674 
In the first branch « ranges from O to + oe. The contribution to 
the frequency in the segment pq between =a, and «=a, (>z,) 
is then 
p 
In the second RER w comes back from + @ to 0. So the con- 
tribution to the en in the same segment pq equals 
on 
a say 
A Te WE) (Al da = NDE 
Vr 2 War av 
“q 
The total eae a the domain in question is therefore found 
to be: 
av 1 Up vy 
q ~+ ] ie | oar 
I, Sh dS sp adr te ~@ daf nt 
War 2V. a 20 J2V av 
. > Up vg vp 
The ideal continuous frequency-curve bounds the area, which 
equals the total frequency. Its equation obviously is 
| 1 
Y= 9, — 9a = 2 I= We, 4 
TH 
Kvidently the axis of y («=0) and the gxis of « (y= 0) are 
asy mptotes. 
The rough frequency-figure accordingly has'@ summit at the limit 
“—=w,=O0 of the domain, and descends towards the other limit 
(C=, SO). 
A frequency-series, which, starting with a very high value, gradu- 
ally diminishes further on, can evidently be explained by a discon- 
tinuity in the ideal frequency-curve, which in its turn is a consequence 
of the many-valuedness of the function z= / (a). 
For convenience’ sake we may suppose, that the two branches 
of the function <= f(«) consist of equal and opposite values, so 
that /, (x) = — f, (x), these values coinciding at the limits «, and 2,,. 
If the frequency-series Y,, Y,,..., Y, gradually descends from 
the highest value Y,, it is natural to join to 2, a finite value z, of 
z and to «x, an infinite value of z. The two branches being sym- 
metrical (by agreement) we must take z, = 0. 
The curve z= f(«) has then a shape as in the adjoining sketch 
(fig. 3). 
