239 
1912-13.] Torsional Oscillations of Metallic Wires. 
of set with simple harmonic motion provided that neither limit was exceeded. 
Such a condition in complete oscillations is unknown in nature ; but it 
was this result which suggested the comparison of the observed data with 
a simple harmonic graph having its zero at the position of set (§ 5). 
§ 10. Hooke’s Law being postulated as applicable on the average, the 
energy lost in the range d(m<pjr) is proportional to the square of m(p 0 
multiplied by the number of groups which break in that range. In this 
way the whole amount of energy dissipated in an oscillation may be 
determined ; but we shall proceed by estimating the force. 
Suppose now that the oscillations have decayed until the last positive 
(right, say) maximum shear was q u <p 0 - No energy is dissipated until the 
subsequent negative (left, say) shear exceeds (1 — q v )<p 0 - Consider the 
negative shear q(p 0 greater than this. Sheared to the left, and therefore 
pulling inwards, there are those groups which were, in the initial state, 
sheared to the left within the limits (l — q)</> 0 and </> 0 , and which have 
therefore broken within the range of shear 0 to qq> 0 . Postulating Hooke’s 
Law and neglecting the constant factor, the pull per unit volume of the 
cylindrical shell exerted by these is got by integrating 
--f ,, (m<f> 0 )d(m<f> 0 ) • (q - m)<f> 0 , m from 0 to q . . . (1) 
Also, in that range are the groups originally there and which were to 
break at (1 — m)<p 0 to the right. These give rise to the pull 
- /"( 1 - m)<f> 0 d( 1 - m)<f> 0 • (q - m)<f> 0 , (1 - m) from (1 - q) to 1, 
or +/"(1 - m)<fi 0 d(l - m)<£ 0 -[(l- 2 ) - (1 - »i)]<#> 0 
or • (1 - q - m)4> 0 , m from (1 - q) to 1 (2) 
Sheared to the left also are those groups originally sheared to the right 
within the range qq> 0 to 0 O , and which broke in the twist to the right 
within the range 0 to (l — q)(p 0 . Let the latter shear be m<p 0 . The total 
shear is therefore (q + m)0 o , and so we obtain 
-f"(?ncf) 0 )d(mcf> 0 )(m + q)<f> 0 , m from 0 to 1 - q . . . (3) 
Finally, sheared to the left, there are those groups which were initially 
sheared to the left within the range 0 to (1 — q)(p 0 , and which would break 
at the shear (1 — m)0 o to the left, (1 — m) lying within q and 1. So 
we get 
-/"(l - m)<h Q d{ 1 - m)<f> 0 • (q + m)<f > 0 , 
or -/"( 1 -m)<f> 0 d( 1 - m)cf> 0 • [(1 +#)-(! - m ) ]<£ 0 , (1 - m) from q to 1, 
-f'Xmcf) 0 )d(m<J> 0 ) • (1 +q - m)cf> 0 , m from q to 1 . . . (4) 
or 
