Lucy Whitaker 
37 
In order that n should be positive, it is needful that 
+ =i(6-2/3, + 2A), 
should be positive. If this is satisfied clearly c will be real because fi^ is always 
positive. Further then 
^^=^^^-^^^^4^ 6-2/3, + 3A 
is always less than a quarter and and q will therefore be real. If the reader 
will turn to Rhind's diagram, Biometrika, Vol. vir. p. 131, he will see that the line 
3 - ySa + /3i = 0 cuts off all curves of Types III, IV, V and VI, and includes a 
portion only of Type I, with a part of its U and J varieties. The binomial 
description of frequency, therefore, is not — considering our experience of frequency 
distributions — likely to be of very universal application. 
(2) Further Limitations. 
Now let us still further limit our binomial by supposing : 
(i) that the unit of grouping of the observed frequencies corresponds to the 
actual binomial base unit c and (ii) that the first of the observed frequencies 
corresponds to the tei'm iVjp" of the binomial*. 
In this case the mean m of the observed frequency measured from the first 
terra of the frequency will be equal to the nq of the binomial and the standard 
deviation of the observed distribution will be equal to 'Jnpq. We have thus : 
p = a'^/m, q = 1 — a"/m, n = m'/{m — a'-) (iii) 
and n and q will both be negative, if m be less than a^. The condition for a 
positive binomial is therefore that a be less than \/m. 
(3) Probable errors of the constants of a Binomial Frequency. 
It is desirable to find the probable errors of p and n as determined by these 
formulae. We have : 
yu-i' = nq, ^2 = '>^'p(l^ 
Syu./ = qhn + niq, Bfx.., = pqSn + nqSp + nphq, 
assuming deviations may be represented by differentials. 
Hence, since dp= — dq: 
^1^-2 — (p — l) ^1^1 = (f^i^ ^"^^ P^fJ^i — ^/^■2 = nqSq. 
Square each of these results, sum for all samples and divide by the number of 
samples, and we have : 
+ (P - lY - 2 (jJ - q) o^a^, r^^^, = r/<7,r 
* The exact nature of these limitations must be fully appreciated. The best fitting binomial to 
a given frequency distribution will usually be far from one in which the first term of the binomial 
corresponds to the first observed frequency. The modes of the binomial and the observed frequency 
will closely correspond, but the "tails" of the binomial may be quite insignificant and correspond to no 
observed frequencies. 
