THE VARIABLE STAR
rj GEMINORUM

'■■tTé'

THE VARIABLE STAR rj GEMINORUM

PROEFSCHRIFT TER VERKRIJGING VAN DEN GRAAD VAN
DOCTOR IN DE V^IS- EN NATUURKUNDE AAN DE RIJKS-
UNIVERSITEIT TE UTRECHT, OP GEZAG VAN DEN RECTOR
MAGNIFICUS DR. B. J. H. O VINK, HOOGLEERAAR IN DE
FACULTEIT DER LETTEREN EN V^IJSBEGEERTE, VOLGENS
BESLUIT VAN DEN SENAAT DER UNIVERSITEIT TEGEN DE
BEDENKINGEN VAN DE FACULTEIT DER W^IS- EN NATUUR-
KUNDE TE VERDEDIGEN OP V^OENSDAG 4 JULI 1928 DES

NAMIDDAGS TE 3 UUR

DOOR

EVERT HENDRIK VOGELENZANG

GEBOREN TE MEPPEL

aan mijn moeder en aan de
nagedachtenis van mijn vader.

Het verschijnen van dit proefschrift biedt mij een welkome gelegenheid allen, die tot
mijne wetenschappelijke vorming hebben bijgedragen, mijn hartelijken dank te betuigen.

In de eerste plaats dank ik U, hooggeleerde Nijland voor de stetm en voorlichting
welke ik steeds, en zelfs reeds voor ik naar de Universiteit ging, van U mocht ontvangen en
niet in het muist voor Uw bereidwilligheid om als mijn promotor op te willen treden.

U, hooggeleerde schoorl, dank ik voor de groote wehvillendheid en voortdurende
belangstelling, die .ik steeds van U mocht ondervinden.

Hooggeleerde de Graaff, Uw voortreffelijk onderricht is mij in de praktijk van zeer
veel waarde gebleken te zijn.

De belangstelling en vriendschap van U, hooggeleerde kolthoff, ondervonden zal mij
steeds in aangename herinnering blijveji.

U, hooggeleerde jaeger, dank ik voor de zvehvillendheid waarmede gij mij in de
gelegenheid hebt gesteld in Uw laboratorium te ^verken.

-iÄ-

ft

T -

CONTENTS.

Page

Introduction...........................i

Chapter I. Determination of the light-curve.

§ i, Material available.....................3

§ 2. Reduction of the observations.................6

§ 3, Reduction of the observations to one observer...........19

§ 4. The light-curve......................21

Chapter II. The maxima and minima.

§ 5. List of the minima deduced from the light-curve..........26

§ 6. The maxima.......................29

§ 7. Elements of the variation..................3^

§ 8. General remarks and summary of the results...........34

INTRODUCTION.

As early as 1843 J. F. J. Schmidt, then in Hamburg, suspected rj Geminorum
to be a variable star, but more than twenty years elapsed before he felt sure of it.
From September 27th 1863 he made observations at Athens and his increased
activity in the observation of the variable stars soon confirmed his suspicion.
On the 24th of October 1865 he writes in his diary: „Der 24. Oktober gilt mir
als Tag der Entdeckung der Veränderlichkeit von r}quot;. Till the eve of his
death, which occurred on February 6th 1884, Schmidt kept the star under regular
observation.

During the period 1884 — 1887 the star was neglected but in 1887 Plassmann
began to observe r] and from that year onward the star has been regularly kept
in view by a sufificient number of observers.

Hoffmeister published a discussion of Plassmann's observations of the period
1887—1913®). He derived the following elements:

Min. = 2410707,4 232,177 E,

which formula represents the minima with a mean error of 29,8 days. He
suggests an analogy of the light-curve with the light curve of an eclipsing binary.

In ,,Geschichte und Literatur der veränderlichen Sternequot; Guthnick discussed
the whole material but as it appears that he used the minima as published by
the observers and by Hoffmeister, it is hardly possible to consider this discussion
as trustworthy. His conclusions with regard to such things as the jump in epoch
of 65 days and the analogy with Algol must therefore be considered as
premature. His formulae are:

Min. = 2402537 231,8 E and 2410715 231,8 E.

In connection with the work of Guthnick the classification by Ludendorff
may be mentioned here. Ludendorff considers t] Geminorum to be the prototype
of a new class of variable stars to which class Vi (R) Sagittae, Vi3(RU) Cephei

1)nbsp;Astronomische Nachrichten Nr. 1687.

2)nbsp;Mitteilungen der V. A. P. 24, 15 (1914).

S) Nr. 378, p. 188.

