65 
COS? » sin 2 1 
LE 
T, Yi 9, 
This can also be represented by a definite multiplical, of which 
the indefinite form is: 
Val (aly? 2 Key B A 
pF an P42) v4] 
For the rational this becomes: 
2 
2 V 14 2dLa o 
I= el ; — a —*? 
i 1 
It is obvious that we can give to S the name of rational length 
of arc. It represents therefore for an arbitrary curve the limit of 
the product of the rational distances taken from point to point; where 
thus FR continually changes into the following form: 
For a line parallel to the X-axis this has the simplest form, viz. : 
ee 
1 zy 
The multiplical mentioned in $ 20 then becomes if y= y, (constant): 
2 x 
2 
Pyare eee 
l DT, 
This represents the rational area of the rectangle determined by 
the above mentioned coordinates. For the rational trapezium bounded 
by y,, y, and a rational we have 
a“ 
: vs 2 
yale — — YY. = Sr Yoo 
ria zy 
— Dr 
when y, is the mean proportional. We can also take that multi- 
plical as a power of a ratio of area, when we write: | 
(Ge) 
LY 
Also for an arbitrary curve that multiplical will be called the 
rational area; it is entirely determined by the limiting coordinates. 
§ 22. From the notion “rational area” is deduced that of “rational 
angle” (already mentioned in § 8). In fig. 8 the rational MB determines 
with M/A and the logarithmic circle a sector whose rational area is 
going to be calculated. The multiplical extended over ABDM is 
