The Proper Motions
of 1418 Stars in and near the

clusters h and %
Persei

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THE PROPER MOTIONS
OF 1418 STARS IN AND NEAR THE
CLUSTERS h AND X
PERSEI

UNIVERSITEITSBIBLIOTHEEK UTRECHT

3168 164 0

The Proper Motions
of 1418 Stars in and near the

clusters h and %
Persei

Proefschrift

ter verkrijging van den graad van Doctor in de Wis- en Sterrekunde aan de Rijks-
Universiteit te Utrecht, op gezag van den Rector-Magnificus, Mr. D. SlMONS,
Hoogleeraar in de faculteit der Rechtsgeleerdheid, volgens
besluit van den Senaat der Universiteit tegen de
bedenkingen van de Faculteit der Wis- en
Natuurkunde te verdedigen op Vrijdag
2 Juni 1911, des namiddags te 4 uur,
door

ADRIAAN VAN MAANEN
geboren te sneek

\'ï.\'. »...

te-

AAN

MIJNH OUDHRS

■ Â \' .

-.

...
^ - • •

Vit-
ä

Wanneer ik, op het punt Utrecht te verlaten, mijn blik laat gaan langs
de schare van hoogleeraren in de Faculteit der Wis- en Natuurkunde, dan zie
ik met vreugde daaronder enkelen, die zich door hun onderwijs of door de mij
bij meerdere gelegenheden betoonde belangstelling, eene blijvende plaats in mijne
herinnering hebben verzekerd. Ik wil hier niet in bijzonderheden treden en slechts
eene uitzondering maken voor U, Hooggeleerde Nijland, mijn Pro7notor.

Menig jaar heb ik onder uwe leiding gewerkt en veel van U geleerd. Uwe
mij zoo veelvuldig betoonde hulpvaardigheid stel ik op hoogen prijs; vooral in
den laatsten tijd kwam ik U bijna dagelijks met mijne vragen en tallooze bezwaren
lastig vallen; steeds vond ik U bereid mij te helpen. Dat gij dit proefschrift hebt
waardig gekeurd om weldra te verschijnen als een deel der annalen van Uw

Observatorium, is mij hoogst aangenaam.

Dal de herinnering aan den op „Zonnenburgquot; door gebrachten tijd tot
de aangenaamste zullen behooren, die ik uit Utrecht medeneem, is voor een groot
deel ook aan U te danken, waarde van der Bilt; moge de goede verstandhouding,
die reeds zoo menig jaar tusschen ons heeft bestaan, van blijvenden aard zijn.

De hulp, mij door U, loaarde Blokhuis, verleend, heb ik dankbaar
aanvaard; de vertaling kon moeilijk in heter handen zijn terecht gekomen; de
belangeloosheid, waarmede Gij die taak op U hebt genomen, maakt het mij
moeilijk mijn dank daarvoor op deze plaats in woorden te brengen.

Hooggeleerde J. C. Kapti-yn. Uw laboratorium te Groningen en Uw
huis hebt gij voor mij opengesteld; niet alleen heb ik van U veel geleerd, neen,
gij hebt mij de wetenschap doen lief krijgen; de hartelijkheid, die Gij mij steeds
hebt betoond, vat ik op als het bewijs van eene vriendschap, welker weldadigen
invloed ik steeds meer heb ondervonden; moge ik die nimmer onwaardig blijken
le zijn.

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CONTENTS

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INTRODUCTION.

When on suggestion of Professor J. C. Kapteyn of Groningen, I under-
took in June 1008 the measurement and reduction of some photographic
plates of the star-clusters h and / Persei, in order to determine the proper
motions of a number of stars in this region, I had a twofold object in view.

Firstly I wanted to try and find the P. M. of the two clusters them-
selves.

Secondly I wished to determine the frequency of the P. M. according
to its amount and to the magnitudes of the stars for this region of the sky,
tlie stars belonging to h and / Persei being of course left out of account.

It would have been impossible to accomplish this task without the help
of several astronomers who all assisted me in the kindest way. I wish to
express my sincere thanks to:

Professor Anders Donner of Helsingfors and Dr. S. Kostinsky of
Pulkovo for the great number of plates, taken with the greatest possible
interval, whicli they put at my disposal;

Professor E. F. van de Sande Bakhuyzen of Leyden and Professor
kiistner of Bonn who were so kind to furnish me with the most recent
meridian observations of some twelve stars, to be used as fundamental stars;

Dr. J. H. ZwiERS and Mr. J. Voute of Leyden and Dr. C. Monnicii-
meyer of Bonn who took much pains in making these observations;

Professor A. Auwers of Berlin who had the kindness to procure Prof.
Kapteyn all the earher observations of these stars, reduced to the equinox
1875,0;

Professor J. C. Kapteyn of Groningen for the interest he has continually
taken in my work, for the advice he gave me on many occasions and particu-
larly for the many marks of friendship I have received from him during these
last three years;

Professor A. A. Nijland of this observatory who not only during the
preparation of this paper assisted me with his valuable advice, but who
moreover determined the brightness of some faint stars which I needed
so much;

Mr. K. Blokiiuis of the Gasworks at Haarlem for translating my paper
into English and for translating a Russian paper, bearing on my subject.

CHAPTER I.

Object of the Investigation.

It has already been remarked in the introduction that the object of
tlie present paper is a twofold one.

In the first place I wished to try and settle whicli stars do and which
do not belong to tlie clusters h and y Persei. It is obvious tliat this may be
decidcd by accurate determinations of the proper motions. Tliis has been
done for two otiier star-clusters, namely the Pleiades and tlie Hyades. In tlie
former case the P. M. afford a very good method, as Newcomb says, ,,of
distinguishing between a star which belongs to the cluster and one which
probably lies beyond it. The amount of tlie proper motion is about 7\' per
century. The first accurate measures made on the relative i)ositions of the
stars of the cluster were tliose of Bessel, about 1830. In recent years several
observers have made yet more accurate determinations. The most thorough
recent discussion is by Elkin \'). One result of his work is that there is as yet
no certain evidence of any relative motion among the stars of the group.
\'Ihey all move on together with their common motion of 7\' per century,
as if they were a single mass.quot;

For the Hyades the P. M. have been determined i)hotographically-).
The plates were taken by Donneu, measured and discussed by Kapteyn
and de Sitter. The adopted final motion of the grouj) is

fjL, = o\'.0!)0
jjij = _ o\'.osf).

Transactions of the As(r. Olxs. of Yale University 1.
Publ. of the Astr. Lab. at (Ironingen, 14.

The question which stars were members of the groups h and y Persei
became doubly important after the publication of: Parallaxes of the clusters
h and 7 Persei, measured and discussed by Prof. J. C. Kapteyn and W. de
Sitter Sc. D. Although a definite conclusion as to a parallax of the two
clusters was not arrived at, yet de Sitter remarked on page 34:

,,In conclusion attention may be drawn to the fact that the foregoing
discussion affords the material from which the parallaxes of the clusters h
and 7 Persei relative to the surrounding stars may be derived with extreme
accuracy, as soon as we shall be able by a discussion of the proper motions to
decide with certainty for each of the stars of the plates, whether it belongs to
either of these clusters or not.quot;

Hence if we could arrive at positive results about the P.M., this would
imply the answer to the important question what position both clusters occupy
in space.

In the second place it is possible for this part of the sky to determine
the frequency of the P. M. according to magnitude and to the amount of P. M.
Material of this kind is unfortunately somewhat scarce as yet. To be sure
Kapteyn has given tables for this 2) and we also possess the data of W. G.
Thackeray but in either case only stars of greater brightness than mag.
9.5 are dealt with. For fainter stars the distribution has until now only been
determined by Turner in „Three notes on the number of faint stars with
large proper motionsquot; Turner goes as far as mag. 12.0, but by the rather
rough methods adopted he only gives proper motions exceeding 0\'.20 and ()\'.15
annually. I hoped that the method followed by me would allow me to advance
further in both respects. My plates contained stars down to mag. 14.0 and
the accuracy reached allowed me to go as far as O\'.OlO annually. Already
a priori I could expect a fairly high accuracy, so that it might be anticipated
that the work would not be undertaken in vain. For, the best of the already

1)nbsp;Publ. of the Astr, Lab. at Groningen 10.

2)nbsp;Publ. of the Astr. Lab. at Groningen 11.

3)nbsp;Monthly Notice.s of the R.A.S. 67, 145.

Monthly Notices of the R.A.S. 69, 57, 69, 491, 71, 45.

available maesuremeiits of the plates for the Carte du Ciel gave ± 0\',237 as
mean error of one coordinate.

If in order to obtain proper motions new plates are taken after some
time, the change of one of the coordinates will be determined with a mean
error of ± 0\'.237 V2 = ± 0\'.334.

This gives for plates taken with an interval of 12 years, in the P. M.
both in a and o, a mean error of ± 0\'.028.

If we have, as in our case, one pair of plates with an interval of 17
years and 2 pairs with a 12 years\' interval, we miglit expect, even starting
from such absolute measurements, in our final results a mean error of ± 0 \'.0I4.

