R A. Fisher, H. G. Thornton, and W. A. Mackenzie 835 



the first factor represents the chance of obtaining a given value of x, 

 and the second, which does not involve di, gives the chance that the 

 sample shall show any particular partition of the total, once the total 

 is fixed. The distribution of any statistic which depends upon this 

 partition, must therefore be independent of m, once x is fixed. The 

 problem of the distribution of v is therefore susceptible of the great 

 simplification, that we need only consider its distribution for given values 

 of X, and that this distribution is wholly independent of m. 



The distribution of this, or any other, statistic, which depends upon 

 a partition of an integer, must necessarily be discontinuous; when, 

 however, .r is large, even for small values of n, the number of possible 

 values of v becomes sufficiently great for its distribution to be represented 

 by a frequency curve. This procedure is the more advantageous in that, 

 by the choice of a new statistic, which shall replace v, we can throw the 

 distribution into a form independent of x, whereas the actual partitions 

 possible in the neighbourhood of equipartition, will necessarily change 

 with the fractional part of ,r. 



The frequency with which any given partition of the total, nx, occurs, 

 is in fact the frequency with which any given series of values are obtained 

 when the total is distributed at random into n cells, the expectation 

 in each being x. It is well known that when this is the case, the 

 statistic 



^ X X 



measures the departure of the sample from equipartition, being equi- 

 valent mathematically to Pearson's test of agreement between observa- 

 tion and expectation. The distribution of ^x^ is well represented by a 

 smooth curve independent of x of the form (Pearson's Type 3) 



1 T' 



^f=;r~~T^ ^~'^^' 



and the frequency with which x^ exceeds successive integral values, has 

 been tabulated by Elderton (4, 1902 and 5, 1914) for values of n from 

 to 30. 



We are therefore in a position to test whether the conditions which 

 lead to the Poisson Series are in fact fulfilled in any given body of 

 bacterial data for which the counts on individual plates are known; it 

 is only necessary to calculate the above index of dispersion (x^) from each 

 set of parallel plates, and to determine whether the distribution of this 



