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RECHERCHES
ASTRONOMIQUES

k

DE L\'OBSERVATOIRE

D\'UTRECHT

VII

■ Ji

utrecht; ,

J. VAN BOEKHOVEN

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ERRATA.

Page 10,nbsp;line 14, read quot;magnitudequot; for quot;magnitudesquot;.

,, 18,nbsp;last line, read quot;a. differencequot; for quot;differencequot;.

„ 24,nbsp;in the table omit quot;H.A. 47quot;.

„ 37,nbsp;in formula (4) read

h

„ 38, line 12, readnbsp;for

„ 4(),nbsp;to the heading of table XIV add quot;in mms.quot;.

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THE HARVARD MAP OF THE SKY

and

THE MILKY WAY

PROEFSCHRIFT

ter verkrijging van den graad van doctor
in de wis- en natuurkunde aan de rijks-
UNIVERSITEIT te utrecht, op gezag van den
RECTOR-magnificus dr. p. van romburgh,
HOOGLEERAAR in de faculteit der wis- en
natuurkunde, volgens besluit van den
senaat der universiteit tegen de
bedenkingen van de faculteit der wis- en
natuurkunde te verdedigen op maandag
15 october 1917 des namiddags te 4 uur door

ISIDORE HENRI NORT

GEBOREN TE GRONINGEN

drukkerij J. van boekhoven - utrecht

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RECHERCHES
ASTRONOMIQUES

DE L\'OBSERVATOIR

D\'UTRECHT

VIT

UTRECHT

j. van boekhoven

1917.

/
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THE HARVARD MAP OF THE SKY

and

THE MILKY WAY

by

H. NORT

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CONTENTS.

Page

Introduction .. • .............................................................................................l

Part I. Star-counts on the Harvard Map of the Sky.

Chapter I. The Harvard Map of the Sky............................................5

Chapter H. The counts ............................................................................U

Chapter III. The determination of the limiting magnitude....................22

Chapter IV. Reduction of the plates........................................................30

Chapter V. Determination of the limiting magnitude by means of

overlapping fields; the reduced plates............................48

Chapter VI. The catalogue of star-density............................................79

Part IL The Milky Way.

\'Chapter VH. The star-density and the galactic latitude....................93

Chapter VHI. The position of the galactic plane......................................104

Chapter IX. The star-density and the galactic longitude......................117

Chapter X. The light of the Milky Way.....................................122

Chapter XL The system of stars down to 11?0......................................130

Summary ............................................................................................134

Appendix L Catalogue of star-density........................................................139

Appendix IL Bibliography.....................................................169

Plates.

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INTRODUCTION.

It is a well-known fact that star-counts by different authors and
the conclusions derived from them sometimes largely diverge. As to the
earlier counts, the circumstance that they are not based on a properly defined
photometric scale, may safely be considered to be the cause. However, there
exist two fairly recent counts, both of them based on a rigorously defined
magnitude-scale, and yet differing greatly from each other not only in
the number of stars of every magnitude, but also in the condensation
towards the Milky Way. I mean the counts by
KapteynI) and by Chapman

and Melotte^).

Although Kapteyn admits that part of this great difference must be
ascribed to the fact that for the fainter stars his counts are based on
Parkhurst\'s visual scale and for the brighter on Pickering\'s photometric
scale and that according to an investigation
by Nijland^) these two scales
diverge rather
considerably, he is still of opinion that for the remaining
difference
Chapman and Melotte must be held responsible. At any rate,
even after reducing
Kapteyn\'s counts of the faintest stars to Pickering\'s
scale there remains a great difference between the results of these two investig-
ations which it would be desirable to explain. So when in June 1915 Prof.
Nijland proposed that I should attempt to elucidate this and some other
unexplained points in various star-counts,
I readily accepted his suggestion
and at once started to collect all the literature on this subject
I could find.
Since a fairly complete list of papers on star-counts was nowhere to be found
- only
stroobanf gives a short list of twenty items in Tome XI, Fascicule II
of the „Annates de I\'Observatoire Royal de Belgique, Nouvelle sériequot; — this
collection took much time. In order to save other workers this trouble

C. Kapteyn, Publ. of the Astr. Lab. at Groningen, No. 18.

2) Chapman and Melotte, Memoirs R. A. S. 60, 145, 1914.

3 A A Nijland. Hemel en Dampkring, Sept. 1916. According to this investigation
the difference between the Harvard and the Parkhurst scales may be very satisfactorily
represented by the formula H-P=0\'^\'22 (H-9-33).

-ocr page 16-

the bibliography I brought together has been added to this paper as
Appendix II.

Under A I have assembled the titles of all papers on star-counts I
have been able to find; a great part of these papers also contain theoretical
considerations on the distribution of the stars. This part of the list is probably
fairly complete up to December 1915; after this date completeness could
not be attained on account of the abnormal conditions resulting from the war.

Under B are gathered the titles of different papers in which the various
magnitude-scales are discusscd. Since the value of any star-count largely
depends on the magnitude-scale used and hence no investigation on the former
subject can ignore the latter, I thought it useful also to publish this part of
the bibliography. Here, however, I have not endeavoured to be exhaustive,
I simply mentioned under this heading the papers consulted by myself.

While collecting this literature I came across an article by M. Selga,
entitled: El mapa celeste de Harvardi). In this article Selga gives a short
summary of a paper by
H. Henie on star-counts, carried out by the latter
on the Harvard Map of the Sky. Though
Henie\'s counts do not include
fainter stars than those of the eleventh magnitude, so that this author does not
nearly go so far as
Kapteyn or as Chapman and Melotte, still it seemed
to me from
Selga\'s short paper that these counts might prove of great
value, not only because the number of fields counted is much greater than in
any other count, but especially because the material used for them is much
more homogeneous.

It is particularly on account of this latter circumstance that I studied
Henie\'s paper 2) in detail. The result is contained in the following pages.
So the original plan, namely to attempt to make
Kapteyn\'s counts agree
with those by
Chapman and Melotte, has not been carried out. However,
I hope to return to it within a reasonable time^).

M. Selga. Revista de la sociedad astronómica de Espana y América, Ano V, Num. 43.

2)nbsp;H. Henie. Lunds Universitets Arsskrift. N. F. Afd. 2. Bd 10. Nr. 1.

3)nbsp;When the present paper was nearly ready for the press, I received Nr. 27 of the
Publications of the Astronomical Laboratory at Groningen, containing an investigation by
Van Rhijn on the difference Kapteyn-Chapman and Melotte. After his elaborate explanation
the investigation planned by myself would be superfluous.

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PART I.

Star-counts on the Harvard Map of the Sky.

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CHAPTER I.

the harvard map of the sky.

In Harvard Circular Nr. 71 Pickering gives a description of a photo-
graphic map of the whole sky which will henceforth for the sake of brevity be
called the Harvard Map. By means of two small anastigmatic lenses, one of them
mounted at Cambridge and the other at Arequipa, and both with an aperture
of one inch (2.5 cms.) and a focal length of about thirteen inches (33 cms.),
a large number of photographs were taken. Each plate, measuring eight by
ten inches (20 by 25 cms.), covers a region of over 30 degrees square. With
exposures of about one hour, stars as faint as the twelfth magnitude were,
in some cases, obtained.

From the plates obtained in this manner a set of 55 was selected,
covering together the whole sky and constituting the Harvard Map. Of
this original at the Harvard Observatory copies on glass have been made
by double contact printing and put at the disposal of astronomers. In table
I the current numbers of the plates are given, followed by the right ascension
and declination of the centre of each plate and its time of exposure. The
plates 1—^21, 24—^27 and 29 were taken at Cambridge, the others at Arequipa.

On the copy of the Harvard Map at Lund counts were carried out by
H. Henie, who gives the following description of this copyi): quot;As regards
quot;the images, these latter are generally very fine in a considerable part of the
quot;centre of the negative, even those of the faintest stars being visible. Towards
quot;the margins the images of the brighter stars are distorted and
those of the

1) H. Henie. Lunds Universitets Arsskrift, N. F. Afd. 2. Bd. 10. Nr. 1, p. 4.

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TABLE L

centres and times of exposure of the plates of the harvard map.

Time

of
expo-
sure

Time

------ _:r~

Time

Plate

R.A.

Dcc.

Plate

R.A.

Dec.

of
expo-
sure

Plate

R.A.

Dec.

of
expo-
sure

1

90°

39quot;

15

150°

30°

56*quot;

29

210°

73°*

2

0

60

63

16

180

67

30

240

} gt;

60

3

45

....

17

210

gt;}

68

31

270

gt;gt;

60

4

90

n

....

18

240

tf

64

32

300

60

5

135

n

59

19

270

,,

56

33

330

f}

60

6
7

180
225

II
gt;1

71

58

20
21

300
330

n
yf

62
69

34

35

0
30

-30

II

60
61

8

270

II

70

22

0

0

58

36

60

68

9

315

II

60

23

30

II

60

37

90

60

10

0

30

57

24

60

»1

71

38

120

60

11

30

62

25

90

II

73

39

150

61

12

60

II

59

26

120

II

64

40

180

67

13

90

II

75

27

150

II

65

41

210

60

14.

120

II

59

28

180

gt;1

65

42

240

II

60

Time

of
expo-
sure

Plate

43

44

45

46

47

48

49

50

51

52

53

54

55

R.A.

Dec.

270\'
300
330
0
45
90
135
180
225
270
315
0
210

-30\'

60quot;
\'61
61
61
64
60
60
72
60
61
60
60
60

-60

-85
-75

\'fainter ones difficult to discover.quot; The precise meaning of Henie\'s remark
(the italics are mine) I have failed to understand, since in my opinion it applies
not only to the marginal but also to the central images. On any photograph
of the sky the images of the stars will come out more faintly as the stars are
fainter themselves. This increasing faintness of the images is gradual and not
sudden, until finally the limit of visibility is reached. So on any star-photograph,
as well at the centre as near the margins, scarcely visible images will occur
and this will also have been the case\'on the copy of the Harvard Map used by
Henie. Only if he had stopped at a certain limit and so had not included
in his counts all stars down to the limit of visibility, his remark could be
understood, in which case it is to be regretted that he has not defined this
limit more sharply, especially in regard to the determination of the limiting
magnitude.

At any rate it appears from Henie\'s words that he was greatly pleased
with this copy of the Harvard Map and that he thought it very suitable for
carrying out counts. Now since the Utrecht Observatory also possesses a

P

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copy of this Map, placed at my disposal through the kindness of Prof. Nijland,
it was not difficult to examine whether this copy also possesses the same
excellent qualities as the one used by
Henie. I examined the plates with special
reference to the shape of the images and to the transparency of the plates.

As to the first point, forty-three out of the fifty-five plates show at the
centre and round it within a distance of about six centimetres perfectly
sharp circular images. On the remaining plates the images of the stars are
elongated, also at the centre, not always in the same direction. Near the
margins of the plates the images are
always elongated and often strongly
distorted. Especially the deformation shown in fig. 1, /e, is very frequent.
On the 43 plates showing sharp round images at the centre, this deformation
of the marginal images is symmetrical with respect to the centre and is there-
fore only a
consequence of the size of the field. On the 12 other plates the
deformation of the images seems to be the combined effect of the size of the
field and of the guiding-error. The different manners in which the deformation
of the images appears are schematically represented in fig. 1. The short lines
in the four quadrants show the
principal direction of the deformation, not
its true
size, this latter often varying as well in the four quadrants of a single
plate as from one plate to another.

d

e

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Fig. 1.

f

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In table II the appearance of the images on each of the 55 plates is
shown. The letters in the column lieaded quot;shape of the imagesquot; refer to the
different types represented in fig. 1. In the column headed quot;transparency
of the platesquot; the Roman figures indicate the four quadrants of every plate.
It has been assumed that the plates lie with the glass side upwards and that
the quadrants are numbered in the order shown in fig. l
a.

TABLE II.

discussion of the plates of the harvard map.

Numb.

Shape
of the

images

1

h

2

a

3

a

4

a

5

h

0

d

7

a

8

d

9

a

10

a

1]

a

12

a

13

b

14

a

15

a

16

a

17

e

18

g

19

a

20

a

21

a

22

a

23

a

24

a

Transparency of the plates.

I and IV fogged at the margins.

I and IV fogged at the margins; also II partly.

Ill partly fogged.

Ill fogged at the margins.

I and IV fogged at the margins.

I and IV slightly fogged at the margins.

nearly clear.

almost entirely slightly fogged,
quite clear.

I and IV slightly fogged.

I, II and III fogged at the margins.

nearly clear.

the margins of I and a great part of III and IV fogged,
nearly clear.

all quadrants partly fogged,
part of III badly fogged,
nearly clear.

fogged everywhere at the margins, III almost entirely
part of III slightly fogged.
Ill very slightly fogged,
nearly clear.

Ill and IV slightly fogged.

