Elastic Constants of Quartz Threads. Ill 



actual length of the active part of the fibre is also uncertain ; 



but of course it is easy to see that almost all the work must 



be done by the thin part, since the resistance to torsion varies 



as the fourth power of the diameter. This enables the results 



to be applied to really uniform cylindrical fibres without 



appreciable error. We can easily estimate the total change 



in the force of torsion induced in any thread by raising its 



temperature one degree. In any instrument, such as a 



gravity-meter, this coefficient of total change is all that is 



required, and, as will be seen, is the same for all fibres of 



similar figure. Let (f>t l be the " moment of forces " exerted 



by the fibre under given conditions at temperature t 1} i. e. 



irnr o 

 9 j , <f>t. 2 be the corresponding value at t 2 . Let T x and T 2 



be the time of a given number of vibrations at the lower and 

 upper temperatures respectively ; and let a be the coefficient 

 of linear expansion of brass. 

 Then 



^ 2 = ^(g)(l + 2a0, 



where t is range of temperature through which the brass 

 vibrator is heated. 

 In the experiments 



^ = 22° C, 

 * 2 =98°-5C., 

 .-. £ 2 -£ 1 = 76°-5 C, 



range through which quartz is heated. 



The brass was heated from 22° to 67° C. ; therefore 

 *=45°, T^ll^^HTs, and T 2 =ll m 36-3125 s . Whence 



^2 =1-01018 = 1 + 76-5e say, or e= '00013307 ; and may be 



called the temperature-coefficient of torsional stiffness of 

 cylindrical fibres. Neglecting the increase in length of the 

 fibres this will be the same as the temperature-coefficient of 

 the modulus of torsion. In order to get the temperature- 

 coefficient of the rigidity one requires to know in addition the 

 coefficient of expansion of quartz, as has been said, and this 

 has not yet been obtained to our satisfaction, t. e. Mr. Pollock's 

 and mine. The degree of reliability that is to be placed on 

 our estimate may be judged from the table (see below) which 

 embodies the result of our researches. The probability,, how- 

 ever, is that by taking a' = '0000017 we shall not be very far 