Astronomische Nachrichten Nr. 5126.

and V5 (V) Ursae majoris are supposed to belong. A relation with the RV
Tauri stars is considered possible by him. In „Handbuch der Astrophysikquot; i)
Ludendorff classes rj among the stars of the Mira Ceti type with light-curve /?4
(symmetric light-curve, very broad maximum with brightness remaining constant
for a considerable time). He writes 2): ,.Unter den Me Sternen kommt die Kurven-
form ß4 nicht vor, und von den Mirasternen der übrigen Spektralklassen sind
nur die Lichtkurven von r] Geminorum (Ma) und S Aurigae (N) mit /?4 be-
zeichnet. Man hat vielfach eine „rj Geminorum-Klasse' unterscheiden zu müssen
geglaubt. Dies ist weder notwendig noch zulässig, denn offenbar ist die Lichtkurve
nur ein extremer Fall einer ß Kurve, und im übrigen scheint sich ri nicht
grundsätzlich von manchen anderen Angehörigen der Mira-Klasse zu unterscheidenquot;.
As the remarks of Ludendorff originate in the results arrived at by Guthnick
the same criticism may be applied to them as to Guthnick.

Spectroscopic evidence does not confirm the reality of the Algol type for
this star, the spectral class being Ma; tj seems to be a spectroscopic binary3).
A comparison of the variation of the radial velocity as found by the Lick
observers with the light-curve will be given in § 8 of this paper. I tried to get
some more spectroscopic work done on this star but without success.

As a treatise on the whole material available has never been published I
undertook this research at the instigation of Prof. Nijland.

Band VI Zweiter Teil (Berlin, Jul. Springer, 1928) p. 99, no, 114.

2)nbsp;p. 130.

3)nbsp;Lick Bulletin 1, 158.

CHAPTER I.

DETERMINATION OF THE LIGHT-CURVE.

§ i. Material available.

The material at hand consists of a small number of photometric measurements
and of a far greater number of estimates made according to well known methods
(Argelander, Nijland, fractional method).

Since only a few series of observations have been published, the greater part
had to be obtained from the observers themselves or from the astronomers in
care of the manuscripts My special thanks are due to Prof Ludendorff, Potsdam,
and Prof. Kustner, Bonn, for copies of the observations made by Schmidt, to
H. Grouiller, Lyons, for a copy of Luizet's work and to F. de Roy, Antwerp,
for the observations made by members of the Variable Star Section of the
British Astronomical Association. The observations made by Plassmann, Knopf,
De Roy, Nijland, Ryves, Scharbe, Landwehr, Molles, Wirtz und Von Stempell
were handed over to me in their original form by the courtesy of the observers
themselves. My own observations completed the material, making a total of 9151
observations, covering the period 1843—1924. In the following table a summary
is given of the publications and manuscripts used.

The photometric measurements available do not appear in this list, since
they are too few in number to be taken into consideration here. It is a well
known fact that the value of observations made by means of the photometer is
often greatly exaggerated and this over-estimation leads to the belief that a few
measurements will suffice to derive a trustworthy light-curve.

Short series of photometric measurements will hardly ever be of any im-
portance in the study of an irregular and difficult object such as rj Geminorum.
The measurements of Kaiser and Scheller and those of Pickering are therefore
excluded from the discussion.

1) Kaiser and Scheller, Astron. Beob. Sternwarte Prag, 2, 23, Pickering, Harvard Annals, 24,254;
46. 237.

Table 1.

Of the observations enumerated in the preceding table 630 had to be rejected
for the following reasons.

Oaks.nbsp;36 estimates all in the form: rj aboutnbsp;i — 7).

Backhouse. 13 estimates in a period of 5 years; differences up to 12 steps are
recorded.

') The symbols lt; and gt; denote fainter resp. brighter than.

Field.nbsp;39 estimates, all with very large differences in steps; differences

of 10 and 12 steps occur repeatedly.
Greenwood, i 2 observations in the form 1] n lt; fi (n — i — 5).
Mitchell. 42 observations in 1906—1907, a series consisting almost entirely
of one-sided estimates in the form junt]] enr]. Nevertheless the
minimum 2417666 is clearly indicated.
Marckwick. 213 estimates in the form: tj f magn. smaller than fi, more than
^ magn. smaller than /u, much smaller than s, not far from s, plainly
smaller than fi, and so on. Large step-differences and one-sided
estimates.

Child.nbsp;64 estimates, 62 of which are one-sided with differences up to 8

steps.

Orr.nbsp;13 estimates, all recorded as follows: between ju and 8, nearer to

ju than to S etc.

Since the observations of Knopf, Rosicky, Worsell, Kopff, Götz, Schiller,
Lohnert and those of Mitchell (1919—1920) form very short series, it seems
obvious that these observers cannot have had the practice necessary for dealing
with the difficulties of r] Geminorum.

