146 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
traction of 87 units at (x, y’, h), a solution is given by 
p= got+e3 I=A+ 4, 
where 
Fo= fit- eceeeers Jock sinha pat re +i) } a a 
ue [{Oneze el ashen S58 (Beat, Be Seta | 
é=[) ate iia oi dee aes ue an 
* Be ca| 
b= sinh kh + 2«h cosh kh Baer eos bee 1 
=| «(sinh 2«h + 2«h) gS ei Bre aes 
The conditions satisfied at the faces by the partial solutions (16) and (17) are 
easily made out. For when ¢, @ are both odd functions of z, then Zz, z are even 
and x odd; but when ¢, @ are even functions of z, then zz, z are odd, z even, as 
is obvious from (5). Hence (16) gives equal values of opposite sign for z at corre- 
sponding points on z= + /; (17) gives equal values of the same sign. 
It follows that (16) is the solution for elements of normal force of 47“ units at each 
of the points (x’, y’, 1), (x, y’, —h), the force being in the positive direction of Oz in 
each case, and therefore a traction on z=h, but a pressure on z= —/; in (17) the only 
difference is that the force is a traction on both planes. 
Hence, also, (16) subtracted from (17) will give the solution for traction on z= —h 
alone. 
Each of the integrals in (16), (17) defines a potential function without singularity 
at a finite distance in the space between the planes z= h, and all the successive deriva- 
tives with respect to x, y, z of any of these functions may be calculated by differentia- 
tion within the sign of integration, provided we are dealing with a point actually within 
the solid, so that -h<z<h. 
The solutions defined by these integrals are therefore formally satisfactory. It is, 
however, a serious objection to them that they do not lend themselves readily to inter- 
pretation, and it is not easy to make out from them any of the simple laws which the 
ordinary approximate theory leads us to anticipate. | 
In particular, the solutions in their present form throw no light on the question of 
the behaviour of the functions and their derivatives at points the distance of which from 
the sources of strain is great in comparison with the thickness of the plate, a question of 
great importance for the application to the thin-plate theory. 
The analytical transformations to which we now proceed reduce the solutions to a 
form entirely free from these objections. Each of the integrals is shown to be composed 
of two parts of very different character. The first part represents a function the value 
of which diminishes with great rapidity as the distance from the source increases, while 
the remaining part is a function of very simple form. ach solution is thus resolved 
into a permanent or persistent element and a local, transitory, or decaying element, the 
latter being insignificant beyond the immediate vicinity of the source. 
