HEREDITY 



509 



but nearer one extreme, then the terms of the binomial should 

 be numerically unequal * (B + 2 R ; ^ + | ; etc.), a case which 

 would fit our illustration had the number of R females been 

 twice the number of B females. 



The ease with which all distributions can be fairly well 

 " fitted " shows beyond a doubt that, even with selection and 

 infertility at work, the final result is largely such as would arise 

 from independent probability, a fact which goes to show that 

 problems in heredity are essentially statistical problems. 



The hopeless tangle in which characters soon become involved 

 through bisexual reproduction shows the utter futility of 

 attempting to infer anything whatever from individuals, and 

 the almost mathematical certainty of being able to detect 

 almost any principle or law of descent by careful study of 

 entire populations. 



1 For the convenience' of the student the formula for expanding a binomial 

 to any power is given here. It is 



(A 



1-2-3 



i + B". 



1-2.3.4 

 This formula gives 



(A + BY = A* + 2 AB + B* 



(A + B}* = A* + 3 A*B + 3 AB 2 + B* 



(A + )* = A* + 4 A*B + 6 A*B* + 4 AB* + B*> 



(A + )* = A* + 6A* + isA*ff*+ 20 A*B* + 15 AW + 6A& + B* 



(A + B)* = A* + 8 A*B + 28 A*B* + 56 A*>B* + 70 A*B* + 56 A*B*> 



Thus in all cases the coefficients form a series like a symmetrical frequency 

 distribution. If, however, the second term be taken as 2B, then the coefficients 

 will be substantially altered, forming a skew. 



Karl Pearson has well established the fact that frequency distributions obtained 

 experimentally can often be fitted better by the terms of the binomial (A + BY 

 when n is not restricted to be a positive integer. In this case the expansion does 

 not terminate, but takes the general form 



to infinity, in which the general, or rth, term is 



n(n - i)(n - 2) (n - r + 2) - 



