20 C. V. L. Charlier 



+ 00 +00 



(42) 



we have now 



I flîx e e^' — ^(Ixe ""^^ cos bx = |/^~ e 



— ■00 — 00 



+ 00 



xmi j 2~ 



(43) « ^(w) = I '^{^) = eke 



— 00 



and 



(43*) 4>-V)=C^e~'^'^'''°'" 

 hence according to (37*) we have 



A (to = —= . = — = e 2 



V2k <t>-'(oü) ]/2kCK 



Using the formula (33) we at last get 



e 



(44) A(?y) = C\ e _ 

 where 



1 rJr 



(44*) = -4= 7^ -.• 0^ = 7c2-Z^ 



(Observe that Jc must be > K). The corresponding value of the density is according 

 to (21*) 



(44**) . nie~'') = C,e 



The density has a maximuu] for y = Mq — -|- 3?>a^. For greater Viilues of 

 y it decreases toward zero. 



9. The expressions for a{x) and ^{x) discussed in the preceeding lecture 

 really have been found by Kapteyn to represent the conditions of the Milky Way. 

 He seems, however, not to have observed that these expressions are nothing else 

 than the usual form of frequency curves of type Ä. In the present lecture I shall 

 make no criticism regarding the rehability of the numerical results of Kaptetn, 

 but simply accept them and apply on them the formulae of the preceding lecture. 



For the frequency I'unclion f^iM) for the absolute magnitudes Kapteyist finds 

 (A.J. N:o 5G6, (19U4)). 



(45; cpJilf ) = S e~ with = 4- 0.385. 



Here we have to substitute the value of H expressed in terms of 3Î. Kaptetn 

 takes the luminosity of the sun = 1 and assuming the sun at the distance = 1, 