-ocr page 23-

Table II. (continued).

Shape

Numb.

of the

Transparency of the plates

images

25

r

nearly clear.

26

a

27

a

28

a

jj gt;gt;

29

/

fogged everywhere at the margins. III to a great extent.

30

a

IV very badly fogged.

31

a

II very badly fogged, also the margins fogged.

32

a

nearly clear.

33

a

II badly. III shghtly fogged.

34

a

II and III fogged at the margins.

35

a

quite clear.

36

a

I and IV rather badly fogged.

37

c

nearly clear.

38

a

II and III almost entirely fogged.

39

c

II, III and IV rather badly fogged at the margins.

40

a

I badly fogged.

41

a

Ill slightly fogged.

42

a

nearly clear.

43

a

))

44

a

II and III rather badly,I and IVslightly fogged at the margins.

45

a

nearly clear.

46

a

Ill slightly fogged.

47

a

nearly clear.

48

a

)} gt;gt;

49

c

II strongly fogged in the corner.

50

a

nearly clear.

51

a

)) )gt;

52

a

I and IV slightly fogged.

53

a

fogged everywhere along the margins.

54

a

I, II and IV almost entirely fogged.

55

a

nearly clear.

By examining a single copy of the Harvard Map it cannot be settled,
of course, what part of the small deficiencies, enumerated in table
II, originated

-ocr page 24-

in the reproduction and what part is already present in the originals, but it
would seem to me that this latter part cannot be small. If this be really the
case, the small defects mentioned in table II will also for the greater part
be present in the plates counted by
Henie, in which case his enthusiasm
about the Harvard Map would seem rather unjustified.

Doubtless these plates were not taken with the pre-conceived purpose
of carrying out counts on them; for other purposes they have proved eminently
serviceable to many astronomers. For star-counts, however, the plates of
the Harvard Map are only suitable in those portions where no deformation
of the images is observed. Obviously, if we want to determine the limiting
magnitude at various distances from the centre of the plates by a
direct method
i.e. by estimating the diameters or the blackening of the images, their imperfect
circularity is a serious drawback. Evidently
Henie has also felt this; in fact,
he determined the limiting magnitudes by an indirect method, namely by
basing the amount of its variation from centre to margin on a somewhat
arbitrary relation between the magnitudes and the numbers of the stars.

It would seem to the present author that this method of determining
the limiting magnitudes is essentially wrong. For it is exactly in order to arrive
at such a formula that, among other things, star-counts are undertaken and
to assume
a priori the validity of such a formula- cannot but render the results
uncertain. Hence photographic plates, if they must serve for starcounts will
have to allow an exact determination of the limiting magnitudes by a direct
method and this condition is only partly fulfilled by the Harvard Map. Only the
central portion of the plates (a circular area of say six cms. radius) can be success-
fully used. Therefore it is important that Prof.
Pickering should have placed
at the disposal of astronomers a Second Harvard Map, composed of 55 plates,
the centres of which appear near the corners of the plates of the first set. By
using these two sets combined and carrying out the counts only in the central
parts, we shall have a most suitable and probably very homogeneous material.

Very likely more reliable results might even be obtained by only
using the central parts of the only available set. Although this would
diminish the number of counted fields, the limiting magnitude might be
determined with much greater accuracy then when using the whole plate.

-ocr page 25-

CHAPTER H.

the counts.

We shall now discuss the manner in which Henie carried out his counts.
He himself says on this point i):

\'Tn order to get the number of stars per square centimeter a most
quot;simple method is applied. The plates are negatives and consequently the images
quot;of the stars are black points on a transparent ground. The plates are therefore
quot;placed on a paper divided into squares of one millimeter, with thick lines for
quot;everv centimeter.

quot;The density may by this means be determined in any part of the
\'\'negative.

quot;Two rectangular axes were drawn on the paper, and the plate was
quot;placed, so that the approximative centre of the photograph coincided with
quot;the origin, and the outlines parallel to the axes.quot;

And later: quot;The counts are all made in good day-light under the same
quot;conditions through a lens magnifying about 3 times.

quot;As the density is not as a rule great the counts are easily made, the
quot;millimeter lines giving an excellent assistance.quot;

The film-surface of the plates measures 19 by 21 cms. If the plate
is laid in the above manner on the squared paper, there will consequently
be 18
x 20 entire fields of a square centimetre. Of these 360 Henie counted
in each quadrant 25 fields, the counted fields lying symmetrically with respect
to the centre, as shown in fig.
1 of Henie\'s paper.

In doing this Henie tacitly assumes that in order to get information
Henie. 1. c. p. 4.

-ocr page 26-

on the apparent distribution of the stars it is not necessary to count the
whole sky, but that it is sufficient to count a number of regularly distributed
fields. Although it appears from the manner in which
star-counts have been
and are still being made that this is the general view of astronomers, I have
not been able to discover that it has ever been tried to determine the amount
of the error which may in this way be introduced into the
star-numbers.

I therefore made complete counts of four plates of the Utrecht copy
of the Harvard Map, covering a surface of 360 sq. cms. In order to obtain
results that would be comparable with
Henie\'s, I followed in the main his
method of counting, using reflected light and working in quot;good day-lightquot;,
although this manner of counting certainly is not the easiest one nor the most
desirable. Without doubt transmitted
artificial light is to be preferred Not
only the trouble of being continually obliged to count in the shadow of
one\'s head is then obviated, but, moreover, it is only in this way that it is
possible so to regulate the intensity of the light-source that there can be any
question of counting under the same conditions, which obviously cannot be
attained when counting by day-light in our climate.

In some other respects however I have departed from Henie\'s method.
So I dit not count on millimetre paper. Not only the greyish tint of this paper
makes an unsuitable background, but also the large number of lines is trouble-
some rather than being quot;an excellent assistancequot;. For one has to be very
careful not to count twice images of stars lying exactly on a line and for this
reason it is important to reduce the number of lines as much as possible.
I found that for plates outside the Milky Way lines at 1 cm. distance suffice
while for the Milky Way itself fields of 0.1 cm^. are not too large at all. The
necessary lines were drawn as fine as possible with Indian ink on white paper;
where images fell on the lines their grey colour contrasted clearly with the
blacker ink.

As to the optical means I had the choice between a BRiicKE-lens
magnifying five times, and an aplanatic magnifier by
Zeiss, magnifying ten
times. The former was kindly put at my disposal by Dr.
Moll, Lecturer in
the University of Utrecht, the latter by Prof.
Lorentz, Director of the
Physical Laboratory of
Teyler\'s Foundation at Haarlem. I wish to express

-ocr page 27-

PLATE 2.

_9 _8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

C:

8 9

40

58

51

100 141

144

144

114 227

158

94 127 119 119

70

68 101

95

10

56

68

80

94

134

133 208 262 238

159

130

162

188 149 128 102 109 102

9

51

74

65

87 116 116 142 113 146

178 164 138 121

160 129 110 117

132

8

96

113

99 129

116

86 146 150 164

199 179

120

152 145 122

96 103 133

7

84 148 148 169 195 120 156 116 179

175

157 195 185 171

153 104

159

163

6

148 196 246 239 219 279 273 177 222

182

180 246 206 165 117

138

189 171

5

208 238 292 258 276 289 320 383 352

245 213 224 201

145

121

125 130 146

4

200 254 279 252 316 402 286 312 225

270 244 349 315 258 153 114

98 161

3

188 193 216 305 307 354 298 231 267

206

159 173 311

386 329 261 217

180

2

207 207 268 254 358 330 388 308 325

261 300 264 240 221 391 277 226 324

1

177 150 180 207 336 351 439 407 370

383 371

452 434 357 316 249 216 184

-1

92 100 138 188 302 319 391 392 431

366 398 422 377 264 242 210 162 251

-2

80

80

98 124 184 158 264 317 339

272 285 354 234 225 196 168 121

152

-3

117

120

90 121

180 176 229 219 310

224 240 258 193 108 151

123

92

113

-4

72

76

62

76

93

81

116 120 160

163

184 175 171

139

85 107

71

80

-5

64

61

62

65

77

67

72

107

91

128 140 109

99

87

76

63

57

48

-6

63

69

64

65

70

66

79

93 101

76

89

93

76

69

57

54

50

68

-7

65

67

78

81

80

62

82

117

114

81

89

94

76

52

49

37

39

52

-8

44

41

46

60

54

54

52

61

57

67

96

47

71

63

56

41

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52

-9

26

36

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34

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50

58

56

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34

-10

Centre at

\\b = -3°
.1 =84°

PLATE

6.

-9

-8

-7

-6

-5

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-2

-1

1

2

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31

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8

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20

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30

30

35

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30

36

18

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7

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12

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26

33

43

36

60

28

31

27

20

17

18

11

6

15

15

23

22

23

23

47

54

50

32

66

55

44

39

33

35

10

8

5

13

19

12

22

31

34

50

47

54

68

74

50

48

39

41

14

17

19

4

14

10

7

20

25

25

51

56

75

62

89

79

66

60

28

25

27

26

3

12

11

24

19

29

55

53

61

64

69

70

62

55

48

29

19

19

28

2

3

9

25

31

37

48

53

60

77

68

82

64

71

55

35

23

23

16

1

14

18

13

29

43

48

84

91

95

83

79

94

72

51

44

32

18

24

-1

14

14

21

16

35

41

47

69

95

67

63

66

65

52

48

19

21

20

-2

14

12

16

22

21

34

60

45

72

74

90

59

67

44

33

22

30

12

-3

14

18

21

41

27

25

37

61

61

57

56

47

46

37

36

15

17

9

-4

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31

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45

45

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57

61

35

40

21

21

18

11

-5

11

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7

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17

11

39

36

53

25

70

83

49

46

33

8

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23

-6

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27

27

21

40

45

32

45

15

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7

-7

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-9

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6

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-10

Centre at

b = 56°
101°

-ocr page 28-

PLATE 34.

9nbsp;13nbsp;10

9nbsp;9nbsp;5

ffnbsp;flnbsp;11

15nbsp;5nbsp;13

ISnbsp;17nbsp;8

19nbsp;21nbsp;12

7nbsp;11nbsp;14

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8nbsp;17nbsp;11

18nbsp;11nbsp;12quot;
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29
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28
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ISnbsp;18nbsp;15
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-9 -8 -7 -6 -5 -4 -3 -2 -1 1

3nbsp;4

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38nbsp;17

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35nbsp;33
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\'42~U

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51nbsp;42

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11nbsp;9
13
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18nbsp;17

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11nbsp;19
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15
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9nbsp;22

12nbsp;28
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10nbsp;10

0nbsp;15

15nbsp;22

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2.3nbsp;25

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15
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28
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44nbsp;28
39
nbsp;39
30nbsp;48

7nbsp;22

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27nbsp;35
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10nbsp;35

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17
nbsp;21

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50nbsp;49

41nbsp;45

47nbsp;44

38nbsp;33

37nbsp;32

20nbsp;24
19nbsp;28
23nbsp;30
22nbsp;27

21nbsp;17

-80°
340°

Centre at

5

0

7

8

9

15

10

12

9

12

10

9

25

10

18

15

9

14

14

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30

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13

24

7

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10

0

28

29

33

19

20

5

29

29

31

23

20

4

35

30

27

21

24

3

41

27

28

21

23

2

29

20

23

28

25

1

32

33

27

28

18

-1

40

42

31

31

25

-2

43

30

29

20

30

-3

25

40

34

15

44

-4

33

31

23

27.

27

-5

20

28

20

18

15

-0

23

20

13

18

18

■ -7

13

28

28

15

8

-8

19

14

13

12

15

-9

17

19

18

13

5

-10

PLATE 50.

10
9
8
7
0
5
4
3
2
1
-1
-2
-3
-4
-5
-0
-7
-8
-9

-9 -8 -7 -0 -5 -4 -3 -2 -1

28 29 22 49 54 41 49 51 65
50 01 58 62 57 GO 71 102 00
62 73 02 08 99 248 405 135 157
71 100 225 147 195 258 190 219 105
81 71 109 109 199 141 179 290 315
78 105 129 219 193 190 274 332 435
105 123 145 209 273 291 394 474 458
102 170 247 307 428 489 497 605 679
199 307 380 357 538 585 583 680 881
188 231 267 339 398 450 618 778 839

44 53 71 67 52 30 31
76
66 203 199 55 60 52 51
86 102
189 251 112 108 74 08 71
188 157 170 229 131 144 123 104 86
337 273 178 162 162 153 J72 165 86

489nbsp;584 465 272 248 232 235 228 167
452 451 393 297 283 287 238 228
454 455 520 463 287 289 243 292 233
359 636 448 528 404 336 283 233 235

490nbsp;517 008 618 553 443 304 247 IQI

176 355 334 369 368 353 359 475 603
239 304 272 442 357 433 504 487 018
331 416 389 423 436 515 688 770 895
393 543 639 593 584 576 71410^ 1138
296 398 542 519 397 287 400 653 888
282 482 528 612 490 319 226 668 815
341 449 497 448 464 406 339 247 458
168 216 243 227 206 174 243 235 251
116 106 137 121 87 121 147 143 127
63 51 87 82 76 79 79 57 68

598 560 672 596 653 GdY^sTE^^
600 094 990 854 730 711 079 529 508
848 807 956 1183 912 817 782 040 510
973 9431025 893 990 779 903 098 466
912 889 670 573 620 728 677 566 394
715 578 269 298 318 316 313 269 269
368 345 250 207 214 276 224 218 168 \'
227 215 214 194 184 166 173 173 99
152 95 135 191 181 163 131 84 38
84 68 90 112 79 104 69 73 55 ■

Centre at •

-ocr page 29-

my sincere thanks to both these gentlemen. After a number of provisional
counts with both instruments the
Zeiss magnifier was. chosen for the final
counts; although its field of view was a little Smaller than with
Brücke\'s
instrument, it was superior to this latter in clearness and was moreover easier
to handle by its much smaller size.

