r^T J\'r7\\

i \\ i^/% J l I V

A \'Ö I

^ M M\' M \\/

I ^ I J

v..

■

■ .......

uf^r.

SS \'yv-feiS-;.:.

ivVff.V

■Lv/:

viv.-;:

•A*\', v^nbsp;■nbsp;A v.-nbsp;■ ■ \' . r-wnbsp;...... .-V \'

iiïVjr-• ^ \'»Vi/:;-\'.-ii^. ,•■ \' . , • \'.J^. \'

STELLINGEN

STELLINGEN.

I

Het karakter der lichtwisseling van R V Tauri tusschen J D 7129
en J D 7687 is onbekend.

II

De uitkomsten aangaande S 5 Cygni, door Whittaker en Gibb af-
geleid met behulp van het periodogram dezer ster, hebben geen waarde.

III

Nijland heeft als kenmerk van zijne interpolatie-methode aangegeven,
dat het schatten der verhoudingen van de in „Stufenquot; uitgedrukte helderheids-
verschillen hoofdzaak is, de absolute waarde van deze verschillen bijzaak.
Men moet echter beide schattingen als even belangrijk beschouwen.

IV

Ten onrechte meent Shapley bij de afleiding van de hchtkromme
eener /g\'o/-veranderlijke, stuf en waarnemingen te mogen verwerpen wanneer
hij tevens over photometrische waarnemingen kan beschikken.

The Astrophysical Journal XXXVIII No. 2.

V

Aan beschouwingen, zooals Kiess, Martin en Plummer en ten slotte
Prager gegeven hebben over R R Lyrae, had een kritisch onderzoek van het
waarnemingsmateriaal vooraf behooren te gaan.

Lick Bulletins VII; Monthly Notices LXXV; Sitzungsberichte der Königl. Preuss. Akademie der
Wissenschaften 1916 VIII.

Het optreden van „stationna.requot; radianten is door W. H. Pickeking

voldoende verklaard.

The Astrophysical Journal XXIX.

VII

De xr^eenrng van Easton dat ons zonnestelsel zich dichter bij C.g^.s dan
^.tA^rh^ndi, heeft door sterren-tellingen steun gevonden.

VIII

De radiale verdeelmg der straling over de zonne-schijf is voor de lange
golven gelijkmatig.

IX

Voor het onderzoek van de wisselende samenstelling van den damp-
kring der aarde zijn stralingsmetingen onontbeerlijk.

X

Het IS niet aan te nemen dat bij het tot stand komen der geluidstilte-
gordels de bovenste lagen van den dampkring een rol spelen.

^ Veisl Kon. Ak. v. Wetlsch. Amsterdan. XXIV. He.el en Dan.pknng

XI

De door Van Lohuizen bij de afleiding van zijne spectraalformule
toegepaste benadering is niet geoorloofd.

^ VAK LoHnizKK. Bifdrage tot de kennis van lijnenspectra. Den Haag X9X..

XII

De beoefenmg der sterrenkunde m Nederland zou er bij ^^
wanneer een der sterrenwachten werd vrijgemaakt van het Universitair

verband.

XIII

De strijd tegen den oorlog zal eer gewonnen worden door de beoefenaars
der natuurwetenschappen, dan door de pacifisten.

^^ , l^ir Li

THE VARIABLE STARS
R SAGITT^ V VULPECUL^

RV TAURI

PART I

AN ANALYSIS OF THE
LIGHT-CURVE OF RV TAURI

PROEFSCHRIFT

TER VERKRIJGING VAN DEN GRAAD VAN DOCTOR
IN DE WIS- EN STERRENKUNDE AAN DE RIJKS
UNIVERSITEIT TE UTRECHT OP GEZAG VAN DEN
RECTOR-MAGNIFICUS DR. ERNST COHEN HOOG-
LEERAAR IN de FACULTEITDER WIS- EN NATUUR-
KUNDE VOLGENS BESLUIT VAN DEN SENAAT DER
UNIVERSITEIT TEGEN DE BEDENKINGEN VAN
DE FACULTEIT DER WIS- EN NATUURKUNDE TE
VERDEDIGEN OP VRIJDAG 7 APRIL 1916 DES NA-
MIDDAGS TE 4 UUR DOOR

JAN VAN DER BILT

GEBOREN TE KAPELLE (Zuid-Beveland)

BIBLIOTHFrK DFR
WJKS\'JNiVtnbsp;17

W T R E C H T

drukkerij J. van boekhoven — utrecht

m

8gt;i \'

i i \\

■ -■»\'Inbsp;^■•S\' --\'^-O\'.V ■nbsp;) ■•■.•

iHT lo m\'fjm/.
igt;^lJA^^7H ^fö awjjiiirjLi

\'M

fTgt;:• ;r]TH■ / ,• -pa?.--mï -ioW.. ^

RECHERCHES
ASTRONOMIQUES

DE L\'OBSERVATOIRE

D\'UTRECHT

VI

utrecht

J. VAN BOEKHOVEN
1916

THE VARIABLE STARS

R Sagittce V Vulpeculce
RV 7 ami

JBY

J. VAN DER BILT

observator at the utrecht observatory

PART I

An analysis of the light-curve oi R V Tauri

S;.

f R.-

^H/crejjjjaAm

n-wi.

CONTENTS.

Introduction
Chapter I.

Chapter II.
Chapter III.
Chapter IV.
Chapter V.
Chapter VI.
Chapter VII.
Chapter VIII.

Chapter IX.
Chapter X.

Appendix.
Summary.

The reduction of the observations to a photometric
scale .......................

The reduction of the observations to one observer......

The magnitudes....................

The light-curve.....................

The maxima and minima.....................

The periodogram .......................

A closer examination of the three periods.............

The three periodicities combined, and the possibihty of

a solution with only two periods...............

