344 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



Any of the family of curves contained in the equation 



V = vx + fax beitlg giveHi a ri £ ht line ma y he f ound > 

 through any point of which, if a right line be drawn perpendicu- 

 larly, cutting the curve in two points, and if the ordinate at each 

 of these points is produced to a point below the axis, until the 

 part below each is equal to the abscissa belonging to the other or- 

 dinate, the two points thus found will always be situated in a pe- 

 riodic curve, given in species and position. 



The line to which the other lines are drawn perpendicular, 

 is at right angles to the line HFG in the figure. 



If we make ax — s/a % — x % -> the curve which is the locus of 



these points is a circle, whose radius is equal to a, and if we 



also make <? x =. b, the original curve is an ellipse. 



a 2 

 If we make uxzz — , and also 9x — b, the equation of the 



x 



c a i 

 given curve is y = - (x -), which is an hyperbola, and 



a 8 

 that of the curve found is y — — , a right-angle hyperbola. 



If in the equation of the right line, 



4> X — \i/ ccX , a, X .4> X — X . -1> U X 



v — w + , 



X — a. X X — a X 



we make w — 0, we have the distance of the point K from A ; 

 let us suppose this to remain constant, then 



T - zrc, or ux .yx — x . y cx.x — c{x — a x) : 



x — ax * * 



the solution of which is ^ x - — • fJ+ " g * X . 



The 



