671 
Prof. J. C. Kapreyy and the author of this paper ') have developed 
a method to derive the so-called “normal function” z= f(«#) from 
the given frequency-distribution, by applying the principle: 
corresponding values of # and z are equally probable. 
Then it appears that 2, as function of z, must be one-valued. 
Two simplifications are besides introduced: 
1. In the whole real domain, we suppose 
fe) >, 
dz 
hereby we prevent that En = /' (e) may vanish, and consequently that 
v 
z can be a many-valued function of «. 
2. The lower limit «, of the real domain corresponds to z—=— o. 
In the following paper we shall expand the thus far developed 
theory by dropping the two simplifications mentioned. 
A. First we drop the latter simplification while retaining the first. 
So we suppose that the extreme limits 2, and x, correspond to 
the values z, and 2, resp. of z, the extreme limits — oo and + cc 
of z being in their turn conjugate to the values w_., and C4» Of x. 
If «4. and w_, do not coincide, then 
~ no part of the real domain is found 
pte =. a. Len between these values ; 80 the real 
+. -_ +domam eonsists of the partial domains 
Lys Eto ANd #_ > … Ly. In fact because 
J' (2) must always be >0, toz,<-+ oo 
must correspond x, <_ #4, and to 2, >—ca@ An >de. 
The segments w,...-+ 0 and —-o«...2,, which do not belong 
to the real domain, are represented together in the segment between 
z, and z, of the axis of z, and as wv, in passing from 2, to z,, con- 
tinually increases (excepting the fall from — oo to — oo), also 2, 
will be less than z, 
Now the quantity z must pass through all values from — a to 
-+- ©; so on the axis of z no segments are found which do not belong 
to the real domain. Consequently the points z, and z, must coincide. 
BEE EES rn Hence we have this situation (fig. 15). 
If the domain between z, and 2, has 
no gap, there is also coincidence of 
7 
Fis.la. 
OO Zn + OO 
PE de and A 
Le) 1 re 
Such a correspondence is generated 
Fac. 1b. 
by the function 
1) J.C. KAPTEYN and M. J. van Uven: Skew Frequency Curves in Biology and 
Statistics, 2nd Paper; Groningen, 1916 Hoitsema, Br. 