A detailed analysis of the three waves

On a possible interpretation of the results of the analysis

The leveled light-curve . .......................................pl^^^nbsp;^

The long periodicity.........................jj^

The curves resulting from periodogram-analysis ............platenbsp;II b c and d

Stars surrounding R V Tauri ................................................pj^^^nbsp;\' jjj

tv ■

Mv iSr, :
Pls.-v-

• öhj\'^moioriq ß Oi ^noitr- Jf. ^dt ïlt;gt; noiîorf^--\' .«ilTnbsp;.1 jiailt;ïAHgt;gt;

■. ■ ■■ ; - \' • ■ \' ■ • • \' - ■ ■ ■ - : .. ^i-itnbsp;shtiatt^

^^ \' • • v ■ .nbsp;. .V .«^TfAoD

iv Mrr., 0 ■ .... . ......... .-.r. ;, ,,,, .-frtjs^iioböngq yrfl^^nbsp;. .r/ a^-r^RD

: ....................: iiagt; i^A

\'.\'.in;\' - ■■■sV., ■

■■■ , .............

INTRODUCTION.

Ten years ago, at a meeting of the Dutch Physical and Medical Congress,
Prof. Dr. A. A. Nijland, Director of the Utrecht Observatory, maintained
the view that the variable stars of the Mira type must be considered as a
physically connected class, not only on account of the well known corre-
lation between their periods and their colours, but also on account of the fact
that the lengths of their periods are grouped around the mean value of 300
days, according to the probability-curve. The number of variable stars and
the knowledge of their elements have been considerably increased since
then, and it is worth while to see whether Nijland\'s view still holds good.
If we take the periods from the latest pubhcation available (E. Hartwig,
Katalog und Ephemeriden veränderlicher Sterne für 1915), and if we consider
all the periods in excess of 100 days, the material consists of 532 periods,
ranging from 100 to 698 days. The mean period is 297, the probable error
64; the first of these quantities (the mean period) enables us to give the
observed number of deviations from the mean, between successive limits; the
second quantity enables us to calculate the same numbers, considering these
deviations to be quot;accidental errorsquot;.

The results are given in the following table:

This table shows that the agreement is all that can be desired, seeing
that the assumed number of 100 days for the shortest period is somewhat
arbitrary, and only the periods, not the hght-curves, have been considered,
whereas it is not certain whether all the stars with periods of about 100
days show a hght-variation of the Mir a type.

Since m the Ephemeris there are 117 stars of the Algol class, 82 Cepheids,
17 and 8 stars belonging to the Lyrae and ? Geminorum types respectively,\'
and 4 stars, which show the characteristic hght-variation of U Gemtnorum,
making a total of 760 stars belonging to definite classes, we may conclude,\'
in accordance with our present knowledge, that 55 % of the variable stars
have not yet been admitted to the typical classes. Nor has there been in the
last 60 years any reason to estabhsh a new class of variable stars.

There is perhaps one exception. At the close of a paper by F. H. Shares
and E. S. Haynes, concerning the light-changes of F = F14 Tauri *) (period
about 79 days), after observations made at the Laws Observatory of the
University of Missouri, between 1906 Nov. 27 and 1908 March 11, the following
interesting remark occurs:

quot;The unusual character of the variation, recorded by these observa-
quot;tions recalls that of the stars R SagtUae and V Vulpeculae, both of which
^\'^\'have been classed as of the Beta Lyrae type, and both of whose light-curves
\'\'have undergone very curious changesquot;

And further on:

quot;Thus far it has not been possible to formulate a satisfactory theory,
quot;accounting for the alterations of light-curve in either R Sagittae or V Vulpe-
yiae. The discovery of a third object presenting some points of resemblance
quot;to these stars is therefore a matter of considerable interestquot;.

In these sentences we find, mentioned in the same connection,
the names of three stars, none of which belong to the Mira variables or to
any of the three standard types of short period variables, the hght-
variations of which however seem to be ruled by the same (apparently com-
plicated) law.

See quot;Astronomische Nachrichtenquot; no. 4705.

Let us first consider the data on which the remarks of Seares and
Haynes are based.

The variabihty of the star R Sagittae was detected in the year
1859 by J. Baxendell Sr., whose first series of observations showed the
occurrence in the hght-curve of a distinctly marked secondary minmium
between each of two principal minima, thus presenting a similarity with
the Lyrae type. But the period was much longer (70^ 88) and the light-curve
far less regular, since the maxima precedmg and following a principal minimum
had unequal brightness; and the elements, computed from observations
extending over a period of 9 minima, could only represent these with residuals
to an amount of 3 days. Thus, Baxendell concluded that quot;this star may
quot;be ranked in the class of moderately regular variablesquot;.

Till 1865 R Sagittae was only observed by its discoverer, but in that
year E. Schönfeld became interested m the star, and his observations soon
led him to the same conviction, that no mean curve can exactly represent
the remarkable hght-variation. In the quot;Astronomische Nachrichtenquot; no. 1857
he deals elaborately with the observations made by Baxendell and himself
up to the year 1871, producing the first, but at the same time the last treatise
on this variable. It had been proved already that the elements derived from
his earlier observations gave the epochs of minimum brightness too early,
and that many principal minima of the years 1869 and 1870 were much
fainter than those of the years 1865 and 1866. Though for his exhaustive
discussion 25 principal minima were available, extending over 58 periods of
about 70 days, and a quadratic term had been involved in the formula, there
remained large residuals.

This formula ran as follows:
Ep. of min. = 1865 Oct. 24.587 42338 (E—31) — 0^0037369 (E—31)^

The residuals have the largest values around the principal minima,
and Schönfeld makes the interesting remark, that they look as if a regular
process of the light-variation is disturbed by causes working for a long time
in the same direction.

quot;Man kann sich dies so vorstellen, dass die störenden Ursachen eine
quot;Zeitlang wenig oder gar nicht wirken, und dass dann ihre Wirkung plötzlich.

^\'^^oder wenigstens sehr rasch anwächst, sich auf dieser Höhe einige Zeit erhält,
quot;und dann ebenso rasch wieder abnimmt.quot;

But, besides these irregular changes, the period seems to have shortened
since the year 1860, and the secondary minimum is no longer, as a rule, ob-
served to fall midway between two principal minima.

The observations of the years 1871 and 1872, did not confirm the shorte-
ning of the period, and those of 1873 gave a slight mdication of its lengthenmg.

