28 



C. V. L. Cliarlier. 



4:o. The quantities X and to are so determined that B^ — B^ — O, whereas c 

 is chosen arbiträr ii t/. 



The method 3:o may seem to he the best one, but has the iucouvenience of 

 giving to (0 very smaU values and to X very large ones, when v.^ is vanishing. Hence 

 it is not applicable when the curve differs little from the normal-form. The 

 following method seems to have a general applicability and has also the advantage 

 of a certain similarity with the process used for the curves of type Ä. 



We begin with choosing a determinate value for the quantity c. In many 

 cases it ivill lie found convenient to identify c ivith the al>scissa of the discontinuous 

 end of the frequency curve. 



When the value of the quantity c is determined (and it must be borne in mind 

 that this determination is to a certain degree arbitrary) we dispose of X and oi in 

 such a manner that the coefficients J?^ and B.^ vanish. According to (14) we 

 thus get the equations of condition 

 K24^ 0-=XtoiJ„" — [x/', 



^ ^ o = x%3^v'-(2>^+i)^"ijV' + iV'- 



For solving these equations we observe that the moments (j-.", which are taken 

 about the point c, may be expressed through the moments \),, about the mean. 

 We have indeed approximately: 



IJ." = ^ + (Î) - 0) [x.-, + {,) {t, - c)^iJ.,_. + . . . 

 As [x.^ = 0 we thus obtain 



[Xj" = (h — c) (X,,, 



= \h + i^' — ^T'lh^ 

 \h" = 1J-.3 + 3 (!> — c) [x^ -f [Ij " (;)''ix,„ 

 \h" = \h + ^ — \h + 6 (?^ — cf[i'2 + - O'hp 



Substituting these values in (24) we get the following values of X and co: 



{b-cY 



(25) 



X 



where '^^(=v^,) signifies the standard deviation. 



As to jBg and B^ they now assume the values: 



li-B, = Bo — 3vi + 5üj^ — Gtovg] 

 Hence we may write the frequency curve in the form 



(27) F[xc. + (0 = ''j ['I' i^) + T3 + 'U 'I' + ■ 



