Temperature on Modulus of Rigidity of Solid Metals. 405 

 where Q is the initial amplitude. The period T is 



T= /T~b 2 



/a 2 

 V I"4P 



If the logarithmic decrement be noted by X, we have 

 By combining the above relations, we have 



2(4t? 2 4-X 2 )IZ 

 n= — 



7T. 



-T 2 R 4 



In our case, X is negligibly small compared with 47? 2 ; even 

 in the most unfavourable case at high temperatures, the 

 neglect of X 2 does not cause an error greater than 0*5 per 

 cent. Hence we get finally 



_SttII 



n ~T 2 n 1 ' 



Thus we can safely use the last equation for the calculation 

 of rigidity. 



3. Results of Experiments. 



The results of experiments for the variation of rigidity 

 and logarithmic decrement with the rise of temperature 

 are given in tigs. 2 to 14. Before entering into a detailed 

 description of these results, we shall briefly refer to the com- 

 plicated phenomena of logarithmic decrement. According 

 to the theory of torsional oscillation above referred to, the 

 logarithmic decrement is independent of its amplitude of 

 oscillation ; but actually this is not the case. The logarithmic 

 decrement of all the metals here investigated increases almost 

 linearly with the amplitude of oscillation. The rate of 

 this increase becomes greater as the temperature, at which 

 the experiment is made, is higher. 



The above discrepancy between the theory and the experi- 

 ment was found to be principally caused by the amplitude 

 of oscillation being not sufficiently small, though in our 

 experiments the angle of twist per unit length of the wire 

 did not exceed eight minutes; for, by increasing the length of 

 the suspended wire and keeping the amplitude of oscillation 

 constant, the rate of the increase of logarithmic decrement 