When selecting the four plates to be counted, I was led by the consider-
ation that they should be representative, as far as was possible with such a
small number, of the different cases met with on the Harvard plates. So the
numbers 2, 6, 34 and 50 were chosen. The first two of these plates were taken
at Cambridge, the other two at Arequipa. The plates 6 and 50 cover regions
of small star-density, the other two parts of the Milky Way.

The results of my counts are schematically represented on pp. 13 and 14and
need no further explanation. It should only be noticed that the numbers for
b
and ^ under each scheme indicate the galactic latitude and longitude of the centre
of the plate and that the itaUcised figures are from fields counted also by
Henie.

It is now easy to ascertain how far the star-numbers which would have
been obtained for each of these plates by
Henie\'s method agree with the
numbers obtained by complete counting. In plate 2 e.g. I counted:

in I 15972 stars
in
II 17761 „
in
III 11945 „
in
IV 13261 „
total 58939 stars

With Henie\'s method we should have counted:
in
I 4425 stars
in
II 5050 „
in
III 3424 „
in
IV 3648 ..

total 16547 stars
which would yield for the whole plate by calculation

59569 stars

i.e. only 630 stars or 1.07 % more than the number actually found.

These results for plate 2, with the corresponding numbers for pi. 6,
34 and 50 have been collected in table III.

-ocr page 30-

TABLE III.

plate

Ca—Co

Calcul-
ated

Co

3648
774
639
11328

58939
10170
7967
119947

59569
9936
8035
117868

1.07%
—2.30
0.85
—1.73

Complete count in

By Henie\'s metl

Plate

I

II

III

IV

I

II

III

2

15972

17761

11945

13261

4425

5050

3424

6

2864

2309

2121

2876

742

632

612

34

2066

1622

1851

2428

567

447

579

50

21617

22115

34356

41859

5805

6090

9518

IV Counted

In the first column of this table the number of the plate is given in the

next four the numbers found by complete counting in each of the four quadrants-

in the following four columns the numbers which would have been found

by Henie\'s method for each of the four quadrants; in the tenth column the

actually counted total for each plate; in the eleventh the total number which

would have been calculated by Henie\'s method. In the last column the percent

age is given by which the numbers based on Henie\'s method exceed the
actually counted numbers.

From table III it is seen at a glance that there is no question of a
systematic difference between the star-numbers obtained by either method For
although one of the Cambridge plates, when counted by
Henie\'s method gives
more and the other fewer stars than when completely counted and although
the same holds for the Arequipa plates, still at Cambridge it is the Milky
Way plate that gives more stars, while at Arequipa this is the other way about

Also the absolute value of the error is very small. If we consider
the whole of the four plates, covering together an area of the sky of more than
3600 square degrees, the total of the stars found by Henie\'s method (in which
only about a quarter of the total surface is actually counted) would be
195408
while the complete count yields 197023 stars. So the error amounts only to
0.82 % of the exact number. This amount is certainly far exceeded by errors
resulting from other sources, viz. from the inaccuracy in determining the
limiting magnitude and from the uncertainty whether a spot on a plate is
caused by
a star or by a dust-particle.

-ocr page 31-

So we may take it as settled that in star-counts it is sufficient to
count a number of separate fields. However, a condition of paramount import-
ance is that these fields are spread as regularly as possible over the celestial
sphere.

When for the above-stated purpose I had completed my counts of
four plates of the Harvard Map, it seemed worth while to compare my
numbers with those found by
Henie for these four plates. The results are
given in table IV. In the first column the number of the plate will be found,
in quot;the second that of the quadrant. The third column contains the numbers
of stars counted by
Henie, the fourth the numbers found by myself. Of my
complete counts only those fields were, of course, retained which were also
counted by
Henie. The last column states, in a percentage of Henie\'s
numbers, the difference H.—N.

TABLE IV.

comparison of counts.

Numbers of stars

H.—N.

Plate

Quadrant

counted

Henie

Nort

H.

2

1

I 1

4469

4101

8.23 %

f}

II

3890

5050

—29.82

tf

III

3080

3384

— 9.87

ft

IV

4821

3284

31.88

6

1 I

591

742

—25.55

f}

II

639

632

1.10

yy

III

560

612

— 9.29

n

1 IV

574

764

—33.10

34

i I

753

567

24.70

gt;gt;

II

626

438

30.03

fgt;

III

718

579

19.36

gt;gt;

IV

993

639

35.65

50

I

7956

5805

27.04

gt;1

II

8223

6090

25.94

fi

III

19171

9518

50.35

ft

\\

IV

19389

11328

41.57

-ocr page 32-

This table reveals a very striking fact. In five out of sixteen cases Henie
counted fewer stars than I, in all other cases more. Bearing in mind the different
power of the optical apparatus used by
Henie and myself, I — from personal
experience — should have been inchned to expect a very different ratio.
The absolute value of the difference between
Henie\'s numbers and mine
varies from
1 % in the second quadrant of plate 6 to 50 % in the third quadrant
of plate
50, and in 11 of the 16 examined cases amounts to more than 25 %.
The cause of these enormous differences may of course lie either in the plates
themselves or in the counts or in both.

As to the first cause, the plates counted by Henie and myself are copies
of one and the same original and it is quite possible that these two copies
are not equivalent. One copy may be weaker than the other, either through
shorter exposure or, more probably, on account of less full development.
This would in itself not be serious. On the weaker copy one would count fewer
stars, to be sure, but on the other hand one would find a limiting magnitude
of less depth. This assertion is only true if the copies have been carefully
made, so as to bear the same ratio of density to the original all over the plate
If this were not the case the reduction of the plates would lead to very meagre
results. For in this reduction we start from the assumption, which probably
holds for the original so far as it is free from local fogging, that the limiting
magnitude is the same at equal distances from the centre. Now, if in taking
the copies, the various quadrants were unequally weakened, the above-stated
condition would no longer be fulfilled and the foundation on which the whole
reduction of the plates is based then becomes unreliable. How far this is true
can only be settled by one and the same observer counting different copies
under conditions as similar as possible. Since, as far as I know, only a single
copy of the Harvard Map is to be found in the Netherlands, namely at the
Utrecht Observatory, I requested Prof.
Pickering to be so kind as to send me
another copy of plates
2, 6, 34 and 50. To my disappointment the circumstances
occasioned by the war made it impossible to forward these plates, so that
this part of the investigation had to be given up for the present.

As a second possible cause of the large deviation between Henie\'s
numbers and mine the counts themselves were mentioned. Not only difference

-ocr page 33-

PLATE 6.

PLATE 6.

counted by mr. g. v.
12 3 4 5 6

biesbroeck.
7 8 9

-9 -8 -7 -6 -5 -4 -3 -2 -1

-9 -8 -7 -6 -5 -4 -3 -2 -1

counted by
2 3 4 5 6

THE AUTHOR.
7 8 9

10

5

7

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3

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24

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26

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30

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25

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33

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28

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16

16

18

34

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32

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46

57

31

42

36

41

32

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17

13

11

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14

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67

63

66

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52

48

19

21

20 ;

_o

9

7

9

18

10

28

33

27

45

50

42

45

45

34

21

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15

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14

12

16

22

21

34

60

45

72

74

90

59

67

44

33

22

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-3

8

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27

29

31

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12

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61

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-4

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22

19

28

11

14

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-5

5

6

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40

21

21

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11

-5

8

6

4

7

9

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23

21

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21

21

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6

7

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-6

11

6

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15

17

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39

36

53

25

70

83

49

46

33

8

10

13

-6

8

6

5

9

4

9

13

17

18

16

13

16

13

17

6

10

10

8

-7

11

9

10

12

11

15

19

27

27

21

40

45

32

45

15

8

14

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7

-7

5

2

7

3

6

9

6

8

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17

10

16

13

14

10

6

3

5

-8

7

2

6

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23

20

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15

9

9

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7

-8

4

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18

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18

21

16

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1 1

-9

2

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8

5

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5

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ID

7

15

11

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2

6

5

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-10

o

-ocr page 34-

in tlie circumstances of illumination and in the strength of vision of the ob-
servers may of course lead to considerable discrepancies, but also their
appreciation of what are star-images and what specks. The part played by
the individuality of the observer may be ascertained by having the same
plate examined by two different persons under conditions as similar as possible
It is owing to the kindness of Mr.
G. van Biesbroeck, astronome adjoint
of the Royal Observatory at Uccle (Belgium), who temporarily resided at the
Utrecht Observatory, that this part of the investigation could be carried out
At my request Mr.
van Biesbroeck counted plate 6 with the same magnifier
and on the same background as were used by me. Page 19 shows the
comparison of our results.

It follows from this comparison that Mr: van Biesbroeck counted
only 2/g of the number found by me and this ratio is about the same in the
four quadrants; in I and II 0.67, in III and IV 0.62. In order to discover the
cause of this very considerable difference we recounted immediately after
each other various fields of plate
6. Although Mr. van Biesbroeck\'s numbers
now came somewhat nearer to mine, which according to him must be attributed
to the fact that during our conference the sky was clearer than when he had
counted alone, still the difference between his results and mine subsisted
So we both made sketches of some fields, containing everything shown by the
magnifier. It then turned out that the fact was not that one of us saw more
or fewer spots on the plate than the other, but that the difference must only
be ascribed to a difference in appreciating which of these spots represented
stars and which not.

That this difference in appreciation may lead to totally different
results is particularly disappointing, as it cannot be made out in practice
whose appreciation is the true one. That two observers, both of them doing
their very best to count accurately, both having a normal strength of vision
and both having sufficient routine for such counts, may arrive at so widely
diverging results, proves in my opinion that the need is pressing that photo-
graphs which must serve for counts, should be so arranged that it is possible
to decide with greater accuracy than with ordinary photographs whether or
not one has to do with a star-image.

-ocr page 35-

A last remark may be made on Henie\'s counts, the results of which
are found on pp.
7—34 of his paper. Henie omits to state whether he includes
star-clusters in his counts. If this be so, it would perhaps afford a partial
explanation of the excessively high number of stars computed by
Henie
from his counts of plate 50. This number namely amounts to 216329, while
by direct counting I found
119947. In the field containing the cluster N.G.C.
3114 Henie counted 678 stars against my 269 and in the field containing the
cluster N.G.C.
3523 he counted 1590 stars against my 990. I cannot state
with certainty, however, whether for these two fields the large difference
must be ascribed to this cause, since also in fields, certainly containing
no clusters, our numbers often diverge very much.

-ocr page 36-

CHAPTER III.

the determination of tiie limiting magnitude.

It is clear that to count the stars in different fields of a photographic
plate for statistic puri)oses is void of meaning unless the limiting magnitude
in those fields is accurately fixed. It is not sufficient to determine the limiting
magnitude at a single point of the plate; especially with plates covering such
a large area of the sky as those of the Harvard Map there is a great difference
between the limiting magnitude at the centre and near the margin. So we must
either determine the limiting magnitude for all the counted fields individually
or we must by means of a formula for the number of stars down to any magni-
tude derive from the limiting magnitude at a few points of the plate and from
the star-numbers found, a relation between the limiting magnitude at any
point of the plate and its distance from the centre.

\'fhis second method compares unfavourably with the former. In the
first place because it compels us to assume an
a priori formula for the number
of stars of every magnitude, which generally is exactly the aim of
star-counting.
Secondly because it is then tacitly assumed that the limiting magnitude has
the same value for all points at the same distance from the centre. Now with
the
original plates this will often be the case, at any rate if the centre was
focused and the plate was at right angles to the optical axis of the telescope.
But for
copies, printed by the double contact process, it is by no means certain
that this condition is satisfied, unless exceptional care has been taken in the
reproduction.

Henie used the second of the methods mentioned for determining the
limiting magnitudes. He himself writes on this account;!) quot;In order to determine
1) Henie. 1. c. pp. 42 and 43.

