MATHEMATICAL THEORY 31 



and taking the sum of the products, we find the probability that a 

 family of n children taken at random from the whole mass, will com- 

 prise s males and r females to be 



)'(0-a)' + C^-a)'(gr + a)'J + A>v}['J] (4) 



This expression may now be developed in even powers of a, the coeffi- 

 cients of the odd powers all vanishing. In the form 



the values of the first two coefficients are 

 A =(h + h')p s q r =PY 



For our present purpose these terms suffice. To investigate uni- 

 sexual deviations it will also lead to no appreciable error to suppose 



P = f l = V-2 



The value of P ( r a \ , that is, the probability that a family of n children 

 will consist of s males and r females now becomes 



We may use this formula to express the probability in question for 

 the case of a family of any number of children, distributed in any way 

 among the two sexes. We shall now form these expressions for fami- 

 lies of various numbers of children. In doing this families in which 

 the sexual distributions are the reverse of each other will be combined. 

 For example, the equal probabilities that a family of five will be wholly 

 male and wholly female will be added into one sum; as will the proba- 

 bilities of 4 of one sex and 1 of the other, whichever sex it may be. 



The pair of probabilities thus combined would be rigorously equal 

 when, and only when, there is an equal probability of male and female 

 children. But not only is the error involved in the assumption of 

 inequality unimportant for the present purpose but, resulting as it does 

 in giving too small a probability for a preponderance of male and too 

 large for the preponderance of females, it is nearly self-compensatory 

 when we combine families of inverselv distributed sexes. 



