DIFFERENTIAL AND INTEGRAL CALCULUS. 93 



i. e. the equation of the chord PQ is 



and the equation of the tangent at P is the limit of this 

 when Q moves up to P, or is 



and the equation of the normal at P is therefore 



Let ty be the angle which the tangent at P makes 

 with the axis of #, then tan ^ ~^j a l s cos 1^ = limit of 



Pn ,. . c Aa; , , r . , chord PQ . 



= limit of -, but the limit of ^ pQ is 1, or rf 



the arc of the curve measured from any fixed point up 

 to P be s, and arc PQ = As, 



,. . A# arc PQ dx 

 cos>|r = limit x -: . = -5- . 

 As chord PQ ds 7 



and similarly sin>|r = -^ . 



If the curve be referred to polar coordinates, $P=r, 

 ASP0^ Q (fig. 20) a neighbouring point whose coordi- 

 nates are SQ = r+ Ar, ASQ= Q + A0, then 



(3^ r + Ar-rcosA0 arc QP 



_ = __ - 



A^ 

 Now (1 - cos A^) = 2 sin 2 ^=- , 



2 



l-cosA(9_ 2 . A(9 



As As IT ' 



or vanishes in the limit ; therefore, if < be the angle SPT 

 which the tangent at P makes with the radius vector, 



dr 