I wish to add some remarks concerning two methods of observation which
are often used, notwithstanding the objections repeatedly raised against them i).
I refer in the first place to the fractional method. The fractional method
(pogson, Pickering) is based upon the erroneous opinion that the photometrically
determined magnitudes of the comparison stars are exact. The exactness of these
values is, however, often not very great, the mean error of the H. p. magnitudes
reaching 0.105nbsp;This is partly to be ascribed to the fact that the colour

error has not sufficiently been eliminated. The observing book of the observer
who uses the fractional method often only contains the magnitudes resulting
from comparisons which are not recorded in detail It is undeniable that this
method takes much less time than the Argelander (resp. Nijland) method, but
even if the observations are recorded more in detail, nothing can be concluded as
to the observer's conception of the interval between the comparison stars used.

The observation by means of the fractional method can easily lead to the
use of would-be steps of „0, i mquot;. Let us consider an interval a—lt;5, photometrically
found to be 0,4 m.; the observer takes of this interval as unit. His estimates

1) cf. Hagen, Die Veränderlichen Sterne, Bnd. I, p. 276—280.
ï) Fetlaar, Recherches Utrecht IX (i) (1923) p. 9.

3) See Hägen, o. c. p. 277/278 and the monthly reports of the American Association of Variable Star
Observers in quot;Popular Astronomy'quot;,

in these „stepsquot; will, as a matter of fact, be affected by the errors resulting from
inaccurate magnitudes of the comparison stars. His value of one step, nominally
o, I m., will be variable in an irregular way. Moreover, he is apt for the sake of
convenience to compare the variable star with only one comparison star, using this
„unitquot;. It will be clear that the difficulties in the discussion of these observations
often appear to be almost insuperable, as, for want of data, it is questionable
whether the assumption i step = o,i m. can be relied upon. One-sided estimates,
occasionally made by observers using the Argelander method of course are
excluded from this criticism.

Of the older observers Schmidt has made an extensive use of the one-sided
method. It is to be regretted that the weight of his numerous observations is
considerably diminished by this procedure and also by his large step-value.

The American Association of Variable Star Observers and the Association
Française d'Observateurs d'Etoiles Variables, the latter working along the same
lines as the American Association, would, in my opinion, materially improve the
results of the work done by their respective members by rejecting all observations
made by the „time-savingquot; methods mentioned above. The A. F. O. E. V. in
particular published observations, such as those by Butterworth, often consisting
of estimates in the form vna, resp. anv {n = i to \2 \ steps). Admitting that
Butterworth's observations are astonishingly good, as Grouiller maintains, I
might emphasize the danger of the example set by him to other observers who
lack the virtuosity necessary for this method and who only wish to have their
names mentioned in the redactional columns as authors of „listes d'observations
extrêmement importantesquot;.

§ 2. Reduction of the observations.

The reduction of the observations consists in the construction of the step-
scale of the comparison stars for each observer; the comparison of this scale
with the magnitudes of the comparison stars as determined photometrically; the
adaptation of the latter values to the individual step-scales; the determination
of the photometric value of one step and finally the deduction of one set of
photometric magnitudes for the comparison stars applicable to all observers.
With the aid of this final photometric scale the observations are calculated.

Before being able to combine into a step-scale the differences observed
between the comparison stars, these differences should be corrected for atmospheric
extinction. This, of course, does not relieve us from the necessity of applying
the same correction in the final computation of the observations.

To the direct application of an extinction table there are several objections.
Muller's table, for instance, has been derived from observations made at Potsdam
and on the top of the Santis and is therefore, strictly speaking, only valid for
these places. The difficulty of each extinction table i) is, that it naturally cannot
take into account the local peculiarities of every place and the meteorological
circumstances of the epoch of the observations. Large discrepancies have been
recorded for instance at Catania, where the value for this correction appeared
to vary with the azimuth

The automatic application of an extinction table should, in my opinion,
always be condemned, unless the stars considered are located in a small field of
a few square degrees and in this case the importance of a correction amounting
perhaps to a few hundredths of a magnitude will hardly be in proportion to the
work involved.

In this paper I have made an attempt to meet the difficulties mentioned
above by proceeding in the following way. For each observer two comparison
stars were selected satisfying the essential conditions of being sufficiently distant
one from the other and at the same time being frequently used throughout the
whole period of observation. The observed intervals between these stars, expressed
in steps, were plotted on squared paper by taking these differences as ordinates
and the corresponding sidereal times as abscissae. At the same time I plotted
the differences corrected for extinction by means of Muller's table, adopting for
the photometric value of one step the value which had resulted from a provisional
reduction, and taking the interval observed at the sidereal time with minimum
extinction as zero. The curves that may be drawn through both sets of points
are expected to coincide. If this is the case the correction for extinction can be
applied unmodified; if not, the extent to which the „theoreticalquot; value has to

be modified can easily be derived.

