by the ,department of ,mathematics at the ,university of ,glasgow47 ,transcriber's ,note3 ,this text is intended for users whose text readers cannot use the real 7unicode_/utf-#h7 version of the file4 ,characters in the ,greek alphabet are represented as follows3 alpha1 beta1 gamma1 etc4 ,in the original text1 the units h and m1 and ordinals th and st were printed as superscripts4 ,for readability1 they have not been represented as such in this file4 ,similarly for the and - signs when used to describe intermediate stellar colours4 ,other superscripts are indicated by the carat symbol1 1 and subscripts by an underline1 4 ,,lectures ,,on ,,stellar ,,statistics ,,by ,c4 ,v4 ,l4 ,,charlier ,,scientia ,,publisher ,,lund #aiba ,,hamburg #aiba ,,printed ,,by ,l,,tcke ,,wulff ,,chapter ,i4 ,,apparent ,,attributes ,,of ,,the ,,stars4 #a4 ,our knowledge of the stars is based on their apparent attributes1 obtained from the astronomical observations4 ,the object of astronomy is to deduce herefrom the real or absolute attributes of the stars1 which are their position in space1 their movement1 and their physical nature4 ,the apparent attributes of the stars are studied by the aid of their radiation4 ,the characteristics of this radiation may be described in different ways1 according as the nature of the light is defined4 7,undulatory theory1 ,emission theory47 ,from the statistical point of view it will be convenient to consider the radiation as consisting of an emanation of small particles from the radiating body 7the star74 ,these particles are characterized by certain attributes1 which may differ in degree from one particle to another4 ,these attributes may be1 for instance1 the diameter and form of the particles1 their mode of rotation1 c4 ,by these attributes the optical and electrical properties of the radiation are to be explained4 ,i shall not here attempt any such explanation1 but shall confine myself to the property which the particles have of possessing a different mode of deviating from the rectilinear path as they pass from one medium to another4 ,this deviation depends in some way on one or more attributes of the particles4 ,let us suppose that it depends on a single attribute1 which1 with a terminology derived from the undulatory theory of ,,huyghens1 may be called the wave-length 7lambda7 of the particle4 ,the statistical characteristics of the radiation are then in the first place3-- 7#a7 the total number of particles or the intensity of the radiation2 7#b7 the mean wave-length 7lambda#j7 of the radiation1 also called 7or nearly identical with7 the effective wave-length or the colour2 7#c7 the dispersion of the wave-length4 ,this characteristic of the radiation may be determined from the spectrum1 which also gives the variation of the radiation with lambda1 and hence may also determine the mean wave-length of the radiation4 ,moreover we may find from the radiation of a star its apparent place on the sky4 ,the intensity1 the mean wave-length1 and the dispersion of the wave-length are in a simple manner connected with the temperature 7,t7 of the star4 ,according to the radiation laws of ,,stephan and ,,wien we find1 indeed 7compare ,l4 ,m4 #daa7 that the intensity is proportional to the fourth power of ,t1 whereas the mean wave-length and the dispersion of the wave-length are both inversely proportional to ,t4 ,it follows that with increasing temperature the mean wave-length diminishes--the colour changing into violet--and simultaneously the dispersion of the wave-length and also even the total length of the spectrum are reduced 7decrease74 #b4 ,the apparent position of a star is generally denoted by its right ascension 7alpha7 and its declination 7delta74 ,taking into account the apparent distribution of the stars in space1 it is1 however1 more practical to characterize the position of a star by its galactic longitude 7l7 and its galactic latitude 7b74 ,before defining these coordinates1 which will be generally used in the following pages1 it should be pointed out that we shall also generally give the coordinates alpha and delta of the stars in a particular manner4 ,we shall therefore use an abridged notation1 so that if for instance alpha #ag;h #ddm4#g and delta #ce4hd1 we shall write 7alphadelta7 7#agddce74 ,if delta is negative1 for instance delta -#ce4hd1 we write 7alphadelta7 7#agddce71 so that the last two figures are in italics4 ,transcriber's ,note3 ,in this version of the text1 the last two figures are enclosed in braces to represent the italics4 ,this notation has been introduced by ,,pickering for variable stars and is used by him everywhere in the ,annals of the ,harvard ,observatory1 but it is also well suited to all stars4 ,this notation gives1 simultaneously1 the characteristic numero of the stars4 ,it is true that two or more stars may in this manner obtain the same characteristic numero4 ,they are1 however1 easily distinguishable from each other through other attributes4 ,the galactic coordinates l and b are referred to the ,milky ,way 7the ,galaxy7 as plane of reference4 ,the pole of the ,milky ,way has according to ,,houzeau and ,,gould the position 7alphadelta7 7#abdebg74 ,from the distribution of the stars of the spectral type ,b ,i have in ,l4 ,m4 ,,ii1 #adb found a somewhat different position4 ,but having ascertained later that the real position of the galactic plane requires a greater number of stars for an accurate determination of its value1 ,i have preferred to employ the position used by ,,pickering in the ,harvard catalogues1 namely 7alphadelta7 7#abdjbh71 or alpha #ab;h #djm #aij1 delta #bh1 which position is now exclusively used in the stellar statistical investigations at the ,observatory of ,lund and is also used in these lectures4 ,the galactic longitude 7l7 is reckoned from the ascending node of the ,milky ,way on the equator1 which is situated in the constellation ,aquila4 ,the galactic latitude 7b7 gives the angular distance of the star from the ,galaxy4 ,on plate ,i1 at the end of these lectures1 will be found a fairly detailed diagram from which the conversion of alpha and delta of a star into l and b may be easily performed4 ,all stars having an apparent magnitude brighter than #dm are directly drawn4 ,instead of giving the galactic longitude and latitude of a star we may content ourselves with giving the galactic square in which the star is situated4 ,for this purpose we assume the sky to be divided into #dh squares1 all having the same surface4 ,two of these squares lie at the northern pole of the ,galaxy and are designated ,,ga#a and ,,ga#b4 ,twelve lie north of the galactic plane1 between #j and #cj galactic latitude1 and are designated ,,gc#a1 ,,gc#b1 4441 ,,gc#ab4 ,the corresponding squares south of the galactic equator 7the plane of the ,galaxy7 are called ,,gd#a1 ,,gd#b1 4441 ,,gd#ab4 ,the two polar squares at the south pole are called ,,gf#a and ,,gf#b4 ,finally we have #aj ,b-squares1 between the ,a- and ,c-squares and #aj corresponding ,e-squares in the southern hemisphere4 ,the distribution of the squares in the heavens is here graphically represented in the projection of ,,flamsteed1 which has the advantage of giving areas proportional to the corresponding spherical areas1 an arrangement necessary1 or at least highly desirable1 for all stellar statistical researches4 ,it has also the advantage of affording a continuous representation of the whole sky4 ,the correspondence between squares and stellar constellations is seen from plate ,,ii4 ,arranging the constellations according to their galactic longitude we find north of the galactic equator 7in the ,c-squares7 the constellations3-- ,hercules1 ,cygnus1 ,cepheus1 ,cassiopa1 ,auriga1 ,gemini1 ,canis ,minor1 ,pyxis1 ,vela1 ,centaurus1 ,scorpius1 ,ophiuchus1 and south of this equator 7in the ,d-squares73-- ,aquila1 ,cygnus1 ,lacerta1 ,andromeda1 ,perseus1 ,orion1 ,canis ,major1 ,puppis1 ,carina1 ,circinus1 ,corona australis1 ,sagittarius1 mentioning only one constellation for each square4 ,at the north galactic pole 7in the two ,a-squares7 we have3-- ,canes ,venatici and ,coma ,berenices1 and at the south galactic pole 7in the two ,f-squares73-- ,cetus and ,sculptor4 #c4 ,changes in the position of a star4 ,from the positions of a star on two or more occasions we obtain its apparent motion1 also called the proper motion of the star4 ,we may distinguish between a secular part of this motion and a periodical part4 ,in both cases the motion may be either a reflex of the motion of the observer1 and is then called parallactic motion1 or it may be caused by a real motion of the star4 ,from the parallactic motion of the star it is possible to deduce its distance from the sun1 or its parallax4 ,the periodic parallactic proper motion is caused by the motion of the earth around the sun1 and gives the annual parallax 7pi74 ,in order to obtain available annual parallaxes of a star it is usually necessary for the star to be nearer to us than #e siriometers1 corresponding to a parallax greater than #j4jd4 ,more seldom we may in this manner obtain trustworthy values for a distance amounting to #aj siriometers 7pi #j4jb71 or even still greater values4 ,for such large distances the secular parallax1 which is caused by the progressive motion of the sun in space1 may give better results1 especially if the mean distance of a group of stars is simultaneously determined4 ,such a value of the secular parallax is also called1 by ,,kapteyn1 the systematic parallax of the stars4 ,when we speak of the proper motion of a star1 without further specification1 we mean always the secular proper motion4 #d4 ,terrestrial distances are now1 at least in scientific researches1 universally expressed in kilometres4 ,a kilometre is1 however1 an inappropriate unit for celestial distances4 ,when dealing with distances in our planetary system1 the astronomers1 since the time of ,,newton1 have always used the mean distance of the earth from the sun as universal unit of distance4 ,regarding the distances in the stellar system the astronomers have had a varying practice4 ,german astronomers1 ,,seeliger and others1 have long used a stellar unit of distance corresponding to an annual parallax of #j4b1 which has been called a ,siriusweite4 ,to this name it may be justly objected that it has no international use1 a great desideratum in science4 ,against the theoretical definition of this unit it may also be said that a distance is suitably to be defined through another distance and not through an angle--an angle which corresponds moreover1 in this case1 to the harmonic mean distance of the star and not to its arithmetic mean distance4 ,the same objection may be made to the unit parsec4 proposed in #aiab by ,,turner4 ,for my part ,i have1 since #aiaa1 proposed a stellar unit which1 both in name and definition1 nearly coincides with the proposition of ,,seeliger1 and which will be exclusively used in these lectures4 ,a siriometer is put equal to #ajf times the planetary unit of distance1 corresponding to a parallax of #j4bjfbfe 7in practice sufficiently exactly #j4b74 ,in popular writings1 another unit3 a light-year1 has for a very long time been employed4 ,the relation between these units is #a siriometer #ae4gi light-years1 #a light-year #j4jfcc siriometers4 #e4 ,in regard to time also1 the terrestrial units 7second1 day1 year7 are too small for stellar wants4 ,as being consistent with the unit of distance1 ,i have proposed for the stellar unit of time a stellar year 7st471 corresponding to #ajf years4 ,we thus obtain the same relation between the stellar and the planetary units of length and time1 which has the advantage that a velocity of a star expressed in siriometers per stellar year is expressed with the same numerals in planetary units of length per year4 ,spectroscopic determinations of the velocities1 through the ,,doppler-principle1 are generally expressed in km4 per second4 ,the relation with the stellar unit is the following3 #a km4_/sec4 #j4baaa sir4_/st41 #j4baaa planetary units per year1 #a sir4_/st4 #d4gcge km4_/sec4 ,thus the velocity of the sun is #bj km4_/sec4 or #d4bb sir4_/st4 7 #d4bb earth distances from the sun per year74 ,of the numerical value of the stellar velocity we shall have opportunity to speak in the following4 ,for the present it may suffice to mention that most stars have a velocity of the same degree as that of the sun 7in the mean somewhat greater71 and that the highest observed velocity of a star amounts to #gb sir4_/st4 7 #cdj km4_/sec474 ,in the next chapter ,i give a table containing the most speedy stars4 ,the least value of the stellar velocity is evidently equal to zero4 #f4 ,intensity of the radiation4 ,this varies within wide limits4 ,the faintest star which can give an impression on the photographic plates of the greatest instrument of the ,mount ,wilson observatory 7#ajj inch reflector7 is nearly #ajj million times fainter than ,sirius1 a star which is itself more than #ajjjj million times fainter than the sun--speaking of apparent radiation4 ,the intensity is expressed in magnitudes 7m74 ,the reason is partly that we should otherwise necessarily have to deal with very large numbers1 if they were to be proportional to the intensity1 and partly that it is proved that the human eye apprehends quantities of light as proportional to m4 ,this depends upon a general law in psycho-physics1 known as ,,fechner's law1 which says that changes of the apparent impression of light are proportional not to the changes of the intensity but to these changes divided by the primitive intensity4 ,a similar law is valid for all sensations4 ,a conversation is inaudible in the vicinity of a waterfall4 ,an increase of a load in the hand from nine to ten hectograms makes no great difference in the feeling1 whereas an increase from one to two hectograms is easily appreciable4 ,a match lighted in the day-time makes no increase in the illumination1 and so on4 ,a mathematical analysis shows that from the law of ,,fechner it follows that the impression increases in arithmetical progression 7#a1 #b1 #c1 #d1 4447 simultaneously with an increase of the intensity in geometrical progression 7,i1 ,i#b1 ,i#c1 ,i#d1 44474 ,it is with the sight the same as with the hearing4 ,it is well known that the numbers of vibrations of the notes of a harmonic scale follow each other in a geometrical progression though1 for the ear1 the intervals between the notes are apprehended as equal4 ,the magnitudes play the same rle in relation to the quantities of light as do the logarithms to the corresponding numbers4 ,if a star is considered to have a brightness intermediate between two other stars it is not the difference but the ratio of the quantities of light that is equal in each case4 ,the branch of astronomy 7or physics7 which deals with intensities of radiation is called photometry4 ,in order to determine a certain scale for the magnitudes we must choose1 in a certain manner1 the zero-point of the scale and the scale-ratio4 ,both may be chosen arbitrarily4 ,the zero-point is now almost unanimously chosen by astronomers in accordance with that used by the ,harvard ,observatory4 ,no rigorous definition of the ,harvard zero-point1 as far as ,i can see1 has yet been given 7compare however ,h4 ,a4 #ejc71 but considering that the ,pole-star 7alpha ,urs ,minoris7 is used at ,harvard as a fundamental star of comparison for the brighter stars1 and that1 according to the observations at ,harvard and those of ,,hertzsprung 7,a4 ,n4 #deah #aiaa71 the light of the ,pole-star is very nearly invariable1 we may say that the zero-point of the photometric scale is chosen in such a manner that for the ,pole-star m #b4ab4 ,if the magnitudes are given in another scale than the ,harvard-scale 7,h4 ,s471 it is necessary to apply the zero-point correction4 ,this amounts1 for the ,potsdam catalogue1 to -#jm4#af4 ,it is further necessary to determine the scale-ratio4 ,our magnitudes for the stars emanate from ,,ptolemy4 ,it was found that the scale-ratio--giving the ratio of the light-intensities of two consecutive classes of magnitudes--according to the older values of the magnitudes1 was approximately equal to #b4 ,when exact photometry began 7with instruments for measuring the magnitudes7 in the middle of last century1 the scale-ratio was therefore put equal to #b4e4 ,later it was found more convenient to choose it equal to #b4eab1 the logarithm of which number has the value #j4d4 ,the magnitudes being themselves logarithms of a kind1 it is evidently more convenient to use a simple value of the logarithm of the ratio of intensity than to use this ratio itself4 ,this scale-ratio is often called the ,,pogson-scale 7used by ,,pogson in his ,catalogue of #ec known variable stars1 ,astr4 ,obs4 of the ,radcliffe ,observatory1 #ahef71 and is now exclusively used4 ,it follows from the definition of the scale-ratio that two stars for which the light intensities are in the ratio #ajj3a differ by exactly #e magnitudes4 ,a star of the #fth magnitude is #ajj times fainter than a star of the first magnitude1 a star of the #aath magnitude #ajjjj times1 of the #afth magnitude a million times1 and a star of the #bast magnitude #ajj million times fainter than a star of the first magnitude4 ,the star magnitudes are now1 with a certain reservation for systematic errors1 determined with an accuracy of #jm4#a1 and closer4 ,evidently1 however1 there will correspond to an error of #j4a in the magnitude a considerable uncertainty in the light ratios1 when these differ considerably from each other4 ,sun -#bfm4#fj ,full moon -#aam4#gg ,venus - #dm4#bh ,jupiter - #bm4#ce ,mars - #am4#gi ,mercury - #jm4#ij ,saturn #jm4#hh ,uranus #em4#hf ,neptune #gm4#ff ,a consequence of the definition of m is that we also have to do with negative magnitudes 7as well as with negative logarithms74 ,thus1 for example1 for ,sirius m -#a4eh4 ,the magnitudes of the greater planets1 as well as those of the moon and the sun1 are also negative1 as will be seen from the adjoining table1 where the values are taken from ,die ,photometrie der ,gestirne by ,g4 ,m,,ller4 ,the apparent magnitude of the sun is given by ,z,,llner 7#ahfd74 ,the other values are all found in ,potsdam1 and allude generally to the maximum value of the apparent magnitude of the moon and the planets4 ,the brightest star is ,sirius1 which has the magnitude m -#a4eh4 ,the magnitude of the faintest visible star evidently depends on the penetrating power of the instrument used4 ,the telescope of ,,william ,,herschel1 used by him and his son in their star-gauges and other stellar researches1 allowed of the discerning of stars down to the #adth magnitude4 ,the large instruments of our time hardly reach much farther1 for visual observations4 ,when1 however1 photographic plates are used1 it is easily possible to get impressions of fainter stars1 even with rather modest instruments4 ,the large #ajj-;inch mirror of the ,wilson ,observatory renders possible the photographic observations of stars of the #bjth apparent magnitude1 and even fainter4 ,the observations of visual magnitudes are performed almost exclusively with the photometer of ,z,,llner in a more or less improved form4 #g4 ,absolute magnitude4 ,the apparent magnitude of a star is changed as the star changes its distance from the observer1 the intensity increasing indirectly as the square of the distance of the star4 ,in order to make the magnitudes of the stars comparable with each other it is convenient to reduce them to their value at a certain unit of distance4 ,as such we choose one siriometer4 ,the corresponding magnitude will be called the absolute magnitude and is denoted by ,m4#d ,we easily find from the table given in the preceding paragraph that the absolute magnitude of the sun1 according to ,z,,llner's value of m1 amounts to #c4d1 of the moon to #ca4b4 ,for ,jupiter we find ,m #bd4f1 for ,venus ,m #be4c4 ,the other planets have approximately ,m #cj4 ,for the absolute magnitudes of those stars for which it has hitherto been possible to carry out a determination1 we find a value of ,m between -#h and #ac4 ,we shall give in the third chapter short tables of the absolutely brightest and faintest stars now known4 #h4 ,photographic magnitudes4 ,the magnitudes which have been mentioned in the preceding paragraphs all refer to observations taken with the eye1 and are called visual magnitudes4 ,the total intensity of a star is1 however1 essentially dependent on the instrument used in measuring the intensity4 ,besides the eye1 the astronomers use a photographic plate1 bolometer1 a photo-electric cell1 and other instruments4 ,the difference in the results obtained with these instruments is due to the circumstance that different parts of the radiation are taken into account4 ,the usual photographic plates1 which have their principal sensibility in the violet parts of the spectrum1 give us the photographic magnitudes of the stars4 ,it is1 however1 to be remarked that these magnitudes may vary from one plate to another1 according to the distributive function of the plate 7compare ,l4 ,m4 #fg74 ,this variation1 which has not yet been sufficiently studied1 seems however to be rather inconsiderable1 and must be neglected in the following4 ,the photographic magnitude of a star will in these lectures be denoted by m'1 corresponding to a visual magnitude m4 ,in practical astronomy use is also made of plates which1 as the result of a certain preparation 7in colour baths or in other ways71 have acquired a distributive function nearly corresponding to that of the eye1 and especially have a maximum point at the same wave-lengths4 ,such magnitudes are called photo-visual 7compare the memoir of ,,parkhurst in ,a4 ,j4 #cf #aiab74 ,the photographic magnitude of a star is generally determined from measurements of the diameter of the star on the plate4 ,a simple mathematical relation then permits us to determine m'4 ,the diameter of a star image increases with the time of exposure4 ,this increase is due in part to the diffraction of the telescope1 to imperfect achromatism or spherical aberration of the objective1 to irregular grinding of the glass1 and especially to variations in the refraction of the air1 which produce an oscillation of the image around a mean position4 ,the zero-point of the photographic magnitudes is so determined that this magnitude coincides with the visual magnitude for such stars as belong to the spectral type ,a#j and have m #f4j1 according to the proposal of the international solar conference at ,bonn1 #aiaa4 ,determinations of the photographic or photo-visual magnitudes may now be carried out with great accuracy4 ,the methods for this are many and are well summarised in the ,report of the ,council of the ,r4 ,a4 ,s4 of the year #aiac4 ,the most effective and far-reaching method seems to be that proposed by ,,schwarzschild1 called the half-grating method1 by which two exposures are taken of the same part of the sky1 while at one of the exposures a certain grating is used that reduces the magnitudes by a constant degree4 #i4 ,colour of the stars4 ,the radiation of a star is different for different wave-lengths 7lambda74 ,as regarding other mass phenomena we may therefore mention3--7#a7 the total radiation or intensity 7,i71 7#b7 the mean wave-length 7lambda#j71 7#c7 the dispersion of the wave-length 7sigma74 ,in the preceding paragraphs we have treated of the total radiation of the stars as this is expressed through their magnitudes4 ,the mean wave-length is pretty closely defined by the colour1 whereas the dispersion of the wave-length is found from the spectrum of the stars4 ,there are blue 7,b71 white 7,w71 yellow 7,y7 and red 7,r7 stars1 and intermediate colours4 ,the exact method is to define the colour through the mean wave-length 7and not conversely7 or the effective wave-length as it is most usually called1 or from the colour-index4 ,we shall revert later to this question4 ,there are1 however1 a great many direct eye-estimates of the colour of the stars4 ,colour corresponding to a given spectrum4 ,sp4 ,colour ,number ,b#c ,,yw- #afa ,a#j ,,yw- #ghh ,a#e ,,yw #aae ,f#e ,,yw1 ,,wy- #bie ,g#e ,,wy #baf ,k#e ,,wy1 ,y- #eeb ,m ,y1 ,y #ie ----------------------------- ,sum 444 #bbbb ,spectrum corresponding to a given colour4 ,colour ,sp4 ,number ,w1 ,w ,a#j #bha ,,yw- ,a#j #cef ,,yw ,a#e #dhb ,,yw1 ,,yw- ,f#c #baa ,,wy ,g#d #bfd ,,wy1 ,y- ,k#a #bhi ,y1 ,y ,k#d #bed ,,ry-1 ,,ry ,k#e #he -------------------------------- ,sum 444 #bbbb ,the signs and - indicate intermediate shades of colour4 ,the preceding table drawn up by ,dr4 ,,malmquist from the colour observations of ,m,,ller and ,,kempf in ,potsdam1 shows the connection between the colours of the stars and their spectra4 ,the ,potsdam observations contain all stars north of the celestial equator having an apparent magnitude brighter than #gm4#e4 ,we find from these tables that there is a well-pronounced regression in the correlation between the spectra and the colours of the stars4 ,taking together all white stars we find the corresponding mean spectral type to be ,a#j1 but to ,a#j corresponds1 upon an average1 the colour yellow-white4 ,the yellow stars belong in the mean to the ,k-type1 but the ,k-stars have upon an average a shade of white in the yellow colour4 ,the coefficient of correlation 7r7 is not easy to compute in this case1 because one of the attributes1 the colour1 is not strictly graduated 7i4e4 it is not expressed in numbers defining the colour74#e ,using the coefficient of contingency of ,,pearson1 it is1 however1 possible to find a fairly reliable value of the coefficient of correlation1 and ,,malmquist has in this way found r #j4he1 a rather high value4 ,in order to facilitate the discussion of the relation between colour and spectrum it is convenient to deal here with the question of the spectra of the stars4 #aj4 ,spectra of the stars4 ,in order to introduce the discussion ,i first give a list of the wave-lengths of the ,,frauenhofer lines in the spectrum1 and the corresponding chemical elements4 ,,frauenhofer line ,element lambda ,a #gei4d ,b #fhf4h ,c7alpha7 ,h 7hydrogen7 #fef4c ,d#a ,na 7sodium7 #ehi4f ,d#c ,he #ehg4f ,e ,fe 7iron7 #ebg4j ,f7beta7 ,h #dhf4b 7gamma7 ,h #dcd4a ,g ,ca 7calcium7 #dcj4h h7delta7 ,h #daj4b ,h7epsilon7 ,ca7,h7 #cif4i ,k ,ca #cic4d ,the first column gives the ,,frauenhofer denomination of each line4 ,moreover the hydrogen lines alpha1 beta1 gamma1 delta1 epsilon are denoted4 ,the second column gives the name of the corresponding element1 to which each line is to be attributed4 ,the third column gives the wave-length expressed in millionths of a millimeter as unit 7mumu74 ,on plate ,,iii1 where the classification of the stellar spectra according to the ,harvard system is reproduced1 will be found also the wave-lengths of the principal ,h and ,he lines4 ,by the visual spectrum is usually understood the part of the radiation between the ,,frauenhofer lines ,a to ,h 7lambda #gfj to #djj mumu71 whereas the photographic spectrum generally lies between ,f and ,k 7lambda #ejj to #djj mumu74 ,in the earliest days of spectroscopy the spectra of the stars were classified according to their visual spectra4 ,this classification was introduced by ,,secchi and was later more precisely defined by ,,vogel4 ,the three classes ,i1 ,,ii1 ,,iii of ,,vogel correspond approximately to the colour classification into white1 yellow1 and red stars4 ,photography has now almost entirely taken the place of visual observations of spectra1 so that ,,secchi's and ,,vogel's definitions of the stellar spectra are no longer applicable4 ,the terminology now used was introduced by ,,pickering and ,miss ,,cannon and embraces a great many types1 of which we here describe the principal forms as they are defined in ,part4 ,,ii of ,vol4 ,,xxviii of the ,annals of the ,harvard ,observatory4 ,it may be remarked that ,,pickering first arranged the types in alphabetical order ,a1 ,b1 ,c1 c41 supposing that order to correspond to the temperature of the stars4 ,later this was found to be partly wrong1 and in particular it was found that the ,b-stars may be hotter than those of type ,a4 ,the following is the temperature-order of the spectra according to the opinion of the ,harvard astronomers4 ,type ,o 7,,wolf-,,rayet stars74 ,the spectra of these stars consist mainly of bright lines4 ,they are characterized by the bright bands at wave-lengths #dfc mumu and #dfi mumu1 and the line at #eja mumu characteristic of gaseous nebulae is sometimes present4 ,this type embraces mainly stars of relatively small apparent brightness4 ,the brightest is gamma ,velorum with m #b4bb4 ,we shall find that the absolute magnitude of these stars nearly coincides with that of the stars of type ,b4 ,the type is grouped into five subdivisions represented by the letters ,oa1 ,ob1 ,oc1 ,od and ,oe4 ,these subdivisions are conditioned by