1337 
where r’ has been replaced by (ls) 7 and where, as to g and s 
we have neglected terms of orders higher than the first. 
Multiplying this by (20), integrating over the cylinder and adding 
the result to the free energy in the state S, of which the value 
has been found already, we obtain for the free energy in the final 
state 
R 
Sap if 9 TU ke : ds \? | EEn ds 2 
wan u 8 En +7 ae Hg +} A Ue eee + ‚) [rw 
8 a 
R 
10° ds ds 
fe (1 + 4s+-r— + 1)-+a+9)+ b (s+ =) | dr . (22) 
l dr dr 
0 
where the original length and radius are denoted by / and R, 
iJ 
(+g) 
-4- 
and the total angle of torsion by 9, so that 9 = 
§ 13. Now @,q and the value s, which s assumes for r— R, 
may be regarded as the parameters upon which external forces can 
act directly. If these parameters are kept fixed, we can determine 
the values of s within the wire by means of the condition that, for 
an arbitrary infinitesimal variation ds given to them, dw must be 0. 
For constant values of @ and q we have by (22) 
R R 
dds 
dp | Gdsdr + eae Ty eenen aA) 
i ls 
0 0 
where 
w =f @, daje oe J Bij Ot 4g Als 0;,0,) zis k (a, SiGe): te 
+1(a, da) (6, —6,)? Hf 2g — Al) (a, + 4,) (a, a, — 4, 6). 
In the case considered in the text the final values of £ and Z (after the 
application of the torsion) are 
§é=—=xe—d8(1i+2_)r2=—=xe+r1+2)2, $'=2z, 
if x, 2 are the coordinates in the original state (before the application of the 
dilatations x, z). Hence 5 
ia dr eve (1 Ae), 8; =O 
If these values are substituted in the expression for &, the coefficient of x-? will 
give us the value of 3 «’. Hence 
wig 422) HKH), 
or, if we replace 2/ by a and observe that, for x=0, z=0, u’ must be 
equal to g, 
u=u(l +22) +a(x Jz). 
