DIFFERENTIAL AND^ INTEGRAL CALCULUS. 99 



d'p 

 pendicular from on the normal at Q = -~~ = Y 1 ; 



d?p 



therefore YY' =p + -j ^ = PQ, or PQ = radius of curvature 



at Pj or Q is the centre of curvature at P. This proves 

 that the centre of curvature at P is the limiting position of 

 the point of intersection of the normals at P, P' when P' 

 moves up to P. 



DIFFERENTIAL COEFFICIENTS OF AREAS, VOLUMES, 

 SURFACES. 



(1) Let U be the area intercepted between a curve 

 y = </> (a?), the axis of #, the ordinate at any point P (fig. 24), 

 and some fixed ordinate aA, 



OM = x, MP= $ (x} , ON= x + Aar, NQ = (x + Ax), 

 then U=AaMP, U+&U=AaNQ- 



therefore A U= area PMNQ, 



which for all forms of the curve, provided (.r) is finite, 

 will lie between <f> (x) A#, <f> (x + Ace) Aa?, when AJ; is 



taken sufficiently small; or - lies between <f>(x), and 



A<3> 



7 rr .x 



<b (x + AJC), or y- = ^> (x), or 7= 1 <^>(u?) Jic where Oa = . 



If Z7 be the area included between the curve, the 

 radius vector to a point P (fig. 25) and some fixed radius 

 vector OA) where 



xOA = a, 0P=r, XOP=e, 0#=r+Ar, XOQ = + &0, 



U= area A OP, U+ A Z7= area .4 #, 

 A f/= area POQ = triangle PO Q in the limit, 



(this means not that they are equal because both vanish, 

 but that their limiting ratio is one of equality), 



= i- (r-f Ar)r sinA0; 