That with the differential method, followed by me, this accuracy has
certainly been suipassed, is fully proved by the results. I shall extensively
discuss this point in Chapter XL

The labour involved in an accurate determination of the P. M. is of
course very considerable. Hence it is impossible to determine the frequency
for so many parts of the sky as could be done by Turner\'s method. And the
distribution extending over a small part of the sky has in itself little value.
However, similar data of equal accuracy may within a reasonable time bo
expected in addition to my plates and then the obtained results will be lielpful
in the solution of the problems, arising in the investigation of the structure
of the stellar universe.

CHAPTER II.

The Material.

In tlie Astronomical Laboratory at Groningen, wliere I did the measure-
ments and ])art of tiic reductions, in the si)ring of 11)09 seven pliotogra])hic
l)lates of h and y Persei were available, taken by Kostinsky at Pulkovo
partly in 1890, i)artly in 1908. After Kostinsky had used them for a paper

on the use of the stereocomparator for the determination of proper motions i)
he had ceded them to Prof. Kapteyn, authorising him to measure them and

to pubUsh the results.

The existence was also known of two plates of the same region of the
sky, taken in 1890 and 1892 by Donner at Helsingfors and measured by Miss
Bronsky and Miss Stebnitzky^). At my request Prof. Donner was so kind not
only to lend me these plates, but he moreover promised to take an additional
number of photographs of h and x Persei in 1909. So I finally possessed the
following plates:

Epoch

Sept. 22
Oct. 1
Oct. 9
Oct. 22
Oct. 24
Oct. 31
Dec. 28
Sept. 5
Sept. 20
Apr. 17
Apr. 17
Aug. 18

1909, Sept. 10
1909, Sept. 13
1909, Sept. 14
1909, Sept. 15

Time

of
expos-
ure

1) Bull, de l\'Acad. de St. l\'étersbourg, 1908, Vie Série, N«. 17.

Mem. de l\'Acad. Imper, des Se. de St. Pétershourg VIIIc serie 2, Xo. 7.

Nos. 7 and 10 had been taken at lower culmination and hence had to
be discarded. Unfortunately all the remaining material could not be worked
up. I therefore selected six plates, all containing good images in approxima-
tely equal numbers and having h Persei at the centre. These were the following:

by Kostinsky: A 103 and A 106 of 1896
B 132 and B 136 of 1908.

by Donner: No. 4 of 1892.

No. 9 of 1909.

I originally intended also to investigate a couple of plates with an
hour\'s exposure, but for this the necessary time lacked. Such plates might
be a valuable contribution towards answering the question whether the stars
of a cluster range through a scale of brightness as wide as the stars in general.
If the clusters h and x Persei had a large proper motion, which will presently
appear not to be the case, it would certainly repay the trouble, with a view
to this (juestion, to scrutinise the plates with a long time of exposure in
regard to P. M. But even in our case some results might be obtained by count-
ings, as was done for the Pleiades by Bailey, for the cluster in Coma Berenices
and for Praesepe by Newcomb^). Possibly 1 shall later have an opportunity
to do this.

After the measurements and reductions had been (piite Ihiished, I
discovered that still two older photographs of the clusters h and / existed.
The lirst of these was taken by the brothers Henry in 1884. A reproduction
of it is found in Sirius, 18, 256. When comparing this reproduction, which is
stated to be a direct rejjroduction of the photograph without any retouching-),
with copies of Donner\'s ])lates of 1892 and 1909, I found such a large num-
ber of deviations, however, from these mutually well according i)lates, tliat I
must doubt tlie correctness either of Henry\'s plate, which is one of their
oldest i)hotographs or of the reproduction.

It is also doubtful whether I should have been able to obtain a second

The St;ir.s, .study of the Univer.sc, i\'.IOt, p. 258 soq.
Cf. Sirius, 18, 210.

plate, taken with the same apparatus in recent years. Yet this would have been
desirable in order to obtain great accuracy in the determination of the P. M.

The second one of the plates mentioned, is by Roberts and was taken
in 1890.1). The time of exposure was three hours and by this circumstance
alone it differed too much from the plates measured by me, so that it could
not have been of great service in the determination of the P. M.

CHAPTER III.

Explanation of the method of measuring.

Until now for the photographic determination of P. M., plates have
mostly been used containing the images of both epochs. I have to explain why
this method was abandoned.

The method applied by Kapteyn and de Sitter in the publications
of the Astronomical Laboratory at Groningen has in comparison with my
method four advantages and one disadvantage.

The first advantage is that the observations are the direct difference
in distance of the two positions of the same star at different times, whereas
in my method each star has to be made to coincide with a scale-division of
the measuring apparatus. The consequence of this is that in the numbers,
yielding by reduction the P. M., to the unavoidable error of pointing on the
star, the error of setting on the scale-divisions is added twice. But this error
is so small that the chief point is the saving of time since tlie number,
yielding in Kapteyn\'s method the P. M. directly by reduction, is obtained
from no less than four numbers with my method.

More important is the second advantage, viz. the small influence of a
distortion of the sensitive layer in the method followed by Kapteyn and de
Sitter, if only the images of different epochs are in close proximity. The

1) Phot, of Stars, Star Clusters and Nebulae, 1893.

result, however, of many measurements by both, has completely confirmed
the opinion already expressed in Publ. of the Astr. Lab. at Groningen 1, that
a distortion of the film does not exist. And this holds not only for the differ-
ential distortion, but generally. For those who fully accept this result the
second advantage consequently vanishes.

Thirdly, if the plate is not accurately plane, this fault will affect my
measures in an uncontrollable way.

As to a fourth error, viz. a possible minute instability of the plate as
fixed in the maesuring apparatus, I shall treat of this in Chapter IV.

On the other hand the method followed until now has a great drawback.
It seems to be difficult to keep the undeveloped plates for more than 6 or
8 years i) and this would limit the interval for the determination of P. M. to
this maximum. A developed plate on the contrary may be kept for an indefinite
time and be compared with a later photograph, at any rate if no distortion
takes place. So in the present case the difference of epoch is with Kostinsky\'s
plates 12 years, with Donner\'s 17. And this is an essential advantage for the
determination of P. M., since the accuracy of the P. M. is proportional to
this difference.

The systematic errors, depending on the position on the plate, which
will be extensively dealt with in Chapter VIII seem to justify the opinion, that
also in the plates used by me there is no cjuestion of a distortion of the film,
at any rate that it is too small to have an apjireciable influence on the P. M.
itself, as long as stars are concerned at a distance of no more than 40\' from the
centre. For stars nearer the edge we are not justified rigorously to maintain this
statement. I am of opinion, that in deriving proper motions photogra])hically
it will in general be safe not to consider stars, lying near the edge of the i)lates.
If we exclude the marginal stars, the advantage of the method here followed
very probably outweighs the advantages of the older method.

Cf. Publ. of the Astr. L:il). at Groningen, 19, X.

CHAPTER IV.

The apparatus and measurements.

The measuring apparatus of the Groningen laboratory which I used
needs no special description, since it is identical with the instrument of which
H. G. van de Sande Bakhuyzen gave a detailed account in the Bulletin
de la Carte du Ciel, 1, 169—173, with the only difference that it was mounted

at an inclination of 45°.

Concerning the periodic and progressive errors of the screw I refer to Publ.

of the Astr. Lab. at Groningen 1. These errors appeared not to be so large that

they could not be explained by errors of observation and even if they had been

real, their influence on the.final results is so much reduced that they can

liave no sensible influence on the P. M. Since for the measurements in a and o

each star was always made to correspond to the same scale-division on all

six plates, also systematic or accidental errors of the scale could not have

any influence on the final results. Besides, the progressive errors of the screw

are entirely eliminated by this process, since for any particular star always

the same part of the screw was used and consequently these errors could only

affect the difference of the readings of the same star on two plates amounting

in maxima to 0\'.852.

The influence of a possible dead run of the screw was avoided by

always turning it in the same direction.

The instrument was as far as possible covered by a case of thick card-
board in order to preclude the effect of radiation from the observer\'s body
on its parts.

Before and after each series of measurements the run of the screw
was determined by measuring the interval between two successive scale-
divisions; for this purpose the divisions 300 and 301 were chosen. The deviations
of the individual values from their mean lay without exception within the prob-
able measuring errors, so that in no case a correction for a changed value of
a part of the screw had to be applied.

In order to discover whether the plate shifted while being measured in

one position, which operation occupied from 8 to 19 days, the distance between
a sharply defined point on the plate and the nearest scale-division was meas-
ured at least three times during each series of measurements.

In order to detect a possible rotation of the plate with respect to the
apparatus, before and after each series of measurements the distances of the
four most suitable stars from the nearest scale-divisions were measured.
These stars should not lie too near the edge, as their images had to be sharp;
on the other hand, in order to reveal any rotation they had to lie as far as pos-
sible from the centre of the plate and preferably symmetrically to it. The
following stars were chosen:

In the first column the number of the star in the tables is given, in
the second and third their positions on the plate in rectangular coordinates.
It should be mentioned that the centre of the plate lay near = 300.0,
y = 100.0.

The result was that neither a shifting nor a rotation manifested
itself of such an amount that the discrepancies should not rather be attrib-
uted to accidental measuring errors than to a real displacement of the
plate. Hence no corrections were applied for it.

That no corrections for a real sliifting were necessary, appeared also
from the following consideration. The twelve fundamental stars already chosen
at the outset were measured 3 or 4 times in each i)osition of the plate, each
star being i)ointed at at least three times; the comparison of the measurements
of the individual stars yielded a mean error of ()\'.0ir)4 on the average. If the
necessary corrections for an assumed shifting are taken into account, this

m. e. works out at 0^0160. The difference is so. small that the corrections

for shifting are undoubtedly unreal.

