K. Pearson 
87 
large, there is no sensible difference. Since y x c gives approximately the ordi- 
nates of (1 + 1)^ , we may expect 
2^y(p!)- 
(2p)! 
y'^c to give the terms of (ii). 
Or, supposing p moderately large and nsing Stirling's Theorem, we may 
anticipate that the terms of (ii) will be given fairly well by 
F - -L ^= e " 21^-^ (iv). 
This is a normal curve of standard deviation V^pc, and accordingly it must 
again be closely given by the binomial + \)'^. I find that the areas of (iv) 
give (ii) closely, but that if we work with ordinates we get a better result from 
^1(^4- I), c as the standard deviation of the normal curve, i.e. neither using 
nor p + but the mean of these values. The following table illustrates the 
closeness of approximation for ^ = 12: 
Normal curve 
Accurate values 
Areas 
Ordinates 
from (ii) 
(o- = n/i7; = 1 •22475) 
[o-=.N/i(p + i) = l-25] 
•0000 
•0000 
•0000 
•0000 
•0001 
•0000 
•0001 
•0001 
•0016 
•0000 
•00-20 
•0019 
•0179 
•0156 
•0185 
■0179 
•090G 
•0937 
•0897 
•0887 
•2320 
•2344 
•2306 
•2318 
•3156 
•3125 
•3181 
•3191 
•2320 
•2344 
•2306 
•2318 
•0906 
•0937 
•0897 
•0887 
•0179 
•0156 
•0185 
•0179 
•0016 
•0000 
•0020 
•0019 
•0001 
■0000 
•0001 
•0001 
•0000 
•0000 
•0000 
■0000 
It is clear that for all practical purposes the normal distribution, and even 
the simple binomial, may replace the expression in (ii) for the distribution of 
determinants in the heterogenic chromosome when p is as large as 12. The 
errors of the actually available samples, 500 to 1000 at most of each generation, 
will be well within the errors of random sampling. Even if p = 4, the agreement 
between (ii) and {\ + \Y is not wildly out : 
■0143 0000 
•2286 ^2500 
•5143 ^5000 
•2286 -2500 
•0143 -0000 
