Material von Adurns benutzt und andererseits alle Zwergsterne und alle Sterne mit groOen Radialgeschwindigkeiten aus. geschlossen. Leider konnte ich die Zwergsterne der Klasse M nicht untersuchen, wegen Mangels an Material. Die Losung der 16 Gleichungen, die sich auf diese Gruppe beziehen, er- gab K = +7 .9 f6 .

Das Vorhandensein eines positiven K ist also nur in zwei Gruppen - bei den Zwergen vom F-Typus und bei den Riesen des K-Typus - zweifellos festgestellt worden; in den anderen Gruppen (naturlich mit AusschluO von B) existiert das gesuchte Glied wahrscheinlich nicht.

The change of the periods of the Cepheid-variables is of considerable theoretical interest. From the standpoint of the pulsation theory these periods must decrease, because of

Es bleibt noch die Frage einer Anderung von K, als Funktion der Entfernung der Region vom Vertex zu unter- suchen, d. h. das, was E. Rreundlich und E. v. d. Pdlttz l) als sspeziellen K-Effektc bezeichnen, indem sie zu beweisen ver- suchen, daO er mit der Annaherung der Region zum Vertex gleichzeitig wachst. Wenn wir die Vertexkoordinaten zu a = 2 7 5", 6 = - I 2" annehmen und die Winkelentfernungen der Zentren der verschiedenen Regionen vom Vertex und die Rest- geschwindigkeiten (ohne das Glied K einzufuhren) ausrechnen, erhalten wir folgendes Bild der Verteilung derselben nach Gruppen :

the gradual increase of the density in the process of evolution, it is impossible theoretically to deduce the order of this decrease, because it depends on the tempo of stellar evolution, unknown

Aus unserer Tabelle ist ersichtlich, daD die Abhiingigkeit derdu vom Winkelabstand der Region voni Vertexnur fiireinzelne Gruppen existiert. Sie ist unzweifelhaft und hat denselben Sinn wie bei den Resultaten von E. Frcundlich und B. v. d. Pablen nur fur die ~Zwerggruppe~ der Klasse F; eine deutliche aber umgekehrte Abhlngigkeit existiert bei den Zwergen der G-Klasse; ein Andeutung fur eine ahnliche Abhangigkeit tritt auf bei den G- und K-Riesen; bei den F-Riesen und K-Zwergen gibt es uberhaupt keine Andeutung einer Veranderlichkeit des K. Unsere Untersuchung hat also im Ganzen die Resultate der Potsdamer Astronomen nicht bestltigt; ubrigens ist in den in Potsdam berechneten einzelnen K die Abhangigkeit von der Winkelentfernung zum Vertex auch sehr bndeutlich ausgedruckt.

Es ist Eaturlich ganz unmoglich, das Vorhandensein des K-Effektes bei den Zwergen vom F-Typus und bei den Riesen vorn K-Typus durch-eine gravitationale Rotverschiebung zu er- klaren. Aufden Oberflachen der Riesensterne wird das Schwere- potential infolge der geringen Dichtigkeit und des Radiations- drucks wahrscheinlich sehr gering sein; aber die Massen der Zwerge sind uberhaupt gering. Als einzige Erkllrung mag nur die kinematische gelten: die Vermutung, daO das ganze System der erwahnten Sterne sich im Stadium einer Ausbreitung be- finde und die Sonne irgendwo in der Nlhe des Zentrums stehe. Der K-Effekt verliert dann ganz seine Ratselhaftigkeit.

Ganz anders steht es mit den B-Sternen. Zweifellos erscheint hier das Glied K als zunehmende Funktion der ab. soluten Helligkeit, d. h. der Masse. Dennoch ist sein mittlerer Wert zu groO, als daR man ihn von der Einsfcinschen Theorie ausgehend erklaren konnte. Nach Gyllenberg ist f i r das ganze System der Heliumsterne K = +4.3. Wenn wir Q = o,IeG

Charkow, Ukraina, Sternwarte, 1923 Oktober.

') A N Band 218. '1 A d 55.