Two settings were always made on the remaining stars and also on the
scale-divisions. I shall return to the error of pointing when discussing the

mean error in Chapter XI.

The measurements were all made with daylight, reflected towards the

plate by a horizontal mirror. On the average 200 stars were measured daily
in one position of the plate. Only during the winter months this number could
not be reached owing to early twilight. It would have been possible to work
with artificial hght then, but I feared that this might introduce a systematic
error in the readings, which has now been completely avoided.

The plates were measured in two positions, with East at the left
and with North at the left. By means of two pairs of stars, each differing
little in S and much in a, the plates were mounted as nearly as possible
according to the parallel of 1900.0. For the second position the orientation
was obtained by turning the plate through 90° by means of the circle of

position.

Since the readings of the micrometer screw increase when the vertical
wire of the micrometer is moved from left to right (i. e. with decreasing a and
decreasing o), we get, after applying the necessary corrections which will be
discussed in the following chapters, calling the star-readings at the first epoch
1 and at the second 2:

Pos. I: (1—2)a = P- M. in a (great circle);
Pos. II: (1—2)j = P. M. in 5.

Other observers usually execute the measurements in four i)ositions by
also turning the plate 180° and 270°; this is done in differential measurements
in order to eliminate the personal error depending on the size of the images.
In my opinion, however, this error is still better eliminated not by meas-
uring the same plate twice, but by taking two different plates. So I did
not hesitate to give up the measurement in four positions and to measure
instead as many plates as possible in two positions.

The six plates that were enumerated above, were combined in the
following three sets:

A 106 — B 132, Interval 12^063,
A 103 — B 136, „ 12\'\'.107,
Donner 4. 1892 — Donner 9. 1909, Interval 16\'\'.984.

The values (1—2)« and (1—2).? are given for these pairs in the tables

anbsp;. 1nbsp;^

under the headings-----— respectively

^ M, M, Manbsp;Ml M, Ma

CHAPTER V.

The method of reducing.

In order to derive the P. M. from the differences M, found in the pre-
ceding chapter the principle was applied, explained by Kapteyn in the „Plan of
selected areasquot; with this difference that also the quadratic terms were taken
into account. If m\'a and m\'j are the components of the annual P. M. of a star
and Q the factor, reducing the P. M. over the interval between the two ejiochs
and ex])ressed in V as unit, to the annual P. M. in 1\' as unit, we may assume
m\'« = [Ma—a—bA;—cy—dA;2—eA;y—fy-J Q,

m\'j = [U^—zi\'—h\'x—c\'y—d\'x--e\'xy—i\'y\'] Q.

It-is evident that in whatever way the twelve constants of these for-
mulae are determined (Q is determined sei)arately) the aj^i^lication of these
corrections to M involves corrections for:
shifting of tlic plates inter se,
rotation of the plates inter se,
alteration of the scale value,
differential refraction,

precession and nutation,
„ aberration,
the reduction of ^ and y to a and o,

the inclination of the plates when taken and when measured

Higher terms than those of the second degree with respect to a
fixed point, in casu the point where the readings of the scale of the measuring
instrument\'were = 300.0 and y = 100.0 are disregarded. The formulae for
these corrections which must be applied, whenever photographic platesare
measured, are fully dealt withe, g. in the Bulletin du Comm. Int. Perm, pour
l\'exécution phot, de la Carte du Ciel. So it would be superfluous to enter

further upon this subject here.

In order to determine my constants, I started from the assumption,

as Kapteyn puts it: „that, for the very faint stars on the plate (after having
excluded those which by a first reduction appear to have a very sensible
p. m.), the mean proper motion both in a and o is the same over the whole of
the plate. Starting from this supposition, the whole of these faint stars will at
once furnish us with very reliable values of the constants b c d e f b\' C d\' e\' f.
The two remaining ones (a and a\') shall be derived from the standard proper
motions determined at the meridian instrumentquot;.

Before proceeding to this determination I want to explain, why I adopted

a differential method for determining the P. M.

An absolute method has the advantage that each observation can
directly be used for determining the P. M. after other plates, taken at a later
date, have been reduced. In my case I might have comi^uted a and ^ for

each star for:

1892.7; 1896.7; 1896.8; 1908.8; 1908.8 and 1909.7.

In order to determine from these tlie position a, o for the epoch 1900.0
and the P. M. I might have combined these epochs into three sets e. g. 1892.7,
1896.8 and 1909.1 and for each star the following equations would then
have been ol)tained:

a — 7.3 {JLa =................

a — 3.2 {Jia =................

a 9.1 |Xa =................

and a similar set for o and jie?. By the method of least squares I could have
determined from these equations a, o, and jxj for 1900.0. Later observations
could easily have been linked to these.

But in order to obtain the second members of the above equations
we should have had to apply to the measurements the following corrections:

1.nbsp;corr. for the systematic and accidental errors in the scale-divisions

of the measuring instrument;

2.nbsp;corr. for the progressive errors of the micrometer screw;

3.nbsp;corr. for the periodic errors of the micrometer screw;

4.nbsp;corr. for the curvature of the cylinder of the measuring instrument;

5.nbsp;corr. for the tilting error (occasioned by turning the microscope
round a cylindrical horizontal axis from star to scale);

6.nbsp;corr. for the inclination of the plate relative to the optical axis

of the microscope;

7.nbsp;corr. for the inclination of the plate relative to the ojitical axis of

tlie refractor;

S. corr. for an erroneous value of a scale-division;

1).nbsp;corr. for the reduction of ;V and y tot a and 5;

10.nbsp;corr. for refraction;

11.nbsp;corr. for aberration;

12.nbsp;corr. for precession and nutation;

13.nbsp;corr. for an erroneous orientation of the plates.

Corr. 1 is usually obviated by referring the star to the preceding and
following scale-division and by moreover measuring the plate in 4 positions.
The labour of measuring is thereby a little more than doubled.

Corr. 2 3, 4, 5 and 0 are usually small if a good instrument is used
and so may either be disregarded or easily taken into account. The measuring
instrument has beforehand to be closely examined in regard to these errors.
In corr. 11 the terms of the second degree may as a rule be neglected.
Corr. 12 and 13 may be combined to a single one.
Corr. 7 cannot be applied in most cases since the inclination of the photo-
graphic plate to the optical axis of the refractor is seldom determined and
depends among other things on the flexure. That it may reach such a value
that it must not be neglected in the determination of P. M., has already
more than once been noticed and is again confirmed by my measurements.

Summing up it is clear that the absolute method requires much more

labour than the differential one.

As to the accuracy which may be reached it already appeared in Chapter I,
that with the absolute method, using six plates, as I did, a mean error in each
of the coordinates of the P. M. may be expected of ± O\'.OU. This holds for
the Carte du del plates (from Potsdam and Paris); each P. M. thus obtained
requires for the Potsdam plates 36 settings, for the Paris plates 48.

The result of my measurements is a mean error in the proper motion
of ± 0\'.008 in each coordinate. And this P. M. was obtained from 24 settings,
two on the star in each plate, two on the scale-division.

In this respect Kapteyn\'s method is a little easier still, giving a m. e.
of O\'.OlO with 12 settings. It is questionable, however, whether my m. e. would
have become K 2 as large, namely O\'.OU, if I had only once set on star and
line, since the error of pointing is not the only cause of the m. e. of the P. M.

Now, since with the differential method not only all sources of errors,
so far as they are functions of the 0th, 1st and 2nd powers of the star-coordi-
nates, are taken into account even such as are at present unknown; but since
also with less labour a higher degree of accuracy is reached, the method I
followed is, in my opinion, fully justified.

If later photographs of the same region of the sky will have to be
compared with the plates I used, it will still be possible, availing ourselves of
Kapteyn\'s investigation on the measuring instrument used, to derive a and
0 from my measurements. But the differential method would then require
the utmost precautions and it is even questionable whether the errors of the
measuring instrument may be regarded as constant in the long run and if
this cannot be stated with certainty, it is absoluty necessary to remoasure
my plates if one wishes to use an accuratc differential method.

CHAPTER VL
The equations of condition.

The stars that may serve for the determination of the twelve constants
a b c d e f a\' b\' c\'d\' e\' f have to satisfy various conditions. The assumption
from which we started on page 14, that the mean proper motion both in
a and o is the same over the whole plate, is only allowed for faint stars^
since with these a possible difference quot;in the parallactic proper motion can only
exist to a very small extent. For this reason the brighter stars must be
rejected at once. But in our case also the stars, belonging to the groups h
or -f had obviously to be dicarded. Besides, the stars with large P. ]\\I.
might have an undue influence in the determination of the constants, so
that these also had to be avoided.

As the magnitudes themselves had not yet been determined, this first
condition was settled by means of the diameters; all stars having a diameter
over 0\'.!)0() were rejected. It appeared later that this maximum diameter
corresponds to mag. 10.o. The only way for excluding the stars belonging to
the groups h and -/ Persei was to determine the extent of the clusters by
counting the number of stars in different parts of the plate. Since it is more-
over desirable to distribute the stars, serving for the determination of the
constants, as symmetrically as possible, all stars had to be e.xcluded, whose
distance from the centre of the plate are less than 32\'.5.

