.5 22 7.36 26 43 43.7

20.0 23 0.30 26 24 41.5 9.6447

20.5 23 53.66 25 5 24.0

21.0 24 47.41 25 45 51,5 9.6404

21.5 15 25 41.56 ——25 26 4.2

(2)

1887 11 l: s o 6 u logA 1887 1. if: u o '6 u logA

May 22.0 15 26 36.12 —25 6 2.5 9.6363 June 23.0 16 34 42.03 ——1 28 7.3 9.6281

22.5 27 31.08 24 45 46.6 23.5 35 49.83 1 10 6.6

23.0 28 26.44- 24 25 16.8 9.6324 24.0 36 57.61 0 52 22.6‘ 9.6316

23.5 29 22.20 24 4 33.5 24.5 38 5.35 O 34 55.2

24.0 30 18.36 23 43 37.0 9 6287 25.0 39 13.06 0 17 44.6 9.6352

24.5 31 14.90 23 22 27.7 25.5 40 20.74 —-0 0 51.1

25.0 32 11.84 23 1 5.9 9 6252 26.0 41 28.36 +0 15 45.4 9.6389

25.5 33 9.17 22 39 32.1. 26.5 42 35.93 0 32 4.6

26.0 34 6.88 22 17 46.7 9 6219 27.0 43 43.45 0 4‘ 6.7 9.6428

26.5 35 5.00 21 5550.1 27.5 44 50.91 1 3 51.4

27.0 36 3.48 21 33 42.8 9 6189 28.0 45 58.30 1 19 18.8 9.6469

27.5 37 2.35 21 11 25.2 28.5 47 5.62 1 34- 28.8

28.0 38 1.58 20 48 57.8 9.6161 29.0 48 12.87 1 49 21.4 9.6511

28.5 39 1.17 20 26 21.1 29.5 49 20.03 2 3 56.6

29.0 40 1.12 20 3 35.6 9.6134 30.0 50 27.12 2 18 14.3 9.6554

29.5 41 1.43 19 40 41.8 . 30.5 51 34.11 2 32 14.7

30.0 42 2.08 19 17 40.2 9.6111 July 1.0 52 41.01 2 45 57.8 9.6598

30.5 43 3.08 18 51 31.3 1.5 53 47.81 2 59 23.5

31.0 44 4.41 18 31 15.8 9.6090 2.0 54 54.51 3 12 32.0 9.6644

31.5 45 6.07 18 7 54.2 2.5 56 1.10 3 25 23.2

June 1.0 46 8.04 17 44 27.0 9.6071 3.0 57 7.58 3 37 57.3 9.6690

1.5 47 10.34 17 20 54.8 3.5 58 13.94 3 50 14.4

2.0 48 12.93 16 57 18.2 9.6055. 1.0 16 59 20.18 4 2 14.4 9.6738

2.5 49 15.82 16 33 37.8 4.5 17 0 26.31 4 13 57.6

3.0 50 19.00 16 9 54.2 9.6041 5.0 1 32.30 4 25 24.0 9.6786

3.5 51 22.47 15 46 8.0 5.5 2 38.16 4 36 33.7

4.0 52 26.20 15 22 19.8 9.6930 6.0 3 43.89 4 47 26.9 9.6835

,4.5 53 30.20 14 58 30.1 6.5 4 49.49 4 58 3.5

5.0 54 31.46 14 34 39.7 9.6021 7.0 5 54.94 5 8 23.9 9.6885

5.5 55 38.95 13 10 49.1 7.5 7 0.26 7 18 28.0

6.0 56 43.67 13 46 59.0 9.6015 8.0 8 5.43 5 28 '16.1 9.6936

6.5 57 48.62 13 23 9.9 8.5 9 10.45 5 37 48.3

7.0 58 53.78 12 59 22.4 9.6011 9.0 10 15.33 5 47 4.6 5.6987

7.5 15 59 59.15 12 35 37.3 9.5 11 20.06 :' 56 5.3

8.0 16 1 4.71 12 11 55.0 9.6010 10.0 12 24.64 6 4 70.5 9.7039

8.5 2 10.46 11 48 16.3 10.5 13 29.08 6 13 20.4

9.0 3 16.39 11 24 41.6 9.6012 11.0 14 33.36 6 21 35.1 9.7092

9.5 4 22.48 11 1 11.6 11.5 15 37.49 6 29 34.7

10.0 5 28.74 ‘ '10 37 47.0 9.6016 12.0 16 41.47 6 37 19.5 9.7145

10.5 6 35.14 1014 28.2 12.5 17 45.31 6 44 49.6

11.0 7 41.68 9 51 15.8 9.6023 13.0 18 48.99 6 52 5.1 9.7199

11.5 8 48.35 9 28 10.5 13.5 19 52.53 6 59 6.3 '

12.0 9 55.15 9 5 12.8 9.6032 14.0 20 55.92 7 5 53.3 9.7254

12.5 11 2.06 8 42 23.2 14.5 21 59.16 7 12 26.2

13.0 12 9.08 8 19 42.3 9.6043 15.0 "3 2.25 - 7 18 45.4 9.7308-

13.5 13 16.20 7 57 10.5 15.5 24 5.20 7 24 50.9

14.0 14 2313 7 34 48.5 9.6057 16.0 25 8.00 7 30 42.8 9.7363-

14.5 15 30.73 7 12 36.7 16.5 26 10.66 7 3 21.6

15.0 16 38.12 6 50 35.5 9.6074 17.0 27 13.18 7 41 47.1 9.7418.

15.5 17 45.58 6 28 45.6 17.5 28 15.56 7 46 59.8

16.0 18 53.12 6 7 7.2 9.6092 18.0 29 17.79 7 51 59.7 9.7473

16.5 20 0.71 5 45 40.9 18.5 30 19.89 7 56 47.1

17.0 21 8.35 5 24 27.0 9.6113 19.0 31 21.84 8 1 22.1 9 7530

17.5 22 16.04 5 3 26.1 19.5 32 23.67 8 5 45.0

18.0 93 23.78 4 42 38.5 9.6136 20.0 33 25.36 8 9 55.9 9.7586

18.5 24 31.54 4 22 4.6 20.5 34 26.92 8 13 55.0

19.0 25 39.34. 4 1 44.7 9.6161 21.0 35 28.35 8 17 42.6 9.7643

19.5 26 47.15 3 41 39.2 21.5 36 29.65 8 21 18.8

20.0 27 54.98 3 21 48.3 9.6188 22.0 37 30.82 8 24 43.8 9.7700.