the varying intensities of the bright bands named above4 ,the due sequence of these sub-types is for the present an open question4 ,among interesting stars of this type is zeta ,puppis 7,od71 in the spectrum of which ,,pickering discovered a previously unknown series of helium lines4 ,they were at first attributed 7by ,,rydberg7 to hydrogen and were called additional lines of hydrogen4 ,type ,b 7,orion type1 ,helium stars74 ,all lines are here dark4 ,besides the hydrogen series we here find the ,he-lines 7#cif1 #djc1 #dab1 #dad1 #ddg1 #dga1 #dic mumu74 ,to this type belong all the bright stars 7beta1 gamma1 delta1 epsilon1 zeta1 eta and others7 in ,orion with the exception of ,betelgeuze4 ,further1 ,spica and many other bright stars4 ,on plate ,,iii epsilon ,orionis is taken as representative of this type4 ,type ,a 7,sirius type7 is characterized by the great intensity of the hydrogen lines 7compare plate ,,iii74 ,the helium lines have vanished4 ,other lines visible but faintly4 ,the greater part of the stars visible to the naked eye are found here4 ,there are #abea stars brighter than the #fth magnitude which belong to this type4 ,sirius1 ,vega1 ,castor1 ,altair1 ,deneb and others are all ,a-stars4 ,type ,f 7,calcium type74 ,the hydrogen lines still rather prominent but not so broad as in the preceding type4 ,the two calcium lines ,h and ,k 7#cif4i1 #cic4d mumu7 strongly pronounced4 ,among the stars of this type are found a great many bright stars 7compare the third chapter71 such as ,polaris1 ,canopus1 ,procyon4 ,type ,g 7,sun type74 ,numerous metallic lines together with relatively faint hydrogen lines4 ,to this class belong the sun1 ,capella1 alpha ,centauri and other bright stars4 ,type ,k4 ,the hydrogen lines still fainter4 ,the ,k-line attains its maximum intensity 7is not especially pronounced in the figure of plate ,,iii74 ,this is1 next to the ,a-type1 the most numerous type 7#aadb stars7 among the bright stars4 ,we find here gamma ,andromed1 beta ,aquil1 ,arcturus1 alpha ,cassiopei1 ,pollux and ,aldebaran1 which last forms a transition to the next type4 ,type ,m4 ,the spectrum is banded and belongs to ,,secchi's third type4 ,the flutings are due to titanium oxide4 ,only #aij of the stars visible to the naked eye belong to this type4 ,generally they are rather faint1 but we here find ,betelgeuze1 alpha ,herculis1 beta ,pegasi1 alpha ,scorpii 7,antares7 and most variables of long period1 which form a special sub-type ,md1 characterized by bright hydrogen lines together with the flutings4 ,type ,m has two other sub-types ,ma and ,mb4 ,type ,n 7,,secchi's fourth type74 ,banded spectra4 ,the flutings are due to compounds of carbon4 ,here are found only faint stars4 ,the total number is #bda4 ,all are red4 #bg stars having this spectrum are variables of long period of the same type as ,md4 ,the spectral types may be summed up in the following way3-- ,white stars3--,,secchi's type ,i3--,harvard ,b and ,a1 ,yellow 3-- ,,ii3-- ,f1 ,g and ,k1 ,red 3-- ,,iii3-- ,m1 3-- ,,iv3-- ,n4 ,the ,harvard astronomers do not confine themselves to the types mentioned above1 but fill up the intervals between the types with sub-types which are designated by the name of the type followed by a numeral #j1 #a1 #b1 4441 #i4 ,thus the sub-types between ,a and ,f have the designations ,a#j1 ,a#a1 ,a#b1 4441 ,a#i1 ,f#j1 c4 ,exceptions are made as already indicated1 for the extreme types ,o and ,m4 #aa4 ,spectral index4 ,it may be gathered from the above description that the definition of the types implies many vague moments4 ,especially in regard to the ,g-type are very different definitions indeed accepted1 even at ,harvard4#f ,it is also a defect that the definitions do not directly give quantitative characteristics of the spectra4 ,none the less it is possible to substitute for the spectral classes a continuous scale expressing the spectral character of a star4 ,such a scale is indeed implicit in the ,harvard classification of the spectra4 ,let us use the term spectral index 7s7 to define a number expressing the spectral character of a star4 ,then we may conveniently define this conception in the following way4 ,let ,a#j correspond to the spectral index s #j4j1 ,f#j to s #a4j1 ,g#j to s #b4j1 ,k#j to s #c4j ,m#j to s #d4j and ,b#j to s -#a4j4 ,further1 let ,a#a1 ,a#b1 ,a#c1 c41 have the spectral indices #j4a1 #j4b1 #j4c1 c41 and in like manner with the other intermediate sub-classes4 ,then it is evident that to all spectral classes between ,b#j and ,m there corresponds a certain spectral index s4 ,the extreme types ,o and ,n are not here included4 ,their spectral indices may however be determined1 as will be seen later4 ,though the spectral indices1 defined in this manner1 are directly known for every spectral type1 it is nevertheless not obvious that the series of spectral indices corresponds to a continuous series of values of some attribute of the stars4 ,this may be seen to be possible from a comparison with another attribute which may be rather markedly graduated1 namely the colour of the stars4 ,we shall discuss this point in another paragraph4 ,to obtain a well graduated scale of the spectra it will finally be necessary to change to some extent the definitions of the spectral types1 a change which1 however1 has not yet been accomplished4 #ab4 ,we have found in #i that the light-radiation of a star is described by means of the total intensity 7,i71 the mean wave-length 7lambda#j7 and the dispersion of the wave-length 7sigmalambda74 lambda#j and sigmalambda may be deduced from the spectral observations4 ,it must here be observed that the observations give1 not the intensities at different wave-lengths but1 the values of these intensities as they are apprehended by the instruments employed--the eye or the photographic plate4 ,for the derivation of the true curve of intensity we must know the distributive function of the instrument 7,l4 ,m4 #fg74 ,as to the eye1 we have reason to believe1 from the bolometric observations of ,,langley 7#ahhh71 that the mean wave-length of the visual curve of intensity nearly coincides with that of the true intensity-curve1 a conclusion easily understood from ,,darwin's principles of evolution1 which demand that the human eye in the course of time shall be developed in such a way that the mean wave-length of the visual intensity curve does coincide with that of the true curve 7lambda #ecj mumu71 when the greatest visual energy is obtained 7,l4 ,m4 #fg74 ,as to the dispersion1 this is always greater in the true intensity-curve than in the visual curve1 for which1 according to #aj1 it amounts to approximately #fj mumu4 ,we found indeed that the visual intensity curve is extended1 approximately1 from #djj mumu to #gfj mumu1 a sixth part of which interval1 approximately1 corresponds to the dispersion sigma of the visual curve4 ,in the case of the photographic intensity-curve the circumstances are different4 ,the mean wave-length of the photographic curve is1 approximately1 #dej mumu1 with a dispersion of #af mumu1 which is considerably smaller than in the visual curve4 #ac4 ,both the visual and the photographic curves of intensity differ according to the temperature of the radiating body and are therefore different for stars of different spectral types4 ,here the mean wave-length follows the formula of ,,wien1 which says that this wave-length varies inversely as the temperature4 ,the total intensity1 according to the law of ,,stephan1 varies directly as the fourth power of the temperature4 ,even the dispersion is dependent on the variation of the temperature--directly as the mean wave-length1 inversely as the temperature of the star 7,l4 ,m4 #da7--so that the mean wave-length1 as well as the dispersion of the wave-length1 is smaller for the hot stars ,o and ,b than for the cooler ones 7,k and ,m types74 ,it is in this manner possible to determine the temperature of a star from a determination of its mean wave-length 7lambda#j7 or from the dispersion in lambda4 ,such determinations 7from lambda#j7 have been made by ,,scheiner and ,,wilsing in ,potsdam1 by ,,rosenberg and others1 though these researches still have to be developed to a greater degree of accuracy4 #ad4 ,effective wave-length4 ,the mean wave-length of a spectrum1 or1 as it is often called by the astronomers1 the effective wave-length1 is generally determined in the following way4 ,on account of the refraction in the air the image of a star is1 without the use of a spectroscope1 really a spectrum4 ,after some time of exposure we get a somewhat round image1 the position of which is determined precisely by the mean wave-length4 ,this method is especially used with a so-called objective-grating1 which consists of a series of metallic threads1 stretched parallel to each other at equal intervals4 ,on account of the diffraction of the light we now get in the focal plane of the objective1 with the use of these gratings1 not only a fainter image of the star at the place where it would have arisen without grating1 but also at both sides of this image secondary images1 the distances of which from the central star are certain theoretically known multiples of the effective wave-lengths4 ,in this simple manner it is possible to determine the effective wave-length1 and this being a tolerably well-known function of the spectral-index1 the latter can also be found4 ,this method was first proposed by ,,hertzsprung and has been extensively used by ,,bergstrand1 ,,lundmark and ,,lindblad at the observatory of ,upsala and by others4 #ae4 ,colour-index4 ,we have already pointed out in #i that the colour may be identified with the mean wave-length 7lambda#j74 ,as further lambda#j is closely connected with the spectral index 7s71 we may use the spectral index to represent the colour4 ,instead of s there may also be used another expression for the colour1 called the colour-index4 ,this expression was first introduced by ,,schwarzschild1 and is defined in the following way4 ,we have seen that the zero-point of the photographic scale is chosen in such a manner that the visual magnitude m and the photographic magnitude m' coincide for stars of spectral index #j4j 7,a#j74 ,the photographic magnitudes are then unequivocally determined4 ,it is found that their values systematically differ from the visual magnitudes1 so that for type ,b 7and ,o7 the photographic magnitudes are smaller than the visual1 and the contrary for the other types4 ,the difference is greatest for the ,m-type 7still greater for the ,n-stars1 though here for the present only a few determinations are known71 for which stars if amounts to nearly two magnitudes4 ,so much fainter is a red star on a photographic plate than when observed with the eye4 ,the difference between the photographic and the visual magnitudes is called the colour-index 7c74 ,the correlation between this index and the spectral-index is found to be rather high 7r #j4if74 ,in ,l4 ,m4 ,,ii1 #ai ,i have deduced the following tables giving the spectral-type corresponding to a given colour-index1 and inversely4 ,,table #a4 ,,giving ,,the ,,mean ,,colour-,,index ,,corresponding ,,to ,a ,,given ,,spectral ,,type ,,or ,,spectral ,,index4 ----------------------------------- ,spectral ,colour-index type index ---------------------------------- ,b#j -#a4j -#j4df ,b#e -#j4e -#j4bc ,a#j #j4j #j4jj ,a#e #j4e #j4bc ,f#j #a4j #j4df ,f#e #a4e #j4fi ,g#j #b4j #j4ib ,g#e #b4e #a4ae ,k#j #c4j #a4ch ,k#e #c4e #a4fa ,m#j #d4j #a4hd ---------------------------------- ,,table #a994 ,,giving ,,the ,,mean ,,spectral ,,index ,,corresponding ,,to ,a ,,given ,,colour-,,index4 ----------------------------------- ,colour-index ,spectral index type ---------------------------------- -#j4d -#j4gj ,b#c -#j4b -#j4hj ,b#g #j4j #j4aj ,a#a #j4b #j4ej ,a#e #j4d #j4ij ,a#i #j4f #a4cj ,f#c #j4h #a4gj ,f#g #a4j #b4aj ,g#a #a4b #b4ej ,g#e #a4d #b4ij ,g#i #a4f #c4cj ,k#c #a4h #c4gj ,k#g #b4j #d4aj ,m#a ---------------------------------- ,from each catalogue of visual magnitudes of the stars we may obtain their photographic magnitude through adding the colour-index4 ,this may be considered as known 7taking into account the high coefficient of correlation between s and c7 as soon as we know the spectral type of the star4 ,we may conclude directly that the number of stars having a photographic magnitude brighter than #f4j is considerably smaller than the number of stars visually brighter than this magnitude4 ,there are1 indeed1 #dgja stars for which m #f4j and #bhgd stars having m' #f4j4 #af4 ,radial velocity of the stars4 ,from the values of alpha and delta at different times we obtain the components of the proper motions of the stars perpendicular to the line of sight4 ,the third component 7,w71 in the radial direction1 is found by the ,,doppler principle1 through measuring the displacement of the lines in the spectrum1 this displacement being towards the red or the violet according as the star is receding from or approaching the observer4 ,the velocity ,w will be expressed in siriometers per stellar year 7sir4_/st47 and alternately also in km4_/sec4 ,the rate of conversion of these units is given in #e4 #ag4 ,summing up the remarks here given on the apparent attributes of the stars we find them referred to the following principal groups3-- ,i4 ,the position of the stars is here generally given in galactic longitude 7l7 and latitude 7b74 ,moreover their equatorial coordinates 7alpha and delta7 are given in an abridged notation 7alphadelta71 where the first four numbers give the right ascension in hours and minutes and the last two numbers give the declination in degrees1 the latter being printed in italics if the declination is negative4 ,eventually the position is given in galactic squares1 as defined in #b4 ,,ii4 ,the apparent motion of the stars will be given in radial components 7,w7 expressed in sir4_/st4 and their motion perpendicular to the line of sight4 ,these components will be expressed in one component 7u#j7 parallel to the galactic plane1 and one component 7v#j7 perpendicular to it4 ,if the distance 7r7 is known we are able to convert these components into components of the linear velocity perpendicular to the line of sight 7,u and ,v74 ,,iii4 ,the intensity of the light of the stars is expressed in magnitudes4 ,we may distinguish between the apparent magnitude 7m7 and the absolute magnitude 7,m71 the latter being equal to the value of the apparent magnitude supposing the star to be