268 Distribution of numbers of mutants in bacterial populations 



do r yy\) ~ 



we have j- L = I! C jr —. — — . (8) 



dm j=l hr (j-l)\ 



Inserting (7) and (8) in (6), and equating coefficients of mP^fj !, we have 



ti+r) C j>r =jC,-i,r-i+(r-l) <W (9) 



With this recurrence relation a table of C jr may be drawn up. Such a table, for values 

 of r ^ 10, is given in the Appendix. 



Evidently C ^ = 7^T)> C r,r = 2 ~ r - ( 10 ) 



From (6) and (7), p = e- m , and for r^ 1 



fr=:£C, >r (^^ (11) 



A generating function for p r 



oo 



Define a function f (x, m) = q + q 1 x + q 2 x 2 + ... = 2 q r x r . (12) 



r=0 



We have %=I,raf- 1 q r and J^ = Zaf ^ . (13) 



ox dm dm 



Multiplying equation (6) by x r and summing for all r we have 



Saf 4 1 + ~ 2> x r ~ x q r = x Saf- 1 a,, + - 2(r - 1 ) a;'- 2 ? r _ ,, 

 dm m 2r a 1 m 



9/" a; 3f a; 2 3/* 



or, using (13), -*- +-J- = x f+- -f, 



cm mox mox 



whence -?- + — (1 -a;) -^- = x, (14) 



dm m dx 



where* </> = log/. This equation is satisfied by 



<f> (x, m) = mifj (x), 

 providing ip + x (l—x) tfj' = x, 



i.e. + / . =- . 



x(l—x) l—x 



Multiplying by x/( l—x) and integrating 



rb^iV 108 * 1 -*'- 1 - 



the integration constant —1 being introduced since when a: = 0, f=q = l, so that ^> = 

 and so i/r = 0. Thus 



0=1+ __i og(1 _ x)= _ + _ + __ + ..., 



whence /= e w, A = e m ( 1 - x) m &-*te, 



so that p r = e~ m q r is the coefficient of x r in the expansion in ascending powers of x of 



of e -.«exp[,„( I ^ + -^ + ...) 



(l-i)" 11 -* or 



* log moans natural logarithm to base e = 2-718... throughout this paper. 

 28 



(15) 



