350 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



or «LL«*/\|,*_ ^d±x\ = dU / ux.d+ux K, 



dux V dx J dx V dux ) 



multiplying both sides by ( -~- — d + ux \ we jj ave 



\ dx dux / 



and since both sides are symmetrical relative to x and u x 

 we have 



— 1 



or 



hence, 



y -£- - x + z (x, ux); 

 d y __ d * 



If we suppose a # — (« n — # n ) n , the class of curves compre- 

 hended in the equation 



lo . r dx 



" J x + z {x y (a n — x*)i} 



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 abscisses, 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 