As, owing to the very complex nature of the phenomenon of atmospheric
absorption, it will never be possible to calculate the required correction exactly,
an approximate method such as the one proposed above, will, in my opinion,
sufficiently meet the needs of the computer. I wish it to be clearly understood
that I am quite aware of the imperfection of this method which, moreover, can
only be used when step estimates are available, estimates made by means of the
fractional method cannot be corrected in this way.

1)nbsp;Recently an extinction lable has been published by H. van der Linden. Annal. Obs. Royal Belg,
3me serie II, i (1928).

2)nbsp;Hagen, o. c. p. 394.

A. The observations by ]. F. J. Schmidt.

These observations have been made, with the naked eye, partly at Hamburg,
Düsseldorf, Bonn, Olmütz, Vienna, Rome, but by far the greater part at Athens.
Schmidt made a large number of estimates by means of Argelander's method,
viz. 3388 in 2877 nights, but it is very much to be regretted that he disregarded
the advice given by Argelander concerning the desirability of comparing
the variable with at least two comparison stars. Schmidt made 31 17 estimates
with the aid of only one, 156 with 2, 114 with 3 and i with 4 comparison stars.

The comparison stars used are ft (3383), « (9), v (227) and i Y\. = Propus
Geminorum (155), the numbers in brackets denoting the number of comparisons
made with each of them. As about 92 % of all observations have been made by
comparing rj with ^ only, the result of an investigation regarding the question
of the correction for extinction cannot be of much value, considering the facts
that the observed intervals suitable for this investigation are scattered over a
period of about 40 years and that the number of these intervals is therefore
relatively small. Moreover any correction for extinction will be superfluous for
all observations in which // is used, since the numerical value of the differential
extinction ft—ri appears to be neglectable during the greater part of the year.
In fact, this differential extinction reaches 0,1 m. only from 12—13 h. S. T.
Nevertheless I decided to study two intervals with a view to obtaining a
step-scale. These intervals are ft—v and y —P, P designing i Fl. Geminorum.

The interval fi—v can be derived from 209 observations in the form fiari \ rihv
giving a mean value of 4,65 steps. Schmidt's step-value shows a slight tendency
to increase in the course of the period of observation, I found for the period

1847— 1867 (63 obs.) fx—v 4«,54 steps
1868—1872 (90 obs.)nbsp;4,63 „

1873—1884 (56 obs.)nbsp;4,35 „ .

The following table contains the intervals «—v arranged in the order of
progressive sidereal time, the column headed fiv calc. containing the value of
the interval fi—v at minimum extinction the calculated differential extinction
(taken from the Potsdam table and converted into steps by adopting the value
of one step as 0.2 m.), D = fiv obs. — fiv calc.

Up to about 8'^30 there is a fair agreement between the observed and the
calculated intervals but in the lower part of the table the deviation from the
calculated values becomes too great to be explained by the accidental errors.
They might be accounted for by admitting for Athens a considerably larger
value for the extinction than Müller's table gives. In that case however, a still
larger deviation should have been found in the first part of the table as the

determination of the light-curve.

Table IL

differential extinction at 0—2 h. S. T. is greater than at 8—12 h. A satisfactory
explanation by means of the extinction seems to me therefore impossible.

For the interval j/ — P 104 observations in the form riav] jjiP are available,
giving a mean value of 0,38 steps for the difference between these stars An
arrangement in the order of progressive sidereal time is given in Table III. This
table contains only the columns S T., v—? obs. and corr. (= correction for
extinction converted into steps). A column v—V calc. cannot be given for reasons
which will appear below.

Table III.

As will be seen from the lower part of this table the estimates of this
interval are affected with a systematic error which cannot be easily explained
by means of the theory of extinction. An explanation might be found in the
variation of the position of the constellation with respect to the horizon. As the
deviations for both intervals studied point to an apparent diminution in the
brightness of the star v in the order of progressive sidereal time, this explanation
seems to be near the truth.

I adopted the following step-scale for the comparison stars used by Schmidt.
P = onbsp;v = o,4nbsp;fi = 4,65

B.nbsp;Observations by E. Schonfeld.

The observations, 123 in number, have been made at Mannheim with the
aid of an opera-glass. They are printed in „VerOffentl. Sternw. Heidelbergquot; i),
where also a scale of steps of the comparison stars is given viz.

V Geminorum = 0nbsp;lt;9 = 4,5nbsp;6=11,3.

The stars e and 0 are used in 92 estimates, 0 and in 10 cases. From a
study of the interval £—0 I found that the correction for extinction can be ap-
plied unmodified for the period 3—8 h. S. T., but from o to 3 and from 8 to 11 h.
the value of the table must be reduced by 50 7o- For the mean value of the
interval e—0, corrected in this way, I find steps.

