71 
2] 
Pa Eu oren 
In connection witb this holds the definition for the “rational 
functions”: 
1 1 
u— 
u+— ey 
chr UZ erhLu = (Ve) u : shr U eshLu ZT (Ve) u ete. 
By the substitution 
i= ay chr. ny sr 
the rational area of TPA (fig. 9) is determined. 
x Le 
Pyle Way: (wor) 
a wie) 
Now the numerator of the 2nd member again represents the rational 
area of PMA so that the denominator is that quantity for PMT 
The argument of the function is therefore determined by the rational 
area of sector MP7Q. Simpler are the relations for a =e; then 
& 
x 1 
Pydlr ey: Ley ; uslLay ; —=L-; 
e 
u y 
out of which again the following relations are formed: 
u 
chru KX shru—=e! ; chru:shru=Ve 
Development of series furnishes 
co 2p! ow (2p—1) 
chrus=etIt ru); shru MH 
1 1 
Lv 2p_ (u). 
§ 24. If the multiplical is calculated for the logarithmic equilate- 
ral hyperbola in the equation on the asymptotes, then these functions 
appear again. 
cs ia 4 
y= ar A Py tle — 5 ‘(als 
= La 
a=e leads to a new form for the logarithm : 
. H x 
é € 
If this (shortened by P,) is introduced as argument, then 
L 
chr Pp = Vay ; shr Pp = |/- 
y 
from which ensue easily the properties; as ia. : 
dir Pr == "(chr Ps) . (or £7). 
The above mentioned curve forms a part of the elementary curves 
in the roottield. The general equation of these “gradation curves” is; 
y= HE) 