,etzen, wobei Q die mittlere Dichtigkeit der Sterne der B-Klasse st, werden wir nach der Einsttinschen Formel absurd groBe Massen erhalten. Zwar hat S. Albrechf z, vor kurzern eine par- :ielle Erkliirung des K-Effektes bei den Heliurnsternen gegeben. [ndem er von neuen Wellenlangenbestimmungen fur Si, 0, N und einige Heliumlinien ausging, hat er gefunden, daO der Fehler in der Berechnung der Geschwindigkeit infolge der Benutzung der alten Wellenllngen fur Bo-B2 + 2 km, fur B3 + I km und fur B5-B8 t o . 3 km betragt. Aber auch wenn wir das Glied K auf +2.3 km verringern, werden wir trotzdem bei der Verwendung der Eirrsftinschen Formel zu groOe Massen erhalten ( M = 2 1 ) . Bei den Heliumsternen wird auf den Gravitationseffekt, welcher bei ihnen unzweifel- haft als Folge der grofien Massen und der nicht geringen Dichtigkeit existiert, im K-Glied nicht nur ein Fehler in der Wellenlangenbestimmung, sondern auch ein gewisser unerkllrter positiver Effekt aufgetragen. Die GroOe des letzten kann man schatzen, indem man von folgenden Voraussetzungen ausgeht : wenn wir die mittlere Masse der Heliumsterne als 7 . 0 an- nehmen (was mit den ziemlich unbestimniten Resultaten der statistischen Untersuchungen genugend iibereinstimnit) und @ = 0. Ieo setzen, werden wir nach der Einsttinschen Forrnel K= + I. I km bekommen. Wenn wir den maximalen Wert der S. Albrechtschen Verbesserung in Betracht ziehen, so wird sich der unerklarte Teil des K-Effektes fur die Heliumsterne gleich + 1 . 2 km erweisen. Dieser unerklarte Teil des K-Effektes erscheint, wahrscheinlich, auch als zunehmende Funktion der absoluten Helligkeit, wie es die Untersuchung iiber die Sterne nach der Lockytnchen Klassifikation verniuten lafit.

R. Gerusintovif.

to us. However w e know already, that with some exceptions no decrease of periods of Cepheids has been established. So it seems, that the pulsation theory is not confirmed. Since long it was known, that the periods of Cepheids (especially of Antalgols) do not remain constant and very often it is possible to obtain a sufficient representation of the epochs of maximum- light only by introducing additional harmonic terms.

The lastyears have brought many news about the question: in Hartwigs Ephemeriden fur I 9 23 we find I 2 Cepheids with harmonic unequalities of the period, without mentioning the number of Cepheids with doubtless, but not yet elucidated unequalities of similar type: the register is not yet complete. It appears thus, that similar unequalities probably seem to be characteristic' for the given group of variables: they mask the expected secular decreases of the periods. It renders highly interesting the investigation of the above said unequalities and their explication from the standpoint of the pulsation-theory.

It is to be regretted that the material at our disposal is still very scanty: to the I 2 stars of Harfwig's Ephemeriden I only could add 3 Cepheids with acknowledged harmonic unequalities: 2 Draconis and RV Coronae bor. (according to M. BlaZko) and RR Lyrae (according to M. Shapley). To this material intentionally I have not added 7 stars of the cluster M. 3, the harmonic terms of which have been indicated by M. Bailey. The fact is, that the newest researches of Larink do not confirm the deductions of the Harvard observers: these stars, as well as the series of their neighbours in the cluster previously mentioned,only show discontinuous changes ofperiods, produced according to a law till unknown to us.

A preliminary discussion of the material at hand has shown the existence of two groups of Cepheids: one with comparatively short periodic variations of the period (during several months) and another with longperiodic harmonic une- qualities (embracing several tens of years). These groups are

' (P is the period of variation, 27 the period of the unequality): Group I.

1' IL AF' R\V Draconis od44 41d5 4'?3 BZaZko (AN 216)

U W Orionis 0.50 62,o I 1.8 Hartwigs Ephemeriden RW Cancri 0 .55 87.0 ? BlaZko (op. cit.) RT Aurigae 3.73 687 4.1 Hartwifi Ephenieriden

S Vulpeculae 67.5 31.8 46.2 B B

XZ Cygni 0-41 51-4 1.2 b b

x Pavonis 9.09 8.0 23.7 >> >>

Group-11. P II n c AP

XX Cygni od14 12a9 8:s 06 I BlaZko (op. cit.)

