192 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
It is now easy to see the significance of the forms of the coetticients in the solutions 
of § 29 and the confirmation of the values there given would obviously present no 
difticulties. 
33. Finite plate under edge tractions. Form of the solution deduced 
by means of Bette’s Theorem. 
We pass, however, to a more important application of the theorems (97). The 
system wv, v, w we still suppose maintained by edge tractions alone, but in addition to 
the external edge the solid may now be bounded by one or more internal edges. For 
%,%,, W,, ete., we take the definite values defined in (79), (82), (83), and in (63), 
(64), (65). 
Thus in (97) uw, 01, , X1, Y;, Z,, and the other displacements and tractions marked 
with suftixes, are known functions of «’, y’, 2’, and the equations give explicitly the 
values of the displacements at any internal point in terms of the displacement and 
stress at the edge or edges. 
The ideal solution would give the internal displacement in terms of edge displace- 
ment alone, or of edge stress alone, but the analytical difficulties are such that we are 
unable to solve the problem thus completely even for the simplest case, that of a single 
infinite plane edge. Meantime, however, we may derive valuable information from 
the expressions of (97), and in the first place as to the form into which any solution 
due to edge tractions alone may be thrown. 
Just as in the case of the original source solutions, we find that the solution, in 
which, of course, the accented letters are now the variables, may be decomposed into an 
extensional and a flexural part, while in each of those parts we may separate a permanent — 
mode from an infinite series of transitory or decaying modes of two types, the \ type, 
characterised by no dilatation or normal displacement, and the 6, ~ type, in which 
there is no molecular rotation in the plane of the plate. 
In the following analysis integrals of the same form as those in (97) occur 
frequently ; the system w, v, w appearing in each case, but associated with various 
other systems. For conciseness we shall refer to the first integral of (97) as the work 
difference from u,, Vy, W,, and similarly in other cases. 
I. Extensional part of the solution. 
(i) Permanent mode. 
In w,, v;, w,, the terms which relate to this mode are the unambiguous terms, even 
in Z, of (83), after these have been divided by 47u(a+1). These, as may be seen from 
a glance at the beginning of § 27, are equivalent to 
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