Mathematical Contributions to Theory of Evolution. [Jan. 24, 



of arguments having more special practical interest.* The theory of 

 the whole system of skew curves and their limiting cases is then 

 discussed. 



The author regrets that while he has obtained a criterion for each 

 species of skew curve, he has hitherto failed to find one which will 

 distinguish a compound curve, e.g., heterogeneous material, from a 

 skew curve resulting from skew variation in homogeneous material. 

 He does not despair, however, of ultimately finding such a criterion. 

 The test of actual fitting is generally sufficient, but is, of course, 

 laborious.f 



PART II. Illustration. 



The second part of the memoir provides the minimum of illustra- 

 tion, which the author considers absolutely necessary, in order to 

 demonstrate that the generalised curves reached are capable of the 

 widest application to every variety of practical statistics. The 

 illustrations show that from the slight amount of skewness usually 

 neglected by statisticians although of vital import when we come 

 to consider variation with growth, as in statistics of child-variation 

 with growth even to extreme cases in which the curve is asymptotic 

 to the ordinate of maximum frequency, a good fitting generalised 

 frequency curve can be found. Although the number of illustrations 

 is considerable, and is only a part of the author's material, yet he 

 hesitates at present to make any dogmatic statements with regard to 

 the relations of skewness in variation to secular evolution ; but he 

 believes that the persistent recurrence of certain types of curves in 

 zoometry and of certain directions of skewness in anthropometric 

 statistics will be found, as sufficient material accumulates, to justify 

 broad generalisations, although at present they can only be treated as 

 suggestions for further investigation. 



The special illustrations given are : barometer variation (Venn), 

 variation in crabs and prawns (Weldon), in height of American re- 

 cruits (Baxter), American school girls (Porter), in length-breadth 

 index of Bavarian skulls (Banke), frequency of enteric fever (Metro- 

 politan Asylums Board), guesses at tints (Gresham College), divorce 

 statistics (Willcox), variation in house- value (Goschen), variation 

 in buttercups and clover (De Yries), variation in pauper percentages 

 (Booth), and resolution of the English male mortality curve (Ogle) 

 into skew components. 



The memoir concludes with some general remarks on the modifica- 

 tions required in the theory of correlation by the use of generalised 

 curves, but reserves for the present its complete discussion. 



* The British Association have kindly given a grant for this purpose. 

 f It is noteworthy that all cases of compounded a ess dealt with hitherto hy the 

 author give 2^ (3/i 2 2 /tt 4 ) +3/u 3 2 positive., 



