ACTIVE FROM INACTIVATED BACTERIOPHAGE 125 



Xj = X— (2) 



where x = multiplicity of infection in the total population, /. = arithmetic mean 

 of the B distribution (average bacterial length). 



The frequency function of the distribution of phage particles per bacterium 

 in the total population (P distribution) is 



f{k) = E n^fm = Z -^—7^ (3) 



i=o j=o k ! 



where Uj is the frequency of the ^th subpopulation in the E distribution 



00 



j=o 



The Poisson distribution of phage particles per bacterium within each homo- 

 geneous subpopulation of bacteria has the following first two moments: 

 First moment about the origin = (;Ul')J■ = a;> 

 Second moment about the mea.n = {1x2) j = Xj 

 For the P distribution in the total population, remembering formula (3), we 

 have a mean 



00 =0 00 00 00 00 



(mi')p = I] ^ D ^hfjik) = S %Z1 k/Ak) = Jlnj{ni)j = X) fhXj-, 



k=0 i=0 j=0 k=0 j=0 3=0 



from formula (2) we obtain: 



(mi')p = X. (4) 



To find the variance (m2)p of the P distribution, let us remember that, in 

 general, 



M2 = M2' — (m/)" (5) 



where M2' is the second moment about the origin. We have then: 



CO GO CO 00 CO 



k=0 ;=0 j=0 A-=0 ;=0 



where (m2')7 is the second moment about the origin of the distribution of phage 

 particles per bacterium in the jth subpopulation. But (M20y = ^y+(^j)^ (since, 

 in general, M2' = M2+(mi')^)> and, for each homogeneous subpopulation, 

 (m2);= itii)} = Xj. We obtain, therefore: 



(m2')p = Z ^jMi = Z «;•»;• + H nj{x,)\ (6) 



Finally, introducing the valaes obtained from (4) and (6) in formula (5) : 



00 QO 



(m2)p = E ^jXj + E ^;(^y)^ - x^- (7) 



y=o y=o 



The first term on the right side of equation (7) equals x\ the other two terms 

 equal the variance of the B distribution, expressed as function of x. We obtain 

 therefore, 



(m2)p = a; + var (B). 



279 



