1335 
1. 
27 (: + 2s+ r _ | i) OP Ob rg od a 
Now to obtain the free energy in the state S that is reached by 
the first two deformations we should have to multiply (19) by this 
expression (20), and then to integrate it with respect to r and /. 
For this calculation however we may replace (20) by 2ardr d/, 
because in the expression for the free energy we shall omit terms 
that are of an order higher than the second with respect to gand s. 
§ 12. To caleulate now the change of the free energy accom- 
panying the third deformation specified in § 11, we shall consider 
the state S as the original one and introduce elastie constants refer- 
ring to it. On account of the preceding deformations determined by 
q and s, these constants are a little different from the values 2 and 
uw introduced into (12). To find an expression for them we regard 
the quantities q and s as infinitely small and neglect their second 
and higher powers. The change we are investigating being propor- 
tional to #*, we obtain in this way terms with 9° and 59? 
A point which in the state S has the coordinates x, y, z and 
lies at a distance r’ from the axis, is displaced by tie torsion 9 over 
the distances 
§ = — Dyz, n= + Bez, Cae 
to which correspond the components of strain 
Ge apa Oy Pea Ue rly ay ‘= Use Deken. 
Let us now consider an element of volume which in the state S 
lies at a distance 7 from the axis and for which «=0, y=r". The 
preceding changes have given to this element the stretches x == s, 
Yos+ ro and 2=g in the direction of the axes, without other 
changes of form. By the torsion it is now further subjected to a 
shear x= — 9r'. . 
It is evident that the change in free energy per unit of volume 
caused by this shear will be obtained by multiplying 4e.” by a co- 
efficient mw’, which is the coefficient of rigidity u as it has been 
modified by the dilatations x, y, z. In calculating this modification 
we may treat X, y, Z as infinitely small. it can be shown that 
w=u(l +2) dax) by, .... . (21) 
where @ and 5 are two constants depending on the nature of the 
material '}. In this way we find for the change of the free energy 
1) In my original paper I had used a wrong formula, in which the term 2 u z 
was wanting, an error that has been pointed out by Mr. Trestine in his paper: 
On the use of third degree terms in the energy of a deformed elastic body, (These 
