12 C. V. L. Chailiur. 



According to the value of f{.:rj we find tliat 



z- 



1 ~ 2 



and, developing this expression into powers of i and integrating, we thus finally 

 find the following equation for determining z^: 



r*) •' = ^."f + • ■ • + 1 + 1^ + ■ ■ ■) + + ■ 



Neglecting ß-, ß,., . . ., and terms of the third order we obtain 



P.S 



^2 = 



1 + 3ß, 



and hence we have 



(9) -. = '' + TTis- 



For ßg = 0 (ß-, and the higher coefficients being neglected) the mean, the mode 

 and tlie median coincide. For frequency curves with small excess (for others we 

 cannot conclude anything definitely from these formulfe) the median is situated be- 

 i/vceii the mean and the mode. 



£,,0 ^3 



o o c- 



o 

 (t> 



The relative position of the mean, the median aiid the mode is first given by 

 Pearson, who has derived it from his theory of frequency curves. For curves 

 with a sensible excess the order of these points may possibly be different. 



III. Numerical determination of the parameters 

 of a frequency curve. 



The calculation of the coefficients ßg, ß^, ... according to the formulas (7) is 

 a fairly simple affair, when the moments of the frequency curves are known. As 

 the calculation of these moments has been thoroughly discussed by Pearson and 

 his disciples, it would not be necessary to expend many words on this matter, 

 were it not that some special points here deserve a closer examination. It ought 



