112 DIFFERENTIAL AND INTEGRAL CALCULUS. 



if 5, i/r increase together, so ~ ; therefore 



ds p 



I ds* ds* dx d*y dyd 2 x l(fd*x\ 

 p dx dy ds dx 2 ds ds* \i \\ds*J 



ds ds 



d 3 x cos\/r sin\|r dp 

 So also -p = f- + f- ~jr , 



d*y _ sin^r cos^/r dp f 

 therefore (g) + (gf)' = 1 [l -f g)] , 



SO ( T ] + ( ~T4 ) == ( ~3 3 ~T ~l a ~7~2^ ) " 



\ds J \ds J \p p as | p ds J 



The centre of curvature may also be defined as the 

 limiting position of the point of intersection of normals at 

 consecutive points. The equation of the normal at (#, y] is 



(1), 



hence for the normal at a consecutive point (x-\- &r, 

 the equation is 



Hence at their point of intersection, making $x indefinitely 

 email, 



dx* 



< 



and therefore X x = 



dx 

 -p- , 



dx* 



