651 
Hence in the point wv =w, also ww) will have an infinite value, 
so that the reaction-curve touches the ordinate-lines «== #, and 
“=, in the points (¢ = z,, n= 0) and (¢ =4,, y= 0) resp. 
C. Three-valued functions z = /(«). 
A frequency-series with a summit at only one or at both extremities 
may, as we have seen, be rather easily connected with a two- 
valued function z= f(w). Likewise a function, which is three-valued 
in a part of the real domain, may be used for examining a frequency- 
series with two discontinuities within the limits. 
Here we suppose a frequency-figure of the 
form of fig. 18. If the discontinuities (infinite 
ordinates in the ideal frequency-curve) are found 
ar ae ve Pe at wa; and «—-,, the function z= f (2) 
Fie. Taye must have branch-points there, so that f'(x) = oo. 
and f'(ac) = oo. 
Besides the curve z= f{(vt) must also 
extend to the left of «) and to the right of a. 
So we arrive, intricate forms being not 
considered, at a curve of the shape of fig. 14. 
The function has three branches /,(7), /,(“) 
and f(x). The lower branch ranges from 
#, (corresponding to z= — op) to #-; In this 
Fie. 14. branch f,'(w) is always > 0. 
The middle branch extends from «#, to «(< #) in a negative 
direction along the axis of «; in this branch /,'(«) < 0. 
The upper branch extends from « to zw (corresponding to 2 — + o); 
in it /(w) is always > 0. 
So the function is three-valued in the segment ay... a. 
In the integral-curve /—= (x), as it follows 
directly from the observations (see fig. 15), 
the ordinate in the interval 2,...«, slowly 
increases from 0 to /, (which is a small value). 
p Te * Then the ordinate suddenly rises, so that in 
Fis: 15% the integral-curve (wy, /y) is a corner, of which 
the left hand tangent is nearly horizontal, the right hand one ver- 
tical. Thereupon the slope decreases to its minimum in the point of 
inflexion to rise again to an exceedingly high value at w,. Here the 
ordinate attains the value /,, which differs but little from unity. 
Finally the ordinate still slightly increases from /. to unity. So the 
integral-curve has another corner at («,, /), the tangent being 
vertical on the negative side, nearly horizontal on the positive. 
