72 
In the supposition 4 >1 we find 
z +1 y +1 
pyle = ay 3 Pet Vey 
1 1 
So that we find, calling the multiplicals for short Pand Dente 
P, eerie 
The gradation curves divide the rootfield in such a way that the 
mutual root of the rational areae measured along the curve has a 
constant value. 
§ 25. An equation, in which besides the variables also rational 
radices or the rationals of the functions appear, is called a rational 
equation. In some cases equations can be solved in which besides, 
the above mentioned quantities still differential coefficients appear. 
I. Required is the curve for which the rational subtangent is 
constant. The equation runs: 
yy Ry =a. 
In succession we write: 
rz |a=tyly 
P(elatte= P(ely)ity 
a,(ela = L{ey): 
y= tl 
This represents the logarithmie curve of arbitrary order. 
II. To find the curve for which the rational radix is proportional 
to the differential coefficient. Out of the condition : 
dy 
follows as answer: 
é 
ye ep and y == = x, 
P 
The last answer is the singular solution and those which by 
means of an integrating power can be reduced to such. Fartheron 
the rationalisation under the multiplicative sign for which it is easy 
to compose the formula. 
$ 26. Some of the above mentioned formulae can be extended, 
as ia. the 3°¢ formula of § 17. 
n 
rt (up) 
VRE (4) = iz. IR) 
