542 



SCIENCE PROGRESS 



both the coefficients c\ and c 2 become zero. It is the funda- 

 mental differential equation representing Pearson's Generalised 

 Probability Curve. 



We are dealing here only with the reasoning by means of 

 which this generalised equation has been obtained, but its 

 further development may very briefly be noticed. The con- 

 stants a, Co, c\ and c 2 must be given numerical values depend- 

 ing on the particular frequency distribution to which the 

 equation is to be fitted, and this is possible by applying the 

 " method of moments," a method by means of which Pearson 

 replaced the " method of least squares." Let the following 

 graph represent a frequency distribution : 



0C.3, Xjj. DCg- 0C 6 



DC 



Fig. 



It is supposed that we are dealing with a numerous series of 

 observations which have been grouped into thirteen classes 

 represented by the rectangles. We further suppose (for sim- 

 plicity) that the distribution is a symmetrical one. The mean 

 is at co. All the observations between # = 0-5 and x = i'$ are 

 contained in the rectangle y 1} and so on. The dotted lines 

 represent mid-ordinates, and we suppose that the area of each 

 group is represented by the product of the mid-ordinate and 

 the base on which it stands. The continuous curved line 



