348 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



found is equal to twice the radius, and the line given by posi- 



tion passes through the centre, making an angle of - with the 



axis. 



ADE being any periodic curve of the second order, and 

 AF, AG any two corresponding abscissae, and ABC being any 

 other curve, whose equation is y — ^ x, required the co-ordi- 

 nates of the point of intersection of its two tangents, at the 

 points B and C. 



Call AF = x' AG = *' 



i 



BF=y CG = y ; 



i 



and let x and y be the co-ordinates of any point in either of 

 the tangents, then 



y = x -f- t -f + y 



ax ax 



d y' x' dy 



- i 



are the equations of the tangents ; and if we call v and w the 

 co-ordinates of the point of intersection P, we have 



> x' dy? 



y — — r# -y • 



V zz 



x' dy' 



dx' I ' dx' 



d x' d x' 



And 



w zz 



J7( y ~d7~) dx'\V dx' ) 



i l 



dx' dx' 



Let 



