D. E. Lea and C. A. Coulson 285 



APPENDIX. (ByC.A.C.) 



(1) It has been suggested that a table of the individual coefficients C jr introduced in 

 equation (7), and which give the expansion of q r in powers of m, might be useful. Such 

 a table, for r^ 10, is shown below. 



Table of C^ 



r mi* 



According to equation (7), q r 2 C jr — 

 j=i ' J ■ 

 r\j 12 3 4 5 6 789 10 



1 I 



2 i | 



q _1_ i 'A 



O 12 6 8 



V A _!_ A i -A_ 



* 20 8 8 18 



5 To e"o 4¥ n 3~2 



ft J_ , «1 , 181 . _S_ -5- _i_ 



u 42 720 2180 12 96 84 



7 -JL. 109 9 7 41 3 5 1 1 



' 58 2520 1440 540 578 32 128 



8 1 853 55 1 17 3 107 1 _7 1 



72 25200 10080 2692 1728 24 384 256 



Q -A- -AS- 13579 1 313 307 203 _7_ _1_ 1 



y 90 700 302400 22880 5184 4320 256 96 512 



(2) It should perhaps be pointed out that the replacement in (3) of n — 1 by n is an 

 approximation whose effect is quite negligible provided that r<^n, as occurs in all 

 experiments. In fact, even for r of the order of n-, the values of q r are seriously in error. 

 As a result of this, and of the fact that it allows r to exceed n (which is manifestly 

 impossible since r is the number of mutants and n is the total number of bacteria), the 

 generating function (15) actually gives an infinite value for all the moments. These two 

 difficulties have been removed in a development of this theory, to be published by Mr 

 D. G. Kendall, of Oxford. But unfortunately his more strictly correct generating function 

 cannot be expanded with any ease to determine the q r . Except for large r or small n, 

 however, it differs insignificantly from our (15). 



(3) Mr Kendall has kindly pointed out to me that the argument in (31) and (32), 

 which was copied from Luria and Delbruck, is not quite valid. For in (32) the complete 



series - + - + — + . . . is not convergent when x = 1 , and in order to get an expression for 



Z O 4: 



the mean value and the variance it was necessary artificially to curtail this series by 

 truncating it at its term x n /n. This device is not a valid procedure, and it appears that 

 although there is no change in the mean r , the variance a 2 of an individual determination 

 of r requires to be multiplied by 2, so that the correct relation a 2 = 2mn. 



45 



