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 



4 



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 = ^|' a;, required the co-ordi- 

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

 points B and C. 



Call AF = a:' AG = x' 



1 



BF=2/' CG=y; 



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

 the tangents, then 



y -"^ dx' dx' ^^ 



dy' x' dy 



d X d X 1 



1 



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

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



^ dx' ^ ^ dx' 



f 

 V = 



d x' d x' 



1 



And 



dl/ ,'dy\ d^f , f ^n 



V) = 



dx' dx 



Let 



