8 



O. V. L. Charlier 



Tlie observed clusters may be arranged into a correlation surface having class 

 ranges equal to dr and db. To each value of r corresponds a certain repartition 

 of the /^-values forming an »r-array» and to each values of b the clusters may be 

 arranged according to increasing values of r into a »/> array». 



Let and /^{r) be the frequency curves of these ai-rays. Then we have, 



apart from a constant factor 



m = 'h{r)'h(rh)r, 



Thus /j and have the same analytical expression, but in /j we have to 

 give to b all values between 0 and x , keeping ?• constant, whereas in h is to 

 be given a definite value and r is varying from 0 to x . 



Consider first the frequency function f^{b). The mean vakie of b is evidently 



00 00 



Mr(b) = ^-ji? 'f,(ß) dB : I'f^l/i) dB 



or 



(2) M.V,) = '^, 



where M{B) denotes the mean absolute diameter of the stars, so that the mean 

 value of the apparent diameter of the stars at the distance r is equal to the mean 

 value of the absolute diameter divided by the distance. A self-evident fact. Similar 

 conclusions are valid regarding the higher characteristics. 



With the fe-arrays the things stand otherwise. The mean value of a b-avvay is, indeed, 



30 00 



(3) il/.(r) = |'f,(r)-^,(rZ.)rV/r:Jx,(;J'f,(r?>)r.^^^ 



0 0 



It is dependent on b, but by no means generally inversely proportional to b, 

 as could be expected. 



We may distinguish between the value of M for very large and for very small 

 values of b. 



We first suppose that cCj and are both frequency curves of the type B so that 



- 'fx(0) + 0 



and 



'f2(0) + 0. 



Puttino- now /> = (), we get ■ - : . 



(4) Jf„(r) = |tpi(r) rf;- : l'fi(r) r t/r ' 



0 0 



— a finite quantity = . 



