MATHEMATICAL THEORY 29 



APPENDIX. 



MATHEMATICAL THEORY OF THE EFFECT OF A UNISEXUAL TENDENCY. 



The statistical theory on which the preceding research is based, being 

 presumably susceptible of other applications than that here made, will 

 now be developed. So far as generality is concerned, nothing will be 

 lost by taking the special problem, considered in section IV preceding, 

 as a basis of investigation. The data of the problem will be as follows : 



1. An indefinite number of pairs of parents, each pair of which may 

 have an indefinite number of children of either sex. The treatment 

 of this subject will include the general case of an indefinite number of 

 causes, each of which may, on each trial, be productive of one or the 

 other of two different effects. 



2. Taking the general average of the whole mass of couples, there is 

 a certain normal probability, p, that a child, taken at random, will be 

 male, and the probability 1 - - p that it will be female. 



3. It may be that this probability is the same for every individual 

 couple of the whole mass. But it may also be that, for some of the 

 couples, the probability is greater than p. In this case it will neces- 

 sarily follow that for certain other couples the probability is less than 

 p, the latter quantity being the average for the whole mass. 



4. In order not to complicate the problem too greatly, we shall sup- 

 pose that each of the individual couples belongs to one of three classes; 

 a class for which the probability of having a male child has the normal 

 value p, another for which it is greater than p by an unknown quantity 

 a, and a third for which it is less than p by the same quantity. We 

 designate these classes by A, B and C; A representing couples with 

 probability p -f- a; B, those with probability p; and C, those with the 

 probability p a. The numbers of classes A and C are necessarily equal. 



Let us put: 



Jt . the fraction of the total belonging to the two equal classes A and C ; 



li' . the fraction of the whole mass belonging to class B. 



We shall then have 



// +h' = -- 1. 



Proceeding according to the method of probabilities, we suppose a 

 parent couple taken at random from the mass. The respective proba- 

 bilities that this couple will belong to the classes A, B and C are 



1/2/1, li and i/oft. 



The probabilities of a male child are, in these several classes : 

 For class A, p -\- a. 



B, p. (1) 



C, p a. 



