Difference Problem 
397 
and Mr Galton takes as a reasonable measure of the prizes lOOd^r^lid-^^ + d^^ and 100c^23/('^i3 + <^23) 
per cent, of the prize money. These are obtained from the last two rows of the table. 
Table of Data fur Two-Prize Ratios. 
n = 
3 
10 
50 
100 
1000 
m 
•43074 
1-28155 
2^05375 
2-32635 
3-09040 
to' 
1-560,6213 
1-244,2739 
2-685,0071 
2-425,7300 
3-527,0311 
- -43074 
-84162 
1-75069 
2-05375 
2-87830 
i 44 / ,uyyo 
0 oUi,yzoy 
0(1) 
•833,910 * 
-524,952 * 
-380,906 1 
-345,992 t 
-274,009 t 
</,(2) 
•833,910* 
-342,013 * 
•222,691 1 
-198,170 t 
-151,399 t 
+ -004,736 
+ -031,971 
+ ^070,072 
+ -084,161 
+ -119,233 
^2 
+ 011,633 
- -005,875 
- ^032,216 
- -042,268 
- -066,830 
Cs 
- -002,055 
+ -000,204 
+ ^002,068 
+ •001,656 
- -001,876 
1+ Cj 4- + Cg 
1-0143 
1-0263 
r0399 
1^0435 
1-0505 
c/ 
+ -004,736 
+ -007,686 
+ •027,170 
+ ^035,035 
+ -055,246 
C2' 
+ -011,633 
+ -002,355 
- •010,553 
-•016,108 
- -030,375 
- -002,055 
- -000,327 
+ ^000,443 
+ ■000,517 
--000,170 
l+Ci' + fj' + Cg' 
1-0143 
1-0097 
1^0171 
1-0194 
1-0247 
Xi/« = d^2 
-8458 
•5388 
•3969 
-3611 + 
-2879 
X-zls = d23 
-8458 
•3453 
•2265 
-2020 + 
-1551 
dn/idis + d.i-^) 
-667 
•719 
•733 
-736 
-741 
diJidi^ + d.^s) 
■333 
•281 
•267 
-264 
•259 
The results are in fairly close agreement with those obtained from Mr Galton's investigation, 
which puts the first and second individuals in the places they would hold if tlie sample of the 
competitive population were actually arranged according to the normal law. His proposition 
that if there be two prizes they should embrace 75 and 25 per cent, respectively of the prize 
money is seen to be a sound rule for practical purposes when n is at all large, and might well be 
impressed vipon the powers that rule such distributions not only in the educational world, but in 
rifle, athletic, sporting and agricultural competitions. 
(7) We may next consider how the divergencies between individual members of an array vary 
when we take the pair close to one end of the array, or nearer to the centre. Let us suppose the 
array to contain 100 individuals ; we already know the diiferences between the 1st and 2ud, and 
the 2nd and 3rd individuals. We will now find the difl'erences between the 25th and 26th and 
the 50th and 51st. In other words we will determine X25(100) and ;^'5q(100). We can easilj' 
find these expressions in the more general case for n fairly large § ; we have : 
, , 2^506,628 / •035,398 •012,327\ 
^i» = ^x — ^ ^) 
and 
3^146,865 /, ^072,942 •026,989\ , 
X. ,(«)=*■ X — 1+ \ — (xxxi). 
* Calculated from (xxv). 
t Calculated from (xxvii). 
X Mr W. F. Sheppard sends me as the values for these constants deduced by quadratures -3594 and 
•2018, -which thus show that our method is sufficiently approximate. 
§ i.e. using (xxvi). 
