.258 Prof. Karl Pearson. Mathematical [Jan. 24. 



If a be the total frequency, and jm r * the rth moment of the frequency 

 curve about its centroid vertical, then for this curve 



This relation must be satisfied or nearly satisfied if a series of obser- 

 vations or measurements is to be fitted with the skew curve, which is 

 related to the skew point-binomial as the normal curve to the sym- 

 metrical point-binomial. For fitting a skew point-binomial we must 

 have 



For the normal curve ^ = 3/t^ 2 . But a great number of statistical 

 returns especially in anthropometry and zoometry give 



Hence they differ from the normal curve in the opposite direction to 

 the skew point-binomial and its corresponding frequency curve. 



After the complete theory of the fitting of skew binomials and 

 this special skew curve has been discussed with examples, the 

 memoir proceeds to the generalisation of the frequency curve by 

 withdrawing the limitation (c) above. Just as the symmetrical bi- 

 nomial and normal curves are illustrated by the tossing of a group 

 of n coins, and the skew binomial and its skew curve by the spin- 

 ning of a group of n wz-sided teetotums, so we can arrive at a series 

 of curves in which the contributory causes are interdependent, by 

 considering the withdrawal of r cards from a pack of ns cards con- 

 taining s suits ; or, again, by drawing a definite amount of sand from 

 ^, vessel containing two kinds of sand. 



For discontinuous series the solution is a hypergeometrical series. 

 If now a curve be formed which is related by the same fundamental 

 geometrical relation to this hypergeometrical series as the normal 

 curve to the symmetrical point-binomial, or the first skew curve to 

 the skew point-binomial, we obtain a generalised frequency curve 

 which contains both those hitherto considered as special or limiting 

 cases. 



It is not suggested that the hypergeometrical series or its corre- 

 sponding curve is the only case in which the a priori condition (c) of 

 dependence of " contributory causes " is replaced by an interde- 

 pendence. But it is suggested that it is one of the most important 

 cases, and one which naturally occurs at the commencement of our 

 investigations. That it is probably quite sufficient is evidenced by 

 the fact that the author has hitherto failed to find any group of 

 homogeneous and skew statistics which cannot be closely expressed 

 by the curves which correspond to the hypergeometrical series. 