From ten estimates the interval d—v was found to be 4,9 steps, the full
correction for extinction having been applied. The star has been used by
Schonfeld only once, giving a difference of 3 steps between and

As the colour of the stars « and 0 differs considerably, e being RG-, 0 WG,
the influence of the moonlight on the interval b—0 was studied by means of 30
estimates. No appreciable influence was found, the mean value coming out at
7,0 steps.

The following step-scale was adopted.

y = onbsp;0 — 4,9nbsp;£ = r 1,9.

C.nbsp;Observations by J. Plassmann.

As I stated in the Introduction to this paper, Hoffmeister has published
an extensive discussion of the observations made by Plassmann in the period
1888—1913.

Up to 1907 Plassmann used 6 comparison stars viz. i Fl., v, e, 0 Gemi-
norum and 0 Aurigae. The observations of the period 1907—1924 consist

1)nbsp;1, 98 (1900).

2)nbsp;]. c. we find v, certainly a misprint.

entirely of comparisons with ft and v Geminorum only and all the observations
have been made with the same instrument (a Steinheil astronomical binocular)
throughout this period i). For these reasons I undertook the discussion of the
848 observations made after 1907, considering a renewed reduction of the obser-
vations before that date superfluous, the light-curve deduced by Hoffmeister
having been placed at my disposal by the courtesy of this astronomer.

From a study of the interval fi—v Hoffmeister found Plassmann s step-value
to be subject to two variabilities, one with a period of about 5 years and the
other with a period of one year giving a maximum for the interval v in April
and a minimum in November. The periodicity first mentioned is most probably
to be ascribed to personal influences, the second to the extinction.

I have studied the interval ^—v starting from the 833 values for this interval
resulting from 848 observations. The secular change is clearly indicated but the
periodicity of 5 years does not persist after 1907. Instead I find a gradually
decreasing value for the interval, the curve shows in the first years a marked
periodicity of 3 years but the amplitude of this variation diminishes rapidly as
will be seen in the following table. In deviation from Hoffmeister I calculated
the seasonal- and not the jj/^ari^-value of the interval. In table IV n denotes
the number of intervals used and d the deviation of the seasonal value from
the mean (10,23 steps).

Table IV.

The following table (Table V) contains the intervals v arranged in the
order of progressive sidereal time, the column /t—r calc, containing the values

1) 1888—1907 Plassm.\nn used feebler instruments.

10,0 calculated differential extinction, D = —v obs.) — C« — v calc.) and
d' = 10,23 — {ju—V obs.).

Table V.

Obviously the application of the correction for extinction will deteriorate
the value of the observations made at o—2 h. S. T.

In connection with the peculiarities of Plassmann's step-value the reduction
of his observations to a constant value for the interval //—v proves necessary.
For this constant value I started with the mean resulting from all observations
viz. 10,23 steps It will be clear from the tables given above that the method
for the reduction can only be an empirical one. Starting from the values given
in the columns d and d' of the tables I proceeded in the following way 2). The
observed differences in steps between t] and the comparison stars are multiplied

10,23

—, for instance:
10,23 —{d d')'

.season 191 2/1913 Sidereal Time 9^30

d-]-d' = {— 0,47) (— 0,10) = — 0,57

the multiplicator is 10,23/(10,23 0,57) = 0,95.

by the factor

1)nbsp;This value is in excellent accordance with the result of the discussion by Hoffmeister of the
observations 1888—1913 viz. 10,26 steps.

2)nbsp;I preferred to use the values d and d' unsmoothed.

3)nbsp;Following Hoffmeister

This method assures homogeneous values for the step-differences between r]
and the comparison stars throughout the whole period of observation.

D.nbsp;Observations by M. Luizet.

Luizet has made 751 observations of n Geminorum covering the period
1898 — 1917. Although the notation used differs somewhat from the usual one
it is clear from the manuscript that he observed according to the Argelander
method.

The comparison stars used are e, ^ and 8 Geminorum. The interval t—
being the one most frequently observed, has been chosen to study the influence
of the extinction on the observations. From 335 observations it was found, how-
ever, that the application of a correction for extinction would not improve the
results, the value for this interval remaining almost constant throughout the
whole year. Consequently the correction for extinction has not been applied. I
wish to add that I tried to explain this remarkable fact by an error caused by
the varying position of the triangle e^r] with respect to the horizon. But in my
opinion it is impossible to separate the very complex causes of this phenomenon
from one another (including the selective absorbtion in the atmosphere and the,
unknown, order of comparison of rj with e and § by the observer) ih a satisfactory
way. Obviously it will make a difference whether the observation is made in the
order e—^ resp. §——e,nbsp;n—«etc. Untraceable personal influences there-

fore play a considerable part

The following step-scale was derived from 115 observations of the interval
68 ofnbsp;335 of e—I, and 55 of l—h.

