32 C. V. L. Charlier. 



It is to be observed that — l) = (j>( — 2) = O, as follows directly from the 

 formula (10). 



We now derive the following values of F{x): 









Fix) 



X 



222'|(x) 















Calc. 



Obs. 



0 



+ 118.1 



+ 16.8 



134.9 



133 



1 



-j- 74.5 



— 22.9 



51.6 



55 



2 



+ 23.6 



— 1.1 



22.5 



23 



3 



+ 4.9 



+ 4.6 



9.5 



7 



4 



+ 0.8 



+ 2.1 



2.9 



2 



5 



+ 0.1 



+ 0.5 



0.6 



2 



The agreement is as coujplete as can be wished. The effect of the second 

 differences is clearly pronounced and is rather considerable for :r = 0 and x=l. 



A great advantage with this method is, that it is only necessary to calculate 

 directly the values of '\>{.>% whereas the values of A'|i, A^t];! ^^'^^ ■ ■ ■ ^^'^ obtained 

 through the numerical differences of '^'{x), which are easily obtained. 



Seventh Example. Variation in the failing percentage of barley. 



In his lectures on the theory of heredity W. Johannsen has given some in- 

 stances of frequency curves that belong to type B. From these I choose the 

 following one relating to the failing of the grains of common barley. 



Mean procent of tailing 2.5 



7.5 12.5 



17.5 22.5 27.5 32.5 37.5 42.5 



47.5 



52.5 



Class 0 



1 2 



3 4 5 6 7 8 



9 



10 



Frequency 53 



131 180 



170 111 50 22 22 7 



2 



1 



From these numbers 



we obtain 



the following values of the moments 







H = 





+ 3.063, 







b = 



+ 2.7.'S7, 



Vg + 4.250, 







+ 35.46i. 



Calculating now co, X and c according to the third method above, we obtain 



(Ü + 1.388, 

 X + 1.591, 

 C + 0.549, 



SO that 



F{x 1.388 + 0.549) = 539.62 '!^{x) + 0.6 A"'];. 



The comparison between theory and observation is seen from diagram 15 

 The second term in the above expression has here been neglected. 