-ocr page 37-

quot;the limiting magnitude I have solely usedHAGEN\'s^^/as Stellarum Variahilium
quot;together with the photometric measurements of Pickering. This atlas,
quot;containing charts of variable stars with the surrounding comparison stars
quot;has previously been used by Prof.
Charlier to determine the limiting

quot;magnitude of\'Cartes du Ciel.

quot;The atlas gives for every chart an index of the magnitudes of the
quot;comparison stars in different scales. The scale used in this discussion is always
quot;the Harvard Scale. I have proceeded in the following manner. When the
quot;part of the negative, that corresponds to the Hagen chart is found, I have for
quot;each comparison between chart and plate noted the magnitude of the star
quot;in the Harvard Scale and the intensity of the photographic image of the
quot;same star, beginning with the brightest. In this way I get two series, one giving
quot;the increasing magnitudes, the other the decreasing intensities of the images.
quot;As the
Hagen charts generally contain fainter stars than the photographs,
quot;I accordingly arrive at a point, where the comparison star has no corres-
quot;ponding image on the plate; then the limiting magnitude is passed.quot;

This description and the accompanying table VP) give rise to the

following remarks:

1.nbsp;Henie explains how he states that the limiting magnitude is
passed, but not how it is determined. This is to be regretted, since it cannot
be made out now whether
Henie counted everything that was just visible
on the plates or whether he put himself a quot;limit of countingquot;. According
to
Van Rhijn^) the plates of the Selected Areas, which were counted at
Groningen, are only complete down to stars 0?6 brighter than the faintest
stars visible on the plate, so that it is necessary to fix a limit of counting if
the numbers of the faintest stars counted shall be reliable.

2.nbsp;In the second column of table VI Henie gives under n quot;the number
of comparisonsquot;. Without doubt the knowledge of this n will be necessary
for an appreciation of the limiting magnitude obtained, but the exact meaning
of this number of comparisons cannot be gathered from his description.

3.nbsp;If one refrains from determining the limiting magnitude for each

1)nbsp;Henie. i.e. pp. 43. 44 and 45.

2)nbsp;Van Rhi jn, Publications of the Astronomical Laboratory at Groningen, Nr. 27, pp. 10,11.

-ocr page 38-

individual field and therefore wants to derive a relation between the limitin

magnitude at any point of the plate and its distance from the centre itquot;!

essential that the number of points for which this limiting magnitude is directly

determmed be as large as possible. In this way, obviously, it is possible to

find such a relation without the intervention of a formula for the number of

stars of any magnitude. Henie evidently has felt this himself for on ^l^e

40 of his paper he says: quot;The table of the limiting magnitude of the different

quot;negatives. Table VI, shows in the cases in which this quantity has been

quot;determined in a series of points on the same plate, that the magnitude decreases

quot;from the centre of the j^late towards the margin. The problem is now to find

quot;the law of this decrease. An attempt to derive this law direct from the obser-

quot; vations failed because of the relative great probable errors in the determinatioi!^^

quot;and because of the observations being too scattered.quot; It is all the more surp^^

ing that Henie did not nearly use all the material at his disposal. For a first

and very striking example I take plate 2 0, since the number of points for which

Henie determined the limiting magnitude is largest for this plate, namely

seven. He used the surroundings of the variables V Vulpeculae, W Vulpeculae

TX Cygni, U V Cygni, S Y Cygni, VX Cygni and V Y Cygni. But besides
these variables we find on this plate:

H.A.nbsp;H.A.nbsp;H.A.

T Y Cygni, 57nbsp;Z Delphini, 57nbsp;S Aquilae 57

^ quot;nbsp;W „ ,63; 47 R Wnbsp;^ gg

T Sagittae, 57

» , 37nbsp;5

W W „ Tnbsp;„ , 57nbsp;R „ 63

Vnbsp;W „ Vnbsp;„ ^ 57; 47nbsp;V „ \' 57

quot;nbsp;Znbsp;„ , 57 RR Vulpeculae, Q3

W X „, 57 V Lyrae , 57nbsp;Rnbsp;37

R W „nbsp;R Unbsp;„ ,57 quot; \'

W Z „nbsp;Snbsp;„ ^ 57

U X „, 57nbsp;R Snbsp;„ ,57

U Y „, 63nbsp;R Vnbsp;,63

Vnbsp;X „, 63nbsp;Unbsp;„ ,57
T X „, 63

T W „ ,57

-ocr page 39-

i.e. 33 variable stars having sequences suitable for this purpose. These sequences
are found in the volumes of the Harvard Annals, mentioned in the second
column. So
Henie might have determined the limiting magnitude at 40 points
of the plate instead of at 7.

As another instance I take plate 24. At a single point only of the plate
the limiting magnitude has been determined, namely in the vicinity of
X Ceti.
But besides this variable we find on the plate:

H.A.nbsp;H.A.

U Arietis, 57nbsp;RX Tauri, 57

R Tauri ,37nbsp;R Orionis, 57

S Tauri ,37

so that the limiting magnitude might have been determined at 6 points.

Besides the three points mentioned there is still another question, passed
over by
Henie, but still of great importance. I mean the fundamental error inher-
ent in the method followed by
Henie — and also by others — of determining
the Hmiting magnitude, namely that the limiting magnitudes of photographic
plates are determined by means of
Hagen\'s visual magnitudes. Calling, as is
customary, the difference: photographic magnitude minus visual magnitude of
a star its colour-index, it is obvious that the above-described method must
have caused errors in every case in which the comparison stars had colour-
indices of a somewhat considerable value.
Kapteyn, who also used this method
of determining the limiting magnitude of plates, pointed this out emphatically.
He says^): quot;The results of the present paper are valid for
visual magnitudes
quot;and this though, for the greater part, the countings have been made onphoto-
quot;graphs. The fact is that the limiting magnitudes have been invariably yzswa/Zy
quot;determined. If we count all the stars equal to or brighter than any determined
quot;star S, then the fact that the countings are not made visually in the sky,
quot;but on photographs, will introduce an error in the case in which the difference
quot;between the photometric and visual magnitude of this star S is
exceptional.

1) J. C. Kapteyn. Publications of the Astronomical Laboratory at Groningen, Nr. 18, p. 5.

-ocr page 40-

2(5

quot;In this way errors must have been introduced in the results for the individual
quot;regions. But as, for the several regions, this error has a purely accidental
quot;character a
systematic influence on the final results need not be feared It is
quot;only the agreement of the separate results which must have suffered. In order
quot;to diminish as far as possible the errors for the individual plates, no single
quot;star has been used to define the limiting magnitude, if its difference in
quot;magnitude with one or more stars in its vicinity, as estimated on the photo-
quot;graph, seemed to be in conflict with the value of this difference
as determined
quot;visually at the photometer. Great part of our determinations are based on
quot;a fairly considerable number of photometrically determined stars. For these
quot;determinations even the accidental error in question must be insignificantquot;
From this quotation it appears that
Kapteyn not only thought it
necessary to point out this error in the method, but also that he attempted
to neutralise it as far as possible.
Henie did neither and this is the more
regrettable since it seems to me that
Kapteyn\'s remark that quot;for the several
quot;regions this error has a purely accidental characterquot; is not quite correct and
that quot;a systematic influence in the final resultsquot; is indeed to be feared, unless
measures are taken, as was done by
Kapteyn, to prevent this influence
My opinion is based on the following considerations. The amount of these
errors depends on the value of the colour-indices of thé comparison stars
Now if equal positive and negative values of the colour-index were equally
frequent in all parts of the sky, the errors would indeed present a purely
accidental character. This supposition, however, is not justified, as appears
from what follows:

In Harvard Ann. 64, 144, Pickering has tabulated the numbers
of stars of the different spectral classes, brighter than visual magnitude 6.25
in terms of their galactic latitudes. The latitudes quoted in the first column
are the average latitudes of each of eight approximately equal zones, beginning
with the one round the north galactic pole and ending with that which surrounds
the south galactic pole. This table I have inserted here as Table V, giving
percentages, however, instead of the original numbers. Of all stars brighter
than magnitude 6.25, occurring in the zone of average galactic latitude 62?3,
1 % accordingly belongs to spectral class B, 33 % to class A, etc. Here

-ocr page 41-

Pickering\'s division of the spectral types is somewhat different from the
usual one. viz.

B = O—B8;
G =F5—G2;

A = B9—A3;
K =G5—K2:

F = A4—F2.
M = K5—N.

TABLE V.

distribution of stars of different spectral classes.

Galactic

Spectra]

i Class

latitude

B

A

F

G

K

M

62?3

1

33

14

11

31

10

41.3

5

33

10

12

31

9

21.0

9

35

11

9

28

8

9.2

20

30

9

9

25

7

— 7.0

16

37

12

8

23

4

—22.2

16

29

12

10

26

7

—38.2

9

25

15

9

32

9

—62.3

6

20

15

13

38

9

Entire Sky

12

31

12

10

28

7

For each of these six spectral classes I computed the colour-indices
from the numbers given by
Pickering on page 7 of Vol. 71 of the Harvard
Annals, as follows:

Spectral class Bnbsp;A F G

Colour-index —0?25 0?04 0?29 0?66

K
1^12

M
1?56

Now firstly it appears that equal positive and negative values of the
colour-index have not only a different frequency, the positive values occurring
in a larger number of cases, but moreover these positive values have in part
much larger absolute values. Secondly we see that the frequencies of the
different values of the colour-index also depend on the galactic latitude. For if

-ocr page 42-

wc combine tlie northern and southern zones of the same galactic latitude we
find that for a galactic latitude of G2?3 thirty percent of the stars, brighter
than Cquot;\'2r), belong to classes A and B (blue stars), while lt;)■}% belong to class
M (red stars). These numbers arc for a galactic latitude of 8fl, respectively
511 % and 51 %.

How far the results here obtained must be modified for fainter stars
cannot be stated with certainty, since for these stars the necessary data as
to the spectrum are lacking. From
Pickering\'s remark^): quot;more than half
quot;of the stars of the ninth magnitude and brighter have spectra of Class A and
quot;the proportion is probably greater for faint stars, especially in the Milky Wayquot;
we should conclude that also for the fainter stars the colour-index depends
on the galactic latitude, that also for these stars positive values of the colour-
index are much more frequent than negative values and that consequently
also for these stars the danger of making
systematic errors in the determination
of the limiting magnitude of photographic plates, if this is done by
Henie\'s
method, really exists.

Although objections may be raised not only to the manner in which
Henie carried out his counts but also to his method of determining the limiting
magnitude and in these respects his counts certainly compare unfavourably
with those by
Kapteyn and by Chapman and Melotte, still for certain
purposes
Henie\'s material may be considered valuable for the following
reasons:

1.nbsp;Henie counted many more separate fields than the other
astronomers mentioned.

2.nbsp;Henie\'s material is much more homogeneous than that used
by
Kapteyn. It is especially this last advantage which should not be under-
rated. Very truly
Eddington^) says as recently as 1914: quot;much statistical
quot;matter that has been used up to now depends on the ingenious adaptation
quot;and correction of data which were initially rather unsuitable.quot;

These considerations induced me to use the results of Henie\'s counts

1)nbsp;Edw. C. Pickering, Harv. Ann. 71, 6.

2)nbsp;A. S. Eddington, Stellar Movements and the Structure of the Universe, p. 185.

-ocr page 43-

for some investigations in stellar statistics. It was necessary, however, to
re-reduce the results given by
Henie on pp. 7—34 of his paper. Since in
this method, of reduction I have departed from the way followed by him,
the manner of reducing will be dealt with in the following three chapters.

-ocr page 44-

CHAPTER IV.

reduction of the plates.

At the end of the preceding chapter it was remarked that the results
of
Henie\'s counts are not directly comparable among themselves. One of the
causes is that a square centimetre on the plate does not correspond to a constant
area on the celestial sphere. Therefore the numbers per square centimetre
have been reduced to numbers per square degree. The method by which this
is done is the same as that explained by
Henie^). We start from the supposition
that in taking the plates the centre has always been focused, that the focal
surface is a sphere and that the plates are tangent to this sphere.

Let r be the radius of the focal surface, du one of its surface-elements
corresponding to a sohd angle
dta and a surface da on the plate; let further
be the distance
oi da from the centre of the plate and qp the angle under
which e is seen from the centre of the focal surface, then

du {r^ 4- gquot;)nbsp;3

da =nbsp;^ {r^ e\') _ {r\'\'

cos tfnbsp;f 1nbsp;y

As the focal length of the lenses used for photographing is 13 inches
(30.00 cms.)2)j as din is expressed in units of solid angle and as 47r of such
units are equivalent to
41253.4 square degrees^) we have

(33^-f-e^f = 0.0000092308 (33=^ ?^......(1).

1)nbsp;Henie, 1. c. p. 36.

