Karl Pearson 
533 
and if TOi, ma, rria, be the moments of i about its zero value, we have : 
vii = P„, m.^ = Pr,o, = P33, iTii = P44, 
where is the uvih product moment of z and x about axes through their zero 
or we can obtain any moment about the zero of i by multiplying the corresponding 
moments of x and z. 
These moments are then transferred by the usual formulae 
M2 = m2 — m^, Mi = m^- Sm^mi + 2^?l^^ M4 = ?/<4 - 4?/i3TOi + 6??;.^ m^^ - Swi^ 
to the mean as origin and the type of frequency calculated in the usual way from 
the corresponding /3i and 
In Greenwood and White's data we have, for the three series discussed below, 
elementary subranges rising by 32, "24 and '16 of a bacillus per leucocyte for 
the distribution of the means of counts of 25, 50 and 100. I find that for such 
distributions, the value of the variate z will only be affected by about a unit in 
the third place of decimals in the worst cases, i.e. the lowest values of y in samples 
of 25, whether we use for z (i) the mean of the inverses of the start and finish 
of the subrange, (ii) the mean of the inverses of all the 32, 24, or 16 hundredths 
in the subrange, or (iii) the inverse of the mid-point of the subrange. I have 
accordingly adopted the last as the simplest value of z for practical purposes. 
(4) Illustration of the method. I. Greenivood and Whites 200 samples of 
100 counts. 
The data are given in Table I. The moments of the frequency distributions 
for y and z as variates about the zero of those variates were then found by tables 
of powers of numbers and a calculating machine*. 
I. Distribution of 40,000 indices for 200 samples of 100 counts. 
For a;: i/j =3-67620, 13-67643, z/^ = 51-50333, z/^ = 196-40357. 
For ^: z/i' = 0-275165, j./ 0-076603, ^'3' = 0-021577, i'/ = 0-0061494. 
These give : 
mi = 1-01156, TO, = 1-04766, m3 = 1-11126, 7724 = 1-20776, 
which transferred to the mean give for frequency constants of the 40,000 possible 
The distribution is accordingly of Type IV. 
* I have cordially to acknowledge help from Alice Lee, D.Sc, Julia Bell, M.A., and Amy Barrington, 
who have each worked out nearly the whole of one distribution for me, and from H. Gertrude Jones, 
who has prepared the diagrams. 
indices : 
= -02440, 
yU3 = -00213, 
/i4 = -00235, 
(7 = -1562, 
/8i=-3123, 
/3o = 3-9472, 
/c = -2651. 
Mean = 101156, 
Mode = -9774. 
Biometrika vu 
68 
