a =! 67 
When asymptotes (rationals) are at hand, the following formulae 
for z, or 7, infinite or zero tend to a definite value 
A=W Ry, andm=y,:2,, “Ry,. 
By rational subtangent of a curve we understand the ratio of the 
abseiss to the absciss of the point of intersection of the rational of 
contact with the axis OX,; it is given by 
yr Ry. 
The envelope of a series of curves is found in the same way as 
in the differential calculus. 
§ 20. There exists an integrating operation which reduces the 
functions obtained by means of rationalization to the original ones. 
It can be regarded as the limiting product of mutual powers, of 
which one of the efficients lies infinitely close to unity. It shall be 
named multiplical(-potence); its form is: 
A Lydl 
tin Ht (y. (1 Se Bia ie 
x 
For an indefinite multiplical a constant factor must be added; e.g. 
n+1 
Pr(eydlz —e dan (25 
P(eia)dla — Lic, x) 
2» A 
Peilen Ss (=) 
é 
P(eroydle = ¢. sra. 
For definite multiplicals the constant disappears; we have to take 
into consideration the following rules: 
3 2 3 
Fb Plts Pydle re dere atd) 
1 1 2 
Hy 
Pyle SP, Pray. tse ee AP) 
Ty Uy ® Yi 
§ 21. A rational is determined by two points: the director exponent 
À = tgp follows out of: 
ts = (2) 
yy v, 
If now (zy, is a point out of which the rational distance (9) is 
measured, then holds 
cos? a sin 9 y 2/2» 2 y 
ba = =o; | and (=) ‘ (+) == '2(@): 
vo Yo ad Yo 
For the ratio of two such distances on the line we find: 
= 3 
o 
