DIFFERENTIAL COEFFICIENT OF AN ARC. 331 



of an arc, namely, that the limit obtained is the same accord- 

 ing to whatever law the polygons be inscribed. 



From this definition of the length of an arc it follows that 

 the ultimate ratio of the length of an indefinitely small arc to 

 its chord is one of equality, that is, 



As 



or 



/( 

 vl 



therefore = * + 



dx 



Tas = J j 1 + (jtf\ , 



309. Since secant PTas 

 we have cos PTx = \, ,.,. = -=- , 



and sin PTa; = cos PUc tan PTx 



dx dy dy 

 ds dx ds 



310. If x and y be expressed in terms of 6 from the 

 equations 



x = r cos 0, y = r sin 0, 

 ds ds dx 



wehave - 



do 



/dy 

 [d0. 



dx Q dr 



But = cos0 



dy /.dr n 



-f Q = sm 6 -ra + r cos 6 ; 

 at/ air 