2)nbsp;Pickering gives about 13 inciies and this rougli estimate of the focal length would
entitle us to only two decimal places in the reducing factors. As however a direct
measurement
of the plates gave 33.00 cms. I have retained 5 decimal places, the last one, of course
being unreliable.

3)nbsp;Following Seeliger I took for a square degree a surface bounded by the parallel
circles Dec. = %° and Dec. = ~ and two hour-circles differing 1° in R.A.

-ocr page 45-

For the coefficient in formula (1) Henie found 0.0000092294. I have
not been able to discover the cause of this difference which appears to have
no influence on the coefficients of reduction.

By means of formula (1) it is now easy to calculate for each of the
360 fields of a plate the number by which the star-number per square centimetre
must be multiplied in order to obtain the star-number per square degree.
These numbers, multiplied by 1000, are found in table VI. The coefficients
for the fields usually counted by
Henie are printed in italics.- Since however
Henie sometimes departed from his usual scheme it was necessary to calculate
these coefficients for
all fields.

TABLE VL

reduction from square c.m. to square degree.

1

2

3

4

5

6

7

8

9

374

375

377

380

383

388

394

401

409

10

365

366

368

371

375

380

385

392

400

9

358

359

361

364

367

372

378

384

392

8

351

352

354

357

361

365

371

378

385

7

346

347

349

351

355

360

365

372

380

6

341

342

344

347

350

355

361

367

375

5

337

338

340

343

347

351

357

364

371

4

335

336

337

340

344

349

354

361

368

3

333

334

336

338

342

347

352

359

366

2

332

333

335

337

341

346

351

358

365

1

Centre of the plate.

The so reduced star-numbers, which I have not inserted here in order
not to become too elaborate, are not yet directly comparable among them-
selves. The limiting magnitude, namely, for which these numbers hold, has

-ocr page 46-

no constant value, it varies on each plate from centre to margin and from
one plate to another. The reduction of all fields to a single magnitude would
be very simple if we knew the limiting magnitude in each of the counted
fields. This is not the case, however. The number of points on each plate for
which the limiting magnitude is known is very limited and indeed far too
small to derive even an approximate law for the decrease of the limiting
magnitude from the centre towards the margin. So this law must de derived
in another way.
Henie proceeds as follows. He assumes that the star-density
shows no systematic decrease towards the margin if this is not the case with
the magnitude. But since the limiting magnitude decreases the density will
do the same. Now if the law of decrease of the density were known and also
the relation between density and magnitude the law for the decrease of the
magnitude could be deduced.

In order to arrive at a law for the decrease of density Henie divides
every plate into seven zones by circles, having their centres at the centre of
the plate and with radii of resp. 2, 4, 6 etc. cms. Four of these zones lie entirely
on the plate, the three others only partially. For all 55 plates the average
star-density is now calculated for each of the seven zones, expressed in the
number of stars per square degree. Several of the fields counted by
Henie
are of course intersected by the circles, in which case Henie considers such a
field as belonging to the zone in which the greater part of it falls. This is not
very accurate and in my opinion not justified if, as
Henie does, the average
densities are given in two decimals. This procedure has moreover the drawback
that zone VII (the outer one) contains only two fields which, for a reliable
result, is not sufficient.

Therefore, following for the rest Henie\'s method, I have calculated
the ratio of the parts of the intersected fields; the results arc for the upper
part of the plate represented in fig. 2. The numbers in the fields, usually
counted by
Henie, are underlined.

-ocr page 47-

-6

-I

-6

-5

-4=

-2

-1

1

.1.

2

3

5

6

7

8

9

Lop_

65y

V

95

/OO

il

63

68

i^—-
SS

98

98

83

___^

63

99

100

/oo

50y

/oo

/oo

66 .

9Z

/oo

/OO

/OO

/oo

/oo

/OO

/oo

/OO

9Z

66

/oo

\\

/oa

Jpo

/oo

^Oy/
/60

/oo

/OO

so

S6

.91

9$

86

80

/OO

/oo

/oo

/OO

too

fo/
/eo

/oo

97

A

M 60

-

/OO

/oo

too

/OO

/OO
_ p

/OO

/oo

/OO

97

a

/oo

/oo

^ /
A

/oo

97

/7S

/OO

96

56
^

z^
SO

B

9S

80

96

fOO

2Z

/oo

\\66

\' 32

/oo

/W

/OO

quot;A

/ay \'

/CO

/oo

/OO

/oo

/oo

/oo

/oo

60\\

/oo

Y
M. \\

so /
k

r-

/oo

96

i

r

7 sz

/OO

A

3/^
69

M

96

/oo

3l\\

96

/oo

\\S0
?b\\

/Oo

fOO

fzj

/oo

56 /

m

/oo

//6

/OO

/oo

/OO

/oo

K\'

/OO

w

«A

/oo

/oo

J
/ 86

/oo

/ so

/oo

3/ /
69

/QO

68y

/32

^

9/

\\

/oo

/oo

so\\

/oo

S6 \\

/oo

^00

93

/oo

9S

/amp;0

96

/OO

y

/ 9/

/oo

/OO

quot; 1

/oo

quot; 1

/Oo

/oo

9S

/oo

Fig. 2.

In computing tlie average density in each zone the parts of the
intersected fields are now related to the zone in which they really lie.
Six observations then fall in zone VII instead of two. The results of these
calculations are given in table VII. In the first column the number of
the plate is given, in the seven following columns the average density in each
of the seven zones, in the last one the average density of the whole plate, all
expressed as the number of stars per square degree.

10
9
8

1
^

5
^

3

2
J

-ocr page 48-

TABLE VIL

the densities in the seven zones and the mean densities of the plates.

Plate

1^5

Do

Dt

1

2

4

5

21 .quot;47
64.08
17.58
11.99
9.37

\'^T\'Tsf

85.42
16.27
14.33
8.69

31!08\'
75.30
17.88
12.26
6.00

26.79
59.93
15.02
8.88
6.15

23.90
51.29
10.17
8;i3
6.50

22.28
42.31
6.96
7.06
8.15

15.81
36.36
3.78
7.42
8.25

25.80
60.15
12.82
9.60
6.92

»

7

8
{)

10

11.13
20.01
14.89
23.54
54.03

\'12.4:5
16.94
13.84
24.80
51.96

10.02quot;

12.49

11.50
20.46
54.:}0

7quot;\'. 74
9.19
8.94
15.87
49.34

7.44
9. .32
8.97
12.14
45.22

7.59
7.55
7.47
9.11
40.28

7.86
8.89
6.00
5.57
42.73

8.6F

10.77
10.04
15.38
48.04

U
12

14
If)

27.83
50.21
51.70
39.87
30.18

24.74\'

53.01

42.71

42.92

29.92

23.45
41.30
;J3.60
;J2.00
31.28

19796
49.87
28.47
25.73
26.85

21.06
62.59
25.14
21.93
24.74

21.72
49.50
23.14
19.29
21.11

23.80
53.25
23.59
16.38
25.51

21.99
52.60
29.80
26.76
26.57

To

17

18

19

20

f4quot;.87
24.34
19.12
27.62
59.87

T7.74
25.79
17.18
23.41
40.20

16.85
24.24
13.61
22.22
29.00

12.82
22.49
11.29
19.13
21.39

13.26
24.90
9.94
18.34
18.57

13.98

21.99
7.95

17.36
15.84

18.05
23.76
8.20
15.04
18.44

14.55
23.90
11.70
19.60
25.00

21
22

23

24

25

33.17

13.95

16.96
28.30
58.30

24.26
16.82
18.16
29.64
64.20

16.44
16.43
25.39
67.29

20.00
19.00
14.94
23.;52
49.69

21.32
20.45
14.11
21.24
41.54

19.22
24.56

11.23
19.09
30.38

17.56
16.32
9.75
13.76
28.37

21.35
19.22
14.66
23.00
48.70

quot;26~

27

28

29

30

18.12
16.97
14.31
20.00
11.63

15.45
21.58
17.73
21.00
13.31

12.7^
21.10
13.13
21.67
11.80

10.42
19.15
10.70
19.01
11.65

9.46
18.86
10.22
17.23
11.18

7.31
17.33
11.30
14.51
14.86

5.53
17.21
10.24
15.10
22.40

10.75
19.26 .
11.89
18.42
12.44

31

32

33

34

35

26.70
15.39
16.09
9.37
19.21

16.85
17.91
16.08
12.49
17.84

21.90
20.10
14.31
11.80
12.88

16.94
16.94
11.77
10.44
11.27

16.93
15.52
12.71
10.69
13.32

15.01
13.08
15.53
11.56
.13.47

9.74
11.10
17.82
12.03
14.52

17.68
16.37
13.64
11.22
13.53

36

37

38

28.35
25.15
62.93

26.04
28.03
62.54

23.15
29.13
58.84

18.15
26..35
40.45

15.17
21.82
31.59

12.56
16.39
20.47

9.52
15.78
15.07

18.26
23.95
40.46

39

40

41.57
13.79

43.71
20.24

34.25
15.28

27.91
10.94

24.82
8.40

21.35
6.36

21.51
5.60

29.35
11.22

41

42

43

44

29.97

43.17
129.27

31.18

29.51
;^7.46
80.19
28.68

29.59
31.63
54.55
28.21

23.39
28.31
38.49
25.59

20.35
28.21
28.10
28.57

17.82
23.24
24.71
27.93

15.17
15.84
23.95
23.46

23.58
29.47
43.49
27.67

45

17.51

14.95

12.35

12.76

12.44

14.47

13.85

13.25

46

47

36.13
15.34

36.32
14.28

29.96
14.72

23.52
12.35

19.06
10.79

14.90
10.15

14.13
8.10

23.56
12.12

48

46.94

43.96

41.22

34.60

25.11

21.76

20.60

31.78

49

107.07

117.77

94.94

58.49

49.52

35.39

34.49

65.72

50 :

337.70

308.45

284.64

193.92

140.66

103.21

69.55

194.09

51

138.51

127.09

87.95

67.93

50.78

33.19

24.32

68.83

52

53

54

55

55.63
21.93
35.89
72.96

55.26
21.91
41.04
64.42

41.62
16.07
39.13
66.26

36.06
14.11
35.59
62.42

.33.27
14.61
28.04
50.48

26.57
12.42
18.09
39.56

22.26
14.72
14.10
41.74

37.13
15.48
32.61
56.34

Since it is our object to derive a law for the decrease in star-density which
applies to the entire set of plates, or, since this is scarcely attainable, for certain
groups of plates, individual peculiarities of each plate must be eliminated as

-ocr page 49-

far as possible. It is therefore desirable to calculate the relative density for
the zones of each plate,
i.e. the average density of a zone divided by the average
density of the whole plate. These relative densities are given in table VIII.

TABLE VIIL

the relative densities in the seven zones.

Plate

di

d2

d3

do

1

0.832

1.077

1.205

1.039

0.926

0.864

0,613

2

1.065

1.420

1.252

0.996

0.853

0.703

0.605 !

;5

1.371

] .269

1.395

1.171

0.793

0.543

0.295 i

4

1.249

1.493

1.277

0.925

0.847

0.736

0.773

6

1.354

1.256

0.868

0.889

0.939

1.178

1.193

(i

1.290

1.440

1.161

0.897

0.862

07880quot;

0.911

7

1.858

1.573

1.160

0.853

0.866

0.701

0.826

8

1.483

1.379

1.145

0.891

0.893

0,744

0.598

9

1.531

1.612

1.330

1.032

0.789

0,592

0.362

10

1.125

1.082

1.130

1.027

0.941

0.8,38

0.889

11

1.265

1.125

1.066\'

0.908

0.958

0.988

r.quot;082

12

0.955

1.008

0.785

0.948

1.190

0.941

1.012

13

1.735

1.433

1.128

0.955

0.844

0.776

0.792 1

14

1.490

1.604

1.196

0.962

0.820

0.721

0.612

15

1.136

1.126

1.177

1.011

0.931

0.794

0.960

16

1.022

1.219

1.158

0.881

0:912

0,961

1.241

17

1.018

1.079

1.014

0.941

1.042

0,920

0.994

18

1.634

1.469

1.163

0.965

0.850

0.679

0.701

19

1.409

1.194

1.134

0.976

0.936

0.886

0.767 1

20

2.395

1.608

1.160

0.8,56

0.743

0.634

0.738 !

21

1.654

1.136

1.022

0.937

0,999

0.900

0.822

22

0.726

0.875

0.855

0.989

1.064

1,278

0.849

23

1.157

1.239

1.121

1.019

0.963

0.766

0.665

24

1.230

1.289

1.104

1.014

0.924

0.830

0.598

25

1.197

1.318

1.382

1.020

0.853

0.624

0.583

26

1.685

1.437

1.184

0.969

0.880

0.680

0.515

27

0.881

1.121

1.096

0.994

0,979

0,900

0.893

28

1.204

1.491

1.104

0.900

0.859

0.950

0.861

29

1.086

1.140

1.176

1.032

0.935

0.788

0.820

30

0.935

1.070

0.948

0.937

0.899

1.195

1.801

31

1.510

0.953

1.239

0.958

0.958

0,849

0.551

32

0.940

1.094

1.228

1.035

0.948

0.799

0,678 1

33

1.180

1.179

1.049

0.863

0.932

1.138

1.306 !

