208 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
41. The problem of given edge tractions for a thin plate, 
The form of the complete solution is exactly known, and the three boundary 
conditions in their exact forms could, therefore, at once be written down. The whole 
strain is compounded of an infinite number of modes of equilibrium of known types, 
and it is obviously suggested as the method of attack that we should try to disentangle 
from the general boundary conditions those special conditions by which each mode is 
separately defined. When the plate is thin we find that within certain limits this 
can be done, and, in particular, the conditions defining the permanent modes, which 
in the case supposed are incomparably the most important, can be found with con- 
siderable exactness. 
We shall understand that the edge traction, or any component of the edge traction, 
is given as a function of x, y,2/h or of s,z/h, where s is the arc of the edge line, so 
that if ¢ be put for z/h the form of this function is completely independent of h. The 
theory may be applied to cases in which the proviso is not fulfilled, but before such 
application the given traction is to be separated into parts of ascending order in h, say, 
for example, fi(w,y,Q)+h A(a,y,QO+h fi(x,y,0)+etc.; then for a first approxi- 
mation we deal only with fi(x,y,¢). The theory does not contemplate such a 
distribution of traction, as, for example, sin (ms/h) ,m being a number, where the rate 
at which the traction varies along the arc is of a lower order in A than the traction 
itself. 
The trace of the cylindrical edge on the middle plane of the plate is the edge lune ; 
the outward normal, and the tangent, to the edge line will be referred to as the 
normal, and tangent simply ; the generator of the cylindrical edge at right angles to 
these at their point of intersection may be called the perpendicular. 
Let J, m be the direction cosines of the normal, 
then —™m, l are those of the tangent. 
The normal displacement is p=lutmv 
and the tangential displacement q=—mu+lv. 
The tractions on the edge in the directions of normal, tangent, and perpendicular, 
are nn,ns,nz or N,S,Z. 
AY, Hextensional strain. 
In this case N, S are even functions, and Z an odd function of z. 
It will be advantageous to express as far as possible the displacements and tractions 
at an edge in the various types of solution in terms of derivatives along the tangent 
and normal. 
Alongside the symbol a we shall use the more familiar o, the relation between the 
two being given by a+1=4(1—c); 3—a=4o. 
