679 
latter belongs to a function, which is two-valued in the zones 
Veeel ANd Zio... A, but one-valued in the part z_….. > Deas it 
has two asymptotes, «=a. for z=—o and «=«#+4,, for 
c=-+ ©. It is usually difficult, from a mathematical point of view, 
to decide between these two forms. 
The branches of the function where /(“) <0 correspond to 
negative values of the reaction-function, hence with negative growth. 
It may be desirable, for biological reasons, to suppose such domains 
of negative growth as small as possible. In this case the second 
form is to be preferred. ; 
The ideal frequency-curve (fig. 9) has now 
two asymptotes, viz. w& =w, and «= 
The integral-curve / = D(e) (fig. 10), directly 
derived from the observations, touches the line 
Et in (@ == 2;,/ == 0). and the line « = #, in 
(facet a Melts DP 
B En If we construct the curves z= f(x) by means 
Fis. 9. of such simple ground-forms, the two values 
of fe) corresponding to the same value 
of « in the two-valued domain have 
opposite signs. Accordingly in the two 
branches we travel along the axis of « 
in opposite directions. 
The contribution to the frequeney in 
a segment between p and q then consists of two parts: 
d ne 
vy vy 
1 ep = 1 a yer 
TEN ie ee fr (z) e1A@P dz and A, J= = fi (x) e Th? dz, 
Up dig 
both of which are positive; they must be added to obtain the total 
frequency. 
For the integral-curve this means that the ordinate / = P(x) is 
considered as the difference of the two ordinates 
BH £ 
1 MN ae OR fe a 
D(z) = aq (AG eTh@P de and @®,(«) = a [ie e—lf2() de. 
The integral-curve now 
assumes either the form of 
fig. 11a, or that of fig. 114, 
depending on the function 
being two-valued in the 
“whole domain or only in 
Fie.lla Fis. 11 ion 