Next year Schönfeld drew attention to a phenomenon of still greater
importance, viz. that of the changing brilhancy of consecutive mmima.

quot;Während in den letzten Jahren die Minimalhelligkeit durchgehends
quot;über der der mittleren Curve geblieben ist, und ins besondere die vom Jahre
^\'^\'1869 um durchschnitthch 6 Stufen übertroffen hat, ist andererseits die
quot;Helligkeit im secundären Minimum stets kleiner gefunden worden als früher,
quot;so dass man, wäre R Sagittae erst neuerdings entdeckt worden, geneigt sein
\'\'könnte die beiden Minima nicht zu unterscheiden, und die Periode nur 35T
\'\'lang- anzunehmenquot;.

The observations made by Glasenapp m December 1875 and during
the year 1876, according to Beljavsky who reduced them, point to the
same fact. He concludes his paper on that subject by saying: quot;Der Stern
quot;R Sagittae scheint einem ausgeprägten Cephei-Typus mit einer Licht-
quot;änderung von 35.3 Tagen Dauer anzugehörenquot;.

Schönfeld, after having been called to the Directorship of the Obser-
vatory at Bonn, was obliged to stop his work on variable stars, in consequence
of which his researches into the character of the light-changes of R Sagittae
came to a premature end.

S. C. Chandler Jr.^ well aware of the peculiar behaviour of this star
m the year 1875, formulated his conclusions in the following sentence:

quot;The results, when charted, exhibit a well marked minimum on Sept.
quot;22, but only a slight depression of the light-curve at about Aug. 23. These
quot;phenomena, which seem to indicate a reversal of the principal and secondary
quot;minima as given by Prof. Schönfeld, are so anomalous, that, as I cannot
^\'^\'deem the observations, on which they rest, though somewhat fragmentary,
quot;to be at fault, I have thought proper to submit them in detail.quot;

And still the full measure of the caprices of this star had not been
observed. Baxendell, who had stopped his observations in the year 1869,
but had taken them up 8 years later, reports on his estimates of the year 1885,
that quot;in August, when a secondary minimum was due, the magnitude remained
\'\'constant {at 8™.8) for a feriod o/ 16 daysquot;.

But after Sch5nfeld, though the star was put on the list of many
observers, none of them has ever tried to detect the true character of its
light-fluctuations. In the meantime, the problem received an increased in-
terest, when in the year 1904 Stanley Williams discovered the varia-
bihty of the star BD 26°3937, V Vulpeculae, the light-curve of which
exhibited, in approximately the same period, the same type as R Sagittae
had shown at the time of its discovery. The period was 75.3 days, the secondary
minimum fell midway between two principal minima and exceeded these
by a full magnitude in brightness. But here too, in the years that followed,
information of a different kind came in, first of all from E. C. Pickering,
who, from a series of 58 photometric measurements made by O. C. Wendell,
remarked: \'^they show that the successive light-curves are not the same,
quot;and that therefore this star cannot be regarded as of the (5 Lyrae type.
quot;The form of the hght-curve appears rather to be of Class II¹) and resembles
quot;i? Sagittae in having a secondary minimum nearly midway between the
quot;principal minimaquot;.

The short photometric series of Wendell was followed by an extensive
hkewise photometric, series by F. H. Seares and E. S. Haynes. Neither did
they find the (i Lyrae type confirmed, but thought that the star had to be in-
corporated with the short period variables, with a period half as long as that
derived by Williams; a result running parallel with Schonfeld\'s remark
about R Sagittae in the year 1875.

What has been said of R Sagittae apphes equally to V Vulpeculae.
The hght-changes of this star also have not been thoroughly examined over a
period of sufficient length. Moreover, the conformity of the character of
the light-variations of both of them has never been proved.

1nbsp; Variables of long peiiod.

Shortly after the discovery of V Vulpeculae, Mrs. Ceraski, examining
the plates taken at the Moscow Observatory by Blajko between 1895 and
1905, found the star BD 25°732 to be variable. Very soon after its dis-
covery, this star, RV^V 14. Taun, happened to be put on the programme
of several keen observers (i. e. Enebo, Haynes, Nijland), and it has
been kept under examination by these three observers in such a regular
manner, that the story of its light-variation - apart from the unavoidable
gaps resulting from the star\'s place near the ecliptic - has been set down
very completely. But of this star also the data of observation have not been
uniformly studied, and what has been published by the separate observers was
derived from their personal work only. The results they have reached, strongly
recall the vicissitudes of R Sagittae and V Vulpeculae, with this exception,
that the order of things is now reversed. In the first instance the star was
considered by both Seares and Enebo to be one of the short-period type,
with a period of about 40 days; but a closer examination revealed the fact
that the light-curve was of the S Lyrae type, havmg a period of about 79
days, which compares with the periods of 70 and 75 days assigned, as we have
seen, to the two stars which were discovered earlier.

Seares and Haynes were struck, moreover, by the appearance of
irregularities in all the phases of the hght-variation; and it was this fact, taken
in connection with the general character of the hght-curve, which led them
to the remark quoted on p. 2 of this work.

After the rejection of the short-period hypothesis, Enebo discussed
his observations in three separate papers. In the first of these he says that the
star probably belongs to the ^ Lyrae type, but that there exists a secondary
variabihty, which finds its expression in the fluctuating values of the maximum
brightness. Taking Blajko\'s photographic magnitudes into account, the
period of this secondary variation seems to him to be about three years.
The mean period of the principal variation is 78.57 days.

In the second paper, the suggestion regarding the long-period variabil-
ity of the maximum, brightness is fully confirmed, the phase of that variation
being in the spring of 1910 the same as in that of 1907. Moreover it would
seem that, roughly speaking, the minimum brightness follows a parallel curve.

The miiiiiTLa observed in the beginning of 1910 differ fro in each other so httle
in brightness, that a strong probabihty arises, that the succession of principal
and secondary minima will shortly be reversed, thus reproducing the same
phenomenon as has been observed in R Sagittae and V Vulpeculae. The mean
period of 79.0 days, given by Seares and Haynes, seems to be more accurate
than that of 78.57 given in his first paper.

