678 
two-branched curve 1 = (x), /= P,(z) from the curve /= @(2). 
Then, by fixing the values of z, conjugate to the different values 
of /, according to the relation 
er 
O (2) =o fetus, 
Jt 
we obtain the required symmetrical curve z= f(«), which for a 
single value of # gives two opposite values of 2. 
Also the reaction-function *) 
ey 
FOF) 
has two opposite values for each value of «. There are two domains 
(overlapping the same segment of the axis of «), one of positive, 
the other of negative growth. lt is exactly this negative growth 
which explains the accumulation at the lower limit. 
1 
We next consider the case that the frequency-domain ends at 
both sides with rising frequeney-numbers, so that the smaller 
frequencies lie inside. 
Now the two limits “, and «, must correspond to finite values 
of z, being at the same time branch-points of the function z= f («) 
which is assumed two-valued. 
Thus the curve z= f (#) must have either the form of fig. 8a, 
or that of fig. 85. The former represents a function, which is two- 
valued in the whole domain, with a single asymptote « —w,. The 
al 
Fic. 6 a. ‘Fie. 6b. 
1) J. G. Kapreyn and M. J. van Uven: Skew Frequency Curves in Biology and 
Statistics, 2nd Paper; Groningen, 1916, Hoitsema Br. 
