394 
Francis Gallon's 
1 = 
P 
n 
We find at ouce : 
^ ni n 
I 2/o<^^ (viii). 
J m 
.(ix). 
It remains to find the successive differentials of u for x=m. Let us write the value of _?/(, at 
x=m, simply and we shall then have 
{da/dx)^ = ij,n, (Paldx''-=y\^, d^aldx^=i/",„, etc. 
We find : 
n J 1 1\ i 12/i*(;i-2») , , ?l3 
(71 — jo)^ 
- 30w3 - ^^^;r^) W^n-^^ymy'my"m+y^ny"'m) 
etc. etc. (x). 
These quantities may be calculated fairly easily when y is known as a function of x, the 
coefficients of the y terms in n and p repeating themselves in each a. 
(4) Let us apply these results to the special case when the distribution from which the 
material is drawn is supposed to obey the normal law. In this case, if s be the standard 
deviation of the material from which the sample is made : 
1 _1^2/<,2 
y-- 
•J' 1 f /I =r \ 
P \P 
f+cc \n 
Xp = cx j a''-P{l-a)Pdx, it c= 