Also stars near the edge could not be used for determining the constants,
since it could not be settled a priori what weight had to be given to the equa-
tions of condition furnished by them, while it might be assumed that this weight
is the same for all the other stars. Hence 1 left out all stars, lying at more than

r)2\'.o from the centre of the plate.

In order to exclude the stars with large i)roi)er motions, which were
not known a priori, a number of simi)lilied ecpiations of condition, containing
only the constants a b c a\' b\' C, were fornred and solved. The P. M. so ob-
tained for the jKiir of plates A 100—B 132 indicated which stars it would be

desirable to drop in the final determination of the constants on account of
their large proper motions. The limit for this rejection I drew at a P. M. of
O\'.OSO annually in one of the coordinates. Finally I rejected all stars which
had not been measured on one or more of the plates or had received a mark
of uncertainty. This gave the advantage that for all pairs of plates, as well
in a as in 0 the coefficients of the unknown quantities were the same in the
conditional and hence in the normal equations. Otherwise the labour of com-
putation which was very considerable as it was would have been greatly
increased.

With all these restrictions 210 \'stars were left. The following table
contains their numbers in the final tables.

Starting from the assumption that for these stars the mean proper
motion is the same all over the plate and denoting this in a and o by A jXa and

A {jilt;? each star furnishes a pair of equations of the form:

Ajxa = (Ma—a—hx—cy—dx-—exy—iy^) Q,
A [X(7 = (Mj—a\'—h\'x—Cy—d\'x\'^—e\'xy—t\'y^) Q.

In practice Ajx^ and Afjtj cannot be separated from the constants
Qa and Oa\' and are combined with them. The constant Q is calculated
separately, ^^\'hen the constants have been determined we find for each star
the components m\'« and m\'lt;? of the P. M. from the tormulae:

m\'a = (Ma—a—hx—cy—dx^—exy—fy^) Q.
mV = {M^—a.\'—h\'x—C\'y—d\'x^—e\'xy—i\'y^) Q.

To the relative P. M. found in this way the corrections A jx\'« and A p.\'j,
deduced from some twelve fundamental stars, have to be ai)plied, by which
the absolute P. M. are obtained.

CHAPTER VII.

Tun normal equations and their solution.

In order to obtain the conditional and normal equations the rectangu-
lar coordinates which in surveying the plate had been found in ()M, p being
very nearly equal to 1were exjiressed in degrees by means of tables I had
nt my disposal in the Astronomical Lalx)ratory at Groningen. These tables give
two decimals. In the same way .v-, y- and xy were determined for all 1418
stars and besides .v^ f, .vy\'-, .v-y, .v^y^, .r\\v, .xy for the 210 stars,
furnishing the equations of condition for the determination of the constants
a b c d e f a\' b\' C d\' e\' f\'.

Since M« and M«? never contain more than three figures (Cf. ]). 10)
and the above mentioned coefficients were only in exceptional cases greater
than l.(Mi, it was always possible to use multiplication tables for making uj)
the normal ecpiations.

It has already been stated in the preceding paragraph that for all
three pairs of plates the same stars have been nsed as well in . as m o; the
consequence of this is that in the normal equations

[1 1] a [1nbsp; [1 .yjc [1nbsp; [1nbsp; [1 -y\']« = [\'

a ..]b .y]c ..^cl nbsp; nbsp;= L^ m

[y .1] a [y b [y .y] c [y d [r ^ •gt;quot;]\' = ^^

] a [.y.x] b [.y.y] c M d nbsp;« [-r-r] f =

J a [y^b [/- .y]c ryquot;- d [y\' ..y] o •/■] f = [y^ .M]

the first men,bers were always the same. This gives (or the determination of
the constants the following sets of normal equations:

H-2lO.OOna-30.2n0b 7.770c .5.5.24.,d 1.080e 42.070f = n .Ml
_ .S0.200a .5.5.24.5b 1.080c - 17.1«0d 0.2r,5c - 2.01.5 = . .M
7.770a ..quot;80b 42.070C n.2«.5d - 2.0.5e \'= y^
245a - 17.100b 0.20.50 22.710d 0.n70c (i.310f = r»^-.MJ
quot;L080a 0.2f,3b - 2.01.5C 0.070d 0.310e nbsp;= .«1

42.070a .- 2.01.5b 1.025C «.Hlod n.02.5e 13.20.5f = [y-.M|

The second members of these equations for the different sets of plates,

arc as follows:

pi8lt;.)2—Dinoo

fJ

13\'.308

—nbsp;1 .333

—nbsp;0 .813
2 .558
0 .352
4 .013

For the solution the method indicated by Encke in the Berliner Jahr-
buch of 1835 was mainly followed. The following table gives a summary of
the results obtained with their mean errors.

D 1892 — D 1909, a

0M»590± 0^0100
0 .0222 ± 0 .0056

—nbsp;0 .0729 ± 0 .0056

—nbsp;0 .0752 ± 0 .0209

—nbsp;0 .0531 ± 0 .0142

—nbsp;0 .0272 ± 0 .0250
0\'.0044± 0\'.0087
0 .0227 ± 0 .0049
0 .0439 ± 0 .0049
0 .0470 ± 0 .0183
0 .0580 ± 0 .0124
4- 0 .0052 ± 0 .0219

—nbsp;0\'.0335 ± 0M)1()7

—nbsp;0 .2273 ± 0 .0060
0 .0009 ± 0 .0060

—nbsp;0 .1574 ± 0 .0224

—nbsp;0 .0290 ± 0 .0152
0 .0033 ± 0 .0268

A 106 — B 132, 6

—nbsp;0^0276± 0 .0090

—nbsp;0 .0006 ± 0 .0050

—nbsp;0 .1064 ± 0 .0050
0 .0423 ± 0 .0188
0 .0320 ± 0 .0127

—nbsp;0 .1061 ± 0 .0224
(r.0537 ± 0\'.0073
0 .0507 ± 0 .0041
0 .0827 ± 0 .0041

—nbsp;0 .0079 ± 0 .0152

—nbsp;0 .0282 ± 0 .0103

—nbsp;0 .0104± 0 .0182
0\'.0222 ±nbsp;0\'.0109

—nbsp;0 .0029 ±nbsp;0 .0061

—nbsp;0 .1722 ±nbsp;0 .0061

—nbsp;0 .0109 ±nbsp;0 .0229

—nbsp;0 .0036 ± 0 .0155
0 .2496 i- 0 .0274

These constants, substituted into the original observations of all
tlie stars, gave (after multiplication by Q) the values for ni\'a and m\'j of
the tables.

0 was determined sei)arately for a and o ,from the differences in a and
0 of three jniirs of stars taken from A. G. Hels, viz.:

The mean distance of two scale-divisions of the measm-ing apparatus,
(ItM-ived from the measurement of 11 intervals, was loMlo a»ul the epoch
differences were:

12.063, 12.107 and 16.984 years respectively.

The Q\'s proved to be mutually concordant for each pair of plates.
Moreover the Q\'s for a and o appeared to be perfectly equal, namely:

for A 106 — B 132 Q = 0.489
„ A 103^ — B 136 Q = 0.487
D 1892 — D 1909 Q = 0.349.

The formation and solution of the normal equations was checked
by double calculation. Moreover, the values found for the unknown quanti-
ties were substituted into the normal equations. Also the check [nn.6]
= [oo] was applied. This gave:

When substituting the constants in the formulae of reduction for
the remaining 1208 stars, suitable checks were applied if feasible. Where this
could not be done the calculation was made twice.

A few constants are greater than O.\'lOO, with Donner\'s ])lates even
f = 0\'.2r)0 occurs. This would seem to show that in the coeflicients y .f- xy y-
more than 2 decimals should have been taken if the computation of the P. M.
is to be carried to 3 decimals. This also explains the differences between
[nn.6l and [oo]. But it should be borne in mind that in order to obtain the
components of the proper motions in seconds i)er year the values

M a—a—b-r—cy—d.x-—e.vy—fy-
and M s—^\'—h\'x—c\'y—d\'x^—ii\'xy—i\'f

must be nmltiplied by Q. Hence in the values m\'a and m\'j, given in the
tables, the 3 decimals are ciuite justified as far as the computation is concerned..

CHAPTER VHL
Systematic errors depending on position.

Although, speaking generally, the method followed by me for determ-
ining the constants of the plate, as Kapteyn puts it i) „seems to promise
a priori a very thorough elimination of all systematic errors depending on the
position of the starsquot; yet the present case is a very unfavourable one, since the
210 stars, furnishing the constants, are not evenly distributed all over the
plate but lie within a ring with radii 32\'.5 and 52\'.0. As well the central as the
marginal parts had to be excluded for the reasons mentioned on page 17.
The great uncertainty resulting from this in the determination of the constants
ad e f a\' d\' e\' f\' is confirmed by the small weights of these constants.
That the rejection of the central stars is a chief reason of this may easily be
seen in the following way. Let us put the imaginary case that to determine
the constants 100 more stars had been used, all of them with = 0 and y = 0,
then the weights of the constants are calculated to be:

Wa =nbsp;112.4

Wb =nbsp;41.3

Wc =nbsp;41.1

Wd =

Wc ---nbsp;(5-2

=nbsp;7.4

M\'a =

Wh =nbsp;J^f.o

Wc =nbsp;.\'»gt;.3

zc/,1 =nbsp;2.8

Wc =nbsp;0.1

WI =nbsp;2.0

Publ. of the Astr. Lab. at Groningen, 1, C4.

