THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PILATE. 227 
If we write F for F,+ F’+ F”, then it follows from (6), (7), (8) that the terms of 
two lowest orders in F, are given by 
es _ ne Ca 
a | ZzNdz— of 2sdz+ F/ Ldz 
F dnJ —n salen Fi 
7a 1 dF 1 dE ds 7 (9) 
a me SS uae 9, ] 
ee a dn op ds We 2h'B(z/h)Sdz | 
As in the extensional case, § 43, the integral with respect to s may be modified by 
: : L : : 
means of integration by parts so that only F and 2 appear under the integral sign. 
Write 
rh h h 
| ZNdz—ING, Sdz=8y, | Ldz=Z, 
—h —h —h 
a (10) 
| 2h? B(2/h)Sdz=Sy,. 
J =<h 
Then if N,, 8,, Z,, S, are continuous functions of s, 
dF BB dS dS, < 
n(tens $2) 
re - 5) =|{- ma peal ds sie tee ds p _ ey) 
Hence any two systems of traction will give the same permanent mode to a second 
approximation, provided the values of N, +& and a +Z)+ = So 
p 
are the same for the two systems. 
Now for the system F,, § 44 (i), 
hee = EOS = BE Zo = ph?9,F, | 
<0) 
ae = he eae, § 45, (iii) (13). | 
Thus, with 3,,9,, 4, defined as in § 44 (1), the boundary conditions are 
Ane 
- "A(R, + a ish <35B, )=N, Bs 
4uh? ae J ae ds, ad 8, ue) 
—(z SF, + 9,F) + — y;h m4 + Z, +— 
3 dsp ds ds p 
When only the principal terms are retained, these reduce to Kirchhoff’s conditions. 
If §, or 8, is discontinuous at any point of the edge, integrated terms will appear 
im equation (11), as in the extensional case, § 43. Thus, if the normal couple S, be 
discontinuous at a point P (s=s’), there will appear on the right of (11) a term 
OA ales 
The method of dealing with such a discontinuity in any actual problem is obvious, for 
by (9) its effect is the same as that of an element | S, | of shearing traction applied at 
P, a result which on the ‘elastic equivalence’ theorv may easily be obtained by a 
trifling modification of the process by which THomson and Tarr reconciled the con- 
ditions of KrrcuHorr and Porsson. 