S = 0 I = 3.9 c = 9»55 ^ = 11.3 steps.

E.nbsp;Observations by F. de Roy.

De Roy made 621 observations, covering the period 1903—1924, by means
of the Nijland interpolation method. The construction of a step-scale is rendered
difficult by the fact that the observed differences between the variable and the
comparison stars do not always have the meaning of steps but are principally
supposed to give the ratio of the differences. Nijland has called attention to
the desirability of giving this ratio in connection with the individual step-values.
Very often, however, this condition is not fulfilled and fi. awih is recorded
in cases in which aiv^h would much better suit the difference between a and b.
De Roy himself says: ... the intervals (i—i or i—2 of course excepted) in

») See also Hagen, o. c. pag, 335.

some measure denote stepsquot;. De Roy has used the following comparison stars
fly e, d, d, y., I, r, v and i Fl. Geminorum. In 541 observations 5 and g are used.
Excluding the observations giving differences between s and ^ of 2 and 3 steps
for the reasons given above, I find from 404 estimates for this interval 5,05 steps
As this value is found to be practically constant throughout the whole year the
influence of extinction on the observations can safely be regarded as imperceptible.
This correction has therefore not been applied.

F. Observations by P. M. Ryves.

These observations have been made by means of Argelander's method but
a somewhat frequent use has been made of one-sided estimates obtained by
comparing ri with ft and c Geminorum simultaneously.

Ryves's work consists of two series of observations viz. 450 from 1905 to
1913 and 72 from 1923 —1924, the place of observation was Zaragoza and the
instrument the naked eye. The comparison stars used are e, d, v and i Fl.
Geminorum and on a few occasions 6 Aurigae. Of these ft and e were used
exclusively in about 63 % of all observations.

The interval between these stars has been studied in five groups, the result
of the investigation being given in the following table.

On behalf of the calculation of the mean value of this interval only the
groups c and d are used, the resulting value is 1,3 steps. An opportunity to
study the interval error is presented by the circumstance that 14 intervals have
been more or less directly (i. e. via t?) observed. The following table contains
the values found for the intervals mentioned in the first column; in the second
column the directly observed differences and in the third the values obtained by
the addition of the intermediate intervals are found.

This table points to the existence of a small negative interval error.

The star ;; has been excluded from the discussion for the following reason.
From the intervals observed between and the other comparison stars the
brightness of y. is found to be about equal to i Fl. whereas H. P. and P. D. give

H. P. (Draper) 3,62nbsp;P. D. 3,75

I Flnbsp;4,30nbsp;4.37

Probably the star has been mistaken for v (H. P. 4,26), the more so as
De Roy did not find any irregularity in this respect.

The influence of the extinction, studied by means of the interval
appeared to be neglectable.

The following step-scale was adopted:

iFl. = 0 y =1,9, 6^=5,0 Ê=io,8 ft=\2,\.

G.nbsp;Observations by A. A. Nijland.

In the period 1904 to 1924 Nijland obtained 494 observations. The estimates
were made at Utrecht with the aid of a field glass and using the comparison
starsnbsp;I, 0, d, A, i Geminorum. 393 observations of the interval c—| are

available to study the influence of the extinction From the discussion of these
observations the following conclusion was drawn. The table-value of the correction
for extinction, must be applied unmodified to the observations made from 4 to 10
h. S. T. For the observations made at 2—4h. and 10 — 13 h. S. T. the theoretical
correction is to be diminished 50 7o, froiquot; o—ah. 75 7o- The construction of the
step-scale for which the correction for extinction, modified in this way, has been
applied, lead to the following result.

t = o 0=1,4. A=2,I C^ = 3.I ^ = 4.8 e=ii,3 ju=ii.7.

H.

The series of observations by von Stempell, Scharbe, Landwehr, Brown,
Vogelenzang, Molles and Wirtz being of less importance than the others, owing
to their rather small number or the short period covered by them, will not be
treated in detail. Only the results of my discussion and a few remarks will be
given here.

16nbsp;chapter i.

«

a. G. von Stempell.

The use by this observer of four different instruments made a reduction of
all observations to one instrument unavoidable, the more so as two instruments
appeared to give values for the red stars smaller by about 2 steps than instrument I
(an army fieldglass) and the naked eye. The following step-scale was derived.

V = o 6= 6,8 fi = 7,4.

d. S. Scharbe (Jekaterinoslaw).

