1887.] deformation of elastic plates. 147 
The surface formed by all those particles of the plate which in 
the natural state lie on the plane midway between the two plane 
bounding surfaces is called the middle-surface of the plate. 
Let any point on the middle-surface be taken as origin, and 
let a, 8 be rectangular coordinates of a point on the middle-surface. 
Let this surface be covered with a network of lines a=const., 
8=const. at distances from each other comparable with the 
thickness of the plate. A prism whose normal section is one 
of these small rectangles will be called an “elementary prism” 
of the plate. 
According to Kirchhoff’s general method we have first to treat 
the equilibrium of one of these elementary prisms. 
Let a, B be the coordinates of the centre P of one of these 
elementary prisms before strain. Imagine three line elements 
of the plate (1, 2, 3) to proceed from P, of which the first two 
are parallel to the axes of a, @ and the third perpendicular to their 
plane before strain. Then after strain these axes will not be 
in general rectangular, but by means of them we can construct 
a system of rectangular axes whose origin is P, to which we can 
refer points in the prism whose centre is P, in the following way. 
The line-element 1 is to lie along the axis of x, and the plane 
of wy is to contain the line-elements 1 and 2. Then the line- 
element 2 will make an indefinitely small angle with the axis 
y, and the line-element 3 will make an indefinitely small angle 
with the axis z. 
Let Q be a point in the prism whose centre is P; and before 
strain, let 2, y, z be the coordinates of Q referred to the axes at P. 
Let & 7, € be the coordinates of P after strain referred to axes 
fixed in space and coinciding with the initial directions of a, 
and the perpendicular to their plane. After strain let 
Dai UY A Onn Ziska tv, 
be the coordinates of Q referred to the (a, y, z) axes at P constructed 
as above described, and let the directions of these axes be con- 
nected with those of the fixed axes (&, n, ¢) by the scheme 
Slog pus 
a m, | n, (1) 
TaN Were era Wa ee ia : 
BN | GOS MOTD. 
then the coordinates of Q after strain referred to the fixed axes are 
E+), @+u)+l(y+v) +1, (2+), 
H+ m,(L+U)+M, (Y+V) +M,(@+W), > veereeeee (2). 
E+ n, (a+) +n, (y +0) +0, (2 + w). 
