214 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
Now U’ is the work difference from the system 
1 
Srph 2 
1 
Sah Sey ae ee 
a am 32mrph ac zi x) 
y= ye $2°y7x) 
From this system let those terms be removed which transmit the resultant stress 
(equivalent to a unit X force through a’, y’, 2’). Then the remaining displacements 
have still the forms discussed in § 42 (i), and we shall use for them, and for the various 
quantities related to them, the notation given there, modified by the addition of a 
suffix 0 in each case. 
The problem is now to balance the edge tractions due to the system w , V9, Wo. 
The principal parts of these tractions, which in this case are simply the terms in- 
dependent of z, are balanced by a solution of the permanent type (which we shall — 
distinguish by the suffix 1) such that at the edge 
dQ, P, ( IOP Pa. | 
Cites 
. 
(- 1, +72-*) + (-1,+-%)<0 | 
These conditions define the solution with suffix 1. 
The residual tractions from the compound solution w+, Vo +v,, Wo+wW, contain — 
the factor z* and are of order h’ as compared with those already balanced. To balance 
these residual tractions, solutions of all types are required, but, as in § 42 (iv), the per- : 
manent solution (which will be marked with sutlx 2) is determined from the integral 
residual tractions; the permanent displacements are of order h, while those from the 
transitory solutions are of order h”, those from the source solution being of order b> 
The displacements of the balanced solution are therefore to terms of order / inclusive 
in the notation of § 42 (i), 
, ‘ d 
ee aia + Po — go022— (Ty + TT) 
, 3 c tar 2 
q = M+Q +Q,+ jo 2 (a, +A,) @) 
B ; 
w= —o2(A,+A,) 
and with these values of p’, q’, w’ 
Ua ae) — | {ow +Sq'+Zw')\dsdz  , ‘ ‘ ‘ (3) 
All the steps of the above process can actually be carried out in the case of a cireular 
plate, and the final formula gives a perfectly definite solution provided merely that 
N,8, Z are functions integrable over the edge. It should be specially noted that, in 
fits sen of the solution, discontinuity of the applied traction gives rise to no difficulty 
whatever. | 
On the other hand, the formula does not give a ready answer to such an important 
question as “ What are the relations between the tractions actually applied, and the 
