392 
Francis Galton's 
Consider first the x' integral, i.e. 
C+x f+x 
1= I dx' l/fl a''-P-\x-x')= da a'^'P-^ix-x'), 
_/ -00 J -00 
and integrate it by parts. It equals : 
["^ix-:^)T + r — dx', 
\_n-p J-oo J -xn-p 
1 1 
= d'^-^dx' = ?7, say. 
n-p J-oo 71— p '' 
^p-i;^\p^ijZ''oU(i-aT~^dx, 
or between limits : 
Thus : 
In 
\n-p \p- 
-|^V(l-n)f-lc;a,by(iii), 
or, taking the value between limits and substituting we have 
In /"+00 
Xp=i ^=-r a^'-^il-aydx (iv). 
\n-p \p } -ao 
This is the complete solution of Galton's diflereiice problem*. 
An interesting theorem which results from this has been given me by Mr W. F. Sheppard ; 
namely : the average differences between successive individuals are the successive terms in 
{a + {l-a)Ydx 
when the subject of integration is expanded by the binomial theorem. 
Given any law of frequency = {x), we must first find a from (i), and then when tables of 
a have been made, calculate Xp by quadratures from (iv). This will be fairly easy, if the distri- 
bution be assumed to be normal, for then tables of a, or tables which readily give a, already 
exist, and quadratures may be used on (iv) to any degree of accuracy required. This has been 
done by Mr Sheppard in the cases cited below for comparison with Mr Galton's results. 
It will be seen that the fundamental difference between the above theory and Mr Galton's 
lies in the assumption of the latter, that the individual results of a special examination give a 
sensibly normal distribution. The above theory only assumes that the competitors are a 
perfectly random sample from material which if it were indefinitely large would obey the law of 
fi'equency >/o=(f>{x). Of course, if we want to compare with Mr Galton's results, we must 
assume this law to be the normal law, but we still have the great generalisation that the actual 
competitors are only a random sample from a great bulk of material following this law. In any 
individual examination, it may be quite possible — especially if the competitors are few — that the 
first man stands anywhere, even below mediocrity, and the chance of this is allowed for in this 
the full mathematical theory. 
* This result is due independently to Mr W. F. Sheppard and myself. I had stated Mr Galton's 
problem to him, and said that I had reduced it to a determination of I A^dx. He sent me, practically 
J -co 
by return of post, the answer in the above notation, suggesting quadratures as the best practical 
solution, and pointing out the theorem referred to in the text. 
