114 



KNOWLEDGE 



[June 1, 1898. 



Advancing cautiously, we found the light came from a 

 sole that had been skinned, and was hanging from a shelf. 

 It had been kept too long, and looked transparent, the 

 bones showing distinctly brighter than the rest. 



Not long since I saw a notice of some " luminous eggs," 

 which probably were not verj' fresh. 



Yours very truly, 



Thos. Blashill, 

 Sujienntemliwi Architect of the L. ('. C. 

 29, Tavistock Square, W.C. 

 April 29th, 1893. 



ON THE DISTRIBUTION OF THE STARS IN 

 SPACE." 



By Prof. .J. C. Kapteyn. 



PROF. KAPTEYN, of Groningen, read a preliminary 

 paper on this subject on the 29th April last year. 

 He has now included in the research all the stars 

 of the Draper Catalogue, which have been observed 

 by Bradley in both co-ordinates. This material 

 embraces 2357 stars, of which 1189 belong to the jiivt, 

 1106 to the seai)i<l, and 62 to the thini spectral type. The 

 material now used permits more trustworthy conclusions 

 than that of April, 1892, not only because it is more 

 extensive, but chiefly because it is more homogeneous, and 

 contains a very great number of smaller stars, possessing, 

 nevertheless, trustworthy proper motions. 



Before entering into a discussion of the results obtained. 

 Prof. Kapteyn spoke upon the question of the best criterion 

 for judging about the relative distances of the fixed stars. 

 As the direct determination of stellar parallax has not as 

 yet given results on anything like the scale wanted for the 

 purpose, we are compelled to resort to the parallactic 

 displacement caused by the motion of the solar system in 

 space. This " motus parallacticus," however, cannot be 

 separated from the real proper motion {nwtua pcculiaris) for 

 individual stars, but there is every reason to think that 

 this " motus peculiaris " shows very little preference for 

 determinate directions, and it may therefore be assumed, 

 with some confidence, that, in the mean results obtained 

 from small groups of stars, these real proper motions will 

 destroy each other, leaving only that part which is due to the 

 solar motion, and it is this parallactic displacement that, for 

 the present at least, gives the most reliable measure of the 

 mean distance of the stars under consideration. If, 

 however, we wish to arrange the imlividual stars approxi- 

 mately in the order of their distances from the sun, we 

 must look out for another measure. This may be found in 

 the total proper motion. It is true that this motion seems 

 only to be approximately proportional to the parallactic 

 displacement (the researches of Ristenpart and of the 

 speaker himself indicate a slow increase of the linear 

 proper motion with the distance), but this is sufficient for 

 the present purpose. There is another difficulty connected 

 with the use of the total proper motion as a measure for 

 distance, which may be very important in some cases. 

 For if we compare by means of their total proper motion, 

 not groups of stars which separately are distributed over 

 the entire sky, but groups situated in different parts of the 

 firmament, then a systematic error will be introduced, 

 depending on the position with respect to the apex of the 

 solar motion. For at the apex and anti-apex the whole 

 proper motion is nothing but the wotits pcrulidiis. At 

 some distance from these points it is the resultant of the 



* Translated bv Prof. Kapteyn from tlie Dutch report of liis 

 communication to the Amsterdam Academy of Sciences, 28th January 

 1893. 



motus peculiaris and the motits jKirallaeticKs. Equality of 

 total proper motion will not, therefore, correspond with 

 equality of iiuitiis peiididvin in different parts of the sky, and 

 it is highly improbable for that reason that even stars of 

 absolutely equal distance would show equal proper motions 

 in different parts of the sky. 



Therefore this measure, too, fails for some detailed 

 investigations of the star-system. In those cases where a 

 systematic error from this source seemed most to be feared. 

 Prof. Kapteyn used another measure, viz., that projection 

 (t) of the proper motion which is independent of the solar 

 motion. As, however, this projection cannot be computed 

 without a knowledge of the place of the solar apex, an 

 assumption had to be made as to the position of this 

 point. 



The adopted co-ordinates for 186.5 are : — 

 a = 276". S=-|-34». 



It was further assumed that the same co ordinates would 

 have been foimd by a separate discussion (a) of the stars 

 in and without the Milky Way; (/)) of the stars of the first 

 and seco)ul spectral type. 



The (at least approximate) truth of the first assumption 

 had been proved by Mr. Bakhuyzen at the preceding 

 meeting ; the approximate truth of the second may be 

 deduced from the fact that Messrs. Stumpe and Porter, and 

 Ristenpart (12th March, 1893), found nearly the same 

 position of the apex from stars of very different amounts of 

 proper motion, a position tolerably accordant with that 

 deduced exclusively from stars of smaller proper motion. 

 As a matter of fact, however, we may say that the more the 

 investigation is limited to stars of greater and greater 

 proper motion, the more it is limited to stars of the second 

 type. This will be proved in Proposition VI. 



Having adopted the position of the apex as given, the 

 total proper motions of all the Bradley-Draper stars were 

 decomposed into two components, viz. : — 



<r The proper motion projected on the great circle 



passing through the star and the apex. 

 T The proper motion perpendicular to the former 

 (which is evidently independent of the solar 

 motion). 



The results of Messrs. Porter and Stumpe prove in- 

 directly for considerable proper motions, and for the sky as 

 a whole, that the quantity t is also a good measure for 

 distance. Directly, it is proved by the consideration of the 

 proper motions equal to or greater than 0"04 of the Bradley- 

 Draper stars ; for, taking the means of the values of t 

 for stars of different amounts of proper motion, we find, if 

 q be the mean value of the solar motion as seen at right 

 angles from the stars under consideration : — 



From this table it is evident, even for the smaller 

 proper motion, that r is nearly proportional to (/, i.e., 

 nearly inversely proportional to the distance. The small 

 discrepancy is in the sense found by Mr. Ristenpart. 



It remains to be proved that, when we come to compare 

 different regions of the sky, equality of /x, or better of r, 

 corresponds with equality of distance. Prof. Kapteyn tried 

 to do this for regions of different galactic latitude, in the 

 following way ; — The small number of stars with proper 

 motions exceeding 0"50 being excluded, we find — 



<il In Tables 1 and 2 the ralues of q furnished by stars whose 

 distance from the apex is less than '10°, and which are necessarily rery 

 uncertain, haye been omitted. 