situated at a distance of one siriometer4 ,the apparent magnitude may be either the photographic magnitude 7m'71 obtained from a photographic plate1 or the visual magnitude 7m7 obtained with the eye4 ,the difference between these magnitudes is called the colour-index 7c m'-m74 ,,iv4 ,the characteristics of the stellar radiation are the mean wave-length 7lambda#j7 and the dispersion 7sigma7 in the wave-length4 ,the mean wave-length may be either directly determined 7perhaps as effective wave-length7 or found from the spectral type 7spectral index7 or from the colour-index4 ,there are in all eight attributes of the stars which may be found from the observations3--the spherical position of the star 7l1 b71 its distance 7r71 proper motion 7u#j and v#j71 radial velocity 7,w71 apparent magnitude 7m or m'71 absolute magnitude 7,m71 spectral type 7,sp7 or spectral index 7s71 and colour-index 7c74 ,of these the colour-index1 the spectral type1 the absolute magnitude and also 7to a certain degree7 the radial velocity may be considered as independent of the place of the observer and may therefore be considered not as only apparent but also as absolute attributes of the stars4 ,between three of these attributes 7m1 ,m and r7 a mathematical relation exists so that one of them is known as soon as the other two have been found from observations4 ,,footnotes3 ,footnote #a3 ,meddelanden frn ,lunds ,observatorium1 ,no4 #da4 ,footnote #b3 ,meddelanden frn ,lunds ,astronomiska ,observatorium1 ,serie ,,ii1 ,no4 #ad4 ,footnote #c3 ,annals of the ,harvard ,observatory1 vol4 #ej4 ,footnote #d3 ,in order to deduce from ,m the apparent magnitude at a distance corresponding to a parallax of #a we may subtract #cm4#dh4 ,to obtain the magnitude corresponding to a parallax of #j4a we may add #a4eg4 ,the latter distance is chosen by some writers on stellar statistics4 ,footnote #e3 ,the best colour-scale of the latter sort seems to be that of ,,osthoff4 ,footnote #f3 ,compare ,h4 ,a4 #ej and ,h4 ,a4 #ef and the remarks in ,l4 ,m4 ,,ii1 #ai4 ,,chapter ,,ii4 ,,sources ,,of ,,our ,,present ,,knowledge ,,of ,,the ,,stars4 #ah4 ,in this chapter ,i shall give a short account of the publications in which the most complete information on the attributes of the stars may be obtained1 with short notices of the contents and genesis of these publications4 ,it is1 however1 not my intention to give a history of these researches4 ,we shall consider more particularly the questions relating to the position of the stars1 their motion1 magnitude1 and spectra4 #ai4 ,place of the stars4 ,durchmusterungs4 ,the most complete data on the position of the stars are obtained from the star catalogues known as ,durchmusterungs4 ,there are two such catalogues1 which together cover the whole sky1 one--visual--performed in ,bonn and called the ,bonner ,durchmusterung 7,b4 ,d471 the other--photographic--performed in ,cape ,the ,cape ,photographic ,durchmusterung 7,c4 ,p4 ,d474 ,as the first of these catalogues has long been--and is to some extent even now--our principal source for the study of the sky and is moreover the first enterprise of this kind1 ,i shall give a somewhat detailed account of its origin and contents1 as related by ,,argelander in the introduction to the ,b4 ,d4 ,b4 ,d4 was planned and performed by the ,swedish-,finn ,,argelander 7born in ,memel #agii74 ,a scholar of ,,bessel he was first called as director in bo1 then in ,hlsingfors1 and from there went in #ahcf to ,bonn1 where in the years #aheb to #ahef he performed this great ,durchmusterung4 ,as instrument he used a ,,frauenhofer comet-seeker with an aperture of #gf mm1 a focal length of #fej mm1 and #aj times magnifying power4 ,the field of sight had an extension of #f4 ,in the focus of the objective was a semicircular piece of thin glass1 with the edge 7 the diameter of the semicircle7 parallel to the circle of declination4 ,this edge was sharply ground1 so that it formed a narrow dark line perceptible at star illumination4 ,perpendicular to this diameter 7the hour-line7 were #aj lines1 at each side of a middle line1 drawn at a distance of #g'4 ,these lines were drawn with black oil colour on the glass4 ,the observations are performed by the observer ,a and his assistant ,b4 ,a is in a dark room1 lies on a chair having the eye at the ocular and can easily look over #b in declination4 ,the assistant sits in the room below1 separated by a board floor1 at the ,thiede clock4 ,from the beginning of the observations the declination circle is fixed at a certain declination 7whole degrees74 ,all stars passing the field at a distance smaller than one degree from the middle line are observed4 ,hence the name ,durchmusterung4 ,when a star passes the hour line the magnitude is called out by ,a1 and noted by ,b together with the time of the clock4 ,simultaneously the declination is noted by ,a in the darkness4 ,on some occasions #cj stars may be observed in a minute4 ,the first observation was made on ,febr4 #be1 #aheb1 the last on ,march #bg1 #ahei4 ,in all there were #fbe observation nights with #ahda zones4 ,the total number of stars was #cbdaih4 ,the catalogue was published by ,,argelander in three parts in the years #ahei1 #ahfa and #ahfbg and embraces all stars between the pole and #b south of the equator brighter than #im4#e1 according to the scale of ,,argelander 7his aim was to register all stars up to the #ith magnitude74 ,to this scale we return later4 ,the catalogue is arranged in accordance with the declination-degrees1 and for each degree according to the right ascension4 ,quotations from ,b4 ,d4 have the form ,b4 ,d4 #bc4agd1 which signifies3 ,zone #bc1 star ,no4 #agd4 ,,argelander's work was continued for stars between delta -#b and delta -#bc by ,,sch,,nfeld1 according to much the same plan1 but with a larger instrument 7aperture #aei mm1 focal length #aicj mm1 magnifying power #bf times74 ,the observations were made in the years #ahgf to #ahha and include #accfei stars4#h ,the positions in ,b4 ,d4 are given in tenths of a second in right ascension and in tenths of a minute in declination4 #bj4 ,the ,cape ,photographic ,durchmusterung#i 7,c4 ,p4 ,d474 ,this embraces the whole southern sky from -#ah to the south pole4 ,planned by ,,gill1 the photographs were taken at the ,cape ,observatory with a ,,dallmeyer lens with #ae cm4 aperture and a focal-length of #ace cm4 ,plates of #cj #cj cm4 give the coordinates for a surface of #e #e square degrees4 ,the photographs were taken in the years #ahhe to #ahij4 ,the measurements of the plates were made by ,,kapteyn in ,groningen with a parallactic measuring-apparatus specially constructed for this purpose1 which permits of the direct obtaining of the right ascension and the declination of the stars4 ,the measurements were made in the years #ahhf to #ahih4 ,the catalogue was published in three parts in the years #ahif to #aijj4 ,the positions have the same accuracy as in ,b4 ,d4 ,the whole number of stars is #dedhge4 ,,kapteyn considers the catalogue complete to at least the magnitude #im4#b4 ,in the two great catalogues ,b4 ,d4 and ,c4 ,p4 ,d4 we have all stars registered down to the magnitude #i4j 7visually7 and a good way below this limit4 ,probably as far as to #ajm4 ,a third great ,durchmusterung has for some time been in preparation at ,cordoba in ,argentina4#aj ,it continues the southern zones of ,,sch,,nfeld and is for the present completed up to #fb southern declination4 ,all these ,durchmusterungs are ultimately based on star catalogues of smaller extent and of great precision4 ,of these catalogues we shall not here speak 7,compare1 however1 #bc74 ,a great ,durchmusterung1 that will include all stars to the #aath magnitude1 has for the last thirty years been in progress at different observatories proposed by the congress in ,paris1 #ahhh4 ,the observations proceed very irregularly1 and there is little prospect of getting the work finished in an appreciable time4 #ba4 ,star charts4 ,for the present we possess two great photographic star charts1 embracing the whole heaven3--,the ,harvard ,map 7,h4 ,m47 and the ,,franklin-,,adams ,charts 7,f4 ,a4 ,c474 ,the ,harvard ,map1 of which a copy 7or more correctly two copies7 on glass has kindly been placed at the disposal of the ,lund ,observatory by ,mr4 ,,pickering1 embraces all stars down to the #aath magnitude4 ,it consists of #ee plates1 each embracing more than #ijj square degrees of the sky4 ,the photographs were taken with a small lens of only #b4e cms4 aperture and about #cb4e cms4 focal-length4 ,the time of exposure was one hour4 ,these plates have been counted at the ,lund ,observatory by ,hans ,,henie4 ,we return later to these counts4 ,the ,,franklin-,,adams ,charts were made by an amateur astronomer ,,franklin-,,adams1 partly at his own observatory 7,mervel ,hill7 in ,england1 partly in ,cape and ,johannesburg1 ,transvaal1 in the years #aije-aiab4 ,the photographs were taken with a ,taylor lens with #be cm4 aperture and a focal-length of #aad cm41 which gives rather good images on a field of #ae #ae square degrees4 ,the whole sky is here reproduced on1 in all1 #bjf plates4 ,each plate was exposed for #b hours and #bj minutes and gives images of the stars down to the #agth magnitude4 ,the original plates are now at the observatory in ,greenwich4 ,some copies on paper have been made1 of which the ,lund ,observatory possesses one4 ,it shows stars down to the #adth-#aeth magnitudes and gives a splendid survey of the whole sky more complete1 indeed1 than can be obtained1 even for the north sky1 by direct observation of the heavens with any telescope at present accessible in ,sweden4 ,the ,f4 ,a4 ,c4 have been counted by the astronomers of the ,lund ,observatory1 so that thus a complete count of the number of stars for the whole heaven down to the #adth magnitude has been obtained4 ,we shall later have an opportunity of discussing the results of these counts4 ,a great star map is planned in connection with the ,paris catalogue mentioned in the preceding paragraph4 ,this ,carte du ,ciel 7,c4 d4 ,c47 is still unfinished1 but there seems to be a possibility that we shall one day see this work carried to completion4 ,it will embrace stars down to the #adth magnitude and thus does not reach so far as the ,f4 ,a4 ,c41 but on the other hand is carried out on a considerably greater scale and gives better images than ,f4 ,a4 ,c4 and will therefore be of a great value in the future1 especially for the study of the proper motions of the stars4 #bb4 ,distance of the stars4 ,as the determination1 from the annual parallax1 of the distances of the stars is very precarious if the distance exceeds #e sir4 7pi #j4jd71 it is only natural that the catalogues of star-distances should be but few in number4 ,the most complete catalogues are those of ,,bigourdan in the ,bulletin astronomique ,,xxvi 7#aiji71 of ,,kapteyn and ,,weersma in the publications of ,groningen ,nr4 #bd 7#aiaj71 embracing #cfe stars1 and of ,,walkey in the ,journal of the ,british ,astronomical ,association ,,xxvii 7#aiag71 embracing #fbe stars4 ,through the spectroscopic method of ,,adams it will be possible to enlarge this number considerably1 so that the distance of all stars1 for which the spectrum is well known1 may be determined with fair accuracy4 ,,adams has up to now published #afdf parallax stars4 #bc4 ,proper motions4 ,an excellent catalogue of the proper motions of the stars is ,,lewis ,,boss's ,preliminary ,general ,catalogue of #fahh stars 7#aiaj7 7,b4 ,p4 ,c474 ,it contains the proper motions of all stars down to the sixth magnitude 7with few exceptions7 and moreover some fainter stars4 ,the catalogue is considered by the editor only as a preliminary to a greater catalogue1 which is to embrace some #bejjj stars and is now nearly completed4 #bd4 ,visual magnitudes4 ,the ,harvard observatory has1 under the direction of ,,pickering1 made its principal aim to study the magnitudes of the stars1 and the history of this observatory is at the same time the history of the treatment of this problem4 ,,pickering1 in the genuine ,american manner1 is not satisfied with the three thirds of the sky visible from the ,harvard observatory1 but has also founded a daughter observatory in ,south ,america1 at ,arequipa in ,peru4 ,it is therefore possible for him to publish catalogues embracing the whole heaven from pole to pole4 ,the last complete catalogue 7#aijh7 of the magnitudes of the stars is found in the ,annals of the ,harvard ,observatory ,t4 #ej 7,h4 #ej74 ,it contains #iaaj stars and can be considered as complete to the magnitude #fm4#e4 ,to this catalogue are generally referred the magnitudes which have been adopted at the ,observatory of ,lund1 and which are treated in these lectures4 ,a very important1 and in one respect even still more comprehensive1 catalogue of visual magnitudes is the ,potsdam ,general ,catalogue 7,p4 ,g4 ,c47 by ,m,,ller and ,,kempf1 which was published simultaneously with ,h4 #ej4 ,it contains the magnitude of #adaii stars and embraces all stars on the northern hemisphere brighter than #gm4#e 7according to ,b4 ,d474 ,we have already seen that the zero-point of ,h4 #ej and ,p4 ,g4 ,c4 is somewhat different and that the magnitudes in ,p4 ,g4 ,c4 must be increased by -#jm4#af if they are to be reduced to the ,harvard scale4 ,the difference between the two catalogues however is due to some extent to the colour of the stars1 as has been shown by ,messrs4 ,m,,ller and ,,kempf4 #be4 ,photographic magnitudes4 ,our knowledge of this subject is still rather incomplete4 ,the most comprehensive catalogue is the ,actinometrie by ,,schwarzschild 7#aiab71 containing the photographic magnitudes of all stars in ,b4 ,d4 down to the magnitude #gm4#e between the equator and a declination of #bj4 ,in all1 #cebb stars4 ,the photographic magnitudes are however not reduced for the zero-point 7compare #f74 ,these is also a photometric photographic catalogue of the stars nearest to the pole in ,,parkhurst's ,yerkes actinometry 7#aiab71aa which contains all stars in ,b4 ,d4 brighter than #gm4#e between the pole and #gc northern declination4 ,the total number of stars is #fgb4 ,during the last few years the astronomers of ,harvard and ,mount ,wilson have produced a collection of standard photographic magnitudes for faint stars4 ,these stars1 which are called the polar sequence1#ab all lie in the immediate neighbourhood of the pole4 ,the list is extended down to the #bjth