20.5 29 2.82 3 2 12.5 22.5 31 31.86 8 27 57.9

21.0 30 10.67 2 42 51.9 9.6217 23.0 39 32.78 8 31 1.2 9 7757

21.5 31 18.52 » 2 23 46.9 23.5 40 33.58 8 3 73.9

22.0 32 26.37 2 4 57.6 ' 9.6249 24.0 41 34.26 8 3 36.2 9.7814

22.5 16 33 34.20 -—- 1 46 24.4 24.5 17 42 34.81 +8 39 8.3

(3)

1887 . 11. . . ,6 ,, 101AJuly 26.0 17 18 86.21 +8 11 80.1 9.7872

26.6 11 36.66 8 43 12.8 _

26.0 45 8.1.76 8 16 46.1 9.7929

26.5 16 86.81 8 17 88.7

27.0 17 86.81 8 19 22.7 9.7987

27.6 48 8.1.66 8 50 57.7

28.0 19 86.10 8 62 28.8 9.8011

28.6 60 86.02 8 68 11.2

29.0 61 81.62 8 61 60.0 9.8108

29.6 62 33.92 8 66 60.6

80.0 68 88.19 8 66 18.0 9.8161

30.6 61 82.86 8 67 27.1

31.0 61 31.41 8 88 1.0 9.8219

81.6 66 8081 8 68 88.0

Aug. 1.0 :7 29.18 8 18 61.1 9 8277

1.5 68 27.89 8 69 8.6

2.0 17 69 26.60 8 69 16.7 9.8886

2.1 18 0 21.00 8 .19 16.8

8.0 1 28.88 8 69 9.0 9.8393

8.6 2 21.66 8 18 66.6

1.0 8 19.88 8 .18 86.7 9.8161

1.6 1 17.90 8 68 9.1

6.0 6 16.86 8 67 86. 8 9.8609

6.6 6 18.71 8 66 58.2

6.0 7 11.16 8 66 18.6 9.8667

6.5 8 9.11 8 5) 23.3

7.0 9 6.66 8 61 27.8 9.8625

7.6 10 1.10 8 68 26.8

8.0 . 11 1.11 8 62 18.9 9.8681

8.6 11 1'850 8 .11 6.8

9.0 12 66.86 - 8 19 19.6 9.8712

9.6 18 62.90 8 18 27.8

10.0 11 «19.87 8 17 0.2 9.8800

10.5 16 16.71 8 16 28.6

11.0 16 1.62 8 18 62.1 9.8858

11.5 17 10.21 8 12 11.2

12.0 18 18 86.82 +8 10 26.0 9.8916

Comparison of part of this ephemeris with an ephemeris

computed by Mr. CHANDLER showed small differences, which

Mr. CHANDLER explained by stating that in the reduction to

apparent place he had used the British Nautical Almanac, in

1 which mutation-terms of short period are neglected.

'2. Oompm'ison-Stars.—The stars used lie between 30°

south declination, and 8° north declination. The positions

of all stars brighter than the ninth magnitude, between ———-2°

and +5°, have been kindly furnished by Prof. Konrazzr of

Nicolajew, and Prof. Boss of Albany, from their asyet un-

published A.G. Zone observations. The positions of the

stars north of +5° had not been reduced, but I am indebted,

to Prof. Bauxs of Leipsic, for the apparent places of these

stars. With a few exceptions, all the positions from the

A.G. zones are the means of two or more observations. I am

MEAN PLACES FOR 1887.0

also indebted to Mr. SKINNER of the U.S. Naval Observatory

for places from various catalogues.

South of —-2°, the southern limit of the Astronomische

Gesellschaft zones, I have collected nearly all the obser-

vations of the stars used; this appears desirable for the

detection of proper motion, and the elimination of accidental

errors. “mile a complete discussion is not warranted, some

system is necessary in combining these miscellaneous obser-

vations. The weights given below were assigned, taking

into consideration the accuracy of observation and the chance

of the effect of a proper motion too small to be detected.

These weights were used when the places depended on one or

two observations; for three to six observations the weights

were increased by one-half ; 1301' seven or more observations

they were doubled, the systematic error of the catalogues

being then probably in excess of the accidental errors.

For the sake of brevity, I have given only the adopted mean

place and the authorities upon which it rests. To indicate

the various catalogues the following abbreviations are used:

A. : Oeltzen’s Argelander. 1Veight 4‘-

A.G. = Ash-0110711980116 Gescllschafi: zones. Abbreviations

are attached to show the places of observations; Albany,

Leipsic and Nieolajew. Weight 1}.

A N. = Star-places in 21.1-17071097118056 Ncwlm'chtcn.

Ar. 2 Second Armagh Catalogue. Weight -_‘-.

B : Bonn Observations, Vol. VI. “reight 4

Br. : Brisbane, Observations at Paramatta. 1 XVeight 31,—.

C, CZ = Argentine General or Zone Catalogue. 1Veight 1.

Cin. 2 Cincinnati Zone Catalogue. Weight 3.

Cp. 2 Catalogues, for the epochs 18:10, ’50, ’60, ’80, of

stars observed at the Cape of Good Hope. 1840—60, weight

5; 1880, weight 1.

D. :: Observations made at Dunsink, 1885.

G. : Gtittingen. 1Veight-1.

Gr. 2 Greenwich. Weight :1.

Weight 4.

1Veight 5‘.

1884—85—86.

“reight 1.

L. : Lamont.

P. : Pulkowa.

R. : Radcliffe,

Si. : Santini. Weight 11.

Sj. : Sehjellernp. Weight 1}.

’1‘. : Tacchini (Washburn Obs. publications).

“1.71. = Washington Zones. 'Wcight ‘3

Y. = Yarnall. “"eight :1.

Comp. indicates that the places were obtained by a com-

parison with a known star.

The Washington zones were only used when no other

authority was available.

Weight 1.

Weight 1‘—

OF THE COMPARISON-Suns.

No. (4 ,3 Authorities. No. a i 6 Authorities.

h in a H h in .

1 15 10 57.34 —290 43 55. 8 A, C, Cp,‘-101'60.'60.’50.Y 4 15 12 30.07 1 —28° 33378 A,

2 11 3.17 30 3 38.6 A, CZ, Y 5 12 51.25 i 29 29 22. 9 OZ

3 15 11 45.8-1- ——30 47 40 8 A, C, Op ’40, ’80 6 15 13 4.02 1—29 41 39.4 A, CZ

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No. a 8 Authorities No. a 6 Authorities

h m s o I Hh m B O l I!

