332 On the Mathematical Theory of Evolution. [Nov. 16, 



normal curve by fitting any series of observations by aid of the area and 

 the first two moments (i.e., the first moment gives the mean, and the 

 second the error of mean square) is justified. The method leads to 

 what is termed the fundamental nonic, every root of which gives a 

 real or imaginary solution of the problem. The best solution is 

 selected by the criterion that it gives the closest approach to the 

 given frequency curve in the value of the sixth moment. l?rom the 

 nonic is deduced a quadratic for the areas of the components corre- 

 sponding to each solution. If both roots of this quadratic are real 

 and positive, we have either a mixture of two heterogeneous specks, 

 or evolution is breaking the homogeneous material up into two 

 families of different magnitudes, different means, and different 

 standard deviations from the mean. If one root of the quadratic be 

 real and positive, and the other real and negative, we have evolution 

 destroying a certain percentage round a certain mean out of an 

 initially homogeneous and normal group. 



Should one of the standard deviations be imaginary, we get the 

 percentage of anomalous and irregular measurements in a homogeneous 

 group. 



6. Class c. The solution here is unique and depends upon the 

 equality of the areas and of the first six moments ; for all odd 

 moments vanish, and we have four quantities to determine, i.e.. the 

 percentages of each group and their standard deviations. The solu- 

 tion depends on a quadratic for the areas, and the same remarks apply 

 as to the quadratic for Class &. 



7. Rules are given for detecting whether we have a mixture of two 

 groups, or whether a differentiation into species of a homogeneous 

 material is going on; and also rales for measuring the amount of 

 asymmetry which is to be considered significant. The former rules 

 are, briefly : 



i. Select the most asymmetrical curve out of the curves for the 

 organs measured ; dissect it into two curves or groups by the 

 method for Class Z>. 



ii. Select the most symmetrical curve out of the curves for the 

 organs measured and dissect it into two groups by the method 

 for Class c, or, if it have significant asymmetry by the method 

 for Class 6 again. Then (a), if the first dissection is possible 

 and the second is not, a real evolution is going on ; (/3), if the 

 first dissection is possible and the second is possible, and both 

 groups give sensibly the same percentages, we have a mixture 

 of two heterogeneous materials and no true evolution, unless 

 the organs be so closely allied that one must vary directly with 

 the other (e.g., length of right and left legs) ; (7), both dis- 

 sections are possible, but give groups with different percent- 

 ages; we have both organs evolving differently at the same time. 