The third paper corrects the 79.0 days\' period, since it was discovered
that the minima of 1911 March and 1912 January feU, respectively 4 and 6 days
earher than was anticipated. A mean period of 78.76 days must be preferred
to that of 79.0 days. In accordance with the three years\' period of the secondary
variation, the maximum brightness reached its greatest value towards the
end of 1911.

Nijland, arguing from a first series of observations, is inclined to
reject the Lyrae type, and, in view of the many different values of the
depth of the consecutive minima, to consider the star an irregular short-
period variable, with a period of about 39 days.

All that has been said and suggested about the light-changes of
R Sagittae, V Vulpeculae and\' V Tauri, justifies a thorough investigation
of the observed light-variation of these stars, from the date of their discovery
till the present time; and the fact that all three have been under
regular observation at the Utrecht Observatory during the last 8 years
{R V Tauri by Nijland, R Sagittae and V Vulpeculae by the writer), provides
an additional reason for undertaking the present work. It embraces the follow-
ing points:

(«) to secure copies of all the series of observations in the form of the
original note-book records.

{§) to adopt a scale of photometrically determined magnitudes of the
comparison stars, and to bring the individual conceptions of the observers,
regarding the differences between the comparison stars, into harmony with
the photometric scale.

(/) to derive from this newly adjusted scale the brightness of the variable,
and to construct a light-curve, which shall be homogeneous, from a consider-

atlon of the systematic differences between the results obtained from the
various observers.

to study the details of the light-curves, to embody in a mathematical

formula the law which governs them, and to seek for a possible physical inter-
pretation.

(0 and, finally, to see whether the three stars under investigation may be
regarded as specimens of a new class of variables.

The latter point is of special interest, since Enebo, m his latest public-
ations, announces that the same kind of hght-variation as is found mRV
Tauri, is to be observed in the stars T F = F 29 Andromedae and SW = V 23
Persei, and probably also in Z = F 16 Ursae majoris, T Y = V 32 Dracoms

R Y = V 17 Laceriae, T X = V 31 Perset, AV =VlQ Cygni and U Y =
V 38 Pegasi.

Though the present writer is unable to estimate the soundness of this
presumption, it appears to him that a warning should be uttered against
any premature association of an observed light-curve with such a compli-
cated type as that of F Tauri.

The material at hand consists of

1938 observations of R Sagittae.

^^^ » ,, V Vulpeculae.
1129nbsp;,,nbsp;^^ R V Taurt.

makmg a total of 3884 observations. Most of these have never been published
before, and were handed to us by the courtesy, either of the observers them-
selves, or of the astronomers in whose care the manuscripts have been placed
Our special thanks are due to Prof. H. H. Turner and Miss M. A.Blagg for
Baxendell\'s valuable observations of R Sagittae, and to Prof F H Seares
for sendmg us, m the most complete form, not only his own observations

but also those made by Messquot; E. S. Haynes and H. Shapley, upon F VulpL
culae and R V Tauri.

In the following table a summary is given of the manuscripts used.

As we have aheady mentioned, the observations of F Tauri have been
very complete. In the series of R Sagittae gaps of several years appear, and in
that of V Vulpeculae unnecessary gaps of several months. Moreover it^proved
possible in the case oi R V Taun alone, to reduce the whole material to one
single observer; whereas the Hght-variation of the other two could only be
studied in separate periods; a fact which, to some extent, affects the uniform-
ity of the results. For these reasons it seemed to be preferable to devote
attention first of all to R V Tauri, rather than to consider the stars in the
chronological order of their discovery.

CHAPTER L

the reduction of the observations to a photometric scale.

A. The polarizing photometer.

In the summer of 1913 the instrumental equipment of the Utrecht
Observatory was enlarged by the purchase of a polarizing photometer. It
was supplied by O. Toepfer of Potsdam, and differs from the original Zöllner
type only in the fact that the lamp is an incandescent one of 2 candle power
(requirmg 1 Ampère at 8 Volts), while the colorimeter is replaced by col-
oured glasses. It can be adjusted both to the 4} inch and to the 10 inch re-
fractors; in the former case the eye-piece has a field of 50\' and a magnifying
power of 49, in the latter the field is 30\' and the magnifying power 78.

The constancy of the current can be kept under control by using a
milhampère-meter. The influence of any change in the current-intensity
on the brightness of the artificial star can easily be expressed in terms of
stellar magnitude. For this purpose, while continually varying the current-
intensity, we have compared the maximum brightness of the artificial star
with that of a few faint stars; and we have moreover carefully noted the
reading of the ampère-meter at which the artificial star became invisible.
The results of several nights were plotted and a correction-table derived
from the curve. This table shows that, from the maximum reading used,
VIZ. I = 104.5 (1.045 Ampère) to I = 85, the change in magnitude may be
considered to be directly proportional with the change in current-intensity,
and that, for values of I smaller than 0.85 Ampère, the brightness decreases
more rapidly than the intensity.

But these low values of current-intensity have seldom been used in
the observations.

The artificial star becomes invisible at a current-intensity of 0.64
Ampère.*).

The light-intensity of the artificial star can also be changed by the

use of different diaphragms. But though there are 8 of them, it proved in

practice to be preferable always to use the same opening of an intermediate
size.

The sector for the coloured glasses contains three openings; one of
them is empty, while the other two contain a red and a blue glass respectively.
If the intention of the constructor was to give a yellow, red or blue tint
to the artificial star, he has not succeeded, since, even with the quot;bluequot; glass,
the colour, as compared with that of a few Potsdam stars, is not appreciably
different from the designation WG of the Potsdam scale. In consequence
of this the whole series of photometric measurements has been made with
the quot;bluequot; glass; but the observation of really blue or white stars has proved
to be very difficult.

