Researches into the theory of probability. 23 



IT 



(10*) ']j (;r) = Je^ cos [X siu co — .no] do), 



0 



which may sometimes be preferable to the series (10). If r be a positive integer, 

 we have 



(11) 



In the following investigation we shall find, that, by suitably choosing the para- 

 meters c, to and X, a frequency curve approximately may be represented by means 

 of the formula 



Hence the function -j;-^ (.r) will give for difîerent values of X the différents forms 

 of the frequency curves of type B. In fig. 13 I have reproduced some of these 

 forms, where it may be observed that only integer values of ./■ are taken into con- 

 sideration. We find that the frequency curves of type B for x = c discontinuously 

 breaks up and possesses a finite value, whereas for j; = co 'f^ (.9?) tends towards zero. 

 With increasing X the curves gradually approach the form of the curves of type A. 



More generally we may write a frequency curve of the type B in the form 



(12) F[x) == B,fy {x) + BM {-'■) + B-A''^ i^-) + ^:A''^ + • • - 



where (»Meddelanden» N:o 27) the coefficients have the following values 



|2i^, = XV- (2X-f l)i./ + li.,', 



|3 ^3 = XVo' - (3X^ + 3X + 2) [./ + 3 (X + 1) - jj,,', 

 |4 if, = X*(j.y' — (4X-^ + 6X2 ^ 8X + Ü) [x/ -f ((3X^ + V2\ + 11) [x,' 

 - (4X + 6) 1J.3' + [J-/, 



and i^o', [X,', [x^', . . . are defined by the formula 



-j- GO 



(12*) [x/ = 2 x'F{x). (.s = 0, 1, 2, . . .) 



— 00 



The parameter X may be arbitrarily chosen. It is possible to introduce two 

 new parameters, if we write instead of (12) 



(13) F{xi. -f c) = B,^\ {x) -f B,^^\ [x] + B,^^ [x) + B,^^ {x) + 



