352 Oil the Error of Counting with a Haemacytometer 
Let us find the limit when N is infinite of the general term of this expansion. 
The (/■ + l)th term is : 
n) • \n) V\ 
1 W 2 \ / J- - 1 
7)1 [m — ??i — . . . m — 
1 
I 
mN - r (inN - r) {mN - r - 1 ) 
1\/ 2\ / 
X 711 ■ 
2 2 7' 1 + l r + s 1 
But when we proceed to the limit -j^, ... ^ and ... — — 
are all negligeably small compared to m so that the expression reduces to 
m- , -..ni^ \ in'' 7n'' 
l-m + --... + (-l)^^...]x^,, = .-™x-. 
That is to say that the expansion is equal to 
j, 7n- m^' ] 
6- + + . .. + - + ...}. 
Hence it is this distribution with which we are concerned. 
The 1st moment about the origin, 0, taken at zero number of particles is 
,„ f 2 Hi,- Sni' i'7if 
^ r + YT + 3T + -+Tr + - 
= '"^"i^ + r!+2! + -+(73iy, + " 
= m X total frequency. 
Hence the mean is at m. 
The 2nd moment about the point 0 is 
1-711- . Z'-m^ . . /•-/«'■ 
1^ 
m- 111'' 2iii^ (r — 1)771'' 
= ' r + n + • • • + (r^Ty:- + • • • + -^-21+- + v-i) i + 
= (ill + X total frequency. 
