346 PROPERTY OF TWO PERPENDICULARS. 



This is the relation we have already shewn to hold at 

 points where the radius of curvature is a maximum or 

 minimum. 



329. In the figure of Art. 284 let SP=r and SY=p\ 

 if p^ denote the perpendicular from S on the tangent at Y 

 to the locus of Y, then will 



P* 



<n = * 

 Pi r ' 



Let x, y, be the co-ordinates of P, 

 x', y, the co-ordinates of Y\ 



v j 



The equation to the tangent at P is 



77 and ff being the variable co-ordinates. 

 Since the point Y is on the tangent, 



The equation to $F is r?=| .......................... ( 3 ) 





But /SFis perpendicular to PY, therefore 



Combining (2) and (4), 



therefore yy + xx =y' 2 + x* .............. ^. . . . (o). 



Differentiate (5), thus 



,dy dy' , dx ,dy ,dx 



y -i L +v-j + x+x-j- = 2y-T-+ 2x -j-' 



y dx y dx dx J dx dx 



