216 MR JOHN DOUGALIL ON AN ANALYTICAL THEORY OF 
Hence, in order that the permanent mode should be absent from the strain due to 
N, 8, Z the following two conditions are sufficient :— 
aN 
N,- 2 Zope Oe 5 eee 
w p ties ds? 2 » ads 
at every point of the edge line or lines. 
Further, all systems of traction for which the left-hand members of (7) have given 
values at every point of an edge will produce the same permanent mode. Now as one 
such system of traction we may take the traction due to, or producing, the permanent 
mode by itself, as given in § 42 (i). This gives at once the boundary conditions satisfied 
by the functions U, V of that section, and these boundary conditions, with the internal — 
equations 
d& _ a 0: OVEN ie 
ie en. hae are 
completely define U, V which are thus determined, to a third approximation in general. 
The defining differential equations and surface conditions being practically of the same 
form as in the familiar first approximation, we need not detail the proof that U, V are 
actually determinate from the conditions, but pass at once to the important conclusion, 
an immediate consequence of this determinateness, that the permanent strain will not 
be absent unless (7) are satisfied, or, in other words, that these conditions are necessary, 
as well as sufficient. From this again it follows that these conditions are fulfilled, to 
the order stated, by each of the transitory modes ; and this remark is valuable, because, 
once it has been verified by direct integration, it obviously leads, by an extension of 
the process of § 42 (iv), to a completely independent method of dealing with the whole 
problem. The method is noticeable for its simplicity and directness, hut a somewhat 
serious defect is the difficulty of adapting it to the case when the edge stress is 
discontinuous. 
This leads us to consider the correction that must be applied to the integral (6) 
when the conditions of continuity stated in connection with it are not fulfilled. It will 
be sufficient to take a case in which breach of continuity occurs at only one pomt E of 
the edge line. 
We have defined the positive direction of an edge line in (4); let the excess of the 
value of f(s) just on the negative side of EK over its value just on the positive side be 
denoted by [f(s)]. Then if (s) be continuous 
[7 Zo@as= sol fo Zeus, 
the integrals, we may suppose, being taken round the edge line from the positive to the 
negative side of HK, and the value of #(s) in the integrated term being taken as at EH. 