Three instruments were used; a reduction to one of them was proved
necessary by the differences in the conception of the brightness of the red stars
found for the separate instruments. The step-scale is

v = o I Fl. =0,4 ^' = 4,5 e=io,o ^=11,0.

c.nbsp;G. Landwehr (Münster).

The following step-scale was found

iFl. = 0 j/= 1,5 €=ri,4 /u = ii,g.

d.nbsp;A. N. Brown (Silchester).

The construction of a step scale being impossible owing to the fractional
method of observation used by this observer, the observations were reduced by
means of the final photometric scale.

e.nbsp;E. H. VoGELENZANG (Hilversum).

The stars e and § were used exclusively throughout the whole period of
observation. The interval between these stars was found to be 4,9 steps.

/. Mölles.

These observations are made according to a fractional method. Mölles
adopted for the interval s 2 steps and for 6 — 1 Fl. 4 steps. 1918 February
18th these values are suddenly changed and from this date /u—i Fl. =8 steps.
A further change occurred in 1921 January when Mölles abandoned this method
for the original Argelander step-method which gave the following step-scale

iFl. = 0 j/=i,4 «=7,4 —

g. C. WiRTZ (Kiel).

The estimates were found to be subject to a large interval-error, the inter-
val —V being 9,1 steps when observed via t} and 13,3 steps when d was used
as an intermediate. The step-scale is

v = o ^ = 6,7 e=i2,8 ^=13,3.

The following table contains a recapitulation of the step-scales deduced from
the observations of each observer separately.

Table VL

Comparison Stars

Observer

I Fl.

Schmidt

Schönfeld

Plassmann

Luizet

De Roy

Nijland

Ryves

Von Stempell

Scharbe

Landwehr

Vogelenzang

Mölles

Wirtz

P. D.

colour

H. P. (Draper)
spectr.

From a preliminary discussion it was found, that the photometric magnitudes
for the comparison stars given by the Potsdam Durchmusterung suited the step-
scales of the majority of the observers much better than the values taken from
the Harvard (Draper) Photometry. There is however one exception viz. v Gemi-
norum. In this case the P. D. gives v lt; i Fl. whereas 4 out of 5 observers who
used both stars find V gt; 1 Fl. Only Scharbe is in accordance with the P. D. but
there may be some bias in his case.

By a graphical method which has been described in detail by several
writers 1), the step-scales were compared with the photometric magnitudes of the

1) Fetlaar, Recherches astr. Utrecht IX (i) (1923) p. 3, 4 fig. i^.

P. D. and the latter values changed to the extent necessary to ensure a constant
step-value throughout the whole step-scale. Table VII contains the result of this
reduction. It will be clear that only the observers who made use of 3 or more
comparison stars are mentioned in this table.

Table VIL

It will be seen from this table that, v excepted, only small corrections had
to be introduced to the original P. D. values. The step-scale of Mölles has not
been taken into consideration for the reasons given above.

The mean values found for the photometric magnitudes are used in the
final reduction of the observations. This is quite justifiable as the values found
for the single observers agree closely. The photometric values for the steps of
each observer as they result from this final photometric scale are given in the
following table.

1) This value seems to indicate, that the sensitiveness of Luizet to yellow is greater than of other
observers.

Table VIII.

§ 3. Reduction of the observations to one observer.

With a star like t] Geminorum an opportunity is presented for reducing all
observations to one observer, the star having been observed over a long period
and frequently on the same night by several observers.

In the case of rj Geminorum however the material falls into two parts viz.
the periods 1847—1884 and 1887—1924, the former consisting of the observations
made by Schmidt and of SchOnfeld's relatively short series. Owing to Schmidt's
defective method of observation a reduction of SchOnfeld to Schmidt can not
be expected to give results of much importance. From 78 observations made
simultaneously I find:

47 positive residuals (in the sense SchOnfeld —Schmidt)

30 negative ,,

to both sides up to 0,39m.

1) Miss J. C. Thoden van Velzen has deduced a formula (Proefschrift, Utrecht, 1928) which enables
her to calculate the necessary reduction for the cases in which a coloured variable has been observed by
means of white comparison stars. In our case however, the comparison stars v, i Fl., s, n and f show more or
less the same colour as r] and as, moreover, the amplitude is very small there is no room here for applying
Miss Thoden van Velzen's results.

Treating these residuals separately for the maximum and minimum phases of the
light-curve I find:

at maximum 41 positive and 2 negative residuals
at minimum 6 „ ,, 28 ,,nbsp;„

Considering the fact, that in the brighter phases of the light-variations Schmidt
compared only with and that, as his large step-value indicates, his eyes were
not very sensitive, not much value can be attributed to Schmidt's estimates at
maximum. A conclusion with respect to an Algol-analogy can ceutainly not be
drawn from Schmidt's observations.

