402 ASTROMETRY 



the real distribution, however, if we can make out which fraction 

 of the stars have a parallax exceeding its theoretical value two, 

 three, four . . . times, for which fraction this parallax is only half, 

 a third, etc. 



We thus have to see whether it be not possible to find out this 

 law of the frequencies. 



Theoretically nothing is easier than to derive it from the data 

 furnished by the stars of which the parallax has been measured. 

 For these objects we know the true as well as the theoretical par- 

 allax, and we may thus determine at once the frequency of any devi- 

 ation of the two. 



We thus see that there is nothing to prevent us from obtaining 

 ultimately a thorough knowledge of the law in question. 



For the present, however, existing materials are quite insufficient 

 for such a thorough determination, and we must provisionally have 

 recourse to a less fundamental course. 



The question is analogous to the other, with which every astrono- 

 mer is familiar : What is the frequency with which errors of a given 

 amount will occur in a determined series of observations ? Every- 

 body knows that, by admitting certain plausible hypotheses, which 

 may be supposed to be approximately satisfied in most cases, the 

 question is reducible to the finding of a single number, to that of 

 the probable error, for instance. 



Something of the same sort may be done here. True, the condi- 

 tions, supposed to be satisfied for the distribution of the errors of 

 observation, are certainly not satisfied for the deviations of the 

 true parallaxes from their theoretical value. For if, for instance, the 

 theoretical parallax is 0" 01, it is evident that a deviation of 0" 02, 

 very well possible in one direction, is impossible in the opposite one. 

 Positive and negative deviations of the same amount are certainly not 

 equally probable. 



We may, however, introduce other conditions, which are cer- 

 tainly satisfied at the limits and which for the rest may be deemed 

 plausible. These will lead to a law of frequency, different from that 

 of the errors of observation, but like that law only dependent on 

 one or only a few constant parameters. I have chosen a form with 

 a single constant. 1 



We may take such a course with the more confidence the smaller 

 the deviations are. For, as these deviations decrease, our independ- 

 ence from the form of the assumed law increases. In our case I find 

 that 70 per cent of all the stars have their true parallax included 

 between 0.4 and 1.6 times their theoretical parallax. 



Still of course it cannot be maintained that the frequency law 

 1 See Publications of the Astronomical Laboratory, at Groningen, no. 8. 



