OF LOCAL THEOREMS AND PORISMS. 349 



Let us now suppose that the point P is always situated in the 

 right line AL, perpendicular to the axis ; we have urrO, and 

 since the curve ADE is a periodic one, whose equation is 

 y — a x, 



, x a ^ x , . a, x ,di> a X _ 



■I x j -i a x A 1 — =r 



ax da, x 



Whence ^ x — , — % { x, ax} 



ax l ' 



xd-^x — tLx. dx dx i- — . 



or -g -__ % | X)a ^j; 



from which 



d x 



y — \x — —x{-^ %{x, ux} 



Now, let a x — (a 4 — x*y, the family of curves are com- 

 prehended in the equation 



y-—xf -I % U, (a 4 — *4)i } ; 



and they possess the following property : 



Any of this species of curve being given, if any two abscissce 

 are taken , the sum of whose fourth powers is equal to the fourth 

 pozver of a line, which may be found, then the tangent drawn at 

 the extremities of the ordinates corresponding to these abscisses, 

 will intersect each other in a line given in position. This line is 

 perpendicular to the axis. 



If we suppose the point P always to be situated in the axis 

 of the abscissae, we have w — 0, consequently, 



dx'V dx' ) dx'\? dx' J-°> 



or 



