584 
As has been shown by Prof. J. C. Kapreyn and the author of 
this paper’), the normal function z= /(v) may be determined from 
the given frequency-distribution, at any rate graphically. 
A normally distributed quantity may be considered as the result 
of growing from an initial value #7, common to all the individuals, 
with increments individually different but distributed round the mean 
increment according to the normal law of error, and independent 
of the instantaneous value of w. 
When spread in a skew distribution, the quantity is built up of 
elementary increments which contain a factor w(v) dependent on 
the « undergoing the increment. Thus the cause of growing being 
supposed to be spread purely accidentally, the reaction upon it is 
proportional to the function (zr), which is called the “reaction- 
function” and is determinate but for a constant factor. 
According to the theory of Prof. J. C. Kaprryn the following 
relation holds between the reaction-function 1 — ®(z) and the normal 
function z= f (a): 
du 1 
n= (2) = EE TH 
Thus far *) some normal functions have been examined analytically, 
viz. that which answers to the normal distribution z= 4 (#—,,) with 
1 
Ie and those which correspond to the so-called “logarithmic distri- 
5 Ld Ed dn Á 
bution”: z= 2 log —— with = —— and 7 = 4 log a 
Em À Vy i 
End . . . . 
i ao (A> 0,2, <a << 2,). In the normal distribution the reaction- 
function 1 is a constant, in the logarithmic distribution 4 is a linear 
function of z. 
In the present paper we shall treat the also logarithmic case that 
the reaction-function is a quadratic function of « Then the normal 
function is of the form: 
t—WHo Um = Xo 
2d eg ( : 1 
Un — d En — Um 
The general method furnishes the values zz; corresponding to the 
n—1 class-limits az. The curve which can be drawn through the 
points (er, zj) is the graph of the normal function Sd) 
1) J. C. Kapreyn andM. J. van Uven: Skew Frequency Curves in Biology and 
Statistics, 294 Paper; Groningen, 1916, Hoitsema Br, 
