MAXIMUM OR MINIMUM CURVATURE. 345 



therefore c = r- r ', 



dp 



dr 



that is, the radius of curvature = r -,- . 



dp 



328. At a point where the radius of curvature is a maxi- 

 mum or a minimum the circle of curvature has contact of the 

 third order with the curve. 



^ s 



Since p = 



dp 



we have, when -f- = 0, 

 ax 



l+f^-' 



_ 

 dx* 



_ 



da?) dx dx* dx 



If in Art. 320 we differentiate the second of equations (2), 

 we have 



^Y <7 3 y 



-t , TV a * 



dY d*Y 



d*Y dX dX* 



= - 



by equations (3) and (5) of that Article. In order that the 

 circle of curvature may have contact of the third order with 

 the curve at the proposed point, we must have 



d 3 Y_d*y 

 dX*~ da?' 



d *y f, , ( d y\*\ o (d*y\* #v 



therefore -^( \ 1 + (-f- }\ = 3 -r^ ~r 



3 \dxj ) \oarj dx 



