Studies iu Stellar Statistics 19 

 + 00 4-00 ^_ 00 



I h{x) I dp A(p) e-P'"^+P I dx è 



— 00 — 00 — 00 



+ 00 +00 



xuii 



(a3-j-p)tui 



— 00 —00 



If now the definition of the conjugate FouKiER-function is extended also to 

 imaginary values of the argument — the legitimacy of which extension must 

 however first be examined — this equation might be written 



(38) }r\iü) = V2i ^-\-{i^^>,)) . . 



Dividing (37) and (38) and changing the signs of co we obtain 



A~^(oj) a~\ — cü) 



Substituting œ = iy and introducing two auxiliary functions <i^{y) and c(p) 

 defined by the equations 



(39) ^'\^^J) = e^^^) ; l^lzM = 



I \-iy) 



we have 



(40) ^(^)_^(^_1) = ,(^) 



a linear difference equation for '{j. When (Jj is known from (40) we get from 

 (39) and ^{x) from (34). Then we get $"'(«) from (37) and $(x) from (34). The 

 problem is thus completely solved. 



8. As an apphcation of the formula of Schwakzschild we shall discuss the 

 following problem. 



Generally speaking we may assume that the frequency curve for the absolute 

 magnitude (il/) of the stars is either of tyj^e A of frequency curves or of type B *). 

 A priori we cannot predict, if the one or the other form is more probable'. (Neither 

 can we say if the magnitude (J/) or the hnuinosity [H) is the best characteristic to 

 take into consideration). As type A is easier to treat analytically we shall assume 

 that <ï> is of this type. For the same reason we assume tliat the frequency function 

 a(m), of the apparent magnitudes also is of that type. Hence we put 



(x-mo)2 (.r-iiro)2 



(41) a[x) = ce ; ^[x)=Ge 



Here and denote the means, k and K the dispersions of these fre- 

 quency curves. Using the well known formula (of Laplace) 



*) Compare Meddelanden Ser. II N:o 4. 



