498 S. E. LURIA AND M. DELBRtlCK 



We thus obtain 



(7) t - to = ln(NtCa). 



Equations (6a) and (7) may be combined to eliminate t — to and to yield 

 a relation between the observable quantities r and N t on the one hand and 

 the mutation rate a on the other hand, to be determined by this equation: 



(8) r = aNJn (N t Ca). 



This simple transcendental equation determining a may be solved by any 

 standard numerical method. In figure 1, the relation between r and aN t is 

 plotted for several values of C. 



I .2 .5 12 5 



Figure i. — The value of aN t as a function of r for various values of C. The upper left hand part 

 of the figure gives the curves for low values of aNt and of r on a larger scale. See text. 



Estimates of a obtained from equation (8) will be too high if in any of the 

 experimental cultures a mutation happened to occur prior to time to. From the 

 definition of to it will be seen that this can be expected to happen in little more 

 than half of the cases. 



While we have thus obtained a relation permitting an estimate of the muta- 

 tion rate from the observation of a limited number of cultures, this relation is 

 in no way a test of the correctness of the underlying assumptions and, in par- 

 ticular, is not a test of the mutation hypothesis itself. In order to find such 

 tests of the correctness of the assumption we must derive further quantitative 

 relations concerning the distribution of the number of resistant bacteria and 

 compare them with experimental results. 