123 16 58 57.16 +2 55 27.4 AG Alb 1451 17 25 8.04 +6 51 3.0 B

124 17 4 30.38 5 2 48.3 AG Leip 146 26 36.76 7 36 0.6 AG Leip

125 :' 7.35 5 31 19.2 “ 147 28 4.41 7 15 29.3 “

126 7 2.80 5 2 43.4 “ 148 28 27.14 811 5.0 “

127 7 18.84 6 ll 1.4 “ 149 34 29.64 7 51 50.0 “

128 8 11.42 5 56 32.6 “ 150 35 12.11 8 32 10.7 “

129 S 47.0? 5 3:" 2.0 Comp 151 36 25.76 8 16 31.4 “

130 10 55.07 5 15 15.5 AG Leip 152 37 32.46 8 6 49.8 .1. (“H-LL)

131 11 21.54 5 58 23.9 Comp 153 38 6.16 8 29 15.4 B

132 13 21.41 6 12 17.2 AG Leip 154 38 20.62 8 3. 59.5 AG Leip

133 14 18.38 6 40 4.2 “ 155 42 8.30 8 35 22.7 “ .

134 15 18.14 6 32 59.1 “ 156 45 39.37 8 33 41.9 “ i

135 15 21.47 6-48 24.4 “ .157 47 29.06 8 50 7.2 B

136 15 54.89 6 32 35.2 L, \V 158 47 30.14 8 43 5.4 B

137 16 48.80 6 46 8.7 AG Leip 159 17 49 59.13 8 55 15.5 A.G. L011)

138 17 50.19 6 ~25 46.0 Comp .160 18 8 29.90 8 56 44.1 \V

139 17 54.30 7 ll 59.9 B 161 8 55.59 8 51 4.3 A.G.-Leip

140 18 55.62 I 6 37 26.7 AG Leip 162 9 30.02 8 56 43.1 “

14], 19 44.46 i 6 31 54.8 “ 163 9 45.65 8 45 20.4 Sj, W

142 20 2.40 l 6 45 40.0 “ 164 12 41.25 8 47 39.4 Comp

143 20 51.72 7 41 42.6 “ 165 15 36.34 8 3- 40.2 A.G-. Leip

144 23 4.19 7 21 45.7 “ .166 18 17 17.22 +8 43 45.4 “

1.15 12‘ 24. 8.62 ' +7 30 36.2 1 “

REMARKS.

3. The Cape Catalogue of 1380 gives a proper motion of +0‘.010

deduced from comparison with the Catalogue of 1840, but the other

observations do not sustain the determination.

7. A provisional proper motion of——0”.OS in (5‘ was used.

18. l’rol‘.1{on'r.\7.7.i used a proper motion for this star and the

preceding one determined by comparisml with Lacaillc; Laeaille

not beng available I have used a provisional proper motion of

+050“) ill (L.

20. The S.1’.D. of BuisnAXI-z 5:} ‘2 requires a correction of —5’.

28. The declination of ‘1'.\u.\'.\i.I. 6425 is incorrect. Prof. Faisnv

has kindly given the correct declination for 1660, —210 39’ ll”.6.

2f). Lalaude not being available, a provisional proper motion of

+0‘.005 in a was used.

3. Observations of the Comet. —On collecting all the

observations published, 3313 were found available. The right-

aseeusions and deelinations given below are the observed

values corrected for parallax and for the adopted placesof

the comparison-stars ; .114 and .48 are the differences (O—C)

between these values and the values given by the ephemeris.

The observations are arranged all.)habetieally with reference

to the place of observation. The name of the observer. the

30. Prof. Boss gives the proper motion as — 05.009 and —0”.50

(AJ. 157).

44. A proper motion, given in the Cincinnati Zone Catalogue, of

—0‘.0075 and —O”.111 was used.

55. There appears to be a small negative proper motion in right

ascension.

66. The proper motion of+0a0112 and —-0”51~l—, given in the

l’ulkowa Catalogue, was used. ~

92. The right-asceusimis given in LL, W, L, Sj, C, I). and .\.G.

Nie. are well satislied with a proper motion of +OS.02S. 'l‘he decli-

nations are very discordant but are best satisfied with a proper

motion of —O".00.

aperture of the instrument used, and the place of the publi-

cation of the observations, are given. In many cases the

character of the instrument used was not stated; in these

cases it has been assumed in accordance with the known

equipment of the observatory in question. 1:3. denotes that

a ring-nucrometer, and filer. that a meridian circle, was

used, in other cases a lilar-micrmneter was employed.

Gr. BLT. a i b’ } .zlu i ,5/5‘ * Gr.M.'l‘. a 0 i [la , .216 I >l<

Albany. Boss. .l3 in. AJ. 157. Algiers. RAMHAUD. AN. 2788; 3.11. Oct., Nov.

.\lu_\' h m a o I H H ' l u ' I m n O s s

13.67719 15 12 20.4.3 '--30 5.14.0 +0.28 ‘— 5.3 | 2 1932693 152-3 237 __2345'47_.1. +1_11 _ 375 14

15-71082 I5 3840= 28 59 26-3 (—0-71 441) l 9 20.37656 23 40.76 26 1019.6 0.31 — 8.5 17

18.72316 20 46.43 27 12 52.6 +0.54 — 5.2 l 13 21.36892 25 27.82 25 3.1 16.7 _.1.9 + 0.1 13

23.71682 20 46.85 23 55 30.0 +0.35 + 0.2 l 27 2337552 29 9_()3 2.1 s) 50.2 .74. __ 3,0 2i

23.75353 15 29 50.83 ——23 53 58.9 +0.21 -— OJ) l 38 24.37555 31 0,34. 23 27 51,0 _05 5.2 23

May 15. Observation doubtful. 25.41920 33 0.70 22 43 4.6 .83 2.6 37

Algiers. RAMBAUD. 50 cm. .4.N. 2788; 13.11. Oct., NOV. 126-40731 _ 34 2434 22 O 0'7 +906 5'7 40

my 28.40828 1:) 38 50.08 —20 3036.3 ——0.16 — 6.3 39

16.46698 15 16 55.30 —-‘28 33 44.5 +0.53 —.ll.6 I 10

18.39274 15 20 12.4.4 -—27 25 5.33 +0.81 —— 7.8 l 11

(6)

Gr. 11:12 a 1 0‘ 2.. 1 216 1 >1< Gr. MT. 1 a 0‘ 1 96 1 .10 i >l<

Algiers. Rum/101). 11.17. 2788; RA. Oct., Nov. Besa11<lz.on 11111110011. 8 in. 0.13. XV. 13.

