270 
Peccavimus ! 
Also we give here additional moments* of the binomial (p + g)" about its mean: 
A^i = 0, = II pq, = vpq (p - q), 
fx^ = njiq {1 + S(n - 2)2)q], fj,,^ npq (p - q) [1 + 2 {rjti - Q)pq}, 
yLt,; = iipq {1 + 5 (5)^ - 6)j)r/ (1 - 4<pq) + 15n{ii - 2)_pYj. 
ix^ = npq (p - fy) {1 + ( ]4« - 15) + jfq;' {105n- - 4{]2n + 360) j, 
/is = |1 + 1pq{l1ii - 18) + l^fq- (35/;- - 154?; + 120) 
+ 1p'q--{mi' - 340/1-' + 1044H, - 720)} 
(D)- 
The values obtained in the above laborious manner for [hmp] agreed as far as 
we proceeded with those obtained by the following or second method. 
(ii) This second method consisted in first summing for hiqp on the assumption 
that was constant, and then summing for hip. This involved some new results 
which will be useful in other problems and are recorded here. 
For constant + hiip : 
Mean {Bn,,)- = f 1 + i^'p " + ^"Z- 
Mean {Bii,J>i.p) = -(l+t^) + ■ 
" J' \ npj Hp np- ' 
Mean (8,1,^^ = f 1 + b) {Tip - n,p) \l - + 38,ip H + t'^ Siip\ 
\ lip / 11.^, ^ { np 11^ ] ?(/ 
Mean (Sn%,Bii.p) = -(l+t^] { 1 - Up -2^ + 3S»,, ^"j + -"i^"^-^- U;; 
Mean {Bii,,p8ri,.pSn,..p) = + g,,^^) (2 - 38/;,,) + 
(E). 
Now the value of this method was at once obvious, for proceeding to the sum- 
mations in the moment-coefficients of hiiip, for constant Tip -\- hiip we found that they 
corresponded with values to be found for the distribution in a sample iip of constant 
size. In other words we reached a conclusion, which should have been obvious at 
first sight, namely that to find the value of 
Mean (BmpY 
all that we have to do is to write down the known value = -|- Biij, constant and 
then sum for Suj,. We might have pulled down the scaffolding in this correctional 
paper and simply started from this residt, but as sevei'al of the means reached in 
processes (i) and (ii) seemed likely to be of value, we have preferred to indicate 
the steps which led us to the final method. 
* A simple reduction formula for the moments of a binomial about its mean was sought in vain. 
After a good deal of energy had been spent on the problem, we believe that fx^ being the sth moment 
about the mean 
=0 
is, perhaps, the easiest expression for reaching those moment-cocl'ticients by successive differentiation. 
