266 PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



We shall deal first with the skew curve of Type III. (''Phil. Trans.' A, vol. 186, 

 p. 373), because its treatment is less complex and leads at once to some general 

 principles which must be borne in mind, whenever natural selection acts upon an 

 organ exhibiting skew variation. 



(15.) Probable Errors and Error Correlations of the Constants of the Generalized 

 Probability Curve of Type y = y l (l -\- - 1 1 e~ yx . 



This is the equation of the curve referred to its mean as origin, where 



7C-OHl)(jj + jy> 



Further, the moments about the centroid vertical are given by, 



p_l 2(p + l) 3(p + l)(p+ 3) 



p.,= -^ , &= -,-- --, /* 4 = r- 



or, 



y = 2p,/^, p - W/W - l ....... (xliii.). 



The criterion for the application of this curve to any frequency distribution is 



2fu (3/v - p 4 ) + 3fi, 2 = 0, 

 or, if we write & = ju, 4 /ju./, ft = pf/pS, 



6 - 2ft + 3/8, = ........ (xliv.). 



Lastly, 



Sk. = the skewness = |^ 3 / (/A 2 ) 3/2 = 



and the modal frequency, 



We require to know the probable errors of p, y, y u y , a- = \/p.>, p 2 , p. 3 , p 4 , and the 

 skewness. We must discover the best physical constants to describe such skew 

 frequencies and we shall at the same time succeed in deducing certain we believe 

 novel properties of normal frequency distributions as limiting cases of this skew type 

 of distribution. 



* See ' Phi], Trans.,' A, vol. 186, pp. 373-4. 