34

0.835

1.113

1.052

0.930

0.952

1.031

1.073

35

1.420

1.319

0.952

0.833

0.984

0.996

1.073 i

36

1.553

1.426

1.268

0.994

0.831

0.688

0,522 1

37

1.050

1.170

1.216

1.100

0.911

0,684

0,659 I

38

1.555

1.546

1.454

1.000

0.781

0.506

0,372 i

39

1.416

1.489

1.167

0.951

0.846

0.727

0,733 i

40

1.229

1.804

1.362

0.975

0.748

0.567

0.499

41 -

1.271

1.252

1.255

0.992

0.863

0.756

0.643

42

1.465

1.271

1.073

0.961

0.957

0.789

0.538 i

43

2.972

1.844

1.254

0.885

0.646

0.568

0.551 1

44

1.127

1.036

1.019

0.925

1.033

1.009

0.848 !

45

1.321

1.128

0.932

0.963

0.9,39

1.092

1.045 i

46

1.533

1.542

1.272

0.998

0.809

0.632

0.600 \'

47

1.265

1.178

1.215

1.019

0.891

0.837

0.668 \'

48

1.477

1.383

1.297

1.089

0.790

0.685

0.648

49

1.629

1.792

1.445

0.890

0.754

0.538

0.525

50

1.740

1.589

1.466

0.999

0.725

0.532

0.358 !

51

2.012

1.846

1.278

0.987

0.738

0.482

0.353

52

1.498

1.488

1.121

0.971

0.896

0,716

0.600 i

53

1.417

1.416

1.038

0.911

0,944

0.802

0.951

54

1.101

1.256

1.200

1.091

0.860

0.555

0.432

55

1.295

1.143

1.176

1.108

0.896

0.702

0.741

-ocr page 50-

From this table it follows not only that the relative density decreases
from the centre to the margin, but also that this decrease is greater for plates
with a large than for such with a small average density (D„). Therefore the
plates are divided into three groups; in the first group we have collected those
for which Do lt; 20; in the second those for which 20 lt; Dolt;
35 and in the
third those for which D„ gt; 35. It must be admitted that the values 20 and 35
are quite arbitrary,but as any other partition would be just as arbitrary I have
herein followed
Henie. In table IX we find under n the number of plates
belonging to each group and in the next columns the relative densities for each
group with their probable errors in each of the seven zones.

TABLE IX.

relative densities in the three groups of plates.

CM

i
0

11

\'li

d2

do

de

dr

i

ii

iii

\'—^—

27
17
11

1.279± 0.036
1.359± 0.058
l.549±0.116

1.292^-0.028
1.287±0.031
i.461±0.0g1

1.134± 0.018
l.i51±0.01{5
1.249±0.041

0.956± 0.009
0.984±0.01]
0.985±0.013

0.90,3± 0.009
0.897±0.014
0.843± 0.030

0.861±0.026
0.780^0.021
0.650± 0.030

0.831±0.042
0.724± 0.028

0.599j-0.043

=

For each of the above-mentioned groups the law has thus been derived
according to which the star-density decreases from the centre towards the
margin. In order now to derive the law by which the limiting magnitude
decreases towards the margin, we must, as has already been stated, assume
a formula for the number of stars as a function of the magnitude.
Henie
adopts the formula given by CharlierI) for the number of stars of magnitude
m in a definite area of the sky, namely

{mnio) ■
2 k^ ~

N —
e

(2)

a (m) =

2 IT

where N, k and m« are constants, to be derived from actual countings.

1) C. V. L. Charlier. Studies in Stellar Statistics. Lunds Universitets Arsskrift
N. F. Afd. 2, Bd. 8, Nr.
2, pp. 32 and 33,

-ocr page 51-

Denoting further by A (w) the number of stars in this definite area, brighter
than m,

im—Wo)quot;

N

gt; \'nbsp;/l\' I / ft TT

,-mnbsp;N

dm

CO

or, putting

= kxnbsp;dm = kdx

m—nia

k

dx

J — cc

X\'

N

A (W) =

1/ 27r

Introducing the probability integral P

r» X

e

—X

px

X\'

a;-

enbsp;dx

dx = —^

1/27,

K 2 TT

we have

N ( ■

A(m) = -jl -P
For
m =11 this becomes

MoM

(3)

k

A(ll) = f|l-P

Dividing (4) by (3) we obtain

Mn}n\\
[ k

(4)

mo—11

1 —P

k

rm,,—m

A (11) = A(m)

1 —P

k

or, if we put

Wo--111
[ k

1 — P

:nbsp;............(5)

Wo—M

1 —p

k

A (11) = A (w). R (w)

(6)

-ocr page 52-

Applying (6) to a definite part of one of our plates for which the limiting
magnitude is
m and dividing by the surface of this part, we get

D(ll) = D.R(w)..................

where D is the density of this area and D (11) denotes the density which
it would show if the limiting magnitude were 11?0.

Let moreover ^ be the limiting magnitude which the plate ought to have
everywhere in order to render the star-density at any point of that plate
equal to its average density Do. We then have •

D(11) = Do. R(/^)..................(8)

From (7) and (8) we find

R(m) = ^ x R(/^)................ (9)

where = is the relative density in the portion of the plate under consider-
ation.

Now (9) gives a relation between the relative density at a point

of the plate and the limiting magnitude at that point and we might use

this formula for calculating this latter quantity from the former if we had

a table for R {m) with m as argument and if we knew the value of i»- for everv
plate.nbsp;^

A table for R {m) with m as argument might be computed by means
of (5), provided we knew the values of the parameters
m^ and k. Charlier
gives the value of these two parameters in 9 areas of the zone Ci) enclosed
by the equator and the parallel circle Dec. = 30°, each area extending
30° in R.A. Since especially the parameter m, largely depends on the
star-density, it seemed appropriate to divide the nine areas of
Charlier
into three groups. In the first were placed the areas C«, C, and C, with small
star-density; in the second C,, C„ C3, C5 and C, with average star-density,
1) C. V. L. Charlier, I.e. p. 40.

-ocr page 53-

the third group containing C4 alone, in which the star-density is very great.
For each of these three groups the average values of Wo and
k have been
tabulated in table X.

TABLE X.

average values of fuo and k.

Group

Areas

nio

k

I

Ce^ C7, Cs

17.66

3.044

II

C] J C2J C3, C5, C9

18.32

3.018

III

C,

20.07

3.119

With these values of m^ and k the coefficient R (w) was calculated

for each of the three groups by formula (5) for w = 9.6,9.7, 9.8........12.6.

From these values those for every 0.01 magnitude were interpolated; they
are found for each group separately in tables XI
a, XI h and XI c.

{

-ocr page 54-

TABLE Xr^

table of r(m) por group i.

r(m)

K(m)

liim)

k{w)

K(m)

k(m)

\\i(m)

0.(50

.()2

.()4
.05
.0(5
.(57
.08
.00

;{.40
3.40
;5.4;i
;5.;}o

.\'$,;5;i

3.27
3.24
3.21

9.00
.01
.02
.03
.94
.05
.00
.07
.08
.09

2.(52
2.60
2.58
2.55
2.53
2.51
2.49
2.40
2.44
2.42

10.20
.21
.22
.23
.24
.25
.26
.27
.28
.29

2.00
1.98
1.90
1.94
1.93
l.OJ
1.80
1.87
1.80
1.84

10.50
.51
.52
.53
.54
.55
.50
.57
.58
.50

1.53
1.52
1.50
1.40
1.48
1.47
1.45
1.44
i.4:i
1.42

10.80
.81
.82
.83
.84
.85
.86
.87
.88
.89

1.18
1.1
1.10
1.15
1.14
1.13
1.12
1.11
1.11
1.10

11.10
.11

0.922
0.015
.120.907
.130.000
0.802
0.885
0.878
0.870
0.803
0.855

11.40
.41

0.724
0.719

11.70
.71
.72
,73

0.574
0.570
0.506
0.502

12.00

0.460

12.30
.31
.32
.33
.34
.35
.36
.37
.38
.39

0.371
0.308
0.306
0.363
0.361
0.358
0.356
0.353
0.351
0.348

.010.457
.020.453

•42 0.713
.43 0.708

.03
.04
.05
.06
.07

0.450
0.447
0.443
0.440
0.437

.14
.if
.10
.17
.18
.19

.44
.45
.40
.47
.48

0.702
0.097
0.092
0.080
0.081

.74 0.558

.75
.76
.77
.78
.79

0,553
0.549
0.545
0.541
0.537

•080.4.34
.090.430

.49 0.075

0.70
.71
.72
.73
.74
.75
.70
.77
.78
.79

3.17
3.15
3.12
3.09
3.00
3.03
3.00
2.97
2.95
2.92

10.00
.01
.02
.03
.04
.05
.06
.07
.08
.09

2.39
2.37
2.35
2.33
2.31
2.29
2.28
2.20
2.24
2.22

10.30
.31
.32
.33
.34
.35
.36
.37
.38
.39

1.82
1.81
1.79
1.77
1.76
1.74
1.73
1.71
1.70
1.08

10.60
.01
.02
.03
.04
.05
.00
.07
.68
.09

1.4C
1.39
1.38
1.37
1.36
1.35
1.33
1.32
1.31
1.30

10.90
.91
.92
.93
.94
.95
.96
.97
.08
.09

1.09
1.08
1.07
1.00
1.05
1.04
1.03
1.03
1.02
1,01

11.20 0.848

11.50
.51

,52

0.670
0.665
0.060
.055
0.050
.044
.639
0.634
580.629
59 0.624

11.80 0.5,33
.81 0.529
•820.525
,830.522
.840.518
.850.514
■860.510
■87 0.500,
•880.503
•890.499

12.10
.11
.12
.13
.14

0.427
0.424
0.421
0.418
0.415
150.412
160.410
17 0.407
180.408
190.401

12.400.346

.21
.22
.23
.24
.25
.26
.27

0.841
0.835
0.828
0.822
0.815
0.809
0.802

.41

.42
.43
.44
.45
.46
.47

0.344
0.341
0,329
0,336
0,334
0.332
0.329

.530
.54
.550
.500

.280.790

.480.327
.490.324

.29

0.789

9.80
.81
.82
.83
.84
.85
.80
.87
.88
.89

2.89
2.80
2.84
2.81
2.78
2.76
2.73
2.70
2.68
2.65

10,10 2.19

10.40 1.07

10.70
.71
.72
.73
.74
.75
.70
.77
.78

1.29
1.28
1.27
1.20
1.25
1.23
1.22
1.21
1.20

11.001.00
.010.992
.020.984

11.30

0.783

11,60
,61

0.619
0 614

11.90
91

0.495
0.491

92nbsp;0.488

93nbsp;0.484
0.481
0.477
0.474
0.470
0.467

12.20
.21
.22

0.398
0.395
0.393

12,50
.51

.52

0.322
0,320
0.318

.11
.12
.13
.14
.15
.16
.17
.18
.19

2.17
2.15
2.13
2.11
2.09
2.07
2.05
2.04
2.02

.41
.42
.43
.44
.45
.46
.47
.48
.49

1.05
1.04
1.03
1.61
1,60
1,58
1,57
1,56
1,54

.310.777
.320.771

.62 0.010
.630.605
.64 0.601
.05 0.596
.660.592
.67 0.587

.03
.04
.05
.06
.07

0.977
0.909
0.901
0.953
0.945

.33
.34
.35
.36
.37

0.765
0.759
0.753
0.748
0.742

.23 0.390
.24 0.387

.53 0.316

.54
.55
.56
.57
.58
.59

0,314
0.312
0.310
0.308
0.306
0.304

.25
.26
.27
.28
.29

0.384
0.382
0.379
0.376
0.374

.080.938
.090.930

.38 0.736

.68
.69

0.583
0.578

.79 1.19

.39

0.730

990.403

-ocr page 55-

TABLE XIV

TABLE OF R(m) FOR GROUP IL

R{m)

R(m)

R(m)

K(m)

R(gt;w)

K(,«)

R(m)

K(m)

R(w)

