264 BOOK OF MATHEMATICAL PBOBLEMS. 



V. Elastic Strings. 



1268. A string whose elasticity varies as the distance from 

 one extremity is stretched by any force; prove that its extension 

 is equal to that of a string of the same length, of uniform elas- 



equal to that at the middle point of the former, stretched by 

 the same force. 



1269. An elastic string rests on a rough inclined plane, with 

 its upper extremity fixed; prove that its extension will lie 

 between the limits 



I' sin (a e ) 

 2X cos! ; 



a being the inclination of the plane, tan e the coefficient of friction, 

 and l t X natural lengths of the string and of a portion of it whose 

 weight is equal to the coefficient of elasticity. 



1270. Two weights P, Q are connected by an elastic string 

 without weight, which passes over two small rough pegs A, B in 

 the same horizontal line at a distance a, Q is just sustained by P 9 

 andAP = 6, BQ = c-, P, Q are then interchanged, and AQ = b', 

 BP c : obtain equations for determining the natural length of 

 the string, its elasticity, and the coefficients of friction at A 

 and 11. 



1271. A weight P just supports another weight Q by means 

 of a fine elastic string passing over a rough circular cylinder whose 

 axis is horizontal, W is the coefficient of elasticity, and a the 

 radius of the cylinder; prove that the extension of the part of the 

 string in contact with the cylinder is 



1272. An elastic string is laid on a cycloidal arc whose plane 

 is vertical and vertex upwards, and when stretched by its own 



