210 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
If P, Q be the values of p, g when z=0, then 
ail dA 
po ee gt O 
p=P- hoz qi 4 Q + $02 -F 
o% 
mx, _4Q_ P_1 i) 
2p Gky “yy ds” p ds 
as dP_Q 1 caf 20 nay 
Iu dsp. Oo Carinae we: 
ae ow at 9 HY 
dy 2p da dy 
v= — ath ne 
ig 2 ay? da? 
a ay 
2p dy dz 
Taking the axes of « and y for a moment along the normal and tangent, these 
give at once by means of (h) 
We gt m _9d dy _ 2 dy 
ds 2u dsdn pds 
eee ns_ Uy, 2dy , ty 
Dee, 2) ae aan de 
ne Cy 
2p dsdz 
The function \ can be expressed as a series of terms of the form 
ules 9) COS, where YY, — Ey, =0. 
Hence in cases where the values of along an edge are given independently of h, 
or generally, when the rate of variation of along an edge is of the same order 
in h as W itself, say order zero, terms of various orders occur in the expressions 
for the displacements and tractions. 
Thus 
2, 
= ; os are of order - 0) 
dy addy d dy 
dn’ ds dn’ ds dz 
ay : 
as atta eee 
dz = 
—1 
It follows that in such cases this type of strain contributes mainly to the tangential 
displacement and traction at an edge. 