Aug. 11 In H O H s H June 3

8.39656 18 11 46.86 + 8 5]. 26.9 0.00 + 4.8 163 18.45995 162426.64 — 40:.2.352.6 +0.53 — 9.,7 88I

9.35318 18 13 35.23 + 8 48 46.21(—-O.93 — 5.8) 164 20.44311 28 55.35 3 1.518 .25 —15.3 1 91

May 16—24. Signs of parallax-factors fora changed. 21‘09811 31 20'4"? 2 23 123-2 .83 ‘1' 0-2 1 9’)

May 26. Simmf 2861166660. 22.1-.9011 33 28.32 1 18 21.9 .80 —_ 8.5 1100

‘ ” 23.47095 35 46.57 1 11 18.2 .68 9.0 1103

24.42892 16 3" 76.1: ._ . 1 _ 7.. _Algiers. 19117119111). 50 em. A..N. 2788; 13.11., Oct., Nov. .1013; l 0 3 O 3! 00') +0' '1 1 I q 1'0

16512891 1' 16 r1 67 98 3_ 11 ,) ( 0 8 19 ) 10 16.45592 17 26 5.88 + 735 1721-071 —— 5.0 146

..«.:..A 9 9.2—; 0 .:.—.7‘—.5 ,.I.,18.36561 20 9.27 27 26 10.4 +0.56 13.0 11 Jul) 10, _1 1\.1.D.1n(.011(.<.t11added.

19.41858 21 59.46 26 46 56.2 .68 8.0 14

20.36493 22 39.54 26 10 41.2 .3- 3.1 17 Bordeaux. Coou'rr. 14 in. 21.N. 2793.

21.36002 25 26.97 7D 3144.0 .61 5.9 18 1““ _ ,

23.35992 29 7.1.3 >110 30.6 .59 7.1 21. 27193,09 10 36 52-2" —'31 15 31-7 0-00 — ’-’ 37

24.36413 30 59.67 23 28 23.5 0.17 9.6 253'- 9.41395 16 411.657—11 5 21.0 +0.55 — 7.1 52

20.40290 ‘ _ 32 59.30 22 44 22.0 1.30 (-4 {)1 10.42858 6 26.09 10 17 46.6 .45 + 1.1 64

28.32227 19 38 47.12 —20 31 48.9 +0.01 —— 7.2 39 16.39597 1:) 46.89 55016.1 .1. 25 _ 86 86

8.877098 18 11. 43.76 + 8 51 18.2 —O.18 — 7. 7 16:3 1740f?” f3?) 1336 '3 733-1 — '71 3’ 5:29.361598118133669 + 71.8 41.9 (—0.98 — 79)|161 111323111 16 if; 3‘31 32110: i 1‘,“ 1‘1"; 5‘5

May 16—24. Signs of parallax-factors for 8 changed. 7 7' I '4" '7') _ d l 76" 0'00 — 8..) 8')

Algiers Sr. 50 cm 1321., Nov “Bordeaux. 17181010111. 14in A.N. 2793, 2803.

7' fl." ' ' v F.— ' ’-

9.3§796 1813 3917 1.1.. 8 L01897 :)1(__0961—- 8.7)116-1 27.1111508 15 3! 020 —21 12 20.0 +0.-h) — 0.1) :31

13.44806 16 13 9.46 — 7 59 32.4 +0.23 — 1.9 66

B01111]. BATTERMANN. 6f refr. A.N. 2808. 267.1,)J5'1 16 4' 0'20 _ 1 04 6'2 +0716 _10'8 1')

1.71-0.71 16 53 56.?0 __ . 79.2 .41 _ .7 12:-

164572811619 5547 547111 +0’71 1—13’3 1 80 12.51011. 17 174-6 89 63.1.9129 +0.29 + $1 1323

‘2417733 ' 3S 418—-— 033 24-9 ( 1901+174) 100 22.17940 38 29:84 8 27.15.:- .49 —. 4.6 150

261337611643 ”-45 +0029 00-0 +1- .9 —106 114 27.48267 ' 48 33.79 50 57.1 .20 + 2.6 159

29.15557 17 52 28.80 -— 55 53.9 +0.16 + 1.4 159

,-, r ,1, - .r-r-gi.‘ Aug.

“lvBmhn. 111701.111“ 910.? 21.N. 2181, 2826. 643196 18 7 1.119 __ 55111.7 +022 +11”) 160

23.11.5273 15 29 17.4.7 —24 6 36.8 +0.56 —- 5 2 27 843611 1150-37 5120-6 -13 2-8 16191.111111; 1 16 37 76 84 .- p 9 ,- 10.4-3007 18 15 39.04 + 8 45 49.2 +0.25 + 7.6 166

"‘1'945061' . 0 ' ~— 0 3' 16") +O"0 '- 1'6 106 July 29. Sign of A N.1’.D. changed.

Besan :01 . GRUEY. 8 ' . 01’. XV..13...0115: 10 1 61, 81 j 9 9111 _; Bordeaux. Bayer. 14 in. 11.N. 2793, 2803.

13.1. 5 1 1 1;.26 —- 75 6 1 0. 50 —22.3 08 M1»; , ... -_ ; 1 , _

14.11.75801 15 28.70 71337.1 1. 2.1 + 3.8 71 23742211 1‘? 2‘ 3222 5311‘" ““01 +07” — 5."! .201621")1'351 19 5478 r) 47 ‘1‘) 6 0 9,3 __ 1- g 90 20.41'le .1.’)3‘1~v’)b.4)4 —21£)9§)1.1.1+1.7'2 —10.:’.) 30

' ’ '7' '7' '( " June

16.198581 20 0.65 o 46 0.7 .14 16.2 80 11.44261 16 3 11.077 __ g) 32 691(+003 __77,7) 65

17.41889 1 16 22 5.88 —— 5 6 77.6 +0.82 —- 7.8 83 12.40966 10 50. 25 8 46 30. 91 + .29 0.9 691Jul\' - 0,. 1. , , 1 .. ,-

8.:39210 17 856.88 + ,- 9: 71.2 +0.1: 6.2 129‘ 14-4737” “'32-‘32 7}?3+5! — '0’? “1“ 6‘

12.710788 17 34.08 6 4.3 21.7 .52 —17.7 140 18.713440 2‘1- 23.04 121- .)].‘.i:i '1' .39 — 5.7 81

16.41407 26 0.08 735 23.9 + .17 0.1- 146 T357453” 4916-36 — 2 2 53-0 + 52 9-‘1' 713‘

.uy

July 16. A N_l’_]). incorrectly added. 2.462141: 16 55 56.48 + 3 2‘1 27.4: +0.42 '1' 2.0 120

6. 431-14- 17 4 47.14 4 57 23.6 .28 ( 43.0) 1261

Besancon.GurLL1N. 7 in. Mer. 0.18. XV. 13. 11ill}? 1% £81; 3 173 3g; '33 21,5 13(1)

17432.150 16 22 5 77 __ 5 6 30 6 +0.05 .1. 5.] __ 13.42092 19 40.19 6 5810.6 .70 10.0 135

1342332 24 2087 4 2;, 32 (5 23 _19_7 _ 19.4.4071 32 17.45 8 5 19.1.49 2.1 148

20212102 28 5], 31 3 5 4.0 9 .30 23.7 _ 24.41942 17 42 25.30 + 8 38 56.1 +0. 24 +11.6 156

21.41984 31 7. 50 2 27 1.3.1 + .14 -—23.9 —— July"... 6 changed 10’.

