St8 EVOLUTE AND INVOLUTE. 



Hence (1) may be written 



which shews that the point (x, y] is situated on the tangent to 

 the evolute at the point (x, y'~]. Also (1) shews that the 

 point (x, y'} is on the normal to the curve at the point (x, y}. 

 Hence the normal at any point of an involute is a tangent at 

 the corresponding point of the evolute. 



331. If p be the length of the radius of curvature at the 

 point (x, y) of a curve, and x, y the co-ordinates of the centre 

 of curvature, we have 



As x and y are functions of x, so also is p ; hence differen- 

 tiating we have 



dx'\ . , (dy 



--- 



By means of equation (1) of the preceding Article this gives 

 , ,. dx , ,, dy dp . . 



fi nt* ft -Y* ' ft 'y* 



\Aj*As UrtAs ( ; ,' 



From equations (1) and (3) of the preceding Article we obtain 



dx' dy { (dx\\ Afy'V ] 



(dx) + \dxj i_ l Ids' 



x -xY+(y'~-y}'} ~ p dx ' 



s' being the length of the arc of the evolute. See Art. 307. 

 Hence, by (1), 



therefore ~ = + ^ (2). 



dx ~ dx 



Since ^ 7 + ^- = 0, we have, by Art. 102, 

 dx 



s + p = some constant, say I. 