magnitude4 ,moreover similar standard photographic magnitudes are given in ,h4 ,a4 #ga1 #he and #aja4 ,a discussion of the colour-index 7i4e41 the difference between the photographic and the visual magnitudes7 will be found in ,l4 ,m4 ,,ii1 #ai4 ,when the visual magnitude and the type of spectrum are known1 the photographic magnitude may be obtained1 with a generally sufficient accuracy1 by adding the colour-index according to the table #a in #ae above4 #bf4 ,stellar spectra4 ,here too we find the ,harvard ,observatory to be the leading one4 ,the same volume of the ,annals of the ,harvard ,observatory 7,h4 #ej7 that contains the most complete catalogue of visual magnitudes1 also gives the spectral types for all the stars there included1 i4e41 for all stars to #fm4#e4 ,miss ,,cannon1 at the ,harvard ,observatory1 deserves the principal credit for this great work4 ,not content with this result she is now publishing a still greater work embracing more than #bjjjjj stars4 ,the first four volumes of this work are now published and contain the first twelve hours of right ascension1 so that half the work is now printed4#ac #bg4 ,radial velocity4 ,in this matter1 again1 we find ,america to be the leading nation1 though1 this time1 it is not the ,harvard or the ,mount ,wilson but the ,lick ,observatory to which we have to give the honour4 ,the eminent director of this observatory1 ,w4 ,w4 ,,campbell1 has in a high degree developed the accuracy in the determination of radial velocities and has moreover carried out such determinations in a large scale4 ,the ,bulletin ,no4 #bbi 7#aiac7 of the ,lick ,observatory contains the radial velocity of #iae stars4 ,at the observatory of ,lund1 where as far as possible card catalogues of the attributes of the stars are collected1 ,,gyllenberg has made a catalogue of this kind for the radial velocities4 ,the total number of stars in this catalogue now amounts to #afdj4ad #bh4 ,finally ,i shall briefly mention some comprehensive works on more special questions regarding the stellar system4 ,on variable stars there is published every year by ,,hartwig in the ,vierteljahrschrift der astronomischen ,gesellschaft a catalogue of all known variable stars with needful information about their minima c4 ,this is the completest and most reliable of such catalogues1 and is always up to date4 ,a complete historical catalogue of the variables is given in ,geschichte und ,literatur des ,lichtwechsels der bis ,ende #aiae als sicher vernderlich anerkannten ,sterne nebst einem ,katalog der ,elemente ihres ,lichtwechsels von ,g4 ,m,,ller und ,e4 ,,hartwig4 ,leipzig #aiah1 #aibj4 ,on nebulae we have the excellent catalogues of ,,dreyer1 the ,new ,general ,catalogue 7,n4 ,g4 ,c47 of #ahij in the ,memoirs of the ,astronomical ,society vol4 #di1 the ,index catalogues 7,i4 ,c47 in the same memoirs1 vols4 #ea and #ei 7#ahie and #aijh74 ,these catalogues contain all together #acbbf objects4 ,regarding other special attributes ,i refer in the first place to the important ,annals of the ,harvard ,observatory4 ,other references will be given in the following1 as need arises4 ,,footnotes3 ,footnote #g3 ,bonner ,sternverzeichnis in den ,astronomischen ,beobachtungen auf der ,sternwarte zu ,bonn1 ,dritter bis ,fnfter ,band4 ,bonn #ahei-fb4 ,footnote #h3 ,bonner ,durchmusterung1 ,vierte ,sektion4 ,achter ,band der ,astronomischen ,beobachtungen zu ,bonn1 #ahhf4 ,footnote #i3 ,the ,cape ,photographic ,durchmusterung by ,,david ,,gill and ,j4 ,c4 ,,kapteyn1 ,annals of the ,cape ,observatory1 vol4 ,,iii-,v 7#ahif-aijj74 ,footnote #aj3 ,cordoba ,durchmusterung by ,j4 ,,thome4 ,results of the ,national ,argentine ,observatory1 vol4 #af1 #ag1 #ah1 #ba 7#ahid-aiad74 ,footnote #aa3 ,aph4 ,j41 vol4 #cf4 ,footnote #ab3 ,h4 ,a41 vol4 #ga4 ,footnote #ac3 ,h4 ,a41 vol4 #ia1 #ib1 #ic1 #id4 ,footnote #ad3 ,a catalogue of radial velocities has this year been published by ,j4 ,,voute1 embracing #bjga stars4 ,first catalogue of radial velocities1 by ,j4 ,,voute4 ,weltevreden1 #aibj4 ,,chapter ,,iii4 ,,some ,,groups ,,of ,,known ,,stars4 #bi4 ,the number of cases in which all the eight attributes of the stars discussed in the first chapter are well known for one star is very small1 and certainly does not exceed one hundred4 ,these cases refer principally to such stars as are characterized either by great brilliancy or by a great proper motion4 ,the principal reason why these stars are better known than others is that they lie rather near our solar system4 ,before passing on to consider the stars from more general statistical points of view1 it may therefore be of interest first to make ourselves familiar with these well-known stars1 strongly emphasizing1 however1 the exceptional character of these stars1 and carefully avoiding any generalization from the attributes we shall here find4 #cj4 ,the apparently brightest stars4 ,we begin with these objects so well known to every lover of the stellar sky4 ,the following table contains all stars the apparent visual magnitude of which is brighter than #am4#e4 ,the first column gives the current number1 the second the name1 the third the equatorial designation 7alphadelta74 ,it should be remembered that the first four figures give the hour and minutes in right ascension1 the last two the declination1 italics showing negative declination4 ,the fourth column gives the galactic square1 the fifth and sixth columns the galactic longitude and latitude4 ,the seventh and eighth columns give the annual parallax and the corresponding distance expressed in siriometers4 ,the ninth column gives the proper motion 7mu71 the tenth the radial velocity ,w expressed in sir4_/st4 7,to get km4_/sec4 we may multiply by #d4gcge74 ,the eleventh column gives the apparent visual magnitude1 the twelfth column the absolute magnitude 7,m71 computed from m with the help of r4 ,the #acth column gives the type of spectrum 7,sp71 and the last column the photographic magnitude 7m'74 ,the difference between m' and m gives the colour-index 7c74 ,,table #b4 ,,the ,,apparently ,,brightest ,,stars4 ,transcriber's ,note3 ,to conserve space 7ad7 is used in place of 7alphadelta74 ---------------------------------------- --------------------------- #a #b #c #d #e #f #g #h ---------------------------------------- --------------------------- ,position ,distance ,name ---------------------------------------- ---- 7ad7 ,square l b pi r ---------------------------------------- --------------------------- sir4 #a,sirius 7#jfdjaf7 ,,gd#g #aie - #h #j4hgf #j4e #b;,canopus 7#jfbaeb7 ,,gd#h #bbi -#bd #j4jjg #bi4e #c,vega 7#ahccch7 ,,gc#b #cj #ag #j4jid #b4b #d;,capella 7#jejide7 ,,gc#e #aca #e #j4jff #c4a #e;,arcturus 7#adaaai7 ,,ga#b #cdd #fh #j4jge #b4g #f;alpha ,centauri 7#adcbfj7 ,,gd#aj #bhd - #b #j4gei #j4c #g,rigel 7#jejijh7 ,,gd#f #agf -#bd #j4jjg #bi4e #h,procyon 7#jgcdje7 ,,gc#g #ahb #ad #j4cbd #j4f #i;,achernar 7#jacdeg7 ,,ge#h #bef -#ei #j4jea #d4j #aj;beta ,centauri 7#acefei7 ,,gc#aj #bhj #b #j4jcg #e4f #aa;,altair 7#aidejh7 ,,gd#a #ae -#aj #j4bch #j4i #ab;,betelgeuze 7#jedijg7 ,,gd#f #afh - #h #j4jcj #f4i #ac;,aldebaran 7#jdcjaf7 ,,gd#e #adi -#ai #j4jgh #b4h #ad,pollux 7#jgcibh7 ,,gc#f #afj #be #j4jfd #c4b #ae,spica 7#acaiaj7 ,,gb#h #bhf #ea 44 44 #af;,antares 7#afbcbf7 ,,gc#aa #cbj #ad #j4jbi #g4a #ag;,fomalhaut 7#bbebcj7 ,,ge#aj #cdh -#ff #j4ach #a4e #ah;,deneb 7#bjchdd7 ,,gc#b #ea #a 44 44 #ai,regulus 7#ajjcab7 ,,gb#f #aif #ej #j4jcc #f4c #bj;beta ,crucis 7#abdaei7 ,,gc#aj #bgj #c #j4jjh #be4h ---------------------------------------- --------------------------- sir4 ,mean 44 44 44 #bc4ej4acd #g4c ---------------------------------------- --------------------------- ---------------------------------------- ------------------------- #a #b #i #aj #aa #ab #ac #ad ---------------------------------------- ------------------------- ,motion ,magnitude ,spectrum ,name ---------------------------------------- -- mu ,w m ,m ,sp m' ---------------------------------------- ------------------------- sir4_/st4 m' #a,sirius #a4cb - #a4ef -#am4#eh -#jm4#c ,a -#a4eh #b;,canopus #j4jb #d4ci -#j4hf -#h4b ,f -#j4dj #c,vega #j4ce - #b4ia #j4ad -#a4f ,a #j4ad #d;,capella #j4dd #f4ch #j4ba -#b4h ,g #a4ac #e;,arcturus #b4bh - #j4hb #j4bd -#a4i ,k #a4fb #f;alpha ,centauri #c4fh - #d4fi #j4cc #c4b ,g #a4be #g,rigel #j4jj #d4gg #j4cd -#g4j ,b#hp #j4be #h,procyon #a4bd - #j4gd #j4dh #a4e ,f#e #a4ag #i;,achernar #j4ji 44 #j4fj -#b4d ,b#e #j4hg #aj;beta ,centauri #j4jd #b4ec #j4hf -#b4i ,b#a #j4de #aa;,altair #j4ff - #f4ig #j4hi #a4b ,a#e #a4ab #ab;,betelgeuze #j4jc #d4dc #j4ib -#c4c ,ma #b4gf #ac;,aldebaran #j4bj #aa4fc #a4jf -#a4b ,k#e #b4fg #ad,pollux #j4jg #j4hb #a4ba -#a4c ,k #b4ei #ae,spica #j4jf #j4cd #a4ba 44 ,b#b #j4hd #af;,antares #j4jc - #j4fc #a4bb -#c4j ,map #c4jf #ag;,fomalhaut #j4cg #a4da #a4bi #j4d ,a#c #a4dc #ah;,deneb #j4jj - #j4hd #a4cc 44 ,a#b #a4db #ai,regulus #j4be 44 #a4cd -#b4g ,b#h #a4be #bj;beta ,crucis #j4jf #b4gd #a4ej -#e4f ,b#a #a4ji ---------------------------------------- ------------------------- m' ,mean #j4ef #c4bf #jm4#fd -#bm4#a ,f#a #a4ac ---------------------------------------- ------------------------- ,the values of 7alphadelta71 m1 ,sp are taken from ,h4 #ej4 ,the values of l1 b are computed from 7alphadelta7 with the help of tables in preparation at the ,lund ,observatory1 or from the original to plate ,i at the end1 allowing the conversion of the equatorial coordinates into galactic ones4 ,the values of pi are generally taken from the table of ,,kapteyn and ,,weersma mentioned in the previous chapter4 ,the values of mu are obtained from ,b4 ,p4 ,c41 those of the radial velocity 7,w7 from the card catalogue in ,lund already described4 ,there are in all1 in the sky1 #bj stars having an apparent magnitude brighter than #am4#e4 ,the brightest of them is ,sirius1 which1 owing to its brilliancy and position1 is visible to the whole civilized world4 ,it has a spectrum of the type ,a#j and hence a colour-index nearly equal to #j4j 7observations in ,harvard give c #j4jf74 ,its apparent magnitude is -#am4#f1 nearly the same as that of ,mars in his opposition4 ,its absolute magnitude is -#jm4#c1 i4e41 fainter than the apparent magnitude1 from which we may conclude that it has a distance from us smaller than one siriometer4 ,we find1 indeed1 from the eighth column that r #j4e sir4 ,the proper motion of ,sirius is #a4cb per year1 which is rather large but still not among the largest proper motions as will be seen below4 ,from the #aath column we find that ,sirius is moving towards us with a velocity of #a4f sir4_/st4 7 #g4f km4_/sec471 a rather small velocity4 ,the third column shows that its right ascension is #f;h #djm and its declination -#af4 ,it lies in the square ,,gd#g and its galactic coordinates are seen in the #eth and #fth columns4 ,the next brightest star is ,canopus or alpha ,carin at the south sky4 ,if we might place absolute confidence in the value of ,m 7 -#h4b7 in the #abth column this star would be1 in reality1 a much more imposing apparition than ,sirius itself4 ,remembering that the apparent magnitude of the moon1 according to #f1 amounts to -#aa4f1 we should find that ,canopus1 if placed at a distance from us equal to that of ,sirius 7r #j4e sir471 would shine with a lustre equal to no less than a quarter of that of the moon4 ,it is not altogether astonishing that a fanciful astronomer should have thought ,canopus to be actually the central star in the whole stellar system4 ,we find1 however1 from column #h that its supposed distance is not less that #cj sir4 ,we have already pointed out that distances greater than #d sir41 when computed from annual parallaxes1 must generally be considered as rather uncertain4 ,as the value of ,m is intimately dependent on that of r we must consider speculations based on this value to be very vague4 ,another reason for a doubt about a great value for the real luminosity of this star is found from its type of spectrum which1 according to the last column1 is ,f#j1 a type which1 as will be seen1 is seldom found among giant stars4 ,a better support for a large distance could on the other hand be found from the small proper motion of this star4 ,sirius and ,canopus are the only stars in the sky having a negative value of the apparent visual magnitude4 ,space will not permit us to go through this list star for star4 ,we may be satisfied with some general remarks4 ,in the fourth column is the galactic square4 ,we call to mind that all these squares have the same area1 and that there is therefore the same probability a priori of finding a star in one of the squares as in another4 ,the squares ,,gc and ,,gd lie along the galactic equator 7the ,milky ,way74 ,we find now from column #d that of the #bj stars here considered there are no less than #ae in the galactic equator squares and only #e outside1 instead of #aj in the galactic squares and #aj outside1 as would have been expected4 ,the number of objects is1 indeed1 too small to allow us to draw any cosmological conclusions from this distribution1 but we shall find in the following many similar instances regarding objects that are principally accumulated along the ,milky ,way and are scanty at the galactic poles4 ,we shall find that in these cases we may generally conclude from such a partition that we then have to do with objects situated far from the sun1 while objects that are uniformly distributed on the sky lie relatively near us4 ,it is easy to understand that this conclusion is a consequence of the supposition1 confirmed by all star counts1 that the stellar system extends much farther into space along the ,milky ,way than in the direction of its poles4 ,if we could permit ourselves to draw conclusions from the small material here under consideration1 we should hence have reason to believe that the bright stars lie relatively far from us4 ,in other words we should conclude that the bright stars seem to be bright to us not because of their proximity but because of their large intrinsic luminosity4 ,column #h really tends in this direction4 ,certainly the distances are not in this case colossal1 but they are nevertheless sufficient to show1 in some degree1 this uneven partition of the bright stars on the sky4 ,the mean distance of these stars is as large as #g4e sir4 ,only alpha ,centauri1 ,sirius1 ,procyon and ,altair lie at a distance smaller than one siriometer4 ,of the other stars there are two that lie as far as #cj siriometers from our system4 ,these are