1 
da oe ae 
& 
where #7,2=5—- 0, t= + Dt = #25 =O) with 2, San) — 0. 
Another example of a generating function is 
(x, — #,)(tn—2) 
1 : log — 
(on wv, )(a —z,) 
a 
where z, = 0. 
Giving up the simplification °; = Fco may lead to an easier 
en 
explanation of frequencies in two ways: 
For one thing: to choose a value ~, conjugate to z= + o within 
the frequency-domain («,< «,, <«#,) may be advantageous if the 
frequencies become exceedingly small somewhere within the domain. 
In this case the theoretical value of the ordinate of the frequency- 
curve: y= —f (we LOF for r =a, is zero (values of f (z,) 
Va i 
of an excessive order of infinity being excluded). 
Moreover to join finite values of z to the limits #, and z, of z 
may help to make high frequencies at the limits admissible, the 
factor e—-'/()" not being infinitesimal at the limits. 
Next we shall examine what happens if we drop the first simpli- 
fication also and accordingly suppose the function z= f(«) (as 
function of #) to be many-valued. So to one value of « several 
de 
values of 2 may correspond, and infinite values of — = f'(x) are 
ar 
admitted for finite values of z. 
On this supposition there must be partial domains where f'(x) <0, 
REE Pen 
Sine, a passing through oo. changes its sign. 
a 
Thus we seem to come into conflict with the condition that the 
observed frequency 
Tz 
1 Se 
= sft (ze fw)? da 
Va) 
Jy 
must be positive. E 
This apparent difficulty is removed by considering that the inte- 
eral may yet turn out positive, provided that we invert the sense 
of integrating. so that we proceed along the axis of 2 in a negative 
sense in those segments, where f' (a) <0. 