22.41866 1 16 33 23.30 1— 1 48 57.1. (—0.13 +27.3) —— July 7. a changed 1'".

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(9)

Gr. M.T. a 6 1 A77 1 J73 ‘ * G11 M-T-i 77 73 i 2177 1 J6 *

Orwell Park. PLUMMER. 10 in. 11!N Nov/.- Rome. CnRULLI. 91n. R. ”AN. 2787, 2801.

Julv h m s 0 I ll Mn’ h In 5 s I/

21.47249 16 36 26.53 + 8 20 58.1 +0.25 — 9.1 154 14.43434 15 13 34.11 —29 41 38. 7 +0.59 —— 7.9 1

24.46698 42 30.92 8 39 7.7 .10 + 9 1 156 15.41784 15 15 9.90 ,—-29 914.2 —0.11 — 2.0 9

27.50462 48 36.83 8 51 12.9 .62 14.4 157 M . - , . - -

_ . - - - - . - 8.36730 17 8 53.19 + 0 3511.1 —0.02 — 0.0 12:)7 , "r . . . 9 -) 7"28.00103 17 50 30.18 + 8 03 51.6 +0.-8 +10... 157 10.38359 1314.42 6 1116.2 + .33 _ 6.8 132

. . . 11.38434 15 22.55 6 27 46.7 — .12 + 1.6 1341 7‘ ‘ '- 3 T ' 9

1. my Pad” Al‘m" ‘ "1' A A' 28‘3 12.39276 17 31.91 6 43 11.3 + .28 — 3.0 140

14.51634 15 13 42.21 ——29 38 59.3 +0.76 — 7.7 1 13.36287 19 35.56 6 5710.5 + .44 — 1.7 142

' 18.47724 ‘20 20.97 27 21 55.5 .66 4.0 11 14.37836 2143.64 7 10 52.4 — .16 + 0.5 144

20.49040 23 53.01 26 5 55.1 .38 8.7 16 15.35634 23 47.53 7 23 11.7 + .40 + 4.5 144

1 20.49040 23 53.24 26 5 51.9 .61 5.5 17 16.42534 26 1.56 7 35 26.3 .24 — 5. 5 146

7 21.45571 25 37.57 25 '28 54.2 .83 5.2 18 17.38343 1 28 1.35 7 45 41.7 .32 (—78.1) 146

1 24.46965 31 7.84 '23 24. 48.3 .39 3.2 32 21.36128 ‘ 36 12,84 8 20 23.0 .18 + 3. 0 151

" 24.46965 15 31' 7.84 —-—23 24 49.9 +0.39 — 4.8 33 22.36046 38 15.28 8 27 24.8 .44. 20.0 154

. .lune ., '. ”1' ,. 7" r .""

7 7 37 7.7.0] 15 39 43.45 _12 41 33.7 +0.39 _ 4.9 51 24.:tl1g902 ll 1:2 25.98 + 8 38 04.4 +0.00 +10.6 100

_,- 8.373 70 16 1 54.37 11 54 21.4 .53 7.1 58 6.32891 , 18 7 49.20 + 8 56 3. 7 (—0.19 +225) 162

' 8.37370 1 54.13 154- 28.3 .29 14.0 581 7.32986 1 18 9 44.21 + 8 53 58. 0 —O.35 +10.6 162

23"]7807 16 55 47‘63 +0'69 97 Mav14 15 MILLOWVICH observci.

j '23'487'37 — 1 10 44.8 _11'1 97 August 6. Observation doubtful.

,1 " Palermo. AGNELLO. 25 cm. A.N. 2788, 2790

7 15$11“le 15 15 10r-(5 _.20 9 2477 +0.77 _11_2 '9 “ SCfll‘bOI'Ollgll. LOI‘ISE. 15.5 111. 1712.737. 4:7.

21 #14685 23 14.27 25 2919.7 (——1.51 68.6) 18 70-19306. 1r 23 {0.7-3 —26 6 38.1 +0.91 _ '- 7- 17-

7 28. 38751 :33 48.50 2031133. 0 +076 6.0 .17 3151395, " 25 5238 25 25 3&9 _ .52| :1 1g

' 30. 37860 42 49.57 1.3) 50 3‘) r. F5 1' 41 i.- 08 ___19 39 30.9 7 , 1

3135750 15 44. 49.30 —18 15 0.6 +1.21 —18.0 4.9 ” ”2’ 8 D "' ' + '“ ‘L 49

May 15. Z0.\'.\, observer. ~ .

. Washington. Fnisnr. 9.6 in. A. J. 157.

P'. ' . “resin 1 ‘ 7 . 6 1 AN. 2788. my

Misgue m” W (“ms in ‘ ’ 14.69101 15 13 59.30 —29 3311.0 +0.93 + 01 1 g -

. 27.3924915 36 50.42 —2 16 20.2 +00.—-—-76 6.5 37 19. 73697 22 3208 26 34- 49.6 — .22 — 3.1 15 .

1 27.4144215 36 52.94 ——21 15 166.5 +0.—70 1.6 37 21.61634 525 55.09 ~—25 2134.8 +0.87— 8.9 18 ’

After the solution of the normal equations there appeared, in

AN. 2835, observations made on ten nights by Herr Kannmnaraxx

at Geneva, and in AN. 2837, observations on eight nights by M.

S'rurvmn'r at Brussels.

7

i

4. E77"073 of Obsearation.—-In order to assign appropriate

5" weights to a series of obsermtions it is neeessarv to consider

‘ the mean, or piobable, errm of the series. This depends

7 only upon the accidental errors; to determine which, 110w-

ever, the differences between the observed and computed

places must first be freed from systematic errors.

Each difference consists of

1. Systematic errors ;

ac, the error of the preliminary ephemeris

3,, the personal equation of the observer.

II. Accidental errors ;

so, the accidental error of observation.

5,, the error of the star-place.

1. Systematic errors.

(7. Error of the preliminary ephemeris.