the two giants ,canopus and ,rigel4 ,even if1 as has already been said1 the distances of these stars may be considered as rather uncertain1 we must regard them as being rather large4 ,as column #h shows that these stars are rather far from us1 so we find from column #ab1 that their absolute luminosity is rather large4 ,the mean absolute magnitude is1 indeed1 -#bm4#a4 ,we shall find that only the greatest and most luminous stars in the stellar system have a negative value of the absolute magnitude4 ,the mean value of the proper motions of the bright stars amounts to #j4ef per year and may be considered as rather great4 ,we shall1 indeed1 find that the mean proper motion of the stars down to the #fth magnitude scarcely amounts to a tenth part of this value4 ,on the other hand we find from the table that the high value of this mean is chiefly due to the influence of four of the stars which have a large proper motion1 namely ,sirius1 ,arcturus1 alpha ,centauri and ,procyon4 ,the other stars have a proper motion smaller than #a per year and for half the number of stars the proper motion amounts to approximately #j4je1 indicating their relatively great distance4 ,that the absolute velocity of these stars is1 indeed1 rather small may be found from column #aj1 giving their radial velocity1 which in the mean amounts to only three siriometers per stellar year4 ,from the discussion below of the radial velocities of the stars we shall find that this is a rather small figure4 ,this fact is intimately bound up with the general law in statistical mechanics1 to which we return later1 that stars with large masses generally have a small velocity4 ,we thus find in the radial velocities fresh evidence1 independent of the distance1 that these bright stars are giants among the stars in our stellar system4 ,we find all the principal spectral types represented among the bright stars4 ,to the helium stars 7,b7 belong ,rigel1 ,achernar1 beta ,centauri1 ,spica1 ,regulus and beta ,crucis4 ,to the ,sirius type 7,a7 belong ,sirius1 ,vega1 ,altair1 ,fomalhaut and ,deneb4 ,to the ,calcium type 7,f7 ,canopus and ,procyon4 ,to the sun type 7,g7 ,capella and alpha ,centauri4 ,to the ,k-type belong ,arcturus1 ,aldebaran and ,pollux and to the ,m-type the two red stars ,betelgeuze and ,antares4 ,using the spectral indices as an expression for the spectral types we find that the mean spectral index of these stars is #a4a corresponding to the spectral type ,f#a4 #ca4 ,stars with the greatest proper motion4 ,in table #c ,i have collected the stars having a proper motion greater than #c per year4 ,the designations are the same as in the preceding table1 except that the names of the stars are here taken from different catalogues4 ,in the astronomical literature of the last century we find the star #ahcj ,groombridge designed as that which possesses the greatest known proper motion4 ,it is now distanced by two other stars ,c4 ,p4 ,d4 #e;h4#bdc discovered in the year #ahig by ,,kapteyn and ,,innes on the plates taken for the ,cape ,photographic ,durchmusterung1 and ,,barnard's star in ,ophiuchus1 discovered #aiaf4 ,the last-mentioned star1 which possesses the greatest proper motion now known1 is very faint1 being only of the #ajth magnitude1 and lies at a distance of #j4dj sir4 from our sun and is hence1 as will be found from table #e the third nearest star for which we know the distance4 ,its linear velocity is also very great1 as we find from column #aj1 and amounts to #ai sir4_/st4 7 #ij km4_/sec47 in the direction towards the sun4 ,the absolute magnitude of this star is #aam4#g and it is1 with the exception of one other1 the very faintest star now known4 ,its spectral type is ,mb1 a fact worth fixing in our memory1 as different reasons favour the belief that it is precisely the ,m-type that contains the very faintest stars4 ,its apparent velocity 7i4e41 the proper motion7 is so great that the star in #ajjj years moves #c1 or as much as #f times the diameter of the moon4 ,for this star1 as well as for its nearest neighbours in the table1 observations differing only by a year are sufficient for an approximate determination of the value of the proper motion1 for which in other cases many tens of years are required4 ,regarding the distribution of these stars in the sky we find that1 unlike the brightest stars1 they are not concentrated along the ,milky ,way4 ,on the contrary we find only #f in the galactic equator squares and #ab in the other squares4 ,we shall not build up any conclusion on this irregularity in the distribution1 but supported by the general thesis of the preceding paragraph we conclude only that these stars must be relatively near us4 ,this follows1 indeed1 directly from column #h1 as not less than eleven of these stars lie within one siriometer from our sun4 ,their mean distance is #j4hg sir4 ,,table #c4 ,,stars ,,with ,,the ,,greatest ,,proper ,,motion4 ,transcriber's ,note3 ,to conserve space 7ad7 is used in place of 7alphadelta74 ---------------------------------------- --------------------------- #a #b #c #d #e #f #g #h ---------------------------------------- --------------------------- ,position ,distance ,name ---------------------------------------- ---- 7ad7 ,square l b pi r ---------------------------------------- --------------------------- sir4 #a;,barnards star 7#agebjd7 ,,gc#ab #ceh #ab #j4eae #j4dj #b;,c4 ,z4 #e;h4#bdc 7#jejgdd7 ,,ge#g #bah -#ce #j4cai #j4fe #c;,groom4 #ahcj 7#aadgch7 ,,ga#a #ace #ge #j4ajb #b4jb #d,lac4 #iceb 7#bbeicf7 ,,ge#aj #ccc -#ff #j4bib #j4ga #e;,c4 ,g4 ,a4 #cbdaf 7#bceicg7 ,,gf#b #cjh -#ge #j4bcj #j4hi #ffa ,cygni 7#bajbch7 ,,gd#b #ej - #g #j4caa #j4ff #g,lal4 #baahe 7#ajegcf7 ,,gb#e #aec #ff #j4djc #j4ea #h;epsilon ,indi 7#baeeeg7 ,,ge#i #cjd -#dg #j4bhd #j4gc #i,lal4 #babeh 7#aajjdd7 ,,gb#d #ace #fd #j4bjc #a4jb #aj,o#b ,eridani 7#jdajjg7 ,,ge#e #afh -#cf #j4agd #a4ai #aa,proxima ,centauri 7#adbbfb7 ,,gd#aj #bha - #b #j4ghj #j4bf #ab,oe4 ,a4 #adcbj 7#aejdae7 ,,gb#i #cad #ce #j4jce #e4ij #acmu ,cassiopei 7#jajaed7 ,,gd#d #ic - #h #j4aab #a4hd #ad;alpha ,centauri 7#adcbfj7 ,,gd#aj #bhd - #b #j4gei #j4bg #ae,lac4 #hgfj 7#baaaci7 ,,ge#aj #ccb -#dd #j4bdh #j4hc #af,lac4 #ajfj 7#jcaedc7 ,,ge#g #baf -#ee #j4afb #a4bg #ag,oe4 ,a4 #aafgg 7#aaadff7 ,,gb#h #ajc #ej #j4aih #a4jd #ah,van ,maanens star 7#jjdcjd7 ,,gd#h #ib -#eh #j4bdf #j4hd ---------------------------------------- --------------------------- sir4 ,mean 44 44 #da #j4bih #j4hg ---------------------------------------- --------------------------- ---------------------------------------- ------------------------- #a #b #i #aj #aa #ab #ac #ad ---------------------------------------- ------------------------- ,motion ,magnitude ,spectrum ,name ---------------------------------------- -- mu ,w m ,m ,sp m' ---------------------------------------- ------------------------- sir4_/st4 m' #a;,barnards star #aj4bi -#ai #im4#g #aam4#g,mb #aa4e #b;,c4 ,z4 #e;h4#bdc #h4ge #ea #i4b #aj4a ,k#b #aj4f #c;,groom4 #ahcj #g4jf -#bj #f4e #e4j ,g#e #g4f #d,lac4 #iceb #f4ij #b #g4e #h4b ,k #h4i #e;,c4 ,g4 ,a4 #cbdaf #f4aa #e #h4b #h4e ,g #i4a #ffa ,cygni #e4bg -#ac #e4f #f4e ,k#e #g4b #g,lal4 #baahe #d4gg -#ah #g4f #i4a ,mb #h4i #h;epsilon ,indi #d4gj -#h #d4g #e4d ,k#e #f4c #i,lal4 #babeh #d4dg #ad #h4e #h4e ,ma #aj4c #aj,o#b ,eridani #d4aa -#i #d4g #d4c ,g#e #e4h #aa,proxima ,centauri #c4he 44 #aa4j #ac4i 44 #ac4e #ab,oe4 ,a4 #adcbj #c4ge #fa #i4j #e4a ,g#j #i4i #acmu ,cassiopei #c4gc -#ba #e4g #d4d ,g#c #f4h #ad;alpha ,centauri #c4fh -#e #j4c #c4b ,g #a4b #ae,lac4 #hgfj #c4ec #c #f4f #g4j ,g #g4e #af,lac4 #ajfj #c4je #ah #e4f #e4a ,g#e #f4g #ag,oe4 ,a4 #aafgg #c4jc 44 #i4b #i4a ,ma #aa4j #ah,van ,maanens star #c4ja 44 #ab4c #ab4g ,f#j #ab4i ---------------------------------------- ------------------------- sir4_/st4 m' ,mean #e4jj #ag4h #gm4#c #gm4#f;,g#h #h4g ---------------------------------------- ------------------------- ,that the great proper motion does not depend alone on the proximity of these stars is seen from column #aj1 giving the radial velocities4 ,for some of the stars 7#d7 the radial velocity is for the present unknown1 but the others have1 with few exceptions1 a rather great velocity amounting in the mean to #ah sir4_/st4 7 #he km4_/sec471 if no regard is taken to the sign1 a value nearly five times as great as the absolute velocity of the sun4 ,as this is only the component along the line of sight1 the absolute velocity is still greater1 approximately equal to the component velocity multiplied by square root#b4 ,we conclude that the great proper motions depend partly on the proximity1 partly on the great linear velocities of the stars4 ,that both these attributes here really cooperate may be seen from the absolute magnitudes 7,m74 ,the apparent and the absolute magnitudes are for these stars nearly equal1 the means for both been approximately #gm4 ,this is a consequence of the fact that the mean distance of these stars is equal to one siriometer1 at which distance m and ,m1 indeed1 do coincide4 ,we find that these stars have a small luminosity and may be considered as dwarf stars4 ,according to the general law of statistical mechanics already mentioned small bodies upon an average have a great absolute velocity1 as we have1 indeed1 already found from the observed radial velocities of these stars4 ,as to the spectral type1 the stars with great proper motions are all yellow or red stars4 ,the mean spectral index is #b4h1 corresponding to the type ,g#h4 ,if the stars of different types are put together we get the table ,type ,number ,mean value of ,m ,g #h #e4c ,k #d #g4e ,m #d #i4f ,we conclude that1 at least for these stars1 the mean value of the absolute magnitude increases with the spectral index4 ,this conclusion1 however1 is not generally valid4 #cb4 ,stars with the greatest radial velocities4 ,there are some kinds of nebulae for which very large values of the radial velocities have been found4 ,with these we shall not for the present deal1 but shall confine ourselves to the stars4 ,the greatest radial velocity hitherto found is possessed by the star 7#jdjhbb7 of the eighth magnitude in the constellation ,perseus1 which retires from us with a velocity of #gb sir4_/st4 or #cda km4_/sec4 ,the nearest velocity is that of the star 7#jajcfa7 which approaches us with approximately the same velocity4 ,the following table contains all stars with a radial velocity greater than #bj sir4_/st4 7 #id4h km4_/sec474 ,it is based on the catalogue of ,,voute mentioned above4 ,regarding their distribution in the sky we find #aa in the galactic equator squares and #g outside4 ,a large radial velocity seems therefore to be a galactic phenomenon and to be correlated to a great distance from us4 ,of the #ah stars in consideration there is only one at a distance smaller than one siriometer and #b at a distance smaller than #d siriometers4 ,among the nearer ones we find the star 7#jejgdd71 identical with ,c4 ,p4 ,d4 #e;h4#bdc1 which was the second star with great proper motion4 ,these stars have simultaneously the greatest proper motion and very great linear velocity4 ,generally we find from column #i that these stars with large radial velocity possess also a large proper motion4 ,the mean value of the proper motions amounts to #a4cd1 a very high value4 ,in the table we find no star with great apparent luminosity4 ,the brightest is the #ajth star in the table which has the magnitude #e4a4 ,the mean apparent magnitude is #g4g4 ,as to the absolute magnitude 7,m7 we see that most of these speedy stars1 as well as the stars with great proper motions in table #c1 have a rather great positive magnitude and thus are absolutely faint stars1 though they perhaps may not be directly considered as dwarf stars4 ,their mean absolute magnitude is #c4j4 ,regarding the spectrum we find that these stars generally belong to the yellow or red types 7,g1 ,k1 ,m71 but there are #f ,f-stars and1 curiously enough1 two ,a-stars4 ,after the designation of their type 7,a#b and ,a#c7 is the letter p 7 peculiar71 indicating that the spectrum in some respect differs from the usual appearance of the spectrum of this type4 ,in the present case the peculiarity consists in the fact that a line of the wave-length #ddh4a1 which emanates from magnesium and which we may find on plate ,,iii in the spectrum of ,sirius1 does not occur in the spectrum of these stars1 though the spectrum has otherwise the same appearance as in the case of the ,sirius stars4 ,there is reason to suppose that the absence of this line indicates a low power of radiation 7low temperature7 in these stars 7compare ,,adams74 ,,table #d4 ,,stars ,,with ,,the ,,greatest ,,radial ,,velocity4 ,transcriber's ,note3 ,to conserve space 7ad7 is used in place of 7alphadelta74 ---------------------------------------- --------------------------- #a #b #c #d #e #f #g #h ---------------------------------------- --------------------------- ,position ,distance ,name ---------------------------------------- ---- 7ad7 ,square l b pi r ---------------------------------------- --------------------------- sir4 #a;,a4 ,g4 ,berlin #acff 7#jdjhbb7 ,,gd#e #ada -#bj #j4jjg #cj4h #b,lal4 #aiff 7#jajcfa7 ,,gd#d #ic - #b #j4jaf #ab4i #c;,a4 ,oe4 #adcbj 7#aejdae7 ,,gb#i #cad #ce #j4jce #e4i #d;,c4 ,z4 #e;h4#bdc 7#jejgdd7 ,,ge#g #bah -#ce #j4cai #j4f #e,lal4 #aebij 7#jgdgcj7 ,,gc#f #aeh #bf #j4jbc #i4j #fec ,cassiop4 7#jaeefc7 ,,gc#d #ih #b 44 44 #g;,a4 ,g4 ,berlin #ahff 7#jeegai7 ,,gd#f #aei - #b #j4jba #i4h #h,w ,lyr 7#ahaacf7 ,,gc#b #ca #ba 44 44 #i;,boss #aeaa 7#jeeibf7 ,,gd#g #bjj -#bj #j4jab #ag4j #ajomega ,pavonis 7#ahdifj7 ,,gd#aa #cjd -#bd 44 44 #aa;,a4 ,oe4 #bjdeb 7#bjagba7 ,,ge#aj #cea -#ca #j4jae #ac4e #ab,lal4 #bhfjg 7#aecgaj7 ,,gb#aj #cbe #cd #j4jcc #f4b #ac;,a4 ,g4 ,leiden #egcd 7#afaacb7 ,,gb#a #ba #de #j4jjb #hi4b #ad,lal4 #cgabj 7#aibicb7 ,,gc#b #cc #f #j4jej #d4a #ae,lal4 #bgbgd 7#adedba7 ,,gb#i #cjh #cd #j4jac #af4b #af,lal4 #egfa 7#jcjbbe7 ,,gd#e #abf -#bh #j4jci #e4a #ag,w4 ,b4 #ag;h4#eag 7#agbijf7 ,,gc#ab #ceh #bj #j4jad #ad4a #ah,lal4 #bciie 7#abdgag7 ,,gb#h #bga #df #j4jab #ag4j ---------------------------------------- --------------------------- sir4 ,mean 44 44 #bc4ij4jda #af4g ---------------------------------------- --------------------------- ---------------------------------------- ------------------------- #a #b #i #aj #aa #ab #ac #ad ---------------------------------------- ------------------------- ,motion ,magnitude ,spectrum ,name ---------------------------------------- -- mu ,w m ,m ,sp m' ---------------------------------------- ------------------------- sir4_/st4 m' #a;,a4 ,g4 ,berlin #acff #j4ed #gb #hm4#i #am4#d ,f#j #i4d #b,lal4 #aiff #j4fd -#fi #g4i #b4c ,f#c #h4e #c;,a4 ,oe4 #adcbj #c4ge #fa #i4j #e4a ,g#j #i4i #d;,c4 ,z4 #e;h4#bdc #h4ge #ea #i4b #aj4a ,k#b #aj4f #e,lal4 #aebij #a4if -#ea #h4b #c4d ,g#j #i4a #fec ,cassiop4 #j4ja -#dd #e4f 44 ,b#h #e4e #g;,a4 ,g4 ,berlin #ahff #j4gf -#dj #i4j #d4j ,f#j #i4i #h,w ,lyr 44 -#ci var4 44 ,md var4 #i;,boss #aeaa #j4aj #ci #e4b -#a4j ,g#e #f4d #ajomega ,pavonis #j4ad #ch #e4a 44 ,k #f4e #aa;,a4 ,oe4 #bjdeb #a4ah -#ch #h4a #b4d ,g#hp #i4d #ab,lal4 #bhfjg #a4ah -#cf #g4c #c4c ,a#bp #g4d #ac;,a4 ,g4 ,leiden #egcd #j4jd -#ce #h4c -#a4e ,k#d #i4i #ad,lal4 #cgabj #j4eb -#cd #f4f #c4e ,g#b #g4f #ae,lal4 #bgbgd #j4gi #cd #h4c #b4b ,f#d #h4i #af,lal4 #egfa #j4hf -#cb #h4j #d4d ,a#cp #h4a #ag,w4 ,b4 #ag;h4#eag #j4fc -#ca #h4f #b4h ,f#a #i4a #ah,lal4 #bciie #j4hh #cj #h4b #b4j ,f#c #h4h ---------------------------------------- ------------------------- sir4_/st4 m' ,mean #a4cd #af4g #gm4#g #cm4#j ,f#i #h4e ---------------------------------------- ------------------------- #cc4 ,the nearest stars4 ,the star alpha in ,centaurus was long considered as the nearest of all stars4 ,it has a parallax of #j4ge1 corresponding to a distance of #j4bg siriometers 7 #d4bf light years74 ,this distance is obtained from the annual parallax with great accuracy1 and the result is moreover confirmed in another way 7from the study of the orbit of the companion of alpha ,centauri74 ,in the year #aiaf ,,innes discovered at the observatory of ,johannesburg in the ,transvaal a star of the #ajth magnitude1 which seems to follow alpha ,centauri in its path in the heavens1 and which1 in any case1 lies at the same distance from the earth1 or somewhat nearer4 ,it is not possible at present to decide with accuracy whether ,proxima ,centauri--as the star is called by ,,innes--or alpha ,centauri is our nearest neighbour4 ,then comes ,,barnard's star 7#agebjd71 whose large proper motion we have already mentioned4 ,as ,no4 #e we find ,sirius1 as ,no4 #h ,procyon1 as ,no4 #ba ,altair4 ,the others are of the third magnitude or fainter4 ,no4 #aj--fa ,cygni--is especially interesting1 being the first star for which the astronomers1 after long and painful endeavours in vain1 have succeeded in determining the distance with the help of the annual parallax 7,,bessel #ahda74 ,from column #d we find that the distribution of these stars on the sky is tolerably uniform1 as might have been predicted4 ,all these stars have a large proper motion1 this being in the mean #c4db per year4 ,this was a priori to be expected from their great proximity4 ,the radial velocity is1 numerically1 greater than could have been supposed4 ,this fact is probably associated with the generally small mass of these stars4 ,their apparent magnitude is upon an average #f4c4 ,the brightest of the near stars is ,sirius 7m -#a4f71 the faintest ,proxima ,centauri 7m #aa74 ,through the systematic researches of the astronomers we may be sure that no bright stars exist at a distance smaller than one siriometer1 for which the distance is not already known and well determined4 ,the following table contains without doubt--we may call them briefly all near stars--all stars within one siriometer from us with an apparent magnitude brighter than #fm 7the table has #h such stars71 and probably also all near stars brighter than #gm 7#aj stars71 or even all brighter than the eighth magnitude 7the table has #ac such stars and two near the limit74 ,regarding the stars of the eighth magnitude or fainter no systematic investigations of the annual parallax have been made and among these stars we may get from time to time a new star belonging to the siriometer sphere in the neighbourhood of the sun4 ,to determine the total number of stars within this sphere is one of the fundamental problems in stellar statistics1 and to this question ,i shall return immediately4 ,,table #e4 ,,the ,,nearest ,,stars4 ,transcriber's ,note3 ,to conserve space 7ad7 is used in place of 7alphadelta74 ---------------------------------------- ---------------------------- #a #b #c #d #e #f #g #h ---------------------------------------- ---------------------------- ,position ,distance ,name ---------------------------------------- ---- 7ad7 ,square l b pi r ---------------------------------------- ---------------------------- sir4 #a,proxima ,centauri 7#adbbfb7 ,,gd#aj #bha - #b #j4ghj #j4bf #b;alpha ,centauri 7#adcbfj7 ,,gd#aj #bhd - #b #j4gei #j4bg #c;,barnards p4 m4 star 7#agebjd7 ,,gc#ab #ceh #ab #j4eae #j4dj #d,lal4 #baahe 7#ajegcf7 ,,gb#e #aec #ff #j4djc #j4ea #e,sirius 7#jfdjaf7 ,,gd#g #aie - #h #j4cgf #j4ee #f 44 7#aaaceg7 ,,gc#f #aeh #c #j4ccg #j4fj #gtau ,ceti 7#jaciaf7 ,,gf#a #add -#gd #j4ccd #j4fb #h,procyon 7#jgcdje7 ,,gc#g #ahb #ad #j4cbd #j4fd #i;,c4 ,z4 #e;h4#bdc 7#jejgdd7 ,,ge#g #bah -#ce #j4cai #j4fe #ajfa ,cygni 7#bajbch7 ,,gd#b #ej - #g #j4caa #j4ff #aa,lal4 #bfdha 7#adbeae7 ,,gb#i #abd -#dj #j4caa #j4ff #ab;epsilon ,eridani 7#jcbhji7 ,,ge#e #aec -#db #j4bie #j4gj #ac,lac4 #iceb 7#bbeicf7 ,,ge#aj #ccc -#ff #j4bib #j4ga #ad,pos4 ,med4 #bafd 7#ahdaei7 ,,gc#b #ef #bd #j4bib #j4ga #ae;epsilon ,indi 7#baeeeg7 ,,ge#i #cjd -#dg #j4bhd #j4gc #af;,groom4 #cd 7#jjabdc7 ,,gd#c #hd -#bj #j4bha #j4gc #ag,oe4 ,a4 #agdae 7#agcgfh7 ,,gc#h #fe #cb #j4bfh #j4gg #ah,krger #fj 7#bbbdeg7 ,,gc#c #gb #j #j4bef #j4ha #ai,lac4 #hgfj 7#baaaci7 ,,ge#aj #ccb -#dd #j4bdh #j4hh #bjvan ,maanens p4 m4 star7#jjdcjd7 ,,ge#c #ib -#eh #j4bdf #j4hd #ba;,altair 7#aidejh7 ,,gd#a #ae -#aj #j4bch #j4hg #bb;,c4 ,g4 ,a4 #cbdaf 7#bceicg7 ,,gf#b #cjh -#ge #j4bcj #j4hi #bc;,bradley #aehd 7#aabicb7 ,,gc#f #beb #bh #j4baf #j4ie ---------------------------------------- ---------------------------- sir4 ,mean 44 44 44 #cj4hj4cdd #j4fg ---------------------------------------- ---------------------------- ---------------------------------------- --------------------------- #a #b #i #aj #aa #ab #ac #ad ---------------------------------------- --------------------------- ,motion ,magnitude ,spectrum ,name ---------------------------------------- -- mu ,w m ,m ,sp m' ---------------------------------------- --------------------------- sir4_/st4 m' #a,proxima ,centauri #c4he 44 #aam4#j #acm4#i 44 #ac4e #b;alpha ,centauri #c4fh - #e #j4cc #c4b ,g #a4be #c;,barnards p4 m4 star #aj4bi -#ai #i4g #aa4g ,mb #aa4e #d,lal4 #baahe #d4gg -#ah #g4f #i4a ,mb #h4i #e,sirius #a4cb - #b -#a4eh - #j4c ,a -#a4eh #f 44 #b4gb 44 44 44 44 #ab4e #gtau ,ceti #a4ib - #c #c4f #d4f ,k#j #d4f #h,procyon #a4bd - #a #j4dh #a4e ,f#e #j4ij #i;,c4 ,z4 #e;h4#bdc #h4ge #ea #i4b #aj4a ,k#b #aj4f #ajfa ,cygni #e4bg -#ac #e4f #f4e ,k#e #g4b #aa,lal4 #bfdha #j4dg 44 #g4h #h4g ,g#e #h4i #ab;epsilon ,eridani #j4ig #c #c4h #d4f ,k#j #d4h #ac,lac4 #iceb #f4ij #b #g4e #h4b ,k #h4i #ad,pos4 ,med4 #bafd #b4bh 44 #h4i #i4f ,k #aj4c #ae;epsilon ,indi #d4gj - #h #d4g #e4d ,k#e #f4c #af;,groom4 #cd #b4hi #a #h4a #h4h ,ma #i4e #ag,oe4 ,a4 #agdae #a4cj 44 #i4a #i4g ,k #aj4e #ah,krger #fj #j4id 44 #i4b #i4f ,k#e #aj4h #ai,lac4 #hgfj #c4ec #c #f4f #g4j ,g #g4e #bjvan ,maanens p4 m4 star #c4ja 44 #ab4c #ab4g ,f#j #ab4i #ba;,altair #j4ff - #g #j4i #a4b ,a#e #a4ab #bb;,c4 ,g4 ,a4 #cbdaf #f4aa #e #h4b #h4e ,g #i4a #bc;,bradley #aehd #a4jf - #e #f4a #f4b ,g #f4i ---------------------------------------- --------------------------- sir4_/st4 m' ,mean #c4db #i4a #fm4#c #gm4#c ,g#f #g4e ---------------------------------------- --------------------------- ,the mean absolute magnitude of the near stars is distributed in the following way3-- ,m #j #a #c #d #e #f #g #h #i #aj #aa #ab #ac ,number #a #b #a #b #a #b #a #d #d #a #a #a #a4 ,what is the absolute magnitude of the near stars that are not contained in table8 ,evidently they must principally be faint stars4 ,we may go further and answer that all stars with an absolute magnitude brighter than #fm must be contained in this list4 ,for if ,m is equal to #f or brighter1 m must be brighter than #fm1 if the star is nearer than one siriometer4 ,but we have assumed that all stars apparently brighter than #fm are known and are contained in the list4 ,hence also all stars absolutely brighter than #fm must be found in table #e4 ,we conclude that the number of stars having an absolute magnitude brighter than #fm amounts to #h4 ,if1 finally1 the spectral type of the near stars is considered1 we find from the last column of the table that these stars are distributed in the following way3-- ,spectral type ,b ,a ,f ,g ,k ,m ,number #j #b #b #e #i #c4 ,for two of the stars the spectrum is for the present unknown4 ,we find that the number of stars increases with the spectral index4 ,the unknown stars in the siriometer sphere belong probably1 in the main1 to the red types4 ,if we now seek to form a conception of the total number in this sphere we may proceed in different ways4 ,,eddington1 in his ,stellar movements1 to which ,i refer the reader1 has used the proper motions as a scale of calculation1 and has found that we may expect to find in all #cb stars in this sphere1 confining ourselves to stars apparently brighter than the magnitude #im4#e4 ,this makes #h stars per cub4 sir4 ,we may attack the problem in other ways4 ,a very rough method which1 however1 is not without importance1 is the following4 ,let us suppose that the ,galaxy in the direction of the ,milky ,way has an extension of #ajjj siriometers and in the direction of the poles of the ,milky ,way an extension of #ej sir4 ,we have later to return to the fuller discussion of this extension4 ,for the present it is sufficient to assume these values4 ,the whole system of the ,galaxy then has a volume of #bjj million cubic siriometers4 ,suppose further that the total number of stars in the ,galaxy would amount to #ajjj millions1 a value to which we shall also return in a following chapter4 ,then we conclude that the average number of stars per cubic siriometer would amount to #e4 ,this supposes that the density of the stars in each part of the ,galaxy is the same4 ,but the sun lies rather near the centre of the system1 where the density is 7considerably7 greater than the average density4 ,a calculation1 which will be found in the mathematical part of these lectures1 shows that the density in the centre amounts to approximately #af times the average density1 giving #hj stars per cubic siriometer in the neighbourhood of the sun 7and of the centre74 ,a sphere having a radius of one siriometer has a volume of #d cubic siriometers1 so that we obtain in this way #cbj stars in all1 within a sphere with a radius of one siriometer4 ,for different reasons it is probable that this number is rather too great than too small1 and we may perhaps estimate the total number to be something like #bjj stars1 of which more than a tenth is now known to the astronomers4 ,we may also arrive at an evaluation of this number by proceeding from the number of stars of different apparent or absolute magnitudes4 ,this latter way is the most simple4 ,we shall find in a later paragraph that the absolute magnitudes which are now known differ between -#h and #ac4 ,but from mathematical statistics it is proved that the total range of a statistical series amounts upon an average to approximately #f times the dispersion of the series4 ,hence we conclude that the dispersion 7sigma7 of the absolute magnitudes of the stars has approximately the value #c 7we should obtain sigma #ac #h 3 #f #c4e1 but for large numbers of individuals the total range may amount to more than #f sigma74 ,as1 further1 the number of stars per cubic siriometer with an absolute magnitude brighter than #f is known 7we have obtained #h 3 #d #b stars per cubic siriometer brighter than #fm71 we get a relation between the total number of stars per cubic siriometer 7,d#j7 and the mean absolute magnitude 7,m#j7 of the stars1 so that ,d#j can be obtained1 as soon as ,m#j is known4 ,the computation of ,m#j is rather difficult1 and is discussed in a following chapter4 ,supposing1 for the moment1 ,m#j #aj we get for ,d#j the value #bb1 corresponding to a number of #ij stars within a distance of one siriometer from the sun4 ,we should then know a fifth part of these stars4 #cd4 ,parallax stars4 ,in #bb ,i have paid attention to the now available catalogues of stars with known annual parallax4 ,the most extensive of these catalogues is that of ,,walkey1 containing measured parallaxes of #fbe stars4 ,for a great many of these stars the value of the parallax measured must however be considered as rather uncertain1 and ,i have pointed out that only for such stars as have a parallax greater than #j4jd 7or a distance smaller than #e siriometers7 may the measured parallax be considered as reliable1 as least generally speaking4 ,the effective number of parallax stars is therefore essentially reduced4 ,indirectly it is nevertheless possible to get a relatively large catalogue of parallax stars with the help of the ingenious spectroscopic method of ,,adams1 which permits us to determine the absolute magnitude1 and therefore also the distance1 of even farther stars through an examination of the relative intensity of certain lines in the stellar spectra4 ,it may be that the method is not yet as firmly based as it should be1#ae but there is every reason to believe that the course taken is the right one and that the catalogue published by ,,adams of #ejj parallax stars in ,contrib4 from ,mount ,wilson1 #adb1 already gives a more complete material than the catalogues of directly measured parallaxes4 ,i give here a short resum of the attributes of the parallax stars in this catalogue4 ,the catalogue of ,,adams embraces stars of the spectral types ,f1 ,g1 ,k and ,m4 ,in order to complete this material by parallaxes of blue stars ,i add from the catalogue of ,,walkey those stars in his catalogue that belong to the spectral types ,b and ,a1 confining myself to stars for which the parallax may be considered as rather reliable4 ,there are in all #fa such stars1 so that a sum of #efa stars with known distance is to be discussed4 ,for all these stars we know m and ,m and for the great part of them also the proper motion mu4 ,we can therefore for each spectral type compute the mean values and the dispersion of these attributes4 ,we thus get the following table1 in which ,i confine myself to the mean values of the attributes4 ,,table #f4 ,,mean ,,values ,,of m1 ,m ,,and ,,the ,,proper ,,motions 7mu7 ,,of ,,parallax ,,stars ,,of ,,different ,,spectral ,,types4 ------------------------------- ,sp4,number m ,m mu ------------------------------- ,b #ae #b4jc -#a4fg #j4je ,a #df #c4dj #j4fd #j4ba ,f #abe #e4fj #b4aj #j4dj ,g #agi #e4gg #a4fh #j4ea ,k #ahd #f4ag #b4ca #j4ec ,m #db #f4jb #b4cj #j4hb ------------------------------- ,we shall later consider all parallax stars taken together4 ,we find from table #f that the apparent magnitude1 as well as the absolute magnitude1 is approximately the same for all yellow and red stars and even for the stars of type ,f1 the apparent magnitude being approximately equal to #fm and the absolute magnitude equal to #bm4 ,for type ,b we find the mean value of ,m to be -#am4#g and for type ,a we find ,m #jm4#f4 ,the proper motion also varies in the same way1 being for ,f1 ,g1 ,k1 ,m approximately #j4e and for ,b and ,a #j4a4 ,as to the mean values of ,m and mu we cannot draw distinct conclusions from this material1 because the parallax stars are selected in a certain way which essentially influences these mean values1 as will be more fully discussed below4 ,the most interesting conclusion to be drawn from the parallax stars is obtained from their distribution over different values of ,m4 ,in the memoir referred to1 ,,adams has obtained the following table 7somewhat differently arranged from the table of ,,adams71#af which gives the number of parallax stars for different values of the absolute magnitude for different spectral types4 ,a glance at this table is sufficient to indicate a singular and well pronounced property in these frequency distributions4 ,we find1 indeed1 that in the types ,g1 ,k and ,m the frequency curves are evidently resolvable into two simple curves of distribution4 ,in all these types we may distinguish between a bright group and a faint group4 ,with a terminology proposed by ,,hertzsprung the former group is said to consist of giant stars1 the latter group of dwarf stars4 ,even in the stars of type ,f this division may be suggested4 ,this distinction is still more pronounced in the graphical representation given in figures 7plate ,,iv74 ,,table #g4 ,,distribution ,,of ,,the ,,parallax ,,stars ,,of ,,different ,,spectral ,,types ,,over ,,different ,,absolute ,,magnitudes4 ---------------------------------------- --------- ,m ,b ,a ,f ,g ,k ,m ,all ---------------------------------------- - - #d 44 44 44 44 44 #a 44 - #c 44 44 44 44 44 44 44 - #b #a #d #a #g 44 #b #ae - #a #b #g #g #bh #ae #d #fc - #j #c #aj #f #cb #dj #aj #ia #j #a #aa #f #g #ad #aa #ej #a #a #c #bj #i #d #a #ch #b 44 #e #dh #bf 44 #a #hj #c 44 #a #cb #cf #b 44 #ga #d 44 #a #e #be #be 44 #ef #e 44 #a 44 #f #be 44 #cb #f 44 #b 44 #c #aj 44 #ae #g 44 #a 44 44 #ad 44 #ae #h 44 44 44 44 #c #g #aj #i 44 44 44 44 #b #d #f #aj 44 44 44 44 44 44 44 #aa 44 44 44 44 44 #a #a ---------------------------------------- - ,total #h #df #abe #agi #aed #db #eed ---------------------------------------- --------- ,in the distribution of all the parallax stars we once more find a similar bipartition of the stars4 ,arguing from these statistics some astronomers have put forward the theory that the stars in space are divided into two classes1 which are not in reality closely related4 ,the one class consists of intensely luminous stars and the other of feeble stars1 with little or no transition between the two classes4 ,if the parallax stars are arranged according to their apparent proper motion1 or even according to their absolute proper motion1 a similar bipartition is revealed in their frequency distribution4 ,nevertheless the bipartition of the stars into two such distinct classes must be considered as vague and doubtful4 ,such an apparent bipartition is1 indeed1 necessary in all statistics as soon as individuals are selected from a given population in such a manner as the parallax stars are selected from the stars in space4 ,let us consider three attributes1 say ,a1 ,b and ,c1 of the individuals of a population and suppose that the attribute ,c is positively correlated to the attributes ,a and ,b1 so that to great or small values of ,a or ,b correspond respectively great or small values of ,c4 ,now if the individuals in the population are statistically selected in such a way that we choose out individuals having great values of the attributes ,a and small values of the attribute ,b1 then we get a statistical series regarding the attribute ,c1 which consists of two seemingly distinct normal frequency distributions4 ,it is in like manner1 however1 that the parallax stars are selected4 ,the reason for this selection is the following4 ,the annual parallax can only be determined for near stars1 nearer than1 say1 #e siriometers4 ,the direct picking out of these stars is not possible4 ,the astronomers have therefore attacked the problem in the following way4 ,the near stars must1 on account of their proximity1 be relatively brighter than other stars and secondly possess greater proper motions than those4 ,therefore parallax observations are essentially limited to 7#a7 bright stars1 7#b7 stars with great proper motions4 ,hence the selected attributes of the stars are m and mu4 ,but m and mu are both positively correlated to ,m4 ,by the selection of stars with small m and great mu we get a series of stars which regarding the attribute ,m seem to be divided into two distinct classes4 ,the distribution of the parallax stars gives us no reason to believe that the stars of the types ,k and ,m are divided into the two supposed classes4 ,there is on the whole no reason to suppose the existence at all of classes of giant and dwarf stars1 not any more than a classification of this kind can be made regarding the height of the men in a population4 ,what may be statistically concluded from the distribution of the absolute magnitudes of the parallax stars is only that the dispersion in ,m is increased at the transition from blue to yellow or red stars4 ,the filling up of the gap between the dwarfs and the giants will probably be performed according as our knowledge of the distance of the stars is extended1 where1 however1 not the annual parallax but other methods of measuring the distance must be employed4 ,,table #h4 ,,the ,,absolutely ,,faintest ,,stars4 ,transcriber's ,note3 ,to conserve space 7ad7 is used in place of 7alphadelta74 ---------------------------------------- --------------------------- #a #b #c #d #e #f #g #h ---------------------------------------- --------------------------- ,position ,distance ,name ---------------------------------------- ---- 7ad7 ,square l b pi r ---------------------------------------- --------------------------- sir4 #a,proxima ,centauri 7#adbbfb7 ,,gd#aj #bha - #b #j4ghj #j4bf #bvan ,maanens star 7#jjdcjd7 ,,ge#h #ib -#eh #j4bdf #j4hd #c;,barnards star 7#agebjd7 ,,gc#ab #ceh #ab #j4eae #j4dj #dag ,lyr ,c 7#aijccb7 ,,gc#b #ca #aj #j4abh #a4fj #e;,c4 ,z4 #e;h4#bdc 7#jejgdd7 ,,ge#g #bah -#ce #j4cai #j4fe #f;,gron4 #ai ,,viii #bcd 7#afahci7 ,,gb#a #bi #dd #j4afb #a4bg #g,oe4 ,a4 #agdae 7#agcgfh7 ,,gb#h #fe #cb #j4bfh #j4gg #h;,gron4 #ai ,,vii #bj 7#afbadh7 ,,gb#b #da #dc #j4acc #a4ee #i,pos4 ,med4 #bafd 7#ahdaei7 ,,gc#b #ef #bd #j4bib #j4ga #aj,krger #fj 7#bbbdeg7 ,,gc#h #gb #j #j4bef #j4ha #aa;,b4 ,d4 #efecb 7#jbabef7 ,,gd#h #ajc - #d #j4aie #a4jf #ab;,b4 ,d4 #eeeha 7#jbacef7 ,,gd#h #ajc - #d #j4ahe #a4ab #ac;,gron4 #ai ,,viii #dh 7#afjdch7 ,,gb#a #bg #df #j4jia #b4bg #ad,lal4 #baahe 7#ajegcf7 ,,gb#e #aec #ff #j4djc #j4ea #ae,oe4 ,a4 #aafgg 7#aaadff7 ,,gb#c #ajc #ej #j4aih #a4jd #af,walkey #fec 7#aeecei7 ,,gb#b #eg #de #j4age #a4ah #ag,yerkes parallax star 7#jbabdc7 ,,gd#h #ajg -#af #j4jde #d4eh #ah;,b4 ,d4 #efecg 7#jbabef7 ,,gd#h #ajc - #d #j4age #a4ah #ai;,gron4 #ai ,,vi #bff 7#jfbjhd7 ,,gc#c #ig #bg #j4jga #b4hj ---------------------------------------- --------------------------- sir4 ,mean 44 44 44 #bg4ej4bdd #j4ii ---------------------------------------- --------------------------- ---------------------------------------- ------------------------- #a #b #i #aj #aa #ab #ac #ad ---------------------------------------- ------------------------- ,motion ,magnitude ,spectrum ,name ---------------------------------------- -- mu ,w m ,m ,sp m' ---------------------------------------- ------------------------- sir4_/st4 m' #a,proxima ,centauri #c4he 44 #aam4#j #acm4#i 44 #ac4e #bvan ,maanens star #c4ja 44 #ab4c #ab4g ,f#j #ab4ie #c;,barnards star #aj4bi -#ai #i4g #aa4g ,mb #h4i #dag ,lyr ,c #a4ge 44 #aa4c #aj4c 44 #ab4e #e;,c4 ,z4 #e;h4#bdc #h4ge #ea #i4b #aj4a ,k#b #aj4fh #f;,gron4 #ai ,,viii #bcd #j4ab 44 #aj4c #i4h 44 44 #g,oe4 ,a4 #agdae #a4cj 44 #i4a #i4g ,k #aj4e #h;,gron4 #ai ,,vii #bj #a4bb 44 #aj4e #i4f 44 44#j #i,pos4 ,med4 #bafd #b4bh 44 #h4i #i4f ,k #aj4c #aj,krger #fj #j4id 44 #i4b #i4f ,k#e #aj4h #aa;,b4 ,d4 #efecb 44 44 #i4e #i4d 44 44 #ab;,b4 ,d4 #eeeha 44 44 #i4d #i4b ,g#e #aj4b #ac;,gron4 #ai ,,viii #dh #j4ab 44 #aa4a #i4c 44 44 #ad,lal4 #baahe #d4gg -#ah #g4f #i4a ,mb #h4i #ae,oe4 ,a4 #aafgg #c4jc 44 #i4b #i4a ,ma #aa4j #af,walkey #fec 44 44 #i4e #i4a 44 44 #ag,yerkes parallax star 44 44 #ab4d #i4a 44 44 #ah;,b4 ,d4 #efecg 44 44 #i4d #i4j 44 44 #ai;,gron4 #ai ,,vi #bff #j4ji 44 #aa4c #i4j 44 44 ---------------------------------------- ------------------------- sir4_/st4 m' ,mean #b4if #bi4c #ajm4#j #im4#i ,k#a #aj4i ---------------------------------------- ------------------------- ,regarding the absolute brightness of the stars we may draw some conclusions of interest4 ,we find from table #g that the absolute magnitude of the parallax stars varies between -#d and #aa1 the extreme stars being of type ,m4 ,the absolutely brightest stars have a rather great distance from us and their absolute magnitude is badly determined4 ,the brightest star in the table is ,antares with ,m -#d4f1 which value is based on the parallax #j4jad found by ,,adams4 ,so small a parallax value is of little reliability when it is directly computed from annual parallax observations1 but is more trustworthy when derived with the spectroscopic method of ,,adams4 ,it is probable from a discussion of the ,b-stars1 to which we return in a later chapter1 that the absolutely brightest stars have a magnitude of the order -#em or -#fm4 ,if the parallaxes smaller than #j4ja were taken into account we should find that ,canopus would represent the absolutely brightest star1 having ,m -#h4ag1 and next to it we should find ,,rigel1 having ,m -#f4ig1 but both these values are based on an annual parallax equal to #j4jjg1 which is too small to allow of an estimation of the real value of the absolute magnitude4 ,if on the contrary the absolutely faintest stars be considered1 the parallax stars give more trustworthy results4 ,here we have only to do with near stars for which the annual parallax is well determined4 ,in table #h ,i give a list of those parallax stars that have an absolute magnitude greater than #im4 ,there are in all #ai such stars4 ,the faintest of all known stars is ,,innes' star ,proxima ,centauri with ,m #ac4i4 ,the third star is ,,barnard's star with ,m #aa4g1 both being1 together with alpha ,centauri1 also the nearest of all known stars4 ,the mean distance of all the faint stars is #a4j sir4 ,there is no reason to believe that the limit of the absolute magnitude of the faint stars is found from these faint parallax stars3--,certainly there are many stars in space with ,m #acm and the mean value of ,m1 for all stars in the ,galaxy1 is probably not far from the absolute value of the faint parallax stars in this table4 ,this problem will be discussed in a later part of these lectures4 ,,footnotes3 ,footnote #ae3 ,compare ,,adams' memoirs in the ,contributions from ,mount ,wilson4 ,footnote #af3 ,the first line gives the stars of an absolute magnitude between -#d4i and -#d4j1 the second those between -#c4i and -#c4j1 c4 ,the stars of type ,b and ,a are from ,,walkey's catalogue4 ,illustration3 ,,plate ,i4 ,,conversion ,,of ,,equatorial ,,coordinates ,,into ,,galactic ,,coordinates4 ,illustration3 ,,plate ,,ii4 ,squares and ,constellations4 ,illustration3 ,,plate ,,iii4 ,the ,harvard ,classification of ,stellar ,spectra4 ,illustration3 ,plate ,,iv4 ,distribution of the parallax stars over different absolute magnitudes4 ,,scientia ,,publisher ,k4 ,g4 ,,malmquist3 ,a study of the stars of spectral type ,a4 ,l4 ,m4 ,,ii1 #bb4 ,price3 #f ,kr4 ,c4 ,v4 ,l4 ,,charlier3 ,vorlesungen ber die ,grundzge der mathematischen ,statistik4 ,price3 #aj ,kr4 ,f4 ,,esscher3 ber die ,sterblichkeit in ,schweden4 #ahhf-aiad4 ,eine ,stabilitts- und ,korrelationsuntersuchung4 ,l4 ,m4 ,,ii1 #bc4 ,price3 #f ,kr4 ,b4 ,f,,nge3 ,on the distances and luminosities of stars of spectral type ,g as derived from their proper motions4 ,l4 ,m4 ,,ii1 #be4 ,price3 #e ,kr4 ,c4 ,v4 ,l4 ,,charlier3 ,introduction to ,stellar ,statistics4 ,price3 #d ,kr4 ,,nils ,h4 ,,rasmuson3 ,a ,research on moving clusters4 ,l4 ,m4 ,,ii1 #bf4 ,price3 #e ,kr4 ,,sigfrid ,,raab3 ,a ,research on open clusters4 ,l4 ,m4 ,,ii1 #bh4 ,price3 #h ,kr4 ,,carl ,,wirtz3 ,flchenhelligkeiten von #eff ,nebelflecken und ,sternhaufen4 ,nach photometrischen ,beobachtungen am #di cm-,refraktor der ,universitts-,sternwarte ,straburg 7,elsa7 #aiaa-aiaf4 ,l4 ,m4 ,,ii1 #bi4 ,price3 #e ,kr4 ,c4 ,v4 ,l4 ,,charlier3 ,stargauges by ,william ,herschel 7,h7 and ,john ,herschel 7h74 ,l4 ,m4 ,,ii1 #cj4 ,price3 #e ,kr4 ,,printed ,,by ,l,,tcke ,,wulff1 ,,hamburg 99 99 99 99 99 ,transcriber's ,note3 ,the following corrections have been made to the original text4 ,page #d3 ,terrestial distances changed to ,terrestrial distances ,page #i3 we must chose1 changed to we must choose1 ,page #ab3 acromasie changed to achromatism ,page #ae3 inparticular changed to in particular supposing1 that changed to supposing that ,page #af3 #cic4d mm changed to #cic4d mumu ,page #bj3 for which stars if changed to for which stars it ,page #bb3 sphaerical changed to spherical ,page #bc3 principal scource changed to principal source ,page #be3 lense with #ae cm changed to lens with #ae cm ,page #bg3 ,through the spectroscopie method changed to ,through the spectroscopic method made to its principal changed to made its principal american manner changed to ,american manner ,page #ce3 many tenths of a year changed to many tens of years ,page #ch3 same appearence changed to same appearance ,page #dg3 red stears changed to red stars dispersion in ,m changed to dispersion in ,m ,page #di3 smaller the changed to smaller than