9.60
.61
.62
.63
.64
.65
.66
.67
.68
.69

10.50
.51
.52
.53
.54
.55
.56
.57
.58
.59

1.60
1.58
1.57
1.55
1.54
1.53
1.51
1.50
1.48
1.47

10.80
.81
.82
.83
.84
.85
.86
.87
.88

3.96j
3.92
3.88
3.84
3.81
3.76
3.73
3.69
3.65
3.62

10.20
.21
.22
.23
.24
.25,
.26
.27
.28
.29

1.20
1.19
1.18
1.17
1.16
1.15
1.14
1.13
1.12
1.11

11.10
.11
.12
.13
.14
.15
.16
.17
.18
.19

0.915
0.907
0.899
0.891
0.883
0.874
0.866
0.858
0.850
0.842

9.90
.91
.92
.93
.94
.95
.96
.97
.98
.99

2.14
2.12
2.10
2.08
2.06
2.04
2.02
2.00
1.98
1.96

2.90
2.87
2.85
2.82
2.79
2.76
2.73
2.70
2.67
2.65

11.40
.41
.42
.43
.44
.45

0.699
0.693
0.688
0.682
0.676
0.670
.665
0.659
480.653
490.648

11.70
.71
.72
.73
.74
.75
.76

0.541
0.537
0.532
0.527
0.524
0.519
0.515

12.00
.01
.02
.03
.04

0.422
0.419
0.415
0.412
0.409
.050.405
.060.4021
0.399
0.396
0.392

12.30
.31
.32
.33
.34
.35

0.331
0.329
0.326
0.324
0.321
0.319

.460
.47

.360.317
.37 0.314

.770.511
.780.507
.790.502

.07
.08
.09

.38
.39

0.312
0.309

9.70
.71
.72
.73
.74
.75
.76
.77
.78
.79

10.60
.61
.62
.63
.64
.65
.66
.67
.68
.69

10.90
.91
.92
.93
.94
.95
.96
.97
.98
.99

10.00
.01
.02
.03
.04
.05
.06
.07
.08
.09

1.45
1.44
1.43
1.41
1.40
1.39
1.37
1.36
1.35
1.33

1.09
1.09
1.08
1.07
1.06
1.05
1.04
1.03
1.02
1.01

11.20
.21

.22
.23
.24

3.57
3.54
3.50
3.47
3.43:
3.39
3.36
3.32
3.28
3.25

2.62
2.59
2.57
2.54
2.52
2.49
2.47
2.44
2.42
2.39

10.30
.31
.32
.33
.34
.35
.36
.37
.38
.39

1.94
1.92
1.91
1.89
1.87
1.85
1.84
1.82
1.80
1.78

0.834
0.827
0.820
0.813
0.806

11.50 0.642
.510.637
.520.631
.53 0.626

11.800.498
.810.494
.820.490
.830.486
.84 0.482
.850.478
.860.474
.87 0.470
.88 0.466
\'.890.462

12.10
.11
.12
.13
.14
.15
.16
.17

0.389
0.386
0.383
0.380
0.376
0.374
0.371
0.368
.365
0.362

12.40
.41
.42
.43
.44
.45
.46
.47 0
.48
.49

0.307
0.305
0.303
0.300
0.298
0.296
0.294
.292
0.289
0.287

.54
.55
.56
.57
.58

0.620
0.615
0.610
0.604
0.599

.250.799
.26 0.793

.27
.28
.29

3.786
0.779
0.772

.180
.19

.590.593

9.80
.81
.82
.83
.84
.85
.86
.87
.88
.89

1.32
1.31
1.29
1.28
1.27
1.26
1.25
1.24
1.23
1.22

11.00
.01
.02
.03
.040
.05
.06
.07
.080

3.21
3.18
3.15
3.12
3.09
3.06
3.03
3.00
2.97
2.93

2.37
2.35
2.32
2.30
2.28
2,25
2.23
2.21
2.19
2.16

10.40
.41
.42
.43
.44
.45
.46
.47
.4S
.49

10.70
.71
.72
.73
.74
.75
.76
.77
.78
.79

1.00
0.991
0.983
0.974
.966
0.957
0.949
0.940
.932
.923

11.30
31

10.10
.11
.12
.13
.14
.15
.16
.17
.18
.19

1.76
1.76
1.73
1.71
1.70
1.68
1.66
1.65
1.63
1.61

0.765
0.758
0.752
0.745
0.739
0.732
0.725
0.719
380.712
390.706

11.60
.61
.62
.63
.64
.65
.66
.67
.68
.69

0.588
0.583
0.579
0.574
0.569
0.564
0.560
0.555
0.550
0.546

11.90 0.458
.910.454
.920.451
.930.447
.94 0.444
.950.440
.960.436
.97 0.433
.980.429

12.200.359
.356
.353
.351
.348
.345
.342
.339
.337
.334

12.500.285

.210
.220,
.230
.240
.250
.260
.270
.280
.290

.51

0.283

.520.281

.53
.54
.55
.56
.57
.58
.59

0.278
0.276
0.274
0.272
0.270
0.267
0.265

.090

.99

0.426

-ocr page 56-

TABLE X19

TABLE OF R(m) FOR GROUP in.

R(m)

li(m)

R(W)

R{w)

R{m)

R(w)

K(w)

R{m)

K(m)

R(wj)

10.20
.21
.22
.23
.24
.25
.20
.27
.28
.29

9.90
.91
.92
.93
.94
.95
.90
.97
.98
.99

3.28
3.25
3.21
3.18
3.14
3.10
3.07
3.03
3.00
2.90

2.35
2.32
2.30
2.27
2.25
2.22
2.20
2.17
2.15
2.13

10.50
.51
.52
.53
.54
.55
.50
.57
.58
.59

4.03
4.58
4.53
4.47
4.42
4.37
4.32
4.27
4.22
4.17

1.70
1.08
1.00
1.04
1.63
1.61
1.59
1.57
1.56
1.54

10.80
.81
.82
.83
.84
.85
.80
.87
.88
.89

9.00
.01
.02

.04
.05
.00
.07
.08
.09

1.23
1.22
1.21
1.20
1.18
1.17
1.16
1.15
1.14
1.12

11.10
.11
.12
.13
.14

0.906
0.897
0.888
0.879
0.870

11.40
.41
.42
.43

0.671
0.665
0.658
0.652

11.70
.71
.72
.73
.74
.75
.76
.77
.78

0.501
0.496
0.492
0.487
0.483
0.478
0.473
0.469
0.464

12.00
.01
.02
.03
.04

0.377
0.374
0.370
0.36\'
0.303
050.300
000.357
07 0.353
080.350
090.346

12.30
.31
.32
.33
.34

0.285
0.283
0.280
0.278
0.276
.350.273
.300.271
.37 0.269
.380.267
.390.264

.440.645

.150.861

.45
.46
.47
.48
.49

0.639
0.633
0.626
0.620
0.613

.16
.17
.18
.19

0.853
0.844
0.835
0.826

.790.460

4.12
4.07
4.03
3.98
3.94
3.89
3.84
3.80
3.75
3.71

10.00
.01
.02
.03
.04
.05
.00
.07
.08
.09

2.93
2.90
2.87
2.83
2.80
2.77
2.74
2.71
2.08
2.05

10.30
.31
.32
.33
.34
..35
.36
.37
.38
.39

2.10
2.08
2.06
2.03
2.01
1.99
1.97
1.95
1.93
1.91

9.70
.71
.72
.73
.74
.75
.70
.77
.78
.79

10.60
.01
.62
.63
.64
.65
.00
.07
.08
.09

1.52
1.51
1.49
1.47
1.46
1.45
1.43
1.41
1.4G
1.38

11.20
.21

10.90
.91
.92
.93
.94
.95
.96
.97
.98
.99

1.11
1.10
1.09
1.08
1.07
1.06
1.05
1.04
LOS
1.02

0.817
0.809
.220.801

11.50
.51
.52
.53
.54
.55
.56

0.607
0.001
0.596
0.590
0.584
0.578
0.573

11.800.455
.810.451

12.10
.11
.12
.13
.14
.15
.16

0.343
0.340
0.337
0.334
0.331
0.328
0.325
17 0.322
180.319
190.314

12.40
.41
.42
.43
.44
.45
.46
.47
.48
.49

0.262
0.200
0.257
0.255
0.253
0.250
0.248
0.246
0.244
0.241

.8;
.8;
.84
.85
.86
.87
.88
.89

0.447
0.442
0.438
0.434
0.430
0.426
0.421
0.417

.23

0.793

.24 0.785

.25
.20
.27
.28
.29

0.777
0.770
0.762
0.754
0.746

.57 0.567

.58
.59

0.561
0.556

3.00
3.03
3.59
3.55
3.51
3.47
3.43
3.40
3.36
3.32

9.80
.81
.82
.83
.84
.85
.80
:87
.88
.89

10.10
.11
.12
.13
.14
.15
.16
.17
.18
.19

2.02
2.59
2.56
2.54
2.51
2.48
2.40
2.43
2.40
2.38

10.40
.41

.42
.43
.44
.45
.40
.47
.48
.49

1.88
1.80
1.84
1.83
1.81
1.79
1.77
1.75
1.73
1.71

10.70
.71
.72
.73
.74
.75
.70
.77
.78
.79

11.001.00

11.30

1.37
1.35
1.34
1.33
1.31
1.30
1.29
1.27
1.26
1.25

0.7.38

11.00
.01
.62
.63
.64
.65
.66
.07
.68

0.550
0.545
0.540
0.5.35
0.530
0.525
0.521
0.516
0.511
690.506

11.90
.91
.92
.93
.94
.95
.96

0.413
0.409
0.406
0.402
0.399
0.395
0.391

12.200.313
.210.310

12.500.239
.51
.52 (
.53 (
.54 (
nn.1

.01
.02

.03

0.991
0.981
0.972

.310.731

0.237
0.235
0.233
0.231
0.229
0.227
0.225
580.223
.590.221

.32
.33
.34
.35
.36
.37
.38
.39

0.725
0.718
0.711
0.704
0.698
0.691
0.084
0.678

.22
.23
.24
.25
.26
.27
.28

0.307
0.305
0.302
0.299
0-296
0.293
0.291

.040.962
.050.953
.060.944
.07 0.934
.080.925
.090.915

.tgt;t)
.56
.57

.970.388
.980.384

.99

0.381

.290.288

-ocr page 57-

As a matter of course the value of ^ will not differ much from the
average limiting magnitude of the plates of the Harvard Map,
i.e. from
a Httle over 11?0. Therefore by means of formula (9) the values of
m
were calculated from the densities of table IX for ^ = 10.5, =11.0,
M = 11.5 and ^ = 12.0. In these calculations the values for R (w) for plates
of the first second and third group were taken from table XI
a, XI b and XI c
respectively. The obtained results, which are considered to hold for the
distance of the middle of each zone from the centre, are found in table XH.

TABLE XH.

values for the limiting magnitude m for different values of

Group

Value of m in

Zone

1

2

1

i 3

4

5

6

7

I

10.5

10.79

10.80

1

10.65

10.45

10.38

10.33

10.29

11.0

11.30

11.32

11.15

10.95

10.88

10.82

10.78

11.5

11.82

11.84

11.66

11.44

11.37

11.31

11.26

12.0

12.34

12.36

12.17

11.94

11.86

11.80

11.75

H

10.5

10.83

10.77

10.65

10.48

10.39

10.24

10.17

11.0

11.34

11.28

11.16

10.98

10.88

10.73

10.66

11.5

11.87

11.80

11.66

11.48

11.37

11.22

11.13

12.0

12.38

12.31

12.17

11.98

11.87

11.70

11.61

HI

10.5

10.91

10.86

10.71

10.49

10.34

10.10

10.03

11.0

11.44

11.38

11.22

10.99

10.84

10.59

10.52

11.5

11.96

11.90

11.73

11.49

11.33

11.07

10.99

12.0

12.48

12.42

12.24

11.98

11.82

11.55

11.47

The numbers of the third column of table XH were now successively
subtracted from the numbers of the next six columns. These differences
were averaged for the same group and for the same distance from the centre,
the averaging being justified by the fact that the differences in the value of
n do not play a great part. The results were used for plotting the curves of
fig. 3, which illustrate for each of the three groups of plates the law of decrease

-ocr page 58-

of the hmiting magnitude from the centre to the margin. The abscissae represent

the distance from the centre in milhmetres, the ordinates the difference between

tlie limiting magnitude and that at the centre, expressed in hundredths of
a magnitude.

Fig. 3.

20 W 60 80 /OO W /iO

-10
-20
-30
-iO
-50
-fiO

:::

-

-

-

-

-

-

-

\\

-

-

-

-

-

-

\\

\\

--

-

-

-

-

-

-

-

-

-

-

_

-

-

-

-

-

-

-

, 1

-

\\

\\

-

\\

\\

\\

\\

gt;

-

ii

-

-

\\

\\

\\

\\

\\

\\

\\

\\

\\

\\

1

iii

O
-10
-20
-30
-iO
-50
-60
-70
-80
-SO
-100

Curve I shows a maximum at a distance of 16 mms. from the centre
In order to test its\'reality the limiting magnitude for Group I was also cal-
culated at25 mms. from the centre by means of a different division into zones.
The point so obtained is marked in the figure and is seen to lie very near the
curve.

