140 
Miscellanea 
material which theoretically should obey the Law of SmaU Numbers, e.g. "Student's" own 
Haemacytometer counts*. Of course the negative binomial may quite conceivably arise from 
other sources I than heterogeneity, but if this be the source of its origin in the material of Bortke- 
witsch, Mortara and McKendrickJ, it is certainly most unfortunate that such material should have 
been selected to illustrate Poisson's limit to the binomial. 
Now we know that the values about the mean of the successive moment coefficients of the 
binomial are 
= ip - r (v) 
= npq(l + 3(71- 2)pq).} 
Further the mean is at a distance nq from the first term p". We shall call this distance m. 
Let fii, jUg', /xg' and jx/ be the moment coefficients round the start of each binomial. Then 
Ml' = M2' = ^Pl + n^<f, \ 
H^' = npq (p - q) + 3npq X nq + n^q^, \ (vi) 
P-i — npq {I + 3(?i - 2)pq) + inpq{p - q)tui+ G{npq)n^q^ + n'^q*.} 
From these equations we deduce 
H-i = M2' - Ml' = « - 1) 1 
^3' - 3ju/ + 2|Lt/ = n {n - 1) (« - 2) q^ V (vii) 
1^/ - G/x/ + 11^/ - 6,./ = n {n. - I) (n - 2) {n - 3) q\j 
Now let aj = /x/ for the combined series, 
a 2 = /X2' - Ml' for the combined series, 
0^3 — Ma' ~ 'V2' for the combined series, 
^4 ~ H-i' ~ ^Ma' + 11^2' ~ ''Ml' for the combined series. 
Then we have, if = vJN, = "2/-^: 
1 
n 
n(n - 1) 
71 (h - 1) 
[71 - 2) 
Xj + Xo, 
^l^l + ^2?2. 
^l9l' + \<bS \ 
^l?l* + ^2 92*- ] 
(viii) 
n (n - 1) (M - 2) {n - 3) 
Multiply each equation by and subtract from that below it and we find : 
9i = ^2 (92- ?l)' ^ 
71 
n (n - 1) (71 - 2) 
«. ( n - 1) 71 
a^qi 
71 (71 
"3 91 
1) 
71 (n - 1) (n - 2) (w - 3) w (71 - 1)(« - .2) 
^292 (92 - 9i). 
^292' (92 - 9i)> 
^292^ (92 - 9l)- 
(ix) 
* Biometrika, Vol. v. p. 356, and see also L. Whitaker's examination of these data. Vol. x. p. 48. 
f Pearson, Biometrika, Vol. iv. p. 209. 
j For an examination of Bortkcwitsch and Mortara's instances see L. Whitaker, Zoc. cit. pp. 49-66. 
McKendrick has recently readied Poisson's series (Proceedings of the London Mathematical Society, 
Vol. xni. (1914), pp. 405 et seq ) without apparently recognising that he was on familiar ground, and 
has suggested its application to tlie frequencies obtained by counts of tlie bacilli ingested by leucocytes. 
He has fitted his series not by moments, but from the first two terms, and has failed to recognise that 
a large proportion of such leucocyte counts give also negative binomials. 
