﻿of a Loaded Spiral Spring* 387 



(#, (f>) are respectively given by* 



where A is the torsional and B the fiexural rigidity of the 

 wire. 



If the spring is hung up in a vertical position with its 

 upper end rigidly fixed, and a mass M attached to its lower 

 end so as to be symmetrical about its axis, then, neglecting 

 the mass of the spring, we have 



Now let the spring be displaced to the configuration (,v + 8x, 

 </> + 8<j>) and released, then the equations of motion are 

 d 2 



d 2 



where Wt? is the moment of inertia of M about the axis of 

 the spring ; that is, 



or 



If the angles of the unstretched and of the stretched spring 

 are both so small that we can neglect ^-, it follows from the 



* Thomson and Tait's 'Natural Philosophy,' vol. i. part ii p. 141. 



2D2 * 



