236 ON SMOOTH < ri;\i:. 



Next suppose the weight of the string taken into a<. 







Take the axis of y horizontal and that of x vertically 

 downwards. The element PQ is acted on by a tension 

 along the tangent at P, a tension at Q along the tangent at 

 Q, the weight of the element vertically downwards, and tin- 

 resistance of the smooth curve which will be ultimately al<m^ 

 the normal at P. Let 6 be the acute angle which the n 

 PN makes with the axis of x, + SO the angle which the 

 normal QN makes with the axis of x. Let s be tin- 1< n_rth 

 of the curve measured from some int up to J'. 



PQ = & ; let T be the tension at P, and T+ BT the tc: 

 at Q; let wf/os be the weight of the element, and //8s the 

 resistance of the smooth curve on the element. Suppose tin- 

 element PQ to become rigid, and resolve the forces acting on 

 it along the tangent and normal at P; \ 



T -(T + ST) cos W-mgfa sin0 = 



(6). 



From (5) we obtain ultimately 



dT 



mgmnO 



(7), 



