1334 
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§ 11. For a few metals the value of 7 can be derived from 
«tog v 
measurements made by Poyntinc.*) This physicist has investigated 
the changes in length and diameter caused by the torsion of a wire. 
We shall shortly discuss this phenomenon, not only with a view to 
the numerical value that follows from it, but also because the 
theory shows a certain analogy with that of the dilatation by heat. 
Let us consider a cylindrical wire, the axis of which we take for 
the z-axis, and let us suppose that, starting from the unstrained 
state, it is subjected- to the following three deformations: 1. a 
homogeneous stretch in the direction of the length, 2. a dispiace- 
ment of the particles in radial direction, so that the distance 7 of 
a particle from the axis changes by sr, s being a function of7, and 
3. a torsion, by which each cross-section normal to the axis is turned 
over an angle 9z about its point of intersection with that line; then 
9 is the angle of torsion per unit of length. 
Supposing the temperature to be kept constant we shall seek the 
free energy of the body in the final state reached by these three 
steps. Assuming it to be 0 in the original state we can caleulate 
its changes by means of (12) or of similar expressions. 
d(s 7) in 
As the second of the three changes produces a stretch 
a 
radial and a stretch s in tangential direction, we obtain the free 
energy that exists per unit of volume after the first two steps if 
ds 
we replace «yy 2: in the first two terms of (12) by s,s + os 
; r 
and g. 
The result is 
ds ds\? ds £ 
ui2s + 2rs— + 15 + q' | +42 (2 +r— + ‚) (LON 
dr dr dr 
A point that originally was at a distance 7 from the axis, has now 
shifted to the distance #/—=(1 4 5)r, while an element of the length 
dl has become dl’ = (1 + q) dl. 
By the first two changes an annular element between two cylindric 
surfaces described about the axis with the radii 7 and r+ dr, and 
further limited by two cross sections at a distance d/ from each 
other, will have taken a volume for which with the approximation 
required for our calculation we may write 
1) PoynriNG, On the changes in the dimensions of a steel wire when twisted, and 
on the pressure of distortional waves in steel, Proc. Royal Soc. (A) 86 (1912), 
p. 524. 