The star to be measured has always been compared with the image
formed by reflexion on the front side of the glass plate. Since care was taken
to keep the light source to the right in every position of the telescope, this
was always the right-handed image. During the observations it was found
convenient always to place the object between the two images of the arti-
ficial star, which are 5\'.4 apart; the line joining the observer\'s eyes was kept
parallel with that joining the star images.

brightness of the comparison stars for R F Tauri had been photometrically
etei mined, the same curve could be derived, with still greater accuracy, by determining from a
large number of measurements, the full brightness (M) of the artificial star at different current
intensities (I). This gave the following results:

Hie distance of the object from the right-handed image of the artif-
icial star was not always the same; for experience showed that, in ascertain-
ing their equal briUiancy, a feeling of confidence in the results was attamed
m the case of the brighter stars, when the images were kept at a distance
of about r; while, m the case of the famter stars, the equality could best be
ascertamed when the images were seen as a relatively close double star.

An quot;observationquot; consisted of 8 readings of the intensity circle two
m each of the quadrants. These two were obtamed(l) by diminishing the
light of the artificial star until it had attained the brightness of the object
(2) after havmg carried on the darkening process nearly to the zero-pomt by
brightening the image again until the equahty of hght was once more verified
The current-intensity was read off before and after each observation and the
changes were taken mto account by using the correction-table mentioned above

Group
Pleiades

BD 20°

{Praesepc)

z/ m
M.K. I B

Before proceeding to the measurements of the magnitudes of the com-
parison stars for R V Tauri, V Vulpeculae and R Sagittae, a small series of
test-measurements was made with the 4} inch refractor on a number of select-
ed pairs of Pleiades and Praesepe stars. The magnitudes of the former were
taken from the Müller and Kempf list in the \'\'Astronomische Nachrichtenquot;
no. 3587; those of the latter from the quot;Potsdam Durchmusterungquot; (PD).

The preceding list contains, in the fifth column, the results of these
test-measurements. Each pair has been measured on one night only; no ob-
servation has been rejected.

The sixth column gives the mean error of the interval B; it shows
that the differences B--M. K. in the last column are not much greater than
could have been expected.

Together with these measurements the number of observations made
with the Zöllner photometer was 341, most of which concern the three vari-
able stars, forming the subject of this paper. To these are added a small
number of measurements of the variables R Leonis and R Boötis, and of
some faint stars mentioned on p. 10.

From a consideration of the different data which can be derived from
these measurements, it appears that the mean error of an quot;observationquot; (i. e.
of the mean of 8 readings) is 0\'^037, that is to say, about half the value of
the mean error given in the preceding table. The intensity-circle shows an
index-error of 0°.5, whereas there is no appreciable effect clue to eccentricity,
or to the personal error alluded to on p. 12.

B. The material at hand.

The following hst contains a complete enumeration of the observations
made on the star under investigation since its discovery, and which were
either available when the work was started, or were sent to me in response
to a request, published in the quot;Astronomische Nachrichtenquot; no. 4693 and
the quot;Astronomical Journalquot; no. 657.

Observer

Mrs. Ceraski
E. Hartwig .
L. Pracka...

S. Enebo . .

E. S. Haynes

E. H. Seares
} ^

H. Shapley . .
A. A. Nijland

J. Voûte ....
A. Brill ....
K. Boda......

Epoch

1895/1905
1905/1907
1906/1907

1912/1915
1906 /1908

1908 /1909
1908

1908
1908/1910
1908 /1915

1908 /1909
1913
1913

Reference

Discovery. Measures on plates taken bj\'
Blajko. Astron. Nachr. no. 4010.

Veröffentlichungen der Remeis-Sternwarte
zu Bamberg. Reihe II. Band I, Heft II.

Beiträge zur Untersuchung des Licht-
wechsels verändei-licher Sterne. 1—2.
Prague 1910. (Bulletin international del!
i TAcadémie des sciences de Bohème ij
14—10—1910).nbsp;I

I Beobachtungen veränderlicher Sterne, an- i
: gestellt auf Dombaas (Norwegen), \'teil p
II; IV; VI. (Archiv for Mathematik og m
Naturvidenskab Bd. XXIX: XXX- Ii
I XXXII.)

^ MSnbsp;•!

! PubHcations of the University of Missouri. \'
Laws Observatory Bulletin no. 14.

IMS ■

Publications of the University of Missouri.
Laws Observatory Bulletin no. 14.

MS

MS

MS. Preliminary results have been publish-
ed in the quot;Astronomische Nachrich-
tenquot; nos. 4404, 4485, 4560, 4642 4765 •
and 4797.nbsp;|

MSnbsp;j

MSnbsp;i

MS

comparison stars.

The estimates of the various observers have, as a rule, been made with

the use of the 10 comparison stars given in the table below. The first column

contains either the star\'s designation after the quot;Bonn Durchmusterungquot;, or

the coordinates which relate them to the variable. The next 5 columns give

the letters, with which these stars appear in the pubhcations or MSS of

the observers. The last column gives the notation adopted in the following

pages and which is also used in the chart, to be found at the end of the
volume.

HartwigI
Pracka j

Nijland| Brill
Voute ; Boda

A

a

h ^ c

d

a

c
e
d

Comparison stars for R V Tauri.

No photometric magnitudes of these stars were available. The exten-
sive series of measurements of the variable, which were made with the wedge
photometer of the Laws Observatory, furnish accurate values for the m-
tervals c—a and f~~c- but none of these 3 stars have been brought into con-
nection with an existing photometric scale.

A complete hst of photometric magnitudes of the comparison stars
being of great value for the reduction of the observations, the necessary meas-

*) Probably by a misprint Pracka gives 26° 733.

urements for this have been made by the writer with the photometer des-
cribed above. The magnitudes so obtained are based on the photometric
values, taken from the Potsdam quot;Generalkatalogquot;, of the two following

stars.

A = BD 26° 759 = PD 2966 Colour WG M = 7-.66
B = BD 25° 720 = PD2816 Colour WG M = 7\'quot;.69

The star BD 25° 731 = PD 2890. M = 8^.08, which, by its cen-
tral position among the comparison stars, lent itself still better to the pur-
pose, could not be used on account of its colour GW- (See p. 11).

Using A and B as a basis, the magnitudes of the comparison stars a—;?,
have been derived by photometric triangulation, after the scheme given
below. This produced 25 equations (considered to be of equal weight), with 10
unknown quantities, which had to be solved by the method of least\'squares.
The faint stars k and I have not been measured, since they have been used
only twice and by a single observer.

