D. E. Lea and C. A. Coulson 269 



/ x x 2 \ m ( x x 2 \ 2 m m 2 ,._. 



,. e . of ^ + (_ + _ + ..J^_ + (_ + _ + ...j«--_ + .... (16) 



Comparing with (11), it is evident that C jr is the coefficient of x r in the expansion of 



(JL+^L+.Y. (17) 



\1.2 2.3 / 



It will be shown later that Cj r is the probability that a culture in which exactly./ muta- 

 tions have occurred shall contain r mutants. If we define 



A>= s c itTt 



r=j 



Dj r is the probability that a culture in which exactly j mutations have occurred shall 

 have ^r mutants. Summing equation (9) over all r between ^ and r leads to the following 

 recurrence relation between the D i<r , 



(r+fiC^JiDj.^-D^), (18) 



or (r+jjD^rD^+jD^^. ^ (19) 



Equation (18) is useful as an arithmetical check during the computation of the C jr by 

 means of equation (9). 



It is sometimes convenient to discuss P r denned as 



P r =Z Pr , (20) 



r=0 



P r being the probability that a culture shall contain 0, 1, 2, . . ., or r mutants, i.e. any number 

 of mutants up to r. P = e~ m ; P r =l for all r at m = 0. The following equations can be 

 readily deduced by summing (5) and (11) over all r between and r: 



p ^ + iM^j)- 



(22) 



Second method 



An alternative method of calculating p r , the probability of a culture having r mutants, 

 is instructive. A mutant appearing by mutation any time after the culture has passed 

 the size \n will not have time to divide by the time the culture size reaches n, and will 

 therefore contribute 1 to the final complement of mutants. A mutant appearing by 

 mutation during the period in which the culture grows from \n to \n will have time to 

 divide once only and will therefore contribute 2 mutants to the final complement. A 

 mutant appearing by mutation in the period in which the culture grows from \n to \n will 

 have time to divide twice only and will contribute 4 mutants to the final complement, 

 and so on. 



Confining attention for the moment to all those cultures in which exactly one mutation 

 occurs during the growth of the culture from one organism to n organisms, in one-half of 

 the cultures the mutation will occur while the culture is growing from \n to n, in one-quarter 

 of the cultures the mutation will occur while the culture is growing from \n to \n, and so 

 on (compare equation (4)). Or, in other words, in one-half of these cultures the mutation 

 occurs at such a time that it gives rise finally to 1 mutant, in one-quarter of the cultures 



29 



