D. E. Lea and C. A. Coulson 



273 



>64 mutants into further classes, e.g. 65-128, 129-256, 257-512 mutants, etc., would 

 involve an impracticable amount of arithmetic. An attempt was therefore made to find 

 asymptotic formulae for p r or P r valid for large values of m. We have not succeeded in 

 finding explicit formulae, but have obtained some information on the form of the function. 

 If we consider P r as a continuous function of the two variables r and ra, then for values 

 of r>l we have approximately P r — P r _ 1 = dP r /dr. Thus equation (21) approximates to 



dm \ ml dr 



which is satisfied by 



K) 

 ■"I 1 



= 0, 



(24) 



•). 



(25) 



where F is any function. 



In Fig. 1 we have plotted P r (derived from Table 2, i.e. based on the recurrence relation) 

 against (r/m — log m) for r = 8, 16, 32 and 64, using five values of m (viz. 4, 6, 8, 13, 15) 



10 



r/m -log m 



Fig. 1. P r , for different r and m, is a function of rjm -log m. The points are plotted for 

 r = 8, 16, 32 and 64, and with wi=4, 6, 8, 13 and 15. 



selected so that the twenty points are conveniently spaced. It is seen that the points lie 

 quite well on a single curve, showing that these values of r are large enough for equa- 

 tion (24) to be a satisfactorily close approximation to equation (21). The smooth curve 

 in Fig. 1 is thus a graph of the function F which enters into equation (25). 



For any given value of m, {r/m — log m) is evidently distributed in a skew distribution 

 about a median 1-24. We have found by trial that the derived variate (r/m -log m + 4-5)- 1 

 is distributed in a distribution rather closely approximating to a Gaussian distribution of 

 standard deviation 0-086. This is shown by the closeness with which the points in Fig. 2 

 lie on a straight line. The points in Fig. 2 are derived from those of Fig. 1 by transforming 



the ordinates to probits, defined by the relation 



V(2 



L_ [ v ~ 



e~ iyi dy = P r , where y is the 



probit corresponding to P r . (Tables of probits are given in Fisher & Yates, 1938.) Also, 



19-2 



33 