The photometer was attached to the 10 inch refractor, and each inter-
val was measured on three separate nights. Though the observations were
commenced on 1914 March 2, when the star group was still conveniently plac-
ed, the unfavourable conditions of the weather did not allow of the series
of measurements being completed in similar circumstances. On a few occas-
ions the measurements had even to be made at zenith distances of 50 to 60
degrees. This should generally be avoided here, both on account of the haze
which lies above the town, and because the observatory is shut in by high
trees, especially on its Western side. Though the corrections for atmospheric
extinction amoxmted to no more than 0™.01 and 0™.02, they were duly applied.

March

18

28

1 .65

1.801

1.84
1.93

2.77
i2.85

12.86

0.97
1.08

0.31

i

0.17

10.29

2.061.911.98
1.01
0.37 0.48

0.65
1.911.971.98

0.99
0.48
0.69 0.86

0.00

mean :0\'quot;.064

Photometric triangulation of the comparison stars for R V Tauri.

The normal equations come out as follows:

5 Bnbsp;- a ^

4A ^ a --- b —
6« — h —

B~
B~
B~

— e~ f^ g —
__ g — /

- d 4:6 - g -

— dnbsp; 5/ — g —

/ 5g —

~ /

These equations can easily be solved by using a method of approx-
imations. When the values of A and B are left as the Potsdam catalogue gives
them, and the values of the measured intervals are applied directly, we get
a set of approximated magnitudes for the comparison stars, from which the
true values can be derived by means of the normal equations. In the present
case the third approximation, which indeed differs but little from the first,
gives the final result.

From these results we learn that the star / must, as regards its photo-
metric brightness, be placed between c and d- but the notations given had
been previously chosen after Enebo\'s hst of comparison stars (so far as his
material carries us), which gives / fainter than d and

The stars g and h may be considered as equally bright. Enebo sees
g lt; h, Nijland g gt; h. Since these differences persist after the reduction
of their hght-scales to that of the photometric, the already adopted notation
for the comparison stars has not been altered.

The observer\'s estimates of the intervals between the comparison
stars give a scale in which the light-values are represented by a sequence
of \'\'stepsquot;, commencing from the arbitrary value of 1.0 for the faintest star.
With the aid of the photometric results, this light-scale must be changed
to a magnitude-scale. This can be done graphically, or by computation.

By the latter method, a set of equations of the form

^ — a, y = p^

had to be solved by the method of least squares. In these equations means
the photometric magnitude of the comparison stars, x the reduced mag-
nitude of one of them, a, the difference, expressed in steps, between ^ and
the other comparison stars, and y the value of one step.

We have slightly changed the above method on the following grounds:

1.nbsp;For certain intervals, the conceptions of the observers frequently
appear to differ from the photometric results to such an extent, that, when
they are admitted into the computation, they are apt to affect\'the accuracy
of the reduction of the other intervals.

2.nbsp;It is evident that the photometric scale is not absolutely correct,
and that the difference between this scale and the observer\'s conception is
not entirely attributable to the latter.

The following method, which is used in the present paper, seems to
meet these objections with some adequacy.

The light-scales of the observers are first constructed and reduced to
the photometric scale. The resulting magnitude-scales are then compared
with this scale; and when, for a certain star, the values taken from the mag-
nitude-scales differ from the photometric value in the same direction for all
observers, this value is changed in the same direction to an amount not
exceeding the mean error.

With this definitive photometric scale, the construction of the mag-
nitude-scales is repeated, and new values for the differences Observer minus
Photometry result. If, for one or more of the stars, this difference exceeds
the somewhat arbitrarily chosen value of 0^.20, such stars are excluded; the
step-values of the rem.aining stars are again reduced to the photometric scale,
and, finally, the excluded stars are inserted, using for this purpose the value
of one step, which has resulted from the final computation.

By this m.eans we obtain this double advantage; that the greater part
of the comparison stars can be reduced with fair accuracy to the photometric
scale; and that a value, best suited to the observer\'s conception can be adopted
for one or two divergent stars, without spoiling the precise results of the others.

For each individual observer, the value of one step has been treated
as a constant quantity throughout the whole range of the Hght-variation.

In the case of R V Tauri the prehminary reduction to the photometric
scale gave the following differences Observer minus Photometry. Unit = 0quot;\'.01

(E = Enebo, L = Laws observers, N = Nijland, V = Voûte).

The star d does not appear in this comparison, since it has been used
by one observer only, and by him on very few occasions.

The differences for the stars 6 and / being all positive, we have made
their photometric magnitudes fainter; while the magnitude of 0 has been
made brighter, to an amount as large as the mean error would permit.

This mean error does not result from the normal equations, as they
have been solved by an approximative method; but it can readily be estimated
from the mean error of the measured intervals (see p. 17 last column). This
comes out at 0-.064, making the mean error of each of the magnitudes forming
0\'°.064

= 0™.045*)

the interval = -

Thus the photometric scale finally adopted, was not that resulting
directly from the solution of the normal equations, but the following:

mnbsp;m

« = 9.37nbsp;= 10.64nbsp;g =z 11.37
b = 9.76 e - 10.91 h = 11.34
_ c 10.36 / = 10.63

*) Since this quot;internalquot; mean error must be smaller than the quot;externalquot; mean error which
would have resulted from the normal equations, we are sure to be on the safe side as to the
corrections, which we now apply.

D. The observations of Hartwig.

The observations of Hartwig have been made at Bamberg with a
10 inch refractor; they are very few in number, one estimate being made in
1905, one in 1906, and 5 in 1907.

It is, sometimes, very difficult to bring the observations of this observer
mto connection with a photometric scale, for the following reasons. They are
recorded in an unusual form, the differences in brightness being very seldom
expressed in steps. As soon as these differences exceed about 0\'^25, three
different symbols are used, two of which are explained in the introduction,
but in terms too vague to be of much use to the computer. The latter has
thus to compare the records themselves for more detailed information; and
this leads to still further confusion. For each variable the adopted magnitudes
of the comparison stars have been given, but without any information as to
their origin; and in the reductions of the observations these magnitudes are
apparently independent of the instrument used.

