673 
We shall. now discuss successively two- and three-valued functions. 
B. Two-valued funetions. 
We first consider functions which are two-valued either in the 
whole real domain or in a part of it. 
At the point(s), where the real domain borders on the imaginary 
one, the two real values of z, which correspond to a single value 
of z, pass into two imaginary values. So at the limits of the rea! 
domain themselves the two values of z coincide. The limits «, and 
v, Of the real domain are the branch-points of the function z= f(a). 
uz 
Now at the limits of the domain — = fia) = 0; the conjugate 
» Av 
value of z may be either finite or infinite. If this value of z is 
finite, the ordinate 
En er 
Va 
of the ideal frequency-curve is infinite at that point. 
If. on the contrary, the corresponding value of z is infinite, this 
ordinate is likely to be infinitesimal or zero. Only for exceptional 
forms of f(x) it might be finite. 
If now the frequeney-table YY... Y,, (Vz individuals lie between 
the class-limits #74 and zz) begins with a very high value )’,, 
decreasing till the last frequencies are zero, we may explain this 
by means of a two-valued function, having a branch-point in «, 
with a finite z, and another in 2, with an infinite 2. 
Let us take as an example 
whence 
2 i! 
4 fy Ge) EN LEL — 
dar 2 
Here ‘the branch-points are 
Coe wy ibn 0, 
Ut, = +o with 2, == o. 
The two branches of the funetion 
are 
. ty li 
AW) =+ Va with Ot 7, 
Zs fe (2c) Ley 
SUG apnea ge 
Pre... ry, ET 
/. itl Mij 1 q — Ì 
4 en U. nn an i ; ETT, 
A@=— Ve with N= Mn zi 
45 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
