292 AN INTRODUCTION TO MODERN GENETICS 



to the average value of the fitness {x^^ to give the "expected" fitness of 

 that genotype. The increments for the genes B and h must clearly be 

 qa and — /)a, so that they will cancel out for the population as a v^^hole. 

 The genetic variance is defined as the mean value of A^, and Fisher 

 shows that this is the same as the mean value oi Xx, i.e. the product of 

 the expected value and the actual value. Now we have expressed the 

 average difference between individuals with B and fc as a , so that to 

 find the value of Xx we have to consider 



p individuals with X = qa and which together have a mean 



x = x^^-\- I a 



and 



q individuals with X = — pa and which together have a mean 

 x= x^^ — i a 



So if we measure both X and x from the average value of x as origin, 

 v/eget Xx = p .qa .^ a -{- q .(— pa) .(— i a) = pqaa, which is the 

 same as the rate of increase of fitness derived above. 



This theorem, of course, can only be regarded as an abstract state- 

 ment of one of the elements in a normal situation. It takes no accoimt 

 of new mutations or migration. It is a statement of the relation between 

 natural selection and variance, the other factors being disregarded in 

 a population which is not in equilibrium with its environment. 



6. Selection in Finite Populations 



According to the Haldane equations, all populations would fairly 

 rapidly get into an equilibrium state balanced between selection and 

 mutation. Thus no evolution will occur unless the environmental 

 conditions change or new genes appear. Completely new genes prob- 

 ably occur very seldom, and the Haldane equations probably find their 

 main appUcation in cases where an advantageous gene has been kept 

 rare by some mechanism, and is then released for selection to start 

 working on. This might happen, for instance, if the gene was originally 

 harmful but became advantageous through some change in environ- 

 mental conditions. 



There is another mechanism, perhaps more important, which can 

 hold back a favourable gene; that is random extinction in a finite 

 population. A certain nimiber of the genes are formed continually by 

 mutation, but if the population is fairly small and is subject to a high 

 death-rate, there is a considerable chance that, after one mutation to 