The brightness of the variable is derived in a way which is not described,
and which in several cases is wholly incomprehensible. Each method consistently
adopted for examining his results, is found to lead to conclusions different
from Hartwig\'s own. Consequently, since the meaning of Hartwig\'s sym-
bols is not clear, and the course which he followed in deriving the hght of
the variable is not explained, a computer finds himself unable to attach to
the observations the weight which they probably merit.

In the few cases, however, in which R V Tauri was observed, we can
derive more or less trustworthy values for the brightness of the variable, if
we adopt the direct photometric magnitudes of the comparison stars.

Here follows, in the notation adopted by us, a summary of Hartwig\'s

estimates; the third column gives the magmtude derived by Hartwig himself;
and the last column contains criticisms, to which, according to the present
writer, these records are hable.

Comparison stars: BD 25° 725 (A) 8^0 / o\'e

anbsp;8.5nbsp;dj 10.0

cnbsp;9.2

Remarks.

1905 Oct. 10 \\a»vyy c»f

1906 March n \\ A gt; vS.5 a\\ 8.3

1907 March 4
1907 Oct. 15

» V = c

a 6 V » c

a » V )gt; c
a»gt; c = v»/

Apparently reduced as y - i
(a c), neglecting the star/.
Then the first two symbols gt;gt;
have values less than 0?5,
contrary to the statement
in the introduction.

Apparently reduced with a
value 0™.06 for one step and
neglecting the comparison
with A.

Value of one step in accordance
with 2.

The comparison with a has been
neglected.

Reduced with the same value
for one step as under 2 and
3; the comparison with c
has been neglected.

See 1.

The symbol »gt; is not explained
in the introduction of Hart-
wig\'s work; in this case it
seems from a comparison
with 4 not to differ from the
symbol ».

Reduced as = c.

E. The observations of Pracka.

Pracka observed R V Tauri at Bamberg, sometimes with the 10 inch
refractor, sometimes with the inch comet-seeker of the Observatory, and
he obtained 1 observation in 1906 and 7 in 1907.

He too has made an extensive use of the symbols gt;, gt;gt; and gt;gt;gt;.

No explanation of these is given, but apparently no definite value
has been attached to them in the reduction of the observations.

For star-magnitudes within the limits and 12™ Pracka arrived
at the value 0quot;\\06 as being the photometric equivalent of one step. In view
of the small number of observations, we have in their reduction applied this
same value, leaving the photometric magnitudes of the comparison stars
unaltered.

F. The observations of Enebo.

These were made at Dombaas in Norway, in 1906 with a 2| inch
refractor, and from 1907 to 1915 with a 4i inch.

The observations made with the former instrument have been reduced
to the photometric scale with the following result:

m

1 Step = 0.097

Smce none of the differences in the last column exceeds 0^.20, a second
reduction (see p. 19) is not necessary.

From a first review of Enebo\'s long series of observations made with

the larger instrument, it appears that his conception of the differences in

brightness between the various comparison stars was a different one for each

season. To illustrate this, we give, in the following list, for the different seasons the

values, expressed m steps, of the star-intervals which have been generally

used. The second column represents the values for the period 1907 Jan 3 to

April 10, and the following 5 columns give those belonging to the observational

seasons between 1907 and 1912. The last column has been constructed

by makmg use of three seasons, the number of observations being too small
to treat each season separately.

Season
Interval^-

Attention was drawn to the differences exhibited in the foregoing
table in the following way. In the course of a prehminary investigation, Enebo\'s
light-scale was deduced from his observations as a whole, and his results
were compared with those of Nijland in the way described on p. 35. From this
comparison it appeared that the remaining errors, which should have shown
a distribution in harmony with the probabihty-curve, as a matter of fact re-
vealed a systematic discordance, whereby errors of an intermediate value
were found to be abundant, and smaller errors were fewer in number than
were expected. For this reason, we have treated the different seasons separately,
with (as will be shown on p. 41) a corresponding improvement in the results!

The reduction to the photometric scale gave the following results:

I. Season 1907. T\' reduction.

li. Season 1907—\'08. L^\' reduction.

Star Lnbsp;Mnbsp;M—Ph

cnbsp;6.19nbsp;10\'quot;.48nbsp; 0quot;M2

^^nbsp;3.42nbsp;10.64nbsp;—0.27

/nbsp;1-00nbsp;10.77nbsp; 0.14

2-\' reduction. The number of stars being only 3, no star can be excluded
from the reduction. Perhaps the most natural way would now be to dis-
card the chosen criterion, and to accept the first reduction, though for . the

value M-Ph is larger than 0^20. But, on the following grounds we have
preferred another course.

The first reduction is not only unsatisfactory for but for c and / as
well. The following pages will show that, for the stars involved in the final
reduction, the value M-Ph can nearly always be kept under 0^.10.

Further: the differences for the various seasons, though m some wav
irregular, show a marked tendency to a gradual diminution of the value of
one step. Now, while for the preceding season this value is 0^.157 and for
the following 0-105, the first reduction for 1907-\'08 would give it as 0-.056-
that IS to say, a value smaller than that acquired even after six years\' obser-
vation (see p. 29). On these grounds we have preferred to adopt for this
season the values for c and / of the preceding one, and to insert thequot; star .
The origmal estimates give the interval directly = 5.43 steps, and the same

^ -nbsp;The mean is 5.07 steps for a magmtude-interval

of 0™.76. Hence 1 step = 0°M50.

The step-estimates give for the stars d, and BD -h 25° 729 = p,
which were only used on a single occasion, 10\'quot;.37 and 10quot;\\39 respectively.
A second reduction is unnecessary.

) ncl

Though its distance from the variable is somewhat great, such a star presents
itself in h, which has been used with great advantage by both Nijland and
Voûte.

10.35

10.80
10.75

0.074: 0.099

0.092

0.072

G. The observations of Haynes, Seares and Shapley.

With the exception of the measurements taken on 1907 July 31; Aug.
3, 10, 18; Sept 12, 18; Oct 4, 5, 8, which were made with a polarizing photo-
meter, all the observations were made with the Zeiss wedge photometer, attach-
ed to the 71 inch equatorial of the Laws Observatory. A description of this
instrument is given in the Laws Observatory Bulletin no. 7, and the method
of observation has been explained in Bulletin no. 8.

