124 Dr. Frank Watson Dyson [April 24, 



peculiar motions, which have made the spreading out much greater 

 in this than in the perpendicular direction. In this part of the sky 

 the streaming happens to be in the direction of and opposite to the 

 solar motion. 



12. Let us now consider the proper motions of the 3700 stars 

 observed by Carrington in the light of these discoveries. The shift 

 of the centre of gravity caused by the solar motion is 1"*44 a 

 century. As we know how far the sun has moved in a century, 

 this gives the average distance of these stars as forty-eight million 

 times tlie distance of the sun from the earth. Turning now to the 

 proper motions in a direction perpendicular to that of the sun's 

 motion which arise from the motus pecuUares of the stars them- 

 selves. Counting these cross proper motions, it is found that : — 



16 are greater than 15"- a century 



25 lie between 10" -0 and 15" -0 



33 



66 



191 



873 



2504 



When suitable allowance is made for the accidental error in 

 these observations, it is found that the number less than any given 

 amount t can be represented by an algebraical formula 



3700 ,^-^ '^, ., 



13. Now suppose that all these stars were actually moving with 

 the same velocity, say 10 kilometres a second, then their distance 

 could be calculated : those with proper motion 1" a century being 

 40 million times as distant as the sun, those with proper motion 

 2" a century 20 million times, those of 4" a century 10 million 

 times, and so on, the larger the proper motion the nearer the star 

 to us. This is only an illustration, for the velocities of the stars are 

 not all the same, but are distributed according to the law of errors. 

 Looking at the question rather differently, let us start with the 

 assumption that the actual accidental velocities of the stars in one 

 co-ordinate are distributed according to the law of errors. If the 

 stars were all at the same distance, their proper motions, found by 

 dividing their velocities by this distance, would also be distributed 

 according to the law of errors. But, actually, they are distributed ac- 

 cording to a different law. The diagram A (Fig. 3) gives the proportion 

 of stars with proper motions between given limits (the area of the 

 shaded portion gives the proportion between 2" and 3" a century) ; 

 the diagram B gives the proportion of stars with velocities between 

 given limits (those between 30 kil/sec. and 40 kil/sec. are given by 



