Theory of Chance to Racial Differentiation. 115 



moments, differed very slightly from her's, and made no 

 difference whatever in the final result *. 



There were, including now those of doubtful sex, 1146 

 skulls ; and using the notation of my first memoir on the 

 Mathematical Theory of Evolution (Phil. Trans, vol. 185. 

 pp. 81-83), the following were obtained : — 



Mean Index = 79*1824. 

 Moments: /x 9 = 19*531932, 

 ^= 35-1(30070, 

 ^=1136*6665, 

 ^ 5 = 5713-07080. 

 Other Constants : \ 4 = 21*5229, 

 X 5 = 3466*2776. 



Hence we have the nonic : — 

 ^ 2 9 -28-6667^ 2 7 + l,854:-345^ 2 6 -121,618-21/? 3 5 -l,188,506-5p 2 4 

 4-16,841,221-5^ 2 3 + 150 ; 411 J 150'2 j? ? 2 2 + 50,071,4. c )6-6p 2 

 -1,889,286,562-1 = 0. 

 Sturm's Functions gave the following results : — 



Sign- 

 i Changes. 



Pz- 



/(*). 



AH 



/*(*)■ 



/.(*)■ 



/*(*)■ 



4- 



/•(*)■ 



3 



— oo 



_ 



+ 



_ 



_ 



+ 



4- 



3 







— 



— 



4- 



+ 



— 



4- 



4- 



3 



1 



— 



+ 



4- 



+ 



— 



4- 



+ 



3 



2 



— 



+ 



+ 



+ 



— 



4- 



4- 



3 



3 



— 



+ 



+ 



4- 



— 



4- 



4- 



2 



4 



4- 



4- 



— 



— 



+ 



4- 



4- 



2 



+ 0O 



4 



+ 



+■ 



~ 





4- 



4- 



Thus there exists only one real root and it lies between 3 

 and 4. Its value was easily determined to be 3*0443. Whence 

 the other constants : — 



p,=S'M65, p x = 2-7745. 

 The quadratic for the distances o£ the means ji and y 2 of 

 the two groups from the mean of the mixed material is thus : 

 7 2 -2-7745y4- 3-0443 = 0. 



This has imaginary roots. In other words, no resolution of 



the Parisian skulls into a male and female group is possible 



on the assumptions: — 



(i.) That we are dealing with a homogeneous race ; and 

 (ii.) That the distributions for either sex are given by the 



ordinary curve of errors. 

 * Dr. Macdonnell found X 4 =24o229, X 3 = 3466-2776, against Miss 



Fawcett's X 4 =24'5727, X 5 =3466 , 1421, differences which are of no 



importance. 



12 