Now, if the P. jM. of the stars, belonging to the clusters h and 7 were
so great that we could make out which stars belong to the cluster and which
do not, which will appear in Chapter XII not to be the case, then, by including
such stars as certainly do not belong to the cluster, the plate constants could have
been determined with much greater accuracy. But even now it is still possible
to improve the values found for the P. M. by determining as well as possible
the systematic errors depending on the position of the stars on the plate
and applying the resulting corrections. This was done in the following
manner. The plate was divided into 21 squares, as shown in the table:

Now it was assumed that the mean P. M. of the stars in each of tiie
squares must be constant. This assumption is entirely analogous to the sup-
position made in ChapterVI and is certainly justified by the large number of stars
in each field. We had to reject again the two clusters h and 7 (this meant the
dropping of the whole square 11 and of the middle of square 10); all stars
with a large P. M. (I chose the limit at 0\'.07r) annually in one of the coordinates);
the stars that had not been measured on all three j^airs of i)lates.

In this manner a larger part of the plate could serve for determining
the constants than formerly. Each field furnishes for every ])air of i)lates
two equations of condition of the form

average m\\ = a \'^x -{y Ix^ zxy Cy-
average m\',- =»\'-[- \'^\'x 7\'y h\'x^ zxy cy,

enabling us to determine the constants

a p Y S s C Y S\'s\' C,
after which to each of the calculated values m\\ and m\'j the corrections

B.xy — Cy-
- s\'jvy - C\'y-

respectively are to be applied.

Weights were assigned to the different fields in the following manner.

Let n be the number of stars to be used in each field, s the mean error
of the average P. M. in a field on each of the three pairs of plates, t the mean
error in the individual P. M. of the stars in the same field on the three pairs
of plates, then the weight to be assigned to each field is given by the fornmla:

w =

S\' I 2/«

It appeared that n as well ass and t did not diverge very much for the
fields 1, 2, 3, 4, 8, J), 13, 14, 18, 10, 20, 21 and this was also the case with
5, (5, 7, 10, 12, 15, 16 and 17. Therefore the fields were divided into two
groups, the marginal and the central fields. I thus found for

For the values given under /, the averages were taken of the mean
errors of the mean of some five stars in each field. As was done throughout
this ])ai)er, the circumstance that the i)air of i)lates by Donner had an epoch
difference of 17 years and both i)airs by Kostinsky of 12 years was taken
into account, by assigning to Donner\'s pair the weight 2, to those of Kos-
tinsky the weight 1.

The 12 constants were redetermined by the method of least squares,
the computation being done twice; for convenience sake the weight of the
central fields in a was taken as 0 instead of 1

On applying the corrections:

— a — [ix — 7y — ox^ — zxy — Cy^
and — a\' — ^\'x — y\'y — — ^\'xy — C\'y^
respectively, to the mean P. M. \'of each of the 20 fields, I found that
the errors s had been much reduced by the above operation viz.:

^in a to 0\'.0062,
marginal fields \\

^nbsp;( „ 0 „ 0quot;.0035,

( „ a „ 0\'.0017,

central fields

I „ 0 „ O\'.OOSl.

The question now arose whether by repeating this process a further
improvement of the P. I\\I. could be achieved. Introducing the new values for
the errors s and retaining the old values for the errors t, which play a much
less important part since they are divided by n, the formula now gave the
following weights:

1, 6, 11 and 9.

The change is none for a and so trifling for o that I felt justified in not
re-calculating the constants. Since, however, the errors s had been so much
more reduced in o than in a, I also re-determined the constants in a, using the
weights 4 and 9 for the marginal and central fields, in this case also, though
there was no a priori reason for doing so. The result was that the errors s
became:

0\'.0058 for the marginal fields,
O\'.OOIS for the central fields.

Since thus also a posteriori there appeared to be no reason at all why
the constants, based on the weights 4 and 9 should be considered more trust-
worthy than those, computed by assigning the weights 1 and 9, the corrections
found in the first determination were applied to the P. M. m\',^ and m\'j.
These corrections generally being small (only near the margins of some
plates the correction amounted to more than O\'.OlO) they could be most
easily applied by means of diagrams. For this j)urpose for each pair of plates
curves in a and 6 were plotted for which the correction amounted to:
......— 0\'.002^ — O\'.OOl® — 0\'.000=^ O\'.OOO® O\'.OOl^ 0\'.002\'\'____,

the scale being such that the correction itself could be read directly for each
separate star with an accuracy of O\'.OOl. Thus the values m«\' and mJ\' in
the tables are transformed into (x\'a and These are the final relative P. M.
They were averaged, assigning the weights 1, 1 and 2, respectively to the 2
pairs of Kostinsky and that of Donner. The results have been tabulated
in the last two columns of the pages on the right (see tables at end of
paper).

CHAPTER IX.

Systematic errors depending on m.\\gnitude.

The i)rincipal errors of this kind are the hour-angle error i) and the
guiding-error. In our case, however, the hour-angle error has been pretty well
eliminated, since the four plates by Kostinsky were all taken very near
the meridian and both plates by Donner at perfectly equal hour-angles.

Immediately after I had measured the six plates, proper motions were
derived without taking into account the (piadratic terms. With the results-
so obtained an investigation was made about the systematic errors depending
on the relative magnitude by determining the mean P. M. in a and o of as many
stars as i)ossible of the same diameter. The curves plotted by means of these
results showed that only with the pair A UH\\ — B 132 in a a magnitude error
could be detected. The error was fairly well proi)ortional to the diameter

and amounted to

—()\'.();J3 X (diameter in revolution —1)^^)^)2).

This value ai)proximately corres])onds to ()\'.()()7 x (mag—13.0).
Hut the number of plates is too small to decide whether this magnitude error
in the i)air A 10(5 — H 132 is real. If it had been possible to distinguish
with certainty whether any given star belonged to the cluster or whether
it did not, these data might have afforded an excellent proof as to whether

Pul)l. of the A.str. Lab. at Groningen, 1. 6Ü.

the differences in the average P. M. with different magnitudes should or
should not be ascribed to a magnitude error, since for these stars the
P. M. should of course have been equal for all magnitudes. Since my
measurements could not give certainty on this point, I thought that I ought
not to take a magnitude error into account, as it is at any rate relatively
small (see Publ. of the Astr. Lab. at Groningen, 19, 35). Similarly its amount
was not re-determined from the final P. M., since the difference from my
determination can never be considerable, the stars of every magnitude
being pretty evenly distributed all over the plate.

CHAPTER X.

The standard stars.

In order to derive the absolute P. M. from the relative P. M. the
corrections A |j.\'« and A [jl\'^, mentioned in Chapter VI are still to be applied to
{x\'a and [x\'e?. From the beginning a dozen bright stars had been selected for
this purpose, namely: A. G. Hels 1938, 1981, 1997, 2057, 2061, 2071, 2088,
2093, 2113, 2150, 2177, 2217. All these stars occurred in various other cata-
logues so that they could furnish good absolute P. M. when combined with
recent meridian observations; they were chosen so as to be distributed as
equally as possible over the whole plate. Prof. Auwers had the kindness
to send me the catalogue positions of these 12 stars reduced to the equinox
1875.0; a summary of them is given in the following table, to which I added
a few positions given by Pihl, Oertel, Schur, and the Misses Bronsky
and Stebnitzky; a star marked a; had no catalogue-number.

The abbreviations are the same as those used by Auwers in .\\.N. 4176.
The star A. G. Hels 1938 (No. 1563 in my list, see end of book) had to
be dropped after all. This star, although situated on the plates at 69\'.5 from
the centre, had been measured, but showed such a considerable mean error
in the P. M. that for the determination of A jx\'« and A (x-j I thought it
better not to compare this P. M. with that derived from catalogues.

Current numbers of Catalogues usedfor Nos. (A.G.Hels):

2150

Br.
Lai.
Pi.
Grb.
Tay. D.
Rob.
12v.
A. be.
12 y.
R. C.
Par. 1
Do. 50

r,y.
W\'ro. 2
Bo. VI
Pu. M.
Pu. M.0.
R. C. 2
Par. 2
Va.
N. 7 y.

P-

Pu. 2.0.

CA.
A.N.3212
A.G.Hels
Re. 1
Be.
Rbg.
Wa. 2
Kam 2

Bo. Hel.
Kü. 1
Strb. 2
L. G.

310
4139
27
485
739

491! 493

322

303

1081 1007 2057
i I 140

328
4311

512
207

207
601
2010

142
110

330

34(5

327

279
2835
1053 1055

318
012 021

ct 64
0 8

400

2003 21 1312150
51

48

551
484

777/8; 770

V 2

12
00
106

Current numbers of Catalogues usedfor Nos. (A. G. Hels):

2113 21502177 2217

Inbsp;i

34!
c

: 796

103i

i 468
161! 162; 165
78\'

1 m

To the values of these catalogues, reduced to the equinox 1875.0
(including the third precessional term) systematic corrections were applied,
in order to reduce them all to the N. F. K. system (A. N. 3927). In doing
this I availed myself as much as ])Ossible of Auwers\' tables in ,,Krgänzungs-
hefte zu den A. N.quot;, Nr. 7, making due allowance for the remark ]niblislied
N. 4200. For the catalogues not mentioned there I used if possible the
systematic corrections communicated by Battermann \').

