176 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
the two particular solutions do not fit, that is, they do not give the same values for a 
displacements and strains on the two sides of the cylindrical surface or surfaces of 
discontinuity. 
On the other hand, the supplementary terms required in order to make the solution 
synectic belong to what we have called the decaying type. They give rise to displace- 
ments and strains of infinitely high order, if we may so speak, in the small quantity h, — 
except very near the surfaces of discontinuity. This being so, we need not be 
surprised to find that the solution (72), (78) is not necessarily the simplest particular 
solution in any one region within which Z is continuously constant. 
Thus, for example, if we pick out the terms which contain z” as a factor, we find 
displacements proportional to 
ioe | 
ie 
dy | 
w=VPF + vB Je + Oe) 
which belong to the type (23), and contribute nothing to body force or face tractions. — 
These terms might therefore be omitted in any problem where the condition of synexis — 
is irrelevant, and in particular when the object is merely to obtain a particular solution — 
for body force and face traction in a problem relating to a finite solid. 
24. Internal force parallel to the faces. 
We will now go on to consider the problem of force applied to the body in a 
direction parallel to the faces of the plate. 
A force of 47u(a+1) units applied at (a’, y’, 2’) in the direction of Ox gives in an 
infinite solid displacements defined, according to (9), by 
es = a+ 1 d q-2ytl 
2 dy Vaz) 
fa oe ad d-*r-l yh ar 
9 dx dz dx dz! 
© IT Sd Bde 
oe 2 dx dz? 
Hence the tractions which such a force produces on z= +A will be neutralised by a 
system 1, 9, » for which 
ay Ce are 
dzg7 ) 
2 
2 -1p-1 
dé, db na 2 Pp _ (ee Se r) ong= +). 
86, 9,84. d(a=I 
d2@ de ~ deb dx 
ar 
2 Slew, AI mae 
5) ee ee, ) 
dz 
