OF ARTS AND SCIENCES. 85 



of the 8th volume of the Bonn observations. Of all these, or nearly 

 all, the proper motion was certain ; of many it was large ; six or seven 

 of them only had been previously used in the problem. 



In studying their motions, I grouped them by tens, assuming those 

 to be equally distant whose proper motions in arc were nearest equal. 

 My result for the apex was not very ditierent from those which pre- 

 ceded ; but there appeared indications that for each such group the 

 average proper motion was inversely proportional to the average dis- 

 tance, or, in other words, that our assumptions of star-distances ought 

 to depend upon proper motion. 



VI. The method I used was substantially Airy's ; but in working 

 out the problem I employed a little device like one I have used to 

 simplify the study of planetary orbits. I selected the point whose 

 right ascension was 259° 50'.8, and whose declination was 32° 29'.1, 

 as a point in the positive direction of the axis of Z ; that of X I located 

 in the equator, in right ascension 349° oO'.S, and that of Y in right 

 ascension 79° oO'.S and declination 57° 30'.9. Thus, so far as Arge- 

 lander's apex is correct, the average stellar motion will be negative, 

 along the axis of Z ; all other motions will, upon the whole, counteract 

 each other. 



VII. So I found it, at least approximately ; and, in addition, that 

 the mean Z''^ for each group of ten were as nearly equal as their prob- 

 able errors would lead us to expect, taking, as before stated, the proper 

 motion of each star as the measure of the reciprocal of its distance. 



VIII. The jjresent paper is for the purpose of showing this relation 

 with respect to the stars earlier investigated by Argelander, Lundahl, 

 and Galloway. 



The details of 0. Struve's paper are not published, and his stars 

 are largely identical with those of the others ; Miidler's investigations 

 have suflFered, as before mentioned, from the many errors in his proper 

 motions, which I have detected over large areas of the heavens, but 

 which it will be impossible to get rid of without an investigation cost- 

 ing enormous labor. I propose to continue these investigations from 

 time to time, as material for them accumulates : the great difficulty in 

 the whole matter arises from the extremely unsystematic way in which 

 it has been the fashion to observe star -places, and work up their results. 

 So soon as the great co-operative zones now in progress are completed, 

 much more will be known regarding proper motions ; so that for the 

 present what I give here may suffice. I should have added my pre- 

 vious results, but have not been able to get the papers from Chicago, 

 owing to the illness of my assistant, who has had charge of them. 



