DIFFERENTIAL AND INTEGRAL CALCULUS. 109 



a, J, p are functions of s determined by these three equa- 

 tions; if we change the position of (a;, y] on the curve, 

 we shall of course alter their value. To investigate the 

 law of these changes, differentiate equation (1), taking 

 account of (2), and we have 



, N da . ,, db dp 



(^> : s*(r-*>S-^f- ........ w> 



and similarly from (2), 



^.^Q^ 

 ds ds ds ds ^ J 



Now the centre of curvature will, as (or, y] moves on 

 the curve itself trace out another curve called the evolute 

 of the former, and the tangent at (a, b) to this curve 



makes with the axis of x the angle tan~M-j-j, which 



by (5) = tan~M ^-J the same as the normal at (x, y). 



But (a, 1}} lies on the normal at (a;, y] (either by equation 

 (2), or because the circle whose centre is (a, b) touches the 

 curve at (a?, y]} r hence the tangent to the evolute at (a, b) 

 is the normal to the curve at (a?, y}. Again by (5) and (2), 

 we have 



(a being the arc of the evolute) 



.da ( j.db dp 



(x - a) -j- + (y - b) -j- p-f 

 x ' ds ^ ' ds r ds 



da\* fdb\* 

 ds) ' h \ds) 



dp do- 



therefore -j- = -y : 



ds ds 7 



therefore p = & -f C. 



This proves the property for which the locus of the 



