390 
Francis Galtoris 
is practically neutral in its effect on the observed results, neither contributing to 
nor conflicting with them in a sensible degree. The curious property of the 
foremost equi-postiles that it discloses, must rest its claims to interest upon its 
own merits and not upon any effective aid that it might be supposed to afford to 
solving the question of the most suitable proportion between the values of first 
and second prizes. 
What I profess to have shown is 
(1) that in the three topmost equi-postiles of a normal series, whose 
measures are a, b, and c, the value of (a — c) is roughly three times as great as 
that of (b — c), almost independently of the number of individuals in the series 
and quite independently of its Mean and of its Modulus of Variability. 
(2) that observation leaiJs to practically the same result as calculation, 
but almost wholly for a different reason. 
(3) that when only two prizes are given in any competition, the first 
prize ought to be closely three times the value of the second. 
I now commend the subject to mathematicians in the belief that those who 
are capable, which I am not, of treating it more thoroughly, may find that 
further investigations will repay trouble in unexpected directions. 
Note on Francis Gallon's Problem. 
(1) The problem proposed by Mr Galton is one of very great interest and, somewhat 
generalised, probably of wide application to a number of important biometrical investigations. 
In its generalised form it seems to open up possibilities of deducing statistical constants from 
comparatively small samples, for it provides us for the first time, I believe, with the most 
probable relatioushi2:)s between the individuals forming a random sample. I would state the 
problem as follows : 
A random sample of n individuals is taken from a popidation of N members which when N is 
very large may be taken to obey any la w of f requency expressed by the curve y^JVcf) (x), ySx being 
the total frequency of individuals with characters or organs lying between x and x + 8x. It is 
required to find an expression for the average difference in character between the and the + 
individuals* when the sample is arranged in order of magnitude of the character. 
I propose to call this general problem : Francis Galton^s Individual Difference Problem in 
Statistics, or more briefly Galton's Difference Problem. It will be seen at once to carry us from 
the consideration of the means and standard deviations of mass aggregates and arrays to the 
average interval between individuals of those aggregates. We may still deal with averages, but 
we fix our attention no longer on the whole population, but on definite individuals in its ordered 
array. This I believe to be a real advance in statistical theory. 
* Clearly a knowledge of the average difference in character of adjacent individuals involves also 
a knowledge of the average difference in character between any two individuals. 
