66 The Ohio Journal of Science [Vol. XVI, No. 2,. 



The Origin. The generating curve is the symmetrical normal 

 probability curve with origin at its center. Since x = when 

 t =0, the origin of the translated curve coincides with that of the 

 base or generating curve. The translated curve may not be- 

 symmetrical so that the mean ordinate may not coincide with 

 the modal ordinate. Because of the relation between corre- 

 ponding areas the ordinate at the origin must continue to- 

 divide the area under the curve into equal parts, that is, the: 

 origin and median always coincide. 



Determination of the Constants. Since the exact position 

 of the median can not ordinarily be determined by inspection or- 

 direct computation there are in reality four constants to be: 

 determined: the distance between the median and the mean,, 

 a, K and X. 



In determining the constants it is usual to compute the- 

 value of the first four moments. The third and fourth moments- 

 are extensions of the idea of the well known formulas for the 

 first and second moments. Denoting the moments about the- 

 median by ju, we have 



1 r+°° 



^^' = N J _. ""^^ 



N 

 Ma 



1 r+°° 

 1 r+'" 

 1 r+°° 



where N is the total area under the curve. 



The values of the yu's are computed from the data* and' 

 equated to the corresponding integrals which of course involve - 

 the four constants. In this way four equations are obtained 

 from which the values of the constants may be determined. 

 Since it is our present object to discuss the solution only of these- 

 equations, merel}' the princi])al results will be given. 



*Elderton, 1. c. 



