1336 
per unit of volume caused by the torsion 
1 
uw (1+ 29+ 2s) + a (9+3s) + (sts ral 9, 
7 
La 
Proceedings, 19 (1916), p. 281). I shall avail myself of the occasion of this trans- 
lation for introducing the corrections necessitated by his remark. 
In deducing the new equation (21) we need not occupy ourselves with the term 
by; we have only to show that 
u'=u(l + 22) + a(X + 3B), 
if y =O. By this latter assumption the problem is reduced to one in two dimen- 
sions, which may be treated as follows. 
Let x, 2 be the coordinates of a point in the original state and «+ £, 2 
its coordinates in the strained state, the displacements £, & being functions of «x, 
z. We shall consider the free energy per unit of volume at the point x- &, 
zt as compared with the free energy which we had originally in unit of 
volume. 
The difference ~ must be a function of the quantities 
fe) we AE OEE 
DH op ea ee oe 
and can be developed in ascending powers of these, the series beginning with 
quantities of the second order and terms of the third order being necessary for 
our purpose. 
As we may assume that the free energy is the same in the body considered 
and in a second body that is the image of the first with respect to a plane 
perpendicular to one of the axes of coordinates, the expansion can contain no 
terms that are of an odd order with respect to bj and bj. Moreover the value 
of W must remain the same when the axes are’ rotated in their plane. These 
considerations lead to the formula 
w =f (a, sd) (6, —6,)* ++ bh @, a, — 6, De) (A A) He 
Bais SN, bi Dt tt Garry AN (ety 074s) 
with six constants /,g,h,k,/,m, which can be easily verified. Indeed it can be 
shown that the values of aj + d3, bj — bg and a 43 — bj bz are not altered by 
a rotation of the axes. 
Let us next suppose the body, strained already in the way determined by 
di, da, Dj, Dz, lo be rotated about O Y through an infinitely small angle w. This 
rotation, which must leave the value of J unchanged, leads to the variations 
dE owlet), doet §), 
O(a, + a,) =o (6, — 6,), 0(6, — b,.) = — 2 0— w (a, + a,) 
Ora —— — es) 
Substituting these values in dy and putting equal to O the coefficients of the 
terms that are of the first and the second order with respect to dh, ay Di, 5, one 
is led to the relations 
heet We yg ad, 
> 
so that 
