G. W. Colles, Jr. — Distance of the Stars. 261 



therefore all values for the solid angle d-'y' are equally prob- 

 able, and the mean projection on r is the same as if r were 

 fixed. Also, it will be seen that v may vary in any way we 

 please, so long as it is not a function of &' or <p' ; its' value in 

 the result being some function of its particular values. The 

 mean value of the projection of v on a plane perpendicular to 

 r is, in like manner, 



J Jo sin B'dB' dtp' 



where v has the same value as before. The ratio of the latter 

 mean value to the former is, therefore, 



M(Q_ = v £"£ 2 l\rt$' d$' d<pl _n 



M(a) *- ~ 2 ' 



«/o «/o 



2 J J cos 5 ' s{Q 5' dB ' d( p' 



Thus we obtain the ratio of the mean velocity of a star across 

 the line of sight to its mean velocity in the line of sight, suppos- 

 ing the direction and magnitude of its total velocity to be at ran- 

 dom, that is, independent of the angles d- and (p, or ■&' and <p'. 

 And if the stars are distributed equably over only one hemi- 

 sphere (that of which the initial line is the pole, suppose), the 

 above ratio remains the same as before ; because one hemisphere 

 is in all respects similar to the other, negative signs having been 

 abstracted. Further, if the stars are distributed equably over 

 any part of the celestial sphere, comprised (suppose) between 

 the angles d- x and # 2 , <p t and <p 2 (which may be functions of #), 

 the above ratio becomes, remembering that negative values are 

 to be reckoned as positive, 



M(ar) " : /•#„ /> 



J &i J 0i C0S 5 ' Sin B ' d $' d P' 



for it will be seen, on consideration, that we may interchange 

 the limits of # and <p with those of ■&•' and <p', the result being 

 the same if we let ■&' and <p' vary between the limits ■& 1 and #„, 

 <p x and <p» and # and <p over the whole sphere. 



Next, let a and /? be the components of the motion of a star 

 {v) in and across the line of sight, respectively ; a its proper 

 angular velocity and d its distance ; M(#), M.(d) and M.(ad) the 