This error may be determined with sufficient accuracy for

purposes of weighting by dividing the observations into

ADDITlONAL OBSERVATIONS.

These observations are referred chiefly to anonymous stars, and

being distributed among six groups, the only etl'ect, probably,

would be slightly to increase the weights.

groups, and taking the means of the differences with refer-

ence to the number of observations. The results are :

‘7‘

MeanDatc J77 .48 No.0bs l):il(el.opt€iflt [JP/7139

s I/ 133." - '

May 22 +0.54 —.".2 92 142

June 12 .51 .28 48 162

20 .42 9.3 66 172

July 11 .38 —l 2 31 192

21 + .27 +3.9 27 202

Aug. 8 ——0. 13 +0.0 6 222

Solving by least squares the six equations of condition

furnished, the errors of the ephemeris at any time 7 days

from June 1 are, for right-ascension and declination respec-

tively,

£0

Ec: —D

I"/I"

05. 52 + 05.00112 t + 05.0000631 t9

.o —-0”.0123t ——-0”.001373t2(1)

l

l.

lg

l

l

l

l

i

(10)

These errors added with changed signs to the differences

will reduce them to the errors arising in observation.

5. Personal equation.

The method used to determine the personal equations was

to compare the constant part of the error of the ephemeris

as given by each observer with that given by the mean of all

the observers, this mean being taken as the standard.

Having corrected each difference for the terms depending

on t as'given by equation (1) and taking the means for

the different observers the results were those given in the

columns headed 21’“ and /]’b‘ in the following table; the next

two columns contain the personal equations in right ascen-

sion and declination given with such signs that by their

addition in the corresponding series the differences are

freed from the personal errors of the observers. The last

column contains the number of observations from which the

mean d'a and 4’6 were obtained.

8,,

11’“ .133 Pic—"hT No. Obs.

Albany +0.31 —— 2.9 +0.17 —4..6 4

{ Algiers:

Rambaud .17 6.0 +0.01 —1.5 11 10

Trépied .416 10.7 +0.05 +3.2 16 5

Berlin :

Battermann 1.21 13.0 —0.70 +5.5 2

Knorre 0.11 6.9 +0.10 —0.6 2

Besaucon:

Gruey 0.11 0.2 +0.10 +1.7 11 5

G-uillin 0.23 16.2 +0.28 +8.7 4

Heriquc 0.67 8.9 -—0.16 +1.4 7

Bothkamp 0.53 6.6 ——0.02 —0.9 6

Bordeaux :

Courty 0.26 6.8 +0.25 —0.7 7

Flaminc 0.45 5.-l +0.06 —2.1 11 3

Rayet 0.53 8.8 —0.02 +1.3 16 9

Cambridge, Mass. :

Chandler 0.52 5.1 —0.01 —2.3 5

\Vendell 0.10 -1.1 +0.11 3.4 10

Cape of G. Hope 0.53 7.4 ——0.02 0.1 6

Dresden 071- 5.6 —0.23 1.9 -:t

Geneva 0.4-7 6.! +0.01- —1.0 15

Gohlis .79 ,8 7— -—0.28 (+1.2 10 9

thttiugen 0.56 155 —0.05 (+8.0) -l-

Greenwich : ‘

Downing 0.31 7.8 +0.20 (+0.3) 1

Turner +0.66 15.2 —0.l5 (+7.7) 1

Hamburg —0.11 0.6 +0.62 —6.9 2

Kiel +0.80 11.7 ——O.29 +4.2 3

‘{ Marseilles 0.26 8.3 +0.25 +0.8 17

Nashville 0.91 6.2 -—O.~10 (—1.3) 28 18

‘ Nice 0.39 4.0 +0.12 ——2.0 13 9

» Nicolajew 0.17 2.0 0.34 5.5 4.

j Orwell Park 0.15 5.2 +0.06 2.3 20 9

Padua 0.55 7.0 -04: —-—0.5 11

Palermo:

Agnello +1.10 —12.0 -—0.59 +1.5 3 2

”it...gl’a .139 (4 8 No. Obs.

Zona +0.77 —11.‘l —0.26 +3. 1

Prague :

Gruss 0.70 1.6 0.19 —5.9 1

Weinek 0.76 6.5 -—0.25 1.0 1

Rome :

Cerulli 0.30 +0.21 13

Millosevich 0.22 5.2 +0.29 2.3 2-

) ’ashington +0.53 ——~L.1 ——0.02 —3.11 3

Mean +0.51 ——-7.5

In taking the above means, each independent observer

was given equal weight; when only a few observations are

made the personal error may, of course, be masked by the

accidental errors, but in the mean these errors are nearly

eliminated by a number of such short series. In fact, taking

the means somewhat with regard to the number of observa-

tions, the results were +0.51 and —-"”.‘l.

In July the comet became elongated north and south, and

difficult to observe in declination. By this change of form

the accidental errors become larger, and there is reason to

suppose that the personal error of observing in declination

may have changed. Hence the observations made in July

and August were not used in determining the personal

equations; and since not enough observations were made

for a. separate determination, no personal equations were

applied to the observations made during these months. i

In obser 'ations made with a ring-micrometer the personal l

equation in declination has opposite effects, according as the

comet is north or south of the star, and is nearly eliminated

in a number of observations. The personal equations cannot

therefore be obtained without a consideration of the relative

positions of the star and the comet. The quantities corre-

sponding to them are inelosed in brackets, and they should

be, when depending upon a number of observations, small.

The Harrow, Kremsmiinster, and Scarborough observa-

tions were received too late to be used in determining the l

personal equations.

Since the observations have been used once with equal

weights for all the observations by an independent observer,

their weights in the formation of the normal places must be

reduced in consequence. When only one observation was

made, it was used only in the determination of personal

equation; when two were made, each was given only half

weight in the formation of the normal places; when three

were made the weight of each was reduced one-third, and so

on until the reduction became insignificant.

II. Accidental Errors.

Errors of observation and star-places. Determination

of probable error.

Each difference between the observed and computed po-

sitions may be reduced so as to contain only these errors

and the personal error, by correcting for the terms involving

1:, as given by equation (1), and for the constant terms,

(11)

05.51 and 7”.5, given by the mean of all the obser 'ations.

These corrections are given for every tenth day, in the

following table : -

am.) Jets) !

3 II

May 12 -——0.51 +7.20 I

22 0.51 7.48 3

June 1 0.51 1.50}

11 0.493 7.24: i

21 0.463 6.70

July 1 0.419 5.901

11 0.364 44.32 ,

21 0.206 3.4-0 ‘

31 0.215 +1.83 3

Aug. 10 —-0.123 -——0.07 f

The corrections from this table being added to the differ-

ences of a given observer, the mean of the resulting differ-

ences will with changed sign, be the personal error of that

observer.

