EVOLUTE AND INVOLUTE. 347 



This by (4) reduces to 



,, - dy 2x x 



therefore -f-, = 7- - . 



ax %y y 



Substitute in (1), and we obtain 



330. DEFINITION. The evolute of a plane curve is the 

 locus of the centre of curvature ; a curve when considered 

 with respect to its evolute is called an involute. 



If x', y', be the co-ordinates of the centre of curvature at 

 the point (a;, y) of a curve, we have by Art. 320, 



(1), 



if j t. 



By means of the equation to the curve y, -/-, and -^ can be 



expressed in terms of x ; hence from the above equations we 

 can, by eliminating x, obtain a relation between x and y 

 which is the equation to the evolute. From the above equa- 

 tions, x and y may be considered functions of x \ differen- 

 tiating the first, we have 



By means of (2) this gives 



dx dx dx 



therefore 1 + ^-^ = ........................ (4). 



dx dx ^ ' 



