1887. ] deformation of elastic plates. 145 
Boussinesq will be found in his memoir (tude sur Pequilibre et le 
mouvement des solides dont certaines dimensions sont trés petites 
par rapport @ @autres), published in Liouville’s Journal in 1871, 
and they are substantially reproduced in de St Venant’s trans- 
lation of Clebsch’s treatise, note on § 73. 
Exactly similar considerations apply to the theory of wires. 
The result will be the establishment of Kirchhoff’s views as to the 
kind of strain which can take place in an indefinitely thin wire or 
plate deformed in such a manner as to remain continuous. 
It seems advisable to recapitulate the general method given 
by Kirchhoff, Vorlesungen, XX VIIT., for the treatment of elastic 
bodies some of whose dimensions are indefinitely small in com- 
parison with others. 
In this method we consider in the first place the equilibrium 
of an element of the body all whose dimensions are of the same 
order of linear magnitude as the indefinitely small dimension. 
When we know the potential energy due to the internal strain 
of such an element, we obtain by integration over the remaining 
dimensions the whole potential energy due to the elastic strain of 
the body. Then taking into account all the forces which act 
on the body we can form the equation of virtual work which will 
lead directly to the differential equations and boundary con- 
ditions of our problem. 
Now let ¢ be a small quantity of the same order as that linear 
dimension of the body which is supposed small; and consider the 
equilibrium of a body all whose dimensions are of the order e. 
Let (a, y, 2) be rectangular coordinates of a point of this body 
and F(a, y, 2) =0 the equation of its surface; X, Y, Z the bodily 
forces per unit of mass applied to the solid, and F, G, H the 
surface-tractions. We write the six component stresses P, Q, 
te SHG, viz. 
on the face yz, P parallel tox, U parallel to y, T parallel to z, 
on the face zz, Uparallel to x, Q parallel to y, S parallel to z, 
on the face zy, T parallel to 2, S parallel to y, & parallel to z, 
and the six strains e, f, g, a, b, ¢, viz.: 
fe Soap Oh, aI 
mies =) Moy I~ oz 
Hee gd he 2 a, Oh MD gill 
dy Gz ° dz Ou ° Om dy’ 
where uw, 2, w are the displacements, in the direction of the axes, 
of the particle originally at x, y, z. The stresses P, Q, R, S, T, U 
are linear functions of the strains e, f, g, a, b, c. 
