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 v = 0, and 

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



, X ci-i' X , , ax .d-l a, X 



■I X J -l a, X -\ } — — 



ax ^ dax 



Whence i x 7^ = % \x, ux\ 



ax ^ ' 



X d4-x — ^x .dx dx (- — ^ 



^"^ y^ - ~ ^ % {x, « x} ; 



from which yz=:^xzz — xj-^ ^ {x, ^] . 



Now, let kxzz {a* — x'i)*^ the family of curves are com- 

 prehended in the equation 



rdx i '/ '. TX » 



■i/-—xj-^ %\x, {a* — x*)* I ; 



and they possess the following property : 



Ani/ of this species of curve being given, if any two abscissce 

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

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

 the extremities of the ordinates corresponding to these abscissce 

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

 perpendicular to the axis,. . ,^^,^,,, 



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

 of the abscissse, we have w — 0, conseqyently^ 



dx'K dx') dx'\H dx' )-^* 



1 1 



or 



