lô 



c. V. L. Charlier. 



of the frequency curve ^). From the ilhistrations we may conclude, that a dis- 

 S3'mmetry corresponding to the value ßg = O.B must be considered as rather high, 

 the frequency curve being then far different from the Jiormal curve. It is to be 

 expected, that in practice the value of will seldom exceed 0.5. The following 

 coefficients in the series may however allow higher values of ßg to occur. 



The effect of the term [i^ cc'^' (j;) may be shown from tig. 4 and 5, in 

 which the normal curve is indicated by a dotted line. 



For ßj^ = -|- 0.1 we obtain a curve similar to the normal curve, but it is 

 directly observable from the figure that the number of individuals between x — h = — a 

 and X — & = -f- o is greater when the frequency curve is characterized by 

 ßj^ = -|- 0.1 than for ß^ = 0, wlien we have a normal distribution. The contrary 

 takes place when ß^ = — O.l, or generally when ß^ lias a negative value. We 

 may conveniently, using an analogous nomenclature proposed by Pearson (Math. 

 Contrib. I 1894), call ß^ the excess of the frequency cimw. 



In the simplest cases — and also the most usual ones — the coefficients ßg 

 and ß^ are sufficient to characterizise the frequency curves, naturally together with 

 the mean {h), the standard deviation (a) and the coefficient A^^ ((J.„), which latter 

 equals the area of the frequency carve. 



The equation (1) of the frequency curve being found it is easy to calculate 

 the values of the mode and the median, which are sometimes used. For the mode, 

 which corresponds to the maximum value of the frequency, we obtain the equation 



0 = F' = A, < + ^3 f^' [x] + A, f (x) + , . . 



If J 3 and A^ are small quantities, as is here supposed, the value of x — b 

 satisfying this equation must be small. We obtain the following equation for the 

 coordinate — x^ — of the mode 



(8**) 0 = - + ßg [3 - (5 ^^,^\ + ß, [~- 15 + 10 - ,e/J + 



• +ß,[-lö + 30,?,^-lö.r,^ + ,e,6J + ..., 



where 



Xy — h 



Retaining only the terms of lowest order, we hence obtain 

 or, if ß^ be neglected, 



(8*) = + 3 aßg. 



A more accurate value is easily obtained from the above equation (8**). 

 The formula (8) may suffice for a general discussion of the position of the mode 



I will also, I'or the sake of hi'evity, call ß.j the skewness of the I'requenc.v curve. 