There remain a few catalogues occurring in neither of the i)ai)ers
named, viz.:

Pu. M. o.: a similar correction was a]^])lied as with Pu. M. 1855.

Bo. Hel.: Krüger\'s Heliometer observations; the position and syst,
corr. from A. G. Hels, (8).

Ru. Ps. ph.: Rutherfurd Photographs of the Stellar Clusters h and /
Persei by S. Young. Contr. from the Obs. of Columbia Un., 24.
Syst. corr. derived from the comparison with Krüger, given there on
page 67.

1) Beob. Ergeb. der Kön. Sternwarte zu Berlin, 12

A.nbsp;N. 3219 and A. N. 3251: Beobachtungen von Vergleichsternen,
angestellt auf der K. Un. Sternwarte in Kasan. The system is that of the

Berliner Jahrbuch.

Pu. 15 N.: Publ. de I\'Obs. Central Nicolas, série II, 15. Syst. corr.

in the introduction itself.

Pihl: The stellar Cluster x Persei micrometrically surveyed, Christiania,
1901. Syst. corr. derived via the comparison by the Misses Bronsky and
Stebnitzky, Mém. de 1\'Ac. de St. Pétersbourg, Série VIII, 2, 126.

Oertel: Neue Ann. der K. Sternwarte in Bogenhausen bei München, 2
Syst. corr. derived from the comparison with Krüger, on page B. 79.

Schur: Astr. Mitt, von der K. Sternwarte zu Göttingen, 6. Syst.
corr. from a comparison with A. G. Hels, page 82.

B.nbsp;and S.: Mém. de 1\'Ac. de St. Pétersbourg, série VIII, 2. Syst. corr.
from the comparisons given there on pp.117 and 119 of the observations
mutually and of tliese observations with A. G. Hels.

I was very sorry not to be able to make use of seven i)ai)ers named
iiereafter, either on account of inaccurate observations, or because il was
impossible to apply a reliable systematic correction, viz.:
Do. 50;
A. N. 3212;
Kam 2;

Lamont, DerSternhaufen// Persei, Ann. der K. Sternwartebei München, 17.
Bkhdichin, Mesures micrométriciues du Groupe de Persée, Ann. de

l\'Obs. (le Moscou, 4, livre 2.

Vogel, Der Sternliaufen x Persei, Leii)zig, 1878.

Ball and Kambaut, On the relative position of 223 stars in the cluster

X l\'crsei, Trans, of the K. Irisli Ac. 30, part 4.

In addition to the i)ositions of the catalogues mentioned I was
fortunate enough to avail myself of some recent meridian observations,
taken at my re(iuest at Leyden and at Bonn.

As to the observations from i.eyden, the right ascensions after applying
tlie correction for magnitude, were based on the N. F. K. system. For the
(leclinations Dr. Zwiers sent me the corrections to be adopted. Finally,

in the observations from Bonn the „Helhgkeitsgleichungquot; (correction for
brightness) had beea directly eliminated by the use of wire gauzes; the system
was that of the N. F. K.

The weights to be assigned to the observations were taken either from
Auwers\' tables in A.N. 3615, 3616, 3844 and 38S8 or derived from the mean
errors by means of Auwers\' table in A. N. 3616.

In the determination of the P.M. of the fundamental stars the method
of least squares was again exclusively followed. The result is given in O\'.OOl in
the following table under the headings C and C for the P. M. in a and o,
while under the heading w the weights are given, resulting from the com-
putation, but expressed in integers.

Under and the P. M. of these fundamental stars are found,
as derived in Chapter VIII, under C—Othe values [la\'C—ix\'aOandfjt-jC—ji\'jO.
These values, averaged by means of the weights given in the table, I used as
the final corrections to be applied to the P. M. in the tables, in order
to transform the relative P. M. and |jl\'c? are into absolute P. M. They are:

Ajx\'c = — 0\'.0()6 ± ()\'.()03(),
= ()\'.(I07 ± ()\'.0{)42.

CHAPTER XI.

The me.-vn error of the Proper Motions.

In order to avoid the tedious process of writing down the squares of
some 8000 numbers, the mean error of the final P. M. jx\'^ and jx\'j was
computed by an approximate method; the result is on the average 0\'.0089 and
O\'.OOSl in a and o respectively.

In order to investigate the difference in reliability of the P. M. for
the stars at the centre and near the edge, the m. e. was separately determined
for the stars within a circle with a radius of 50\'.2, inside the ring with radii
50\'.2 and 50\'.3 and inside the ring with radii 5G\'.3 and 62\'.0.

The result was:

The increase in m. e. when approaching the edge is evidently, not-
withstanding the bad images of the marginal stars, so gradual that no sharp
line can be drawn for the stars that have to be rejected for further examination.
Consequently I used all stars, having a weight of at least 3, as well for determ-
ining tlie P. M. of tlie star-clusters as for determining the frequency of
the P. M.

That the accuracy, corresponding to the m. e. of 0\'.0085 cannot be
readied by an absolute method has already been shown in the Chapters I
and V. This oi)inion was based on the m. e. of the Potsdam and Paris
astrograi)hic catalogues, to which I shall now add a few other mean errors:

From this summary it is evident that in our comparison (Chapter I)
of the accuracy attainable by the absohite and differential methods we

remained on the safe side.

One of the unavoidable sources of errors is the error of pointing. We

shall therefore investigate its influence on the m. e. in the obtained P. M.

For this purpose the error of pointing was determined on each plate
for 40 to 60 stars of different diameters. A summary is given in the following
table :

The large error of pointing with stars of diameters gt; r.THO on Donnkr\'s
plates is to be explained by the fact that there- the diameters are considerably

..............\' ■nbsp;u

greater than on Kostinsky\'s and often exceeded 3^000. For convenience
sake these have all been collected under the heading gt; r.750.

The error of pointing on a scale-division was repeatedly determined
and found to be on an average 0\'.0037; if in order to facilitate calculation
the error of pointing of a star is assumed to be on an average 0\'.0106, we
find 0\'.0112 for the definitive total error of pointing in one of the numbers
M in the tables, derived from the mean of two pointings on two stars
and two scale-lines (Cf. Chapter IV).

It will be easily perceived that by the pointing on the scale-divisions
which is necessary in our method but is avoided in Kapteyn\'s method the
accuracy of the final results is not perceptibly diminished and that con-
sequently the assertion made on page 8 is fully justified.

The value found for the m. e. in the numbers M gives by multi-
plication with 0.244 the m. e. in seconds of arc, caused by the error of
pointing in the average P. M. of the three plates. It works out at 0\'.0027.

It is evident that the error of pointing constitutes only a small
fraction of the m. e. resulting in the P. U. If I had not pointed twice but
only once, as well on the stars as on the scale-lines, this would have increased
the m. e. in the final P. M. only from 0\'.0()89 and 0\'.0081 in a and o

respectively to 0\'.0093 and 0\'.0085.

Hence repeated pointing cannot increase the accuracy of the final
results in proi)ortion to the increase in labour.

The slight amount of the m. e. of the average P. M. (Cf. Chapter VIII)
proves that, the distortion of the film being exceedingly small, the principal
source of the final m. e. nuist be sought in irregularities of the images
themselves. This o])inion has already been exj^ressed by other observers, among
others by Kapteyn,W\'ilsing and Perkine and is once more confirmed here.

To what extent the greater m. e. in the final P. M. and at the same
time the greater m. e. in (he average P. M. of the marginal fields, discussed in
Chapter VIII nmst be ascribed either to erroneous values of the i)late con-
stants, or to the greater irregularities of the images or to a distortion of the film,
cannot be decided by my measurements. It would be i)ossible to decide this by
a reduction, (piite analogous to the present one, in which the plates of the same

epoch were combined, since then, with more precise constants, an irregularity
of the images would not introduce systematic errors depending on the position
on the plate, whereas distortion would do so. The labour of reduction would
have been doubled by such an investigation.

CHAPTER XII.

The group stars.

Already at the first reduction it became apparent that the stars of the
groups \'h and -/ Persei had such a small P. M. that it would prove impossible
to make out with certainty which stars are and which are not members of

the groups.

In order to determine the P. M. as accurately as circumstances per-
mitted I took the densest parts of both\' groups. After a first approximation
the stars whose P. M. differed to an appreciable amount from tlie minute
group drift were excluded. So I obtained for the average relative P. M.:

The next step was to try and find out a possible relative movement
among the group-stars. For this purpose I drew the vector diagram of the
P. M. of the stars probably belonging to h and -/ (which could be decided by
countings) for areas of 5 minutes square.
The result was negative.

Starting from the P. M as found above and from a probable error in

the total P. M. of ± u\'.015, I found the following actual and theoretical
numbers of stars, probably belonging to h and x, as classed according to the
deviation of the P. IL from the general group drift of the clusters:

I would not venture to draw from this a conclusion concerning a
swarming P. M., as de Sitter did for the Hyades (Publ. of the Astr. Lab.
at Groningen, 14, 20), the less so, since it was found that for /j Persei as well
as for the surrounding stars the P. M. showed a preference for two different
values, instead of one, viz:

= — O\'.OOf);nbsp;|x\'j = — O\'.OOl

= _ O\'.OOl;nbsp;[iV = O\'.OOS.