Let 0,,1‘2, . . . r, be the residuals of the individual dif-

ferences from this mean, then

0 I! -)

IU- : E“. + 85-

If a, be the mean ‘error of observation in an observation

depending upon a single comparison,

a _ [713,20]

1 _ 71.1—

where [715050] 2 a, :02, + 722 so"2 + . . . . + a, 2091

Substitution gives

2 __ [at v] —— [as5 as]0 .- __ '

( ) 1 i—l ’

and the mean error of an observation depending upon a

number of comparisons a is

\_ ”1“ 2

(3) s_\’<n+e,)

IVhen the star-places are carefully determined, a, will

usually be small in comparison with 5., and the probable

error will then be approximately given by

(4, My]-7‘ : 0.67-’15 — , ’n(z—l)

which is the formula commonly usedfi-i'

There is however an objection to weighting strictly in

accordance with this, or any other similar formula. On each

night there are undetermined instrumental and personal

errors peculiar to that night; such errors are not diminished

by an increased number of comparisons, hence the insuf-

ficiency of a formula in which the probable error depends

solely upon the number of comparisons. I have used (4)

*The accurate formula is, however, not much more dilllcult in

application, and when the error of observation is small, the dis-

.crepancics arising from the use of the approximate formula may

become appreciable. The mean error would be determined from

(2) by assigning to each observation a value of a, in accordance

with the mean error of the catalogue on the authority of which the

place of the comparison-star rested; then, for each observation,

r=0.6745 ii 5971+ s,

. 9, being assigned as before.

'to be independent of the aperture.

when the number of comparisons did not exceed sixteen,

but I have assumed that the probable error was not diminished

by a number of comparison greater than this. This limit is

probably too great, while the comet was strongly condensed.

The calculated probable errors of a single comparison,

arranged in order of the size of the instrument, are:

l’lace anti Observer Probable Error Aperture

3 II cm

Algiers, Rambaud 1.02 _ 6.6 50

“ Trépicd 0.92 10.8 50

Cambridge, Wendell 0.50 6.3 38

Nice, Charlois 0.18 { lg: 38

Bordeaux, Courty 1.03 10.8 36

“ Flamme 0.45 9.9 36

1 ”O 7'6‘4 133,01; 0... {1,3 36

Marseilles, Borrelly 0.31 6.7 26 .

Orwell Park, Plummer 0.39 { 13g 25 l

Geneva, Kammermann 0.82 8.7 25

Rome. Cerulli 0.47 6.8 23 R

Besaneon, Gruey 0.94 (328.0) 20

“ I-Ierique 0.67 11.9 20

Patina, Abetti 0.44 5.:' 18?

Gohlis, Winkler 0.65 26.0 15 R

Kremsmiinster, Scliwab 0.61 9.3 15 R

Nashville, Barnard 0.61 8.4 15 R

* Adopted probable error, 20”. When two errors are given, the latter refers to the obser~

vations made in July and August when the comet was more

difficult to observe in declination; in most cases there was

no difference. All the Rome observations were made during

these months. As a guide to assigning probable errors in

accordance with the size of the instrument used, I have

grouped these results.

_ . . Mean ). . No. of .Aperture Aper. l 1101). lnior Obs. !

eln eIn cm s N

15 ——- 18 16 0.58 12.3 4

20—26 I 21 0.60 10.1 5 , 4

30 —38 l 37 0.50 3.1 l 5

The solution of the equations of condition, taken with

equal weights, shows that the probable error in declination

may be nearly represented by the equation

7': 10”.3 + (25—a) 0”.2O , (5)

for apertures between 15”" and 40“”, a being the aperture in

centimeters. The probable error in right-ascension appears

The probable error of each observer was calculated by (4)

when the number of obser 'atious was seven or more. In

other cases it was, the aperture of instrument being the only

available guide, assigned by (5) and taken as 05.60 in

right-ascension. The weight was then calculated by

_ 9'3 _ (5”.5)?

P — 0‘2 *— ,3

(12>

a” 5 being assumed as the probable error of a single eom- The corresponding normal places are :—

parison in a standard observation.

. . . ~ . Dit ‘

5. Formatton of Normal Places.-——We1ghts being given I e a b

to each observation, and personal equation applied, in ac- 1337 May 18.0 229052117792 _27o39'23:,37

cordanee With the preceding results, the residuals were di- 26.0 233 31 43.66 22 11 58.58

vided into groups, and their weighted means taken. June 13.0 343 2 12.91 8 19 54.68

22.0 248 629.18 — 2 511.54

. July 10.0 258 5 2.11 6 4 43.29. VI 1 — . - . -

Group *ng: JO 01],. Ngbs‘ft Weight ' 22.0 264 22 33.8.7 3 21.1 30 87

Aug. 8.0 272 4511.59 + 8 52 17 07

May 12—21 17.8 +0.620 — 6.65 48 152 180

Mar 22—31 26.0 .503 6.80 42 146 168

Jun§e 1—16 12.9 502 7.770 63 62 214 .136 6. Formation of Normal Equations and Deter7717'71att'on

June 17.30 21_3 _335 8.57 68 206 135 of Deflnitwe Elementa—The coefficients of the variations

July 1—14 10.5 377 — 1.32 34 33 104 75 of the elements were computed by the formulas given by

July 15—31 232-2 303 + 2-04 27 26 115 43 Orromnnfi' Whence were formed the following

Aug. 1—10 8.4 +0 141 + 1.20 6 25 14 °

1 *Lehrbuch zur Bahnbestimmung. chlter Baud, pp. 405—406.

EQUATIONS on CONDITION (coefficients logarithmic).

Right—Ascension.

_9.1014 61"+ 719.7910si11i’652’ '1' 0.5091671' + 0.6943 Slogq + 718.6336 (ST + 719.47020‘e

08439 9.3201 719.7866 0.5296 0.6382 718.6698 719.3811

0 8722 9.5725 71.9.6509 0.5267 0.3633 71.8.6929 718.6523

0 7601 9.6108 719.4889 0.4993 0.0396 718.6719 8.8019

0.7465 9.5663 718.5065 0.4145 719.9454 718.5856 ~ 9.3627

0.6557 9.4678 9.0244 0.3502 710.2324 718.5105 9.4675

0.3323 9.2351 + 9.3578 + 0.2622 + 710.3689 + 718.3957 + 9.5169

Declination.

