396 
Francis Gallon's 
Here the term in a.^ is generally the largest, a^ the next and the least. 
We can write the terms in the curled brackets : 
c 
. ^\n — p)'p in- , ..V 
^ 8a2^ n^{n-p)p ?/,„ ® »^ 
^ AQa^ " 11^ {71 — pY p^ n'^{n-p)p y. 
m 
And thus : 
^{n—pf+p^ 13ot2-4 ^ (ti - 'Zp) {n — p) p m{Qm^-l) 
_ , (>.-jt.)>^ 31»i^-101mH28 
Ui {n-pY-PpP /{n-p)p 1 , , , , , , , , 
Xp = s-. — ^=-n v27r {I+C1 + C2 + C3+...} (xxv). 
The solution of the problem is now purelj' arithmetical, although of course laborious. 
(5) We may note some special cases. 
Corollary (i). Siippose both n and p large and not nearly equal. 
Since if q be large 
[g^=«/27rg q-'ie-i, 
we have 
Xp=s 7V{1 + Ci + C2 + C3+...} (xxvi), 
a much simpler form. 
Corollary (ii). Suppose n large and p small. 
^¥irppVe-P 1 , , / "N 
= ' — T + + + (^vu). 
[jT "-ym 
Corollary (iii). Suppose ?i large, and that we consider Mr Galton's special problem of the 
ratio of the distance between the first and second to the distance between the second and third 
in a graduated ariuy. Then 
Y, e ?/'„ 1 +c, + C9 + c, + ... , .... 
i-' = — ;= ' , , (xxvu)), 
X2 2V2 Ijm 1+Ci +C2 +C3 + ... 
where undashed letters refer to quantities for p= \ and dashed letters to the same quantities 
when p = 2. 
(6) As a first series of illustrations, let us apply these results to Mr Galton's consideration 
of the proportion of money to be given in prizes, supposing only two prizes, for the cases n = 3, 
10, 50, 100, 1000. 
The following table contains the chief values*. We write : 
Xp = sx^(P)(1+Ci + C2 + C3+...) (xxix). 
Then, if d„i be the difference measured in variability units between the r^^ and r't^ individuals, 
drr' = {Xr + Xr + l + Xr + l+ +>r/-l}A) 
* I owe to Dr Alice Lee, not only a careful revision of my numbers, but an extension of this table. 