Although I would not assert that this i)lienomenon is caused by a real
P. .M. of the stars of h Persei and although the small amount of the P. M. of
both clusters does not enable us to settle whether it is caused by a number of
stars, not physically belonging to h, but possibly to y, or to what extent it
must be ascribed to systematic errors in the P. M., yet I thought it necessary to

mention it here.

Since the small amount of the P. M. does not i)ermit to make out with
certainty which stars do and which do not belong to the groups, 1 refrained
from any attempt at determining the parallax of the clusters, by means
of Kapteyn\'s and de Sitter\'s results mentioned in Chapter L We might surely
exclude with certaintv the stars with a large P. M. as not belonging to h or x
Persei, but inversely a small P. M. in the same direction as that of h or x does

not necessarily prove that a star is a member of either of the two clusters.

The final result is, that a conclusion as to this vital question will
have to be postponed till the P. M. have been determined with a consider-
ably greater degree of accuracy than was reached here, unless other means,
e. g. the radial P. M. of these stars, will furnish the necessary data.

CHAPTER XIIL
Determination of the magnitudes.

The diameters of all stars, except a few very faint ones, had been
measured on one of the plates, viz: A 106. Magnitudes were derived from
them by a graphical process. The bright and faint stars were treated
differently, while of part of the stars the magnitudes were determined in both
ways.

For the brighter stars (233) the magnitudes of the Bonn Durch-
musterung were used, after applying a correction to reduce them to the H. P:
system. For this purpose the magnitudes of the stars of this region of the
sky, occurring in Annals of the Astr. Obs. of Harvard College 54,

a = 2* 0quot;\' to 2\'^ 24\'quot;, o = 50° to 60°,
were compared with those of the B. D. The following tabic shows that the
systematic difference H. P. — B. D. is nearly constant for the rather
small area under consideration:

y gt; It does however depend on the magnitude, as will be seen from the
next table.

For the fainter stars (diameters lt; r.400) the magnitudes were deduced
from the numbers of stars of different diameters, counted on the plate, by
means of Kapteyn\'s tables (Publ. of the Astr. Lab. at Groningen, 18). Of
course the groups had to be excluded from the discussion, their borders being

again found by countings.

Besides Prof. Nijland had the kindness to estimate the brightness

of 14 stars of mag. ± 13.5 and of 6 stars of mag. ± 11.2 in the Utrecht
refractor (aperture 26 c.iM.) and in its finder (aperture 7.5 c.M.), whose limits
of vision had been formerly found to be 13.9 and 11.5 respectively, in the H. P.
system (A. N. 4116). The method used consisted in estimating how many
steps a certain faint star was above the limit of the instrument. Although
these observations are as a matter of course rather difficult. Prof. Nijland
is yet of oi^inion that the m. e. in a magnitude determined by him in this
way, based on at least three observations, does not exceed 0quot;\'.2. A small
correction for atmospheric extinction was applied, taken from the tables in

Publ. des Astroi)h. Obs. zu Potsdam, 3, 285.

The corrected magnitudes derived by means of the three methods just
mentioned were plotted as ordinates with the diameters as abscissae and a
smooth curve was drawn through all the points thus obtained, discarding however
tlie B.I), stars witii magnitudes fainter than 9\'quot;.0, since it has already repeatedly
been found that among them numerous stars occur, whose magnitudes are
considerably (even as much as two classes) fainter than given in the B. D.

The agreement between the magnitudes, obtained by the widely
different methods was rather good on the whole. Still there are some dis-
crepancies, which may be explained in the following way.

From 8\'quot;.5 to 12\'«.() the magnitudes derived from the countings are on

an average 0\'quot;.2 fainter than those indicated by the final curve. This deviation
may be largely if not totally explained by the supposition that, although I
tried to exclude the clusters, yet a number cf stars belonging to them has been
embodied in the countings.

On the other hand we find for magnitudes fainter than 12\'quot;.0 that the
magnitudes from the countings are brighter than those following from
Nijland\'s estimates. The reason is that there are very many small stars,
presenting such vague discs that there could be no question of a regular
measurement and which were consequently rejected from the beginning;
therefore the numbers of stars with diameters gt; 0\'.600 come out far too
small, as will appear from the following table, containing in the first column
the Hmits of the diameters, in the second the numbers of stars on a surface
of 1.38 square degree, in the third the ratios of these numbers.

These ratios are at first somewhat irregular, which must certainly be
ascribed to the small number of stars used, but they become fairly well con-
stant for the diameters IMOO to O\'.GOO. With the smaller diameters the
decreasing ratio undoubtedly indicates that not all the stars with dia\'meters
between the assigned limits were taken into account. If the constant ratio 1.04
be applied throughout the table, the magnitude of the faintest stars is to be
estimated 13.5 from the countings, whereas Prof. Nijland finds 14.0. The
latter value has finally beend adopted.

Though of no importance for the present paper, it is worth while to
remark that the photometric determination of Parkhurst used by Kap-
teyn almost exclusively for the magnitudes 14.0 and fainter are by no
means homogeneous with those of Pickering.

It was already pointed out by van der Bilt in „Recherchss
Astronomiques de I\'Observatoire d\'Utrecht, 3, that there is a systematic
difference between Parkhurst\'s and Pickering\'s magnitudes in the case of
the comparison stars for the variable stars U Geminorum and Nova Aquilae.

I therefore examined this systematic difference in the case of a few other
variable stars, viz.: T Andromedae, S Cygni and S Comae Berenices, whose
comparison stars are to be found in H. A. 37 and in Parkhurst\'s Researches
in Stellar Photometry, 1906.

To these I added the comparison stars of W Andromedae and Y Cassio-
peiae, the brightness of which, expressed by Nijland in the H.P. system, could
be compared with the values given by Parkhurst. The result is that Park-
hurst\'s and Pickering\'s observations agree well at 8quot;\'.0 and 9\'quot;.0, but that
for fainter stars the difference is considerable; at 12\'quot;.2 the difference Park.-
PiCK. even amounts to — 0\'quot;.04 (from 17 stars). Unfortunately the material
on which this conclusion is based is small and contains no more than 35 stars.

Besides the already mentioned systematic deviation in the magnitudes
of the stars of the B. 1). with mag. fainter than 9.0,1 still found large deviations
ill magnitude with some 10 brighter B. D. stars. These have been collected
ill the following table:

The stars B. D. 56° 497, 547, 551, 583, 595 and 597 are not found
on Rutherfurd\'s photographs i), measured by Young. This points to a pos-
sible great difference between the photographic and visual magnitudes.

B. D. 56° 583 and 563 are called reddish by Vogel^). Nijland
assigns to the former a colour-shade 4^3. (In Schmidt\'s scale). The latter is
not shown in the table; photographically it is 9-7, while the corrected
visual mag. (B. D.) is 9.4\'. Lohse^) has already drawn attention to the
remarkably small difference between the visual and photographic magnitude

for these this coloured star.

Also B. D. 56° 547 and B. D. 56°551 are called yellow by Nijland,

shade 3^7.

CHAPTER XIV.
The frequency of the Proper Motions.

While for the determination of the P. M. of the clusters h and y. Persei
1 had to start from stars, pretty certainly belonging to either of the groups
and consequently had to restrict myself to their densest parts, on the contrary
for the determination of the frequency of P. M. according to magnitude and
amount of P. U. I was only to use those stars which very likely were no mem-
bers of either of the two clusters.

Different ways could now be followed.

1.nbsp;All the stars could be used which visually appeared to fulfil this
condition.

2.nbsp;I might start from the stars whose P. M. differed from the P.M.
of the groups by more than 2 or 3 times the probable error in the P. M.

3.nbsp;I might take only the stars which for both reasons did not belong

either to h or to y Persei.

An enormous drawback of the two last methods is that a great part

of the stars whose P. M. lie between ()\'.()()()—()\'.0()9 and O\'.oio—()-.ol9 and

1)nbsp;Contr. from the Obs. of Cohnr.bia Un. 24, 51.

2)nbsp;Vogcl, Der Stcrnhaufen li Persei, Leipzig, 1878,12.

3)nbsp;A.N. 2650.

even part of those stars with P. M. 0quot;.020—0\'.029 is excluded, which have the
P. M. in common with the clusters h and y but are not physically connected with
them. If the P. M. is large as with the Hyades, this drawback is much less felt.
In our case we should have had to limit the discussion to P.M. exceeding 0\'.030
or make up for the deficiency in the smaller P.M. by introducing hypotheses
about the frequency of a certain value in various directions. This would have
led to all sorts of difficulties by which after all in the P. M. less than 0\'.030
a great uncertainty in the frequency would remain. Although I also followed
these ways, I shall here only give the results found by the first method. It is
quite possible of course that yet a number of stars were included that must
be reckoned to h or y Persei, but I think this difficulty has been sufficiently,
overcome by taking fairly wide limits for the stars which visually are no
group-stars. The number of stars is hereby considerably lessened, but the
certainty is much inci-eased. Thus 763 stars remained, the frequency of which
is given in the following table. The P. M. have been reduced to the
N. F. K. system.

