1888.] Mr G. H. Bryan, On the Stability of Elastic Systems. 207 
with F’, G, H, the latter will still be so large as to produce “ set,” 
unless the strain variations be infinitely small in comparison with 
those of the displacements. 
But this is precisely what happens when such a thin wire or 
shell is deformed by pure bending or twist. 
In Kirchhoff’s theory of the bent wire*, the wire is supposed 
divided up into a number of small elements by planes perpen- 
dicular to the middle line at distances apart of order e, where e is 
comparable with the thickness of the wire; & 7, € are supposed to 
be the coordinates of the centre of any such element. With this 
centre as origin, Kirchhoff takes a system of rectangular axes(a, y, 2), 
that of z being the tangent to the middle line, and that of # being 
a fixed transverse of the wire. These axes are supposed to move 
with the element. He also introduces three quantities p, q, 7, 
depending on the bending and twist, which we shall denote by 
«,A,T. If the wire is but slightly bent x, A, 7 represent the rate of 
change, per unit length of the middle line, of the angles through 
which the axes in the various elements are rotated by bending. 
Supposing 6 to be a quantity of the same order of magnitude as 
x, X, T, then by considering a finite length of wire, it 1s evident 
that the component rotations of the axes in any element, as well 
as the component translations of its centre (&, 7, €), are quantities 
of order #6. The total displacement of any point in the element is 
therefore compounded of a rigid-body displacement of order @, 
together with the displacement of the point relative to the moving 
axes of a, Y, 2. 
Now if we suppose that instead of the wire being purely bent 
the extension of the mean fibre is o, the expressions for these small 
relative displacements are to a first approximation 
2 Y a® 
U=— ope + TyZ—NG— Du x + Kacy 
10) Oe IM oh eke 21 
V=—opy — 720+ «5 + es — uy a 
W=o2+ TW, + ANZ — KYZ 
where «=ratio of lateral contraction to longitudinal elongation, 
and w, being found, as in Saint-Venant’s problem, is a small 
quantity of order e’. 
If o =0, or likewise if o is a small quantity of order ¢ in com- 
parison with «,X, 7, the expressions will be quantities of order ¢’6, 
the strains therefore will be of the order ¢0, and the displacement 
* Vorlesungen, No. 28. 