RR Lyrae 0 .57 16.5 17.8 2.5 Shpley (ApJ43) 2 Uraconis 1.36 28.2 34.7 1.8 Blazko (op. cit.) q Aquilae 7.18 161.5 164.9 18.5 Luizef (AN 193) CGeminorum 10.15 223.9 223.7 334 HarfwigsEphem

Out of these two groups there are only two Cepheid: with acknowledged harmonic unequalities:

RVCoronaebor. 0.33 14.2 12.6 6.6 n b b

AY Sagittarii P = 6d7 II = 13od3 2 Cancri I0 483 -

The first group shows a distinct dependence between and n, sufficiently expressed by a parabolic formula. It

iffers from the second group by the smallness of LI and he comparatively great amplitude of the period of oscillation - AP - which surpasses several ten times the respective alue of AP for the stars of the second group. In the second ;roup the relation between P and ZI is much simpler: it is ,asily expressed by the following linear formula :

The fourth column for group I1 gives the value n,, calculated rccording to this formula. The agreement between obser- ration and calculation is good enough, however it seems un- loubtedly necessary to add to the left part of the formula I) a small harmonic term relative to P, with a period of 3-9 days. The calculation of this term till now is difficult 3ecause of the scanty material.

The amplitudes of the periods of oscillation for the stars of group I1 are very small and increase together with the Deriods. The fraction APIP is of the order of IO-LIO-~. Unfortunately AP is determined but very unprecisely by obser- vation; for RV Coronaeand SGeminorum they are probably wrong.

The changes of the periods may be of two types: either :he whole curve of light changes, remaining similar to itself, 3r its aspect is undergoing some changes. The Cepheids dout- !essly haveoscillations ofthe second type: theirlight curves under- go considerable changes. For the Antalgols they are investigated 2omparatively well: they are reduced to changes of the sharpness of maximum with minimums invariable in position and in form. Of course there results a change of the time of light increase TM-T,,,. To the sharp maximums corresponds a decrease of TAtf- T,#, and to the obtuse - an increase of this interval. So we may treat (at least for the best investigated stars of the second group) the change of period in this manner: the light maximum with an unchanged minimum makes pendulum- like oscillations about. its middle position. Thus it seems possible, relying upon data relating to stars of the second group, to calculate the amplitude of the oscillation of T,l- T, during the period I7, according to ihe formula:

in parts of the period P. For '1 Aquilae these movements ofthe maximum have been

observed by Sh.utono#in Tashkent. For this star one oscillation of the maximum happens during 8087 and A( TM- TMI) = od9 (computed). It agrees well enough with the result of Wylic'), who finds, that during 20 years (.190o-1920) TM- T,lI of this star diminished by od3.

Because the curves of the radial velocities of the Cepheids appear like reflected images of the light curves, we must expect relying upon the above said, long-periodic changes of the spectroscopic elements of these variables.

For Y Sagittarii, the oscillations of the spectroscopic elements of which have been studied by y. Duncan'), we must [according to formula (I)] expect a period of these oscillations of r32?4 (P= 5d88); on the base of dw (variation ofperiastron), discovered by Duncan, we find a period of 150 years - a plausible concordance, considering the great probable error i n the determination of dw.

2 I. 5 Pd)+ 5.5 = n(a) . (1)

A ( T M - - TIN) = AP*II/2P2

According to the researches of A. BeZopoZsRy, the results of which were kindly communicated to me before publication, the element w in the case of a Ursae minoris changes from o to 276 in a period of 7 years. According to the period P of this variable, refering it to the I group, we find that the theoretical period of the change of periastron would be 6%.

The harmonic unequalities of the periods of the Cepheids (of both types) appear as a characteristic feature of these variables. I t is not astonishing, that the secular changes of the periods masked by harmonic ones, remained unnoticed till now.

The presence of the two groups of Cepheids with two sharp expressed types of harmonic unequalities proves the existence of two causes, which produce them. It is not impossible, that in some cases these causes operate together and produce long as well as short periodic oscillations of the same variable (apparently in the case of KR Lyrae). Then we shall expect a better expressed and clearer longperiodic unequality. It is impossible to explain it from the standpoint of a disturbed Keple- rian motion, even inaresisting medium, because of relation (I). Perhaps it will be better to explain it by the pulsation-theory, considering it almost as an hypothetical explanation.