Most striking is the great difference between the curves I and III
It is a well-known fact that star-counting gives rise to systematic errors
depending on the star-density, as this density, in visual work, affects the
estimates of magnitudes to a considerable amount. In counting on a photo-
graph, however, the density cannot possibly affect the estimation of magnitude.
How then shall we explain the large difference between both curves? Two
causes may have been active. In the first place the supposition on which
the law of decreasing density — and consequently also that of decreasing
hmiting magnitude — is based, may perhaps not hold for the plates of the
third group. Secondly it seems possible that the colour of the stars plays
an important part. I shall briefly explain my view on these two points.

-ocr page 59-

As to the first, it was pointed out on page 32 that in deriving the density
law the star-density was supposed not to show any systematic decrease
towards the margin if the limiting magnitude was the same all over the plate.
Now this assumption will certainly not be correct for plates of the dimensions
of the Harvard Map; on these two fields may be situated at the same
distance from the centre and yet differ as much as 30° in galactic latitude.
On account of the galactic condensation we cannot expect the star-density
to be the same in two such fields and so the galactic condensation may influence
the decrease of the star-density and hence also of the limiting magnitude
towards the margin. On closer consideration it will be clear that this influence,
if present, must be greatest on such plates as have their centre exactly on the
galactic circle. This is very nearly the case with plates 2, 3, 13, 20, 37, 42, 49
and 50. Now the fact that only three of these plates belong to group HI is
in itself sufficient to prove that the influence of galactic condensation cannot
be the cause of the great difference between the curves I and IH.

As to the second point — the influence of the colour of the stars —
the following may be remarked. That on a plate the limiting magnitude near
the margin is smaller than at the centre is caused by the fact that the images
near the margin are of a larger size than those of equally bright stars at the
centre. This larger size depends on the shape of the focal surface and this
again is closely related to the refractive index of the star-light. Now since
it appears from table V that the percentage of blue stars increases as we
approach the Milky Way, and since the plates of group III, except plate 25,
lie on the whole much nearer to the Milky Way than those of group I, it is not
impossible that it is the colour of the stars which causes the great difference
between the curves I and III. I regret that the numerical data at my disposal
do not enable me to investigate this interesting point further. No more can
I explain the maximum in curve I.

As soon as the law, governing the decrease in limiting magnitude
towards the margins has been derived for each group of plates, we may reduce
the star-density in each field to the limiting magnitude ll?0. This reduction
is carried out in the following manner.

-ocr page 60-

Each hmiting magnitude determmed by Henie for a certain distance

from the centre gives, by means of one of the curves of fig. 3 the cent a

hmiting magnitude of the plate under consideration If several 1

found for the limiting magnitude at the centre, their mean is taken

way the limiting magnitudes for the centres of 39 plates have been calclted
and collected in table XIILnbsp;calculated

TABLE XIIL

i

2

3

4

5

6

7

8

9

94

96

98

100

104

110

114

120

126

10

84

86

88

92

96

100.

106

112

120

9

74

76

78

82

86

92

98

106

112

8

64

66

70

74

78

84

92

98

106

7

54

56

60

64

70

78

84

92

100

6

44

48

52

56

64

70

78

86 ,

96

5

36

38

44

50

56

64

. 74

82

92

4

26

28

36

44

52

60

70

78

88

3

16

22

28

38

48

56

66

76

86

2

8

16

26

36

44

54

64

74

84

1

-ocr page 61-

For each group we may now find from one of the tables XI a, XI h
or XI
c the coefficients R (w) by which the star-numbers in the several fields
must be multiplied in order to obtain the numbers down to the eleventh
magnitude. In this way the 39 plates, enumerated in table XIII, were reduced.
The results of this reduction will be given in the next chapter.

It would take too much space and would moreover be of secondary
importance to mention all the details in which my method of reduction
deviates from that used by
Henie. May it suffice to remark that Henie,
although he had found that the density-law and the parameters of Charlier\'s
formula are different for the three groups mentioned, still with respect to
the decrease of the limiting magnitude towards the margin treats all plates
as equivalent. To me this seems inadmissible and I believe that the curves
of
fig. 3 sustain this view.

-ocr page 62-

CHAPTER V.

determination of the limiting magnitude by means of overlapping

fields. the reduced plates.

In table XIII the limiting magnitude at the centre has been quot;
for
30 plates. For the remaining 16 plates this quantity could not be calc^Id
m the usual way since for these plates Henie gives no data. He savs on
this account): quot;The reason why no determination has been made
on Lse
^ pla es
is that no Haoen chart exists for variables more southern than about
-30 and as for the few northern the magnitudes in some cases are not dven
quot;m the Harvard Scalequot;. On page
52 he says further: quot;For these plates I have
quot;made use of those parts of the negatives that are covered by the more north^^
quot;ones with known limiting magnitude and I have adopted those values of the
\'\'hmitmg magnitude, that agree best. My mode of proceeding has been that
quot;I have supposed one value of the magnitude of the plate in question and
quot;calculated the reduced densities on this supposition. The part in common
quot;between the plate in question and the more northern one at once shows if
quot;the supposed value \'quot;s correct. If this had not been the case, the value has
quot;been modified until the best agreement has been obtained.quot;

The present author used a more direct method that will be best
elucidated by the following example. Plate 27 is partially covered by plates
14, 15, 16, 26, 28, 38, 39 and 40. Of plate 40 the limiting magnitude is un
known. With 15, 26 and 39 plate 27 has no counted fields in
common- with
28ji^and with 14,16 and 38 one each. For these coinciding fields the equatorial
1) H. Henie, I.e. p. 46.

-ocr page 63-

co-ordinates, tlie numbers {n) of stars counted on each plate and the distances
(r) from the centre of plate 27 are tabulated in table XIV*.

TABLE XIV*.

number of stars per square degree in overlapping fields.

n

Dec. R.A.

( 13.5 on 27

jjhQTn

11 0
11 0
11 0

—14 11 0

9 0
11 0
9 0

I 6.6

a

28

j 13.2

}f

27

j 9.6

28

10.9

}gt;

27

6.8

if

28

13.1

27

7.3

ti

28

j 15.6

gt;i

27

j 8.4

28

20.0

27

12.0

14

j 15.5

ty

27

j 16.9

ty

16

18.4

yy

27

18.8

yy

38

— 1°

—nbsp;4

—nbsp;7
—11

84 mms.
88
96
106
120
120
126
126

14
16
—16

The limiting magnitude in the various fields of plates 28, 14, 16
and 38 is known. By means of the formula

(10)

which can be derived directly from (6) and in which the index 1 refers to plate
27, Ml can now be calculated in the eight different fields of this plate. Using
the curves of fig. 3 we get eight independent determinations of the limiting
magnitude at the centre. These values are

-ocr page 64-

12?60
11.96
12.17
12.32
12.36
11.69
12. 07
11.59

12?17

weighted mean

The braclceted figures indicate the weights to be assigned to each of

these values on account of the accuracy with which on the overlapping

plates the central limiting magnitude has been determined. This accuracy

depends on the number of fields for which the limiting magnitude has been

determined and on the character of the sequences of the comparison stars

used. In order not to render the matter too complicated, fields for which

Henie gives a probable error 0^1 received a weight 2 and those for which his

probable error amounts to 0?2 or 0?3 a weight 1. To fields for which the limiting

magnitude was determined by overlapping, a weight i was assigned each

further overlapping halving the weight again. In this way we find e.g. for the

weight of plate 40 the value 2f. In two fields of this plate the limiting

magnitude is determined by the overlapping plate 41, in two by plate 39

in one by plate 27 and in one by plate 29. Of these plate 27 had its limiting

magnitude determined by overlapping, 41, 39 and 29 had not. Accordingly

the weight of plate 40 is 2 x i 2 x i- l xi l x i =2f. Table XV contains the
weights of all
55 plates.

Plate
28
28
28
28
28
14
16
38

(4)
(4)
(4)
(4)
(4)
(2)
(2)
(1)

TABLE XV.
weights of the plates.

o
cO

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1

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29

2

36

5

43

3

50

2

8

9

11

16

2

23

2

30

5

37

1

44

2 ■

51

4

3

2

10

6

17

1

24

2

31

4

38

1

45

5

52

8

u

§

4

5

11

3

18

3

25

5

32

4

39

3

46

2i

53

5

2

12

H

19

3

26

3

33

5

40

2i

47

i

54

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2

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20

12

27

4

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8

41

5

48

3

55

I-

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49

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-ocr page 65-

Finally table XVI contains the values of the central limiting magnitudes
of all 16 plates that are lacking in table XIII.

TABLE XVI.

Central limiting magnitude on 16 plates determined by
means of overlapping fields.

Plate

Lim.
Mag.

Plate

Lim.

Mag.

Plate

Lim.
Mag.

Plate

Lim.
Mag.

12

12.89

40

i 10.54

48

11.48

52

11.69

22

11.33

43

11.59

49

11.67

53

10.72

27

12.17

46

11.27

50

12.49

54

11.10

35

10.97

47

10.61

51

11.26

55

11.30

These plates can now also be reduced to the eleventh magnitude by
the method discussed at the end of the preceding chapter. The results of
this reduction for the whole set of plates are given in the following pages.

PL.\\TE 1.

-9 -f

3 -7

-6 -5

-4 -3

-2 -1

1

2 3

4 5

6 7

8 9

21.7

14.3

44.1

23.7

10

15.1

17.8

23.3

32.2

32.2

9

15.5

19.6

26.2

41.3

8

10.9

20.7

20.3

36.2

50.6

7

29.8

15.1

25.5

42.5

26.5

6

1

10.0

20.3

22.0

43.5

31.2

5

1 21.2

22.4

26.1

30.3

26.8

4

14.0

15.0

24.0

38.8

23.2

3

13.7

21.4

12.2

19.7

45.6

2

15.9

14.1

14.3

26.0

28.6

1

22.1

10.0

11.4

12.2

17.7

-1

12.0

8.9

8.4

17.5

28.0

-2

10.8

10.9

15.4

15.5

-3

8.3

13.3

11.6

15.3

20.3

-4

27.4

10.3

10.2

15.5

16.4

-5

7.3

9.0

9.8

15.3

14.4

-6

11.9

11.7

13.0

17.9

11.5

-7

12.8

12.8

14.3

13.1

8.8

-8

11.9

13.1

15.1

7.3

-9

13.7

10.2

10.2

8.3

6.4

-10

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PLATE 4.

PLATE 5.

-9 -i

-7 -6

-5 -4

-3 -2

-1

1 2

3 4

5 6

7 8

9

-9

; -7 -6

-5 -4

-3 -2

-1

1 2

3 4

5 6

. 7 8

9

I

11.1

7.7

24.9

24.6

8.8

10

.

10

7.1

16.5

12.6

28.9

25.7

9

6.6

10.8

9.3

8.1

24.2

9 i

6.4

12.0

16.9

11.3

13.5

8

16.8

8.9

13.2

8.4

16.2

8

5.1

10.1

13.9

18.0

23.4

7

15.3

13.7 ,

7.1

8.4

19.5

7

13.0

8.1

15.8

18.0

11.8

6

17.2

6.9

12.0

9.2

8.8

6

2.6

17.2

18.9

14.4

10.5

5

8.0

9.0

6.2

7,6

23.6

5

8.8

17.4

13.3

26.1

19.5

4

6.1

14.7

13.1

6.2

10.9

4\'

1

8.9

15.1

24.4

16.7

18.6

3

6.9

7.0

9.4

5.4

17.4

3

1 9.8

17.0

13.1

35.8

11.0

2

9.5

17.4

14.4

8.2

30.0

2

1

11.5

14.0

10.8

19.7

17.4

1

r

7.1

11.3

9.0

7.9

21.0

1

n.8

16.8

16.1

18.5

21.5

-1

i .

i 10.6

10.0

6.3

12.4

15.1

-1

9.2

15.0

19.5

19.5

15.6

-2

i

7.2

8.2

9.6

5.6

13.5

-2

14.4

19.8

11.3

18.8

20.7

-3

1 10.8

1

5.0

7.7

8.4

9.5

-3

18.2

15.2

19.3

29.6

18.2

-4

!

t

6.6

4.5

9.6

5.7

15.7

-4

2L6

24.6

15.4

19.8

18.0

-5

6.3

7.6

3.3

5.9

8.0

-5

17.4

15.7

18.6

25.1

7.6

-6

8.8

4.3

5.6

7.5

25.4

-6

18.6 .

1

12.1

• 15.6

25.7

15.3

-7

7.7

4.5

4.6

7.5

3.6

1

12.7

16.7

12.4

37.6

7.2

-8

11.2

1.7

6.6

8.9

24.1

12.4

14.1

7.6

21.3

10.1

-9

12.5

6.3

5.9

10.8

8.4

-9i

i

|l-

19.5

16.4

14.7

30.1

16.3

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7.4

6.3

4.9

8.1

10.6

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