The measurements yield, not only the differences in magnitude
between the variable and one or two of the comparison stars /, c, and but also
an extensive series of mutual differences between the latter. These differences

we have closely examined m the case of each observer separately, with the
following results:

Haynes
d m : obs.

50
82

Since the three values of each interval differ by so little, the mean
value can be taken without having recourse to weights. The result is:

c —a = 0™.72nbsp;f-~c = 0™.41

These values, together with the photometric magnitudes of a, c and /,
give 5 equations, from which the magnitudes to be used for reducing the Laws
observations can be derived by the method of least squares. The solution gives:

H. The observations of Nijland.

These were made at Utrecht, the instruments used being the 10 inch
refractor of the University Observatory, and its 3 inch finder.

The method of observation is the direct interpolation method, designed
by the observer himself.*)

The following table gives the results of the reduction to the photo-
metric scale.

Lnbsp;Mnbsp;M—Ph

1 step = 0™.085 (10quot; refractor)
,, - 0.087 (3quot; finder)
A second reduction is not necessary.

quot;Astronomis\':he Nachrichtenquot; no. 3695.

I The observations of Voûte.

These were made at Leiden with the 6 inch refractor of the Univer-
sity Observatory; the method of observation is Nijland\'s interpolation
method.

The observations have been reduced to the photometric scale with
the following result:

A second reduction is not necessary.
On two days in February 1909 Voûte saw the variable brighter than
On these occasions he interpolated it between a and BD -h 26° 750 which
he estimated to be 6 steps brighter than a on Feb. 1 and 6^steps on Feb. 5.

Using the value of one step as derived above, the resulting magnitude of
BD 26° 750 is 8quot;\\68.

K. The observations of Brill and Boda.

These have been made with the 8 inch refractor of the Astronomical
Observatory of the quot;Physikahscher Vereinquot;, Frankfurt a. M. The variable
has been compared with c and/only, after the direct Argelander method.¹)
The mterval between these comparison stars has been estimated at from 0 to 6
steps, which difference the observers themselves are inchned to ascribe to a
variabihty of the star c. This, however, is not evident from the long series
of estimates made by the other observers.

The mean value of the interval c—/is 3.36 steps for Brill, and 3.21 for
Boda, from which, retaining the photometric magnitudes of these stars, the
value of one step becomes 0quot;\\089 and 0\'quot;.094 respectively. With these values
the brightness of the variable has been deduced from the direct photometric
values of the comparison stars.\'

1nbsp; The observations contain one single comparison with the star e.

L. Summary.

The following table gives a summary of the reductions discussed in

the preceding pages, exclusive of the stars which have been used only by one
observer.

01

02
35
04

—30

In Enebo\'s case we have taken the mean value for the two instruments
and for the different seasons, in Nijland\'s the mean value for the two
mstruments. The first column gives the designation of the stars and the
following 4 columns the differences: Adopted minus Photometric magnitude,
O-^.Ol being the unit. (See also p. 20.)

From this table it appears that the star / is, for all the observers, a fainter
object than is given by the photometric measurement, which is practically
the same for the Laws and the Utrecht photometers. Differences of this kind
which have often been remarked in other cases, must probably be ascribed to
the difference in colour between the real and the artificial stars.

CHAPTER IL

The reduction of the observations to one observer.

With a star hke R V Tauri, which has been observed over a long period
by more than one observer, and frequently on the same night, the opportunity
is presented of reducing all the observations to the standard of one observer,
after a careful investigation of the systematic differences between the various
results. For the purpose of this reduction, that observer should obviously
be chosen, whose contributions have been most continuous, and have spread
over the longest period. In the case oi R V Tauri this observer is Nijland.
Thus our next task will be to give the results of the following comparisons:
The observations of Nijland with the 3 inch finder, and those
made with the 10 inch refractor. Abbreviation N —n.

2.nbsp;Enebo compared with Nijland (N—E).

3.nbsp;The Laws observers compared with Nijland (N—L).

4.nbsp;Voûte compared with Nijland (N — V).

5.nbsp;The Frankfurt observers compared with Nijland (N — F).

In certain cases, allowances have been made for the difference in Green-
wich M.T. at which the observations were taken. As will be seen on pp. 76—-77
the change in magnitude was, in some extreme instances, about 0\'quot;.01 to 0\'quot;.02
per hour; and, had no allowance been made for the time-difference, this would
have led to erroneous results.

Since the observations of Hartwig and Pracka are very few in number,
and fall entirely beyond Nijland\'s period of observation, they have been
excluded from this final reduction.

I N — n. The first table gives, in the second column, the magnitudes
as observed in the finder, and, in the third, the differences N — n, expressed

in O-\'.Ol as the unit. From the second table, it appears that these differences
do not depend upon the brightness. The mean value is N—n = —

Date

1908nbsp;Nov. 4

6
28

Dec. 30

1909nbsp;Jan. 8

15
15

23
7

7

8
]0

Aug. 10
22

Sept. 16
19

Oct. 6

24

n = —0™.17

Interval

9.40—10.00
10.00—10.20
10.20—10.40
10.40—10.60

2. For the argument E, the table contains not only the direct differences
N — E, but also those resulting from (n — O\'^.H) — E, on the nights
when Nijland used the finder only. Here, too, there is no systematic change
corresponding with that of the brightness. The mean value is N—E = —0®.20.

N — E = — 0quot;\\20

3. The number of nights, on which both Nijland (N) and one of the
Laws observers measured the star\'s brightness, being relatively small, the
number of comparisons has been augmented by adding the indirect compar-
isons: (E-0\'quot;.20)-L and (n-0-17) - L. The table contains both the
direct and indirect comparisons. There being no perceivable relation between
difference and brightness, the result is the mean: N^L = ^O^^.OO.

The first and the last numbers have been rejected for the present
purpose.

Date

1908nbsp;Oct. 21

27

28

Nov. 8
I Dec. 19
21

1909nbsp;Febr. 10

16
19
25

April 1
8

1910nbsp;March 2
11