From the standpoint of the theory of free oscillations of gas-spheres, given already by R. Emden'), , the period of the oscillation, corresponding to a spherical function of the given order n, is completely determined by the average density of the sphere Q, following the formula

P, = I 1897 I/( I/@) cgs . Thus the period does not depend on the constitution of the gas-sphere, i. e. on the so called polytropic index R, which enters in the relation between the density and the pressure within the sphere, according to the formula

p = const:@'. But the result of R. Emdm seems only as a first approximation, because he neglects the compressibility of the gas. In the second approximation, if we do not neglect the compressibility, the period of free pulsation *) must be a function of the polytropic exponent k .

From the standpoint of the pulsation theory the harmonic unequalities ofPmay be explained by the oscillation ofquantities, which determine the period, i. e. of and (in the second approxi- mation) R. A longperiodic oscillation of Q is excluded, because Po.z is the greatest of the periods of free oscillation. From the standpoint of the pulsation theory it is only possible to explain the longperiodic oscillation of P by an oscillation of k.

In AN 2 1 9 . 2 9 1 hat Lciner eine sorgfaltige Reihe von Beobachtungen des Bedeckungsveranderlichen X Trianguli mit- geteilt, die in Verbindung mit den hiesigen photometrischen

If in the interior of the star processes, destroying theradi- ative equilibrium and the constancy ofthe energy flux take place, directed from the centre to the surface, the type of the poly- tropic equilibrium must change. If the quantity of energy, ascending to the surface, increases, the average temperature- gradient of the star must decrease and, consequently, the poly- tropic index R must increase; the decrease of the energy flux produces also a decrease of k. We suppose, that, because of some yet u n k n o w n reasons, the flux of energy is a periodic function of the time with period n. Since the polytropic index, which determines the period of free oscillation PI must be also a similar function of the time, Pwi l l be a periodic function with a period IT, if the gaseous sphere is in state of free pulsation. The oscillation of k may be very small.

The presence of an oscillation of k in the Sun is doubtless, longperiodic changes of the solar constant testify, that at the time of the maximum of the spots, the flux of energy, perhaps owing to the combined action of convection and conduction, reaches its maximum: in the epoch of minimum it is, in the contrary, minimal. Consequently, the polytropic index of the Sun must undergo longperiodic oscillations, increasing and decreasing simultaneously with Wovs numbers. The investi- gation of the variation of the intensity distribution upon the Sun's disk confirms this supposition. According to R. Emden the distribution of intensity on the disk .of a star of class m = I/(&- I ) , consisting of gray material, is subjected to the following law: A ( j ) = ~(cOsj )4 ,~(+1 where i is the angle ' between the normal of the element and the radius of vision. Thus to the increase of the limb-centre contrast corresponds an increase of R, to the decrease of contrast - a decrease of k. Indeed according to the data of the Smithsonian observers the contrast limb-centre increases in the time of maximum of spots and decreases in the time of minimum.

Making use of a graphical extrapolation of the data given for the second group of Cepheids, it is possible to calculate the period of sunspots on the ground of the theoretical period of free pulsation of the Sun.

According to Emdens formula for the Sun we have Po.z = oI'08 and we obtain 3, with formula ( I )

II= 11:s . I am not inclined to c:onsider this result as an acci-

dental one and I believe, that the harmonic unequality of the Cepheids, explained by the pulsation theory, will open new ways for the thermodynamics of the stars.

hlessungen bereits ein gesichertes Bild der Verhaltnisse in dem System ergeben. Die Reduktion der von Lriner in Stufen gegebenen Lichtkurve auf photometrische GroOen wird sehr

Charkow, Ukraina, Astronomical Observatory, I 9 2 3 March. B. Gemsi~~iovii~.

') Gaskugeln 1907. 3 In my paper .The mathematical theory ofcepheidsn, which shall be printed in vol. IV of the .Annals of the Chief Russian Astro:

physical Observatory., I have analysed this question more extensively. Supposing, that the gaseous polytropic sphere with the index k undergoes pulsations, following the same polytrope, I have found that (in the second approximation) the period of oscillation, according to a spherical function of order 2 is equal to P = PO.^ v ( 2 / e ) (Po.z period of Emden), where z is the single positive root of the cubic equation eJ-a/sye'+(8+'Gj3y+'!3y')z-~24+20y+'1/3y~ = o where y = 6(K- I). If k changes, r' changes very little.

*) My more precise formula for the convenient k ( K / 8 - 6 / 4 ) gives P = 0.06 and n = 11a2.