218 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
With reference to the equivalence of mere discontinuities to line elements and — 
doublets one or two remarks may be made. Discontinuity in the applied force will not 
produce infinite displacement at a line where it takes place, but a line element of load, 
and, a fortiori, a doublet, will do so. The permanent mode may therefore contain 
infinities at the edge which do not exist in the exact solution. There is really no 
difficulty in this, since the permanent mode does not purport to represent the strain, 
even approximately, in the immediate vicinity of the edge. The point may be 
illustrated by the permanent part‘of the infinite solid solution fora single force. This 
becomes infinite on the perpendicular through the source in a totally different way from 
the exact solution. A good deal of discussion took place at one time over a similar 
point in the flexural solution. This will be referred to again, but the considerations we 
have adduced seem to remove the chief part of the difficulty. 
44, Flexural strain. 
In this case N, S are odd functions, and Z an even function of z. 
(i) Permanent mode, 
This mode is defined in terms of one function F of (a, y) satisfying V*F =0, and 
may be referred to simply as an F strain. 
@ = —-(2F -}y2F); 6 = 2F- 427k — 2h2zyF 
— 4(1 —o0)(2F — be y?F) + 2(428 — hz) VF 
s 
ll 
4(1 —o)(F — $22y?F) + 2(2? — h?) vy? 
b) 
For shortness in writing out the stresses, we shall work with symbols 9,, 9,, 3, 
denoting operations of differentiation applied to F, and defined by the equations 
sate 0(hd 4 2) 
d= 40 -o( Fs = a) 
I, = 4i 
Then 
= ape {tee tt (Lam) Layer 
