22 



O. V. L. Charlier 



well as the dimensions of the Milky Way should, practically, be infinite! Indistinct 

 contradiction to the usual view of our starry system *). 



But we have still to complete our expression for fQ{M). We have already 

 found Mq = 9'".476, but it remains to determine the dispersion (Z). We get from (45*) 



and 



^2 = 0.16 = 0.0616 •.• K=2"\Sb {K^^S.12) 



(49) ^^(j^) _ 2(2.«.)^ 



(two thirds of the stars should hence have a magnitude [M) between 6"'.63 and 

 12"'.33. The sun (with M — 5™.6) should be a rather bright star. 

 From (44) we now have 



(y -l-4i.rO'-' (y - yo) 



(00) Hy)=C,e =G,e 



For the density D{e~^^) we get a similar formula with the same dispersion 

 (5™.60) and with a mean value [y^] of y (according to (44**)) equal to 



9.5 — 51.0 + 30(5.60)2= + 1.83 



The corresp. value of r being 

 = e "^^^ = = 0.4426 tt = = 0['232 (max. density at that distance), 



where is the distance for maximum density. 



10. Let us next draw the conclusions from the results of Kapteyn, (48) 

 and (49), with respect to the mean distances of the stars. For this purpose we first 

 deduce the value of some general integral expressions. Let 



(51*) 9, =e ; ^,= e 



and consider the integrals 



(51**) I 'i\ ?2 ^i^d I^=j dxfj^ 



GO —00 



qx 



After some easy reductions we obtain 

 (51) I,^—^ß^V2^e + 



(52) I, = -7=4= ' 



of which (51) evidently is a special case of (52). 



+ 1 :r +; 



*) Form (47) the total number of stars in the Milky Way can be computed. It should 

 amount to nothing less than 49 700 000 000 000 stars! 



