﻿366 Mr. L. II. Wilberforce on the Vibrations 



mercial " copper and that which is sold by chemical supply 

 houses under the label " chemically pure." 



From the tables it will be seen that the plates are practically 

 clear except for the impurity lines, which are very weak, many 

 of them not showing on a silver print. In any case a table 

 of the impurity lines and " ghosts " might accompany each 

 map. A few years hence, when the spectra of the metals are 

 more completely measured, such a table will be easily made. 



North-Western University, 



Evanston, Illinois, U.S.A. 

 July, 1894. 



XLIY. On the Vibrations of a Loaded Spiral Spring. By 

 L. R. Wilberforce, M.A., Demonstrator in Physics at the 

 Cavendish Laboratory, Cambridge*. 



IT has been pointed out by Profs. Ayrton and Perry f that, 

 by comparing the axial elongation and the twisting pro- 

 duced in a spiral spring of finite angle by the action of an 

 axial force, we can deduce the ratio of the torsional and 

 flexural rigidities of the wire or strip of which the spring is 

 made, and hence obtain the ratio of the rigidity to the 

 Young's modulus of its material. 



This method is very interesting and instructive ; but as it is 

 not easy to produce springs of convenient and yet sufficiently 

 uniform angles, nor to determine accurately a small axial 

 elongation, it seemed to me that it might be worth while to 

 modify it by attaching a mass to the spring and observing 

 the periods of the vibrations which it executes when dis- 

 placed. In this case it will be found convenient to use a 

 spring of an angle so small that its square may be neglected. 



Apart from their use in comparing moduli of elasticity, the 

 vibrations of such a system present some rather interesting 

 features, of which a detailed consideration may not be out of 

 place. 



If we have a spiral spring made of a length / of wire, 

 and wound on a cylinder of radius r, so that the distance 

 between the ends of the spring is x, and if (/> is the angle 

 between the planes through the axis of the spiral and the two 

 ends of the wire, the force and couple required to produce a 

 deformation from the equilibrium state (x 0} <£ ) to the state 



* Communicated by the Author, 

 t Proc. Roy. Soc. vol. xxxvi. p. 311. 



