262 G. W. Colles, Jr. — Distance of the Stars. 



mean values of a, d and ad; then in the case of distribution 

 over a whole sphere or hemisphere, 



/3 = ad, M (/?) = M(ad) =— M(«) . 



]STow let «j, # 2 , . . . ., d„ d„ .... refer to the particular stars, 

 and let n be their number; we have then 



,,. 7 , ^"(rcc?) a/7 a «? a.a*, a„c7 n 



M(ffflf)=— ^ — i— -i_ 1 +_B-^4-.-i- 3 + .... + -?-", 

 ?i n n n n 



M(a)M(d)=,?yS^=^S3 + ^iS+2aS + 



aJVEJoO 

 w 



Subtracting the second of these equations from the first, we 

 have 



M(ad)-M(a)M(d)=±{a 1 [d i -M(d)]+a i [d i -M(d)) + 



+ a n [d u -Mffi]\" 



Now the sum of the coefficients of a x , a 2 , etc. is 2(d) — riM.(d) 

 = ; so that some of the terms within the braces will be posi- 

 tive and some negative. And since positive and negative 

 values for a 1 [d 1 —M.(d)~\, a i {d i — 'M.(d)'], etc., are equally prob- 

 able, the sum of these terms will be some small finite quantity 

 e, and the last equation becomes 



M(ad) -M(a)M.(d)=— . 



If n is practically infinite, as we have supposed it, the right- 

 hand member of this equation vanishes; giving us, by the 

 previous results, 



_M{ad)_n M(a)_7t 2(a) 



M(d): 



M(a) "~2 * M(a)~"2 ' 2(a) ' 



thus reducing the required mean distance d to a simple func- 

 tion of the mean angular velocity and the mean velocity in the 

 line of sight. 



Although we cannot employ an infinite number of stars in 

 calculation, yet the error will be quite small if a very large 

 number of stars be used, provided their motions really are at 

 random, i. e. show no "drift" in any particular direction. 

 But, as previously hinted, because of the comparatively small 

 number of stars whose velocities in the line of sight and angu- 



