354 On the Error of Comiting with a Hamiacytometer 
is never as much as 1, bemg- about '8 for the term 1000 e ^ —. which is 175 5 
° 0 ! 
+ 
against 176'3 from the binomial. 
Diagram I compares 1000 e ' ( 1 + 5 + + ... - 
1000 + Jy)"" which of course differ, but not by very much. 
DiAGEAM I. Comparison of the exponential and binomial expansions. 
with the binomial 
5'- 
Firm line represents 1000 e ' |1 + 5 + ... + j^-j + 
(19 1 ) 1'^" 
Broken line represents 1000 ^Oy 
etc. 
180 
170 
160 
150 
140 
130 
120 
no 
100 
90 
80 
70 
60 
50 
40 
30 
20 
10 
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 
In applying this to actual cases it must be noted that we have not taken into 
account any " interference " between the particles ; there has been supposed the 
same chance of a particle falling on an area which already has several particles as 
on one altogether unoccupied. Clearly if m be large this will not be the case, but 
with the dilutions usually employed this is not of any importance. 
It will be shewn that the actual distributions which were tested do not diverge 
widely from this law, so we will consider the probable error of random sampling on 
the supposition that they follow it. 
We have seen that ^u, = m. 
Hence the standard deviation = ^Jm. 
