OF LOCAL THEOREMS AND PORIS?MS. 345 



The class of curves comprehended in the equation 

 ■^ X = — • -^^ 1 '- , are possessed of the followine 



a.X'PX-i-fccJC "^ ° 



property : 



Any curve of the kind being given, a point (K) may be found, 

 through which, if a tight line (KG) be drawn in any direction, 

 cutting the curve in two points (F and G), if the ordinate at one 

 of these points be prolonged to a point below the axis, until the - 

 part below is equal in length to the abscissa corresponding to the 

 other point of intersection, then the point thus found is always si- 

 tuated in a periodic curve of the second order, given in species 

 and position. Fig. 5. 



If in the equation of the line, we make v = 0, we have 



ctX . ^ X — X .■<i' a, X .- 



«'= — ^x—-4,ax ' ^"" " ^^ suppose this to remain 



constant, we in fact fix the point K in the axis of the abscissae. 

 This gives the equation for determining t^, 

 {kx — c) <l>x = {x — c)4'»x; 



whose solution is •>// a: = . %\ ^ "* — L. 



ccX — X 



Let AFG be any periodic curve. Fig. 6., and AD, AE any 

 two corresponding abscissae. Required the equation of all 

 curves, such that the line joining the summits of the two or- 

 dinates, raised at the points D and E, shall be constant. 



The equation of the periodic curve being y=.a.x, let that 

 of the family of curves sought be t/— •^ar, then we have 

 AD =: X, AE — a.x, BD — ^x, CE —A'o.x, and the equation 

 determining the form of ^ is 



(4- « .r — ^xf -{-[(tx — xf — c^ . (o) 



X X 2 This 