7108228136 : 710.5027 + 719.7115 + 9.0392 + 9.9016 + 716.7537 + 7.5274

710.8325 710.5168 719.9688 9.3296 9.4857 71.7.4650 71.8.1708

710.8865 710.4630 770.2538 9.7602 710.1533 117.9423 717.9174

710.9330 710.3902 720.3102 9.8723 710.3483 118.0196 8.1229

710.1399 710.1846 710.3282 9.9686 710.4771 118.0424 8.6720

0.3032 110.0168 710.3278 ' 9.9782 110.4762 717.9908 8.7276

0. 6149 719. 7208 + 71.0. 2965 + 9. 9526 + 71.0.4242 + 117.9062 + 8.6984

After multiplying each equation by the square root of its weight,

introduction of the factors

and rendering all the equations homogeneous by the

a' = 1.0303 M t = 1.7852 8 log (1

y = 1.3754 633’ sin 7‘ a = 9.8581671

3 = 1.6919 0‘7z' w = 0.7159 8e

residual unit 2 2.0374

the normal and elimination equations were obtained and checked in the usual way.

+3.2495 .7; +2.43% y +0.03-l4 z +0.6054t ——0.0712 n +0.0262 w =

+2.43% +3.5101 —l.5012 +0.3-l-76 +1.3998 +0.3285 =

+0.0344 —1.5012 +3.8590 +1.8300 —3.7560 ——0.‘:l-‘712 =

+0.6054 +0.3476 +1.8300 +2.8651 —1.7-lv99 +1.5851 =

—0.0712 +1.3998 —3.7560 —1.7499 +3.66-l-8 +0.3049 :

+0.0262 +0.3285 —0.3742 ——1.5851 +0.3049 +1.53-16 =

+0.51181a‘

NORMAL EQUATIONS (natural numbers).

ELIMINA’I‘ION EQUATIONS (coefficients logarithmic).

+0.38607 _7/ +8.53656 :4 +9.7820-lt +718 .85248 17 +8.41830 10

0.22763 71018384 71902366 0.16230 9.48982

0.39412 0.23756 710.38766 118.97864

0.18772 8.75435 710.1773-

7.74036 7.14613

.7.69897

Direct solution of these equations gave

log .7; : 0.3653 61" = + 5. 92 log t = 0.6181 «S'logq

log y = 0.4387 653’ : —33. 13 log a = 0.5185 8T

log 3 = 0.1488 671’: + 3.12 log 10 = 0.6212 08

3.2908

+1.3221

+:3_.2702

+2.5]87

—3.1962

——0.5268

HH

HII

HII

+0.0000360

+0.002418

+0.0004249

WWW“-

..._h

(13)

The small coefficients in the equations involving 71 and it)

alone, show that the values obtained by direct solution are

highly uncertain. An attempt was made to diminish this

uncertainty by expressing the other unknown quantities in

terms of 10 and an absolute term. Accordingly,

a: = 9.93648 + 0.16283 70

y : 9.39094 + 710.47594 w

z = 0.71350 + 710.57532 w

t : 8.47124 + 0.61496 10

a = 0.63985 + 710.02695 w

the coefficients being logarithmic.

Substitution in the equations of condition and solution of

the fourteen resulting equations for 10 gave

(1) log 70 = 0.6604;

the corresponding changes of the elements are given in the

column numbered I in the table below. Comparison with

the normal places gave —— 9”.86 as the sum of the weighted

residuals, showing that the uncertainty of solution had not

been wholly eliminated. To determine by trial what varia-

' tion of the eccentricity would best distribute the residuals,

an assumption was made, —

(2) log 70 = 0.6814,

which gave the changes of the elements numbered II. Com-

parison with the normal places gave the residuals numbered

II. Interpolation between the values (1) and (2) gave

(3) log 10 : 0.6788

The results of this assumption are given in the columns

numbered III. They show no improvement on the results

of the second hypothesis. The residuals given by' direct

comparison being so nearly equal, it was considered safe to

interpolate between them, on the assumption that the sum

of the weighted residuals should be zero. The weighted

values are numbered (IV) and the sum of their squares

being slightly smaller than for the other assumed variations

of the eccentricity, the value

(4) log 10 = 0.6801

was taken as the definitive value.

CHANGES on ELEMENTS.

I II III

5‘ T + 0.002344 + 0.002301 '1' 0.002307

H H H

871' + 2.33 ‘+ 1.88 + 1.94

81" + 6.27 + 6.47 + 6.45

II III

—38'.'55 ——38."33

+ 0.0000413 + 0.0000411

+ 0.0004880 + 0.0004852

11

——36.59+ 0.0000393+ 0.0004650

68'

Slogq

8e

RESIDUALS (Normal— Computed).

Unweightcd. Weighted

II III II (IV) III

Right-Ascension.

11 , 11 H I II

+1.08 +0.91 +1.33 +1i23 +1.12

—0.59 1 —0.57 —0.72 —0.71 —0.70 ,

+0.21 ‘ +0.17 +0.31 +0.28 +0.25

—0.89 —0.94 —.1.28 -——1.31 —1.35 -

+0.61 +0.53 +0.62 +0.58 +0.54

0.72 0.64 0.77 0.73 0.68

+0.80 I +0.76 +0.40 +0.39 +0.38

Declination. .

—0.60 —0.84 —0.80 —0.96 —1.13

+0.59 +0.25 +0.76 +0.54 +0.32

+0.75 +0.41 +0.88 +0.68 +0.48 ,

—1.43 ———1.77 ——1.66 —1.86 ——2.05

+0.81 +0.53 +0.70 +0.58 +0.46

+0.75 +0.34 +0.50 +0.36 +0.22

—1-27 —1.67 —-0.47 —0.54 ——-0.62

Sum +1.54 ——1.25 +1.36 —0.01 -—1.40

All the weights have been divided by 100 for convenience.

The sums of the squares of the residuals are:

.II'I (1y) III

Unweightcd 10.02 10.11 10."50

“Weighted 10.87 10.72 10.96

Transforming the preliminary elements from the ecliptic

to the equator, and adding the changes interpolated between

II and III, the resulting definitive elements of Comet 1887

IV are :

T = June 16.663384 Gr. M.T.

71’ : 257° 4’ 4”.38

9" = 313 54 12 .14 E Mean Equator 1887.0

1" = 22 2O 9 .02

log (I = 0.1442046

e 2 0.9960879

CONSTANTS 19011 Tun EQUATOR.

7' [9.9830763] sin (349° 18’ 5”.82+71)

7' [9.9843699] sin (254 50 26 .90+0)

7‘ [9.5798234] sin (303 9 52 .24+11)RQ

H0

ll

Leander 11100077111071 Observatory, Univ. Virginia, 1888 May 26.