OF LOCAL THEOREMS AND PORISMS. 341 



point in the curve, and also the arc GF, together with the arc 

 BF, will be equal to the quadrant AF of the curve. 



Let ABC be any periodic curve of the second order, whose 

 equation is y := ax, and DB, EC any two corresponding or- 

 dinates, it is required to find another curve AFG, such that 

 the sum of its two ordinates, at the points D and E, may be 

 always a constant quantity. 



From the nature of the curve ABC, we have AD ■=. x, and 

 AE — DB =: a, X ; also ct^x = x. 



Let y :^-^ X be the equation of the curve AFG, then 

 FD — -^x, and GE — 4- (AE) — ^ax, and by the prescribed 

 condition, we have for the equation determining ^ 



'\, X -\- '^ a X "=. c ; 

 the general solution of which is, 



c(p X 



4/x 2z 1 ; 



^ (px-j- <p kx 



the function <p is perfectly arbitrary, and of the class of curves 

 comprehended in the equation y — -^ , we may enun- 



ciate the following porisms. 



Any of this family of curves being given, aperiodic curve of the 

 second order (ABC), may always be found such, that if we take 

 any two abscissce in the curve given, respectively equal to any two 

 corresponding ordinates of the curve found, and draw ordinates 

 to the given curve, and if we prolong either of these ordinates 

 (EG) above the curve, until the part above (GH) is equal to the 

 first of the two ordinates, the extremity of the ordinate, thus in- 

 creased, will always be situated in a right line given by posi- 

 tion. Fig. 3. 



The line thus given by position is parallel to the abscissae, 

 and situated at the distance denoted by c. 



If 