The reduction of the observations to one observer has therefore been

f

restricted to the observations of the period 1887—1924.

For the purpose of this reduction it would be natural to take the observer
whose contributions have been most continuous and spread over the longest
period namely Plassmann, but the peculiarities of this observer which we have
already mentioned suggest a comparison with the next candidate, viz. De Roy.
The results of the comparisons are given in Table IX. The second column gives
the number of the observations made simultaneously, the third the resulting
difference Plassmann—Observer ; the fourth and fifth columns give the results
of the comparison with De Roy.

Tabic IX.

The mean error resulting from the comparisons with De Roy being smaller
than the m. e. of the column PI.—O., De Roy has been chosen for the final
reduction.

Before proceeding to discuss the light-curve based on the newly reduced
observations I wish to apply a final criterion as to whether I now have the right
to consider this curve as homogeneous. Following J. van der Bilt and starting
from the 697 differences between De Roy and the other observers, I calculated
by means of the m. e. found, the number of cases which can be expected to be
found within certain limits, if these differences had been purely accidental errors.
The following table shows that there is a satisfactory accordance between the
calculated and the observed numbers

From this table it appears that the homogeneity of the light-curve is satis-
factorily established.

§ 4. The light-curve.

A description of the general features and of the details of the light-curve
derived from the observations made by Schmidt will be given in connection
with the list of the minima in the following chapter.

In order to derive the light-curve 1887—1924, in which of course gaps of
at least 100 days are expected to occur owing to the star's place near the
ecliptic, the values of the brightness of t] were condensed into a number of means
representing the mean brightness at intervals of about 10 to 20 days. These mean
values were plotted on squared paper, on a scale of i mM. equalling 2 days
and 0,01 magn., and a smooth curve was drawn through the points.

Total number of observations

» points........573

Number of points above the curve .... 229
. below „ „ .... 216
^ n on „ „ .... 128
Recurrences of the sign of the deviation . . 213
Changes „ „ „ . „ „ • • 232
Largest positive deviation....... 0,17 m.

J, negative „ ....... 0,12 m.

The mean' deviation of the points above and below the curve does not
exceed 0,03 magn.

The ordinates of the light-curve were read off for intervals of 20 days
(Greenwich mean noon). During periods of rapid change the readings were made
every lo days. The results are given in the following list. Curves representing
the light-variations during a few selected seasons are reproduced on the Plate.

Sharp and flat maxima and minima alternate in an irregular way with periods
of approximate constant brightness. The maxima become more distinct in pro-
portion to the number of the observers contributing to the light-curve. The
amplitude of the variations varies from o, i to 0,8 magn. The ascending as well
as the descending branches in some instances seem to show a secondary curvature,
displaying degenerated minima or maxima. A noteworthy feature of the light-
curve is the recurrence of the same brightness at maximum viz. 3,35 m. The
existence of a long periodicity in the sense found for V 14 (RV) Tauri is thereby
rendered very improbable. Apart from this constancy of the maximum brightness
the most regular phenomenon of the light-curve is the recurrence of the greatest
number of minima at periods of about 235 days.

In the light-curve 1887—1924 (12890 days) the results of observations covering
about 7410 days are embodied. As 33 gaps occur amounting to a total loss of
5480 days, the light-variation during this period is only known to the amount
of 58 7o- It would therefore be premature to attempt to explain the irregularities
of the light-curve by a second or even a third periodicity, the more so in con-
sideration of the long period and small amplitude of the principal variation.

The Light-curve.

•aAgt;in3-xHon anx .io NOixvNiK^axaa

CHAPTER II.

THE MAXIMA AND MINIMA.

§ 5. List of the minima deduced from the light-curve.

From the observations made by Schmidt during the period 1843—1865
only two minima could be derived viz.

J. D. 2394682 and J. D. 2394925

A.third one at 2399020 is indicated but the observations are too few in number
to include this minimum in the final list. The two minima mentioned above
however are sufficiently established by series of 28 and 36 observations respectively.

The following list contains all minima I was able to derive from the
material discussed in the preceding chapter. The first column contains the
reference-number, the second the Julian Date, the third the magnitude at minimum,
the fourth the duration reckoned as the time between the preceding and the
following full brightness, the fifth the diminution in brightness with respect to
the preceding maximum, the sixth the type for the designation of which I adopted
the following scheme:

I. descent and ascent both smooth.

II. descent smooth, ascent disturbed.

III.nbsp;descent disturbed, ascent smooth.

IV.nbsp;descent and ascent both disturbed

a.nbsp;M—m lt; ^ (M = preceding maximum).

b.nbsp;M-m = i.

c.nbsp;M—m gt; i.

The columns headed E and O—C contain the results of § 7 and will therefore
be referred to later on.

the minima and minima.

Tabic X.