738 
If the sum passes into an integral, this formula becomes: 
var 
er (vdo ete > 
From the preceding we can gather that in all respects the field 
of difference with the corresponding functions represents the loga- 
rithm of the field of difference with the corresponding rational 
functions. This can be carried further consistently as regards angles 
and areae. By means of a simple “transformator” we pass from one 
field to another, thus we arrive by substitution of 
ae y =eY¥ anda= eA 
in the equation of the logarithmic hyperbola 
Oe pee as 
at that of the ordinary hyperbola. 
Also ambiguous fields can be considered; i. a. the “semi-rational 
field’, in which the absciss ascends with differences, whilst the 
ordinate changes by ratios. Thus the consideration of the semimulti- 
plical : 
Pade == ® 4 
0 é 
has led me to the equation: 
p 
bmp: rep! 
one thing as well as the other in connection with the “geometrical- 
arithmetrical series”. 
§ 27. The rational in fig. 8 brought through a point M'(a,b) 
equidistant with MB, for which we write MB’) || WB, has as equation: 
CREE 
(dje b 
If a logarithmic circle is drawn having M’ as centre, r as 
radius, then the rational area of the sector, described by M’B’ and 
MX’ (|| MX,) proves to be: 
Yo) a \- 
a ) 
On); tone ( 5 
hence the rational angle is 
This is -as large as the one between MB and MC, the equation 
of MB being 
Cn . 
Fa 
a 
tan—! r e?. 
Beke? 
So we can also see, that: 