13.0—14.0\'Total

The remarkable phenomenon is now at once noticed that most P. M.
do not lie between the limits O\'.OOO and 0\'.009 but between O\'.OlO and O\'.OlO.
This is also the case with the frequency of the P. M. according to magnitude
and amount of P. M. for stars, brighter than 6-.5, given by Kapteyn in Publ.
of the Astr. Lab. at Groningen, 11, 8. There the phenomenon is not found
with the stars of 6^.5 to O\'^.S, but then the numbers given for these are not
directly derived from observations; the subdivision of the P. M. O\'.OOO-O\'.OOO
was, as Kapteyn puts it: „made by the aid of certain plausible conditions, which
are centainly or probably fulfilled by the numbers of small proper motionsquot;.
This phenomenon of relatively few small P. M. can be explained in

various ways.

1.nbsp;The corrections A {x\'« and A ii\'e may be inaccurate. They are
based on no more than 10 stars and have m. e. of resp. O\'.OOSO and 0\'.0042.
So it is not impossible that the corrections A jx\', and A jx\'j would come out
different if we started from a larger number of fundamental stars. But I doubt
whether this change would be such that the zero point of P. M. would shift

sufficiently to cause the phenomenon to disappear.

2.nbsp;It is perhaps better to base the P. M. on another system than the
N. F. K. In Publ. of the Astr. Lab. at Groningen, 9 Kapteyn reduces the
P. M. to a system which is practically equivalent to that of Newcomb. In
order to reduce my P.M. to this system, assuming that the systems
„Auwers—Bradleyquot;, „Fundamental-Katalog der A. G.quot; and „N. F. K.quot;
do not appreciably diverge, as far as P. M. are concerned, the absolute

P. M. should be corrected by

0\'.00035 0\'.00035 sin a tg o

and

0*.0053 cos a

in a and o respectively.

Since these corrections, amounting in the i)iesent cas: to
0\'.0053 and 0\'.()043,
do not shift the zero point in the right direction I thought it advi5a1)lc not
to apply them until more certainty about the best system will have been
obtained.

3. The parallactic P. M. may be of influence. In order to find out
whether the relatively small number of stars with P. M. less than O\'.OOO
may be ascribed to parallactic motion, I determined for 755 of the stars used
(the eight P. M. exceeding O\'.lOO were discarded for practical reasons) the
frequency of the components u and x parallel and at right angles to the great

circle h Persei_Apex. We may assume symmetry in both cases, in the first

case with respect to the parallactic P. M. of the stars of my average magnitude,
in the second to the P. M. O\'.OOO. For the direction of the parallactic motion
the position-angle was taken = 133°, corresponding to the Apex

a = 269°.7, 8 = 30°.8,

this being the latest determination from Prof. Kapteyn\'s data, which he
was so kind to give me. (Pos. values for u are counted towards the Antapex

(S. E.), for - towards N. E.)

The real frequency of and - is given in the following table:

From this table symmetrical curves were drawn; the maxinmm of
the u-curve lies at —0quot;.Oil in the direction of the Apex, that of the --curve
at _0\'.002 in the direction S. W. Hence the phenomenon cannot be explained
by parallactic motion either, since for stars of average magnitude irquot;.7, as we
have here, the latter is 0\'.0()05 in the direction Antapex.

None of the above given possible causes seems sufficient to explain the
small number of small P. M. and it remains an open question whether indeed
vcrv small P. M. are less numerous than larger ones. Much material will jirob-
ably be required before a definite answer to this ([uestion can be given.

CHAPTER XV.

Remark.-^ble Proper motions.

The number of stars on the plates with large P. M. is very small. W\'hile
Turner i) among 13.000 stars finds 123 with an annual P. M. gt; 0M50,1 among
703 stars only find 8 with an annual P. M. gt; OMOO. These have been collected
in the next table:

In col. r,, 0 and 7 the numbers of these stars are given, as occurring in
.Kostinsky, Ueber die h:igenbcweging der Sterne in der Umgebung der
Sternhaufen h und x Perseiquot;, A. N. 4300, and the fx\'« and jx\'j, derived

from the values there given.

These values do not very satisfactorily agree with mine. But since these

P. M. were measured by Kostinsky by means of a stereocomparator and

each of them is only referred to two faint, synunetrically situated comparison

stars, and since K(.stinsky cmi)hasises that he looks upon his results as

l)reliminary only, these differences did not seem to render it necessary to

discuss the relative values of my method and that of the stereocomparator.

M. N. of K. A. S. 71, no.

Kostinsky derives from his results the existence of two drifts, forming
an angle ofnbsp;= 130°.6 and f = 103°.3),

I must confess that neither his figures nor his diagram are, in my opinion,
very convincing; for, the p.m. of the first so-called drift vary from 0\'.030 to 0\'.414
annually, of the second from 0\'.044 tot 0\'.175. Now a great part of this P. M.
should be ascribed to the parallactic motion, for which p = l33°. There is little
reason to see in these stars two physically connected groups, although it is
possible that some of them form a physical system, as e. g. 1024 and 1500,
for which Kapteyn and de Sitter also found fairly equal parallaxes, namely
0\'.12 and 0\'.13 i).

CHAPTER XVL
Final conclusions.

summarising the obtained results, I think it may be said tnat in
chapters V and XI it has been sufficiently proved that the method here followed
may give very accurate results and compares favourably with other methods.

Kapteyn\'s method, in which the images of both epochs are contained in. one
plate seems to be, at the first glance, the method giving the greatest possible
accuracy with the least possible labour. But on the other hand the non-
developing of the plates implies the use of an interval that has not yet begun
and therefore the method can never avail itself of earlier plates, whereas in
my method any early plate after the ,,Carte du Cielquot; pattern may even now
be combined with any other plate of the same region as soon as it is taken.

That I could not make out which stars belong to the clusters h and
y Persei (Cf. Chapter XII), must be ascribed to the very small P.M. of these
groups themselves.

The second aim of this investigation, the determination ot the frequency

1) Publ. of the A.str. Lab. at Groningen, 10, tabic 5.

of the P. M. according to brightness and amount of P. M., may be said tot have

been reached for 763 stars.

In Chapter XIV I drew attention to the low number of small P. M. It
was also investigated there whether this phenomenon may be ascribed to :

1.nbsp;inaccurate values of A[i.\'ct and Afi-\'J;

2.nbsp;an incorrect system, to which the P. M. were reduced;

3.nbsp;the parallactic motion;

and it was proved that none of these three possible explanations accounts for

the shifting of the zero-point of P. M.

I further examined whether these small P. M. possibly lie in the
direction of one of Kapteyn\'s two star drifts. For the centre of the plate the
position angle (as asually counted from N. to E.) of the drifts are 122° and 240°.
whereas the P. found by me, besides having a maximum, pretty well
coinciding with the direction of the parall. P. M. (for which = 133°.) show
a second maximum for p = 302°. Hence also the two drift theory cannot

account for this second maximum.

Although it is still possible that this curious phenomenon is caused by

a magnitude error, since Apt\'« and Ajx\'j were determined from the P. M. of
very bright stars (which point cannot be settled with the material at my
disposal) and although consequently it cannot be said with certainty that
this maximum of P. M. is real, it deserves to be mentioned all the more since
it is in a direction parallel to the Milky Way and approximately towards
the Apex. It is nmcli to be desired that more certainty on this point could be
obtained unless it should appear that the above mentioned causes singly or
combined are the true reason. It is therefore to be hoped that within a reason-
able time similar material may be available for other parts of the sky in order
to be able to settle whether perhaps we have here a drift of faint stars (the
average mag is 11.7). not coinciding with the drifts known at present, or
whether we must think of a rotation parallel to the Milky Way, as was already

suggested by Schoni-eld\')•
1) V. J. S. 17. 255.

Explanation of the tables.

The tables will require little explanation. Next to the current numbers
the diameters are given, from which the magnitudes were derived in
Chapter XIII.

The columns 3—8 contain the values M^ M, M3 (See Chapter IV);

9—14 „ „ „ m, m^ ms ( „ „ VI);
22—27 „ „ » V-2 1^3 ( „ ,, VIII),
which, averaged with the weights 1, 1 and 2, furnish the final relative
P. M.\' fx\'« and [x\'j. In order to obtain the absolute P. M. the corrections
A {x\'a and A [x\'j which were calculated in Chapter X must be applied. These
corrections are given at the foot of each page of the tables. The sign î after
these stars indicates an uncertainty in the measurements, generally caused

by vagueness or oblongness of the images.

Under a; and y the rectangular coordinates of the stars are found,
referred to the middle of the plates: a = 2* 12quot;\' 3-5, S = 56^33\'. These

coordinates are given to OM, j) being about 1\'.

Under the heading B. D. or Br. and St. we find either the number
in the Bonn Durchmusterung or, for stars not included in this catalogue,
the current numbers, to be found in „Les Positions des étoiles de h et ■/
Persei et de leurs environsquot; by IMiss Bronsky and j\\Iiss Stebnitzky (Mém.
de l\'Ac. Imp. des Sciences de St. Pétersbourg, série VIII, 2, nr. 7). The B. D.
stars occurring also in this paper are marked with an asterisk. Under a 1900.0
and 0 1900.0 we finally have the positions of the stars for the equinox 1900.0,
a being given to T and 0 to 0M. As far as these were measured by the Misses
Bronsky and Stebnitzky, a and 0 have been derived from their values;
for the remaining stars they were calculated from A;andy. Since the coordinates
a and o are only given to identify the stars, great accuracy was not wanted.

TABLES

M,

Ms

2/1 19«! r
f 0^094
185
9
72