350 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



dux \ ax / dx V decx J 



multiplying both sides by (^— — ^p!^\ we have 



^'C^y'—^-C^)" 



a, X 



and since both sides are symmetrical relative to x and a x^ 

 we have 



— 1 



or 



hence, 



\ X (-jy) — x = x,{x, ux) 

 dx 1 /~ \ 



1/ J- = X + X, K^^ '^^>i 



dy _ dx 



y X + y^{x^ a.x) 



If we suppose « x = (a" — *") ", the class of curves compre- 

 hended in the equation 



''^y - J x + ^{x,(a^^^r^Y. } 



possess the following property : 



If we take any two abscissa AD, AE, the sum of whose nth 

 powers is equal to a given nth power, then the tangents BP, CP, 

 drawn to the curve at the extremity of the ordinates correspond- 

 ing to those ahscissce, will always intersect each other on the axis 

 of the abscissce. Fig. 10. 



In the same manner, a class of curves may be found, and 

 ordinates may be drawn, in such a manner, that the tangent 



at 



