Elastic Constants of Quartz Threads. 107 



Let a be coefficient of expansion of brass (linear). 

 aj „ „ quartz (linear). 



m be mass of vibrator of brass. 

 k be radius of gyration of vibrator. 

 / be length of fibre. 



where n = rigidity of quartz. 

 r = radius of fibre. 



Let T : be time of a complete oscillation at temperature t 1 : 

 then 



' 1 = 2^!!^ = 27ry / 



mkH . 2 



Let the temperature of both brass and quartz change to t 

 and the time to T 2 ; then 



V ni 



r.2 



where k', I 1 ', /, n denote the corresponding values of these 

 quantities. Whence 



Now // = /j{l + «(/ 2 -g[and r , =r{l + a , (t 2 -t^}. 



\1 + (|«'-«) Qt-h)} 



Now neglecting any possible change in n, i. e. putting 

 n = n', T 2 will be greater than T x if J- u x — « is negative. If, 

 on the other hand, the rigidity changes, 



Hl) 2{i - (3 *'- 2a) fe -*' )} - 



If the temperature of the brass is less than the temperature 

 of the fibre when both are heated, as was the case in some of 

 the experiments, we have 



'h 



T, 



=\Z^[ 1 +i^-^)--ft-o], 



allowing us to estimate the change in n when a and «' are 

 known. 



