here 0' denotes the natural length of that portion of the string 

 between the lowest point and the point (or, y). 



Hence for tan ^ in (5) and (7) we may put - , and make 

 corresponding substitutions for sin ^ and cos ^r. Thus (7) 



(). 



As an example of these formulas suppose that a heavy 



trine hangs in equilibrium over two smooth pen 

 in a horizontal plane, and let it be required to find the '. 

 of the ends of the string below the vertex of the curved 

 portion. 



'in (3) the tension at any point of the curve is 



Let f be the natural length of the portion which hangs over 

 one of the pegs ; then the weight of this portion is m'gf. 



ioto the unstretched length of the portion between the 

 vertex and one peg ; then by equating the two expressions for 

 the tension, we nave 



thus from (8) and (9) 



(10). 



Suppose I to be the length to which a string of natural 

 length T hanging vertically would be stretched; then by 



Art. 1't'i 



Art. lyy, 



By (10) and (11) 



/ m^r\ 



r \ l -A } 



u,+*. 



("). 





