STELLAR ASTRONOMY 403 



thus provisionally adopted represents the facts of nature accurately. 

 We have not sufficient data for testing the matter. But we may 

 safely assume that, the constant parameter being determined from 

 the observations themselves, we must get enormously nearer to the 

 truth than by adopting for the parallax of any star its theoretical 

 value. It appears that even from existing determinations of parallax 

 this one parameter is already obtainable with some precision. 



In a few years more extensive results of observation will surely 

 enable us to get a better approximation, and we may well hope that 

 the time is not far distant when accumulated data will furnish such 

 knowledge of this frequency-law as leaves little to be desired. 



Having once found the law of the frequencies, we may determine 

 the true frequency of the distances with the same ease as we deter- 

 mine the frequency of determined errors of observations as soon as 

 the probable error is known. The following example will illustrate 

 the whole process. 



From the observations of Bradley, which embrace about two thirds 

 of the whole sky, we learn that somewhat less than 10 per cent of 

 the stars of the sixth magnitude have centennial motions ranging 

 from 4" to 5". The total number of the sixth magnitude stars in the 

 whole sky is 4730. 



We conclude that in the whole sky there must be almost 10 per 

 cent of 4730, in fact, 461 stars of the sixth magnitude having proper 

 motions ranging from 4" to 5". 



Our formula gives almost exactly 0"01 for the theoretical paral- 

 lax of the stars of this magnitude and proper motion. 



With these data our frequency-law leads at once to the following 

 distribution of the true parallaxes : 



