464 MR ROBERT WORTHINGTON. 



view, as affording a means of directly testing the homogeneity 

 of any material yielding a skew frequency curve. 



In a note at the end of the memoir, the author mentions that 

 he has been able to split up the frequency curve for the 

 cephalic index in the ' Eow-Graves ' skulls into two components, 

 one of which is practically identical with the curve for the 

 skulls of modern South-Germans ; thus sliowing that the 

 theory that an evolution has taken place in this index of the 

 skull of South-Germans since primitive times is baseless. 



In the memoir on ' Skew Variation in a Homogeneous 

 Medium,' a generalised curve is obtained which expresses all 

 known groups of homogeneous statistics. To quote from the 

 abstract in the Proceedings : — 



" In the deduction of the normal curve of frequency from the 

 symmetrical point binomial, three conditions are usually 

 assumed to be true : — 



{a) The chances of any ' contributory cause ' giving its unit 

 of deviation in excess or in defect are presumed to 

 be equal. 



(&) The number of ' contributory causes ' are supposed to be 

 indefinitely great. 



(c) The 'contributory causes' are all supposed to be inde- 

 pendent. 



(c) amounts to the assumption of the binomial form {p + qy\ 

 (a) to the equality of p and q, (&) to the indefinitely great value 

 of n." 



If all three of these conditions hold, the curve is given by 



If {a) is removed we require a more generalised form (p + qY, 

 which is skew. 



If (c) is removed the series required is known as a 'hyper- 

 geometrical series.' This bears the same relation to the 

 generalised frequency curve as (i + i)" to the normal curve. 



Frequency curves are divided in this paper into five distinct 

 types, depending on their symmetry or skewness and their range 

 (limited or unlimited in one or l)Oth directions). Type IV. is of 

 special interest to us here. Professor Pearson says, " But so 

 far as I have gone, in both anthropometric and biological 



