224 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
If we write F for F+K’, we get from (4), (14) 
neglecting the term in AF’, which is of order /? relative to F. 
This may be written 
(5, +t hf L$.) = 0 a 
Similarly from (6), (16) we obtain 
a 
d 384 a Il == 
(9+ Ft wt Ene - zal, satin i 
We may regard (17) and (18) as the equations giving F to a second approximation. — 
If we combine the results of the three cases of this article, we obtain 
J,F= - os [ zNdz 
aera ae ge (2) 
d 3 bs Cele 
SF+2 5,F— — ca | tere) Se \ 
These are the equations usually referred to as Kirchhoff’s boundary conditions. The 
extension of the more approximate conditions (17), (18) to the general case will be 
given in the next article. . | 
46. Flexural strain under given edge tractions. The Green’s function method — 
for the permanent mode. 
The displacement at (a’, y’, 2’) due to tractions N, 8, Z is defined in § 33, II. (3), in © 
terms of the work difference from the system 
b= —(3/32rph*) (zx — bevy) 
¢ = (3/32rph*) (zx — te y’y — 229°) . 
From this system let the terms conveying resultant stress be removed; the residue is” 
still an F strain with F=F, say, and F, is of order h~*. a 
We have to balance F, at the edge, and the edge displacements in the balanced 
solution being p’, q’, w’, the work difference required (F, of § 33) is 
[fo + Sq’ + Zw’ )ds dz. 
‘I'he problem is to determine p’, q’, w’ as closely as is practicable. 
The tractions to be balanced are 
N/2p= — 29K 
8/24 = —29,F, 
Z/2p=4(2 —l)I4¥, - 
with terms in 2? 
