180 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
26. Approximate values of the displacements. Lagrange’s equation for flexure 
to a second approximation. 
The unambiguous terms in (83), as in (82), fall naturally into two classes, in the 
first of which ¥/, 6’, 6’ are odd functions of z, while in the second they are even. Of the © 
displacements derived from the first class, wu and v are odd, and w even in z, and the 
strain may be described as flexural. In the other class wu, v are even, and w odd in zg, 
and the strain may be described as extensional. A force X at (x, y’,2’) acting along 
with a parallel but oppositely directed force X at (a’, y’,—2’) would give rise to flexural 
strain only ; equal and similarly directed X forces at these two points to extensional 
strain only. ‘This follows at once from the fact that the terms of W’, 6’, ¢’, which are 
odd in z, are also odd in #, and wee versa. 
The distribution of force being X(a, y, 2’) per unit area at (x, y) on z=2’, let 
P= “| [X(@,y,2)x(R)de ay. 
Then from the flexural part of (83), 
3 1 
= woe 2) 
Pan spade aC eed 
ashe 
32rphe 
p= —2y'F Serhs cs fe (ae euse3) 2. 
fe es, each multiplied by SI PSN hed — 12 ) 
1 with 
2 (a = eve - ahiey'F ) 
These lead to 
d | 
Dae 
= 3 a+5 a+) a+21 
= o 2 Bote es 2p! 
Le BDaruhd 7 (a+ 1)ccF +k ( 6 Je + 6 Z'3z — 5 he'z ) 
~ dy 
3 a= ae , a+5, ; 
w= Siaal® | (a+1)¢F+y?F (5 Set hea — 6 Za) . (86) 
. . d , ‘ , U , 
For Y force the same expressions hold if we take F=7, / Y(x', y', #) x(R)datdy’, 
These formule, with all of (72) but the last terms of wu, v, and with the odd parts 
in z arismg from the ambiguous terms, give to a second approximation the displace- 
ments of the flexural mode under any forces. The differential equation satisfied by @, 
the normal displacement of the mid plane, or value of w, for z equal to zero, is important 
in the history of the approximate theory. We can now write it down to a second 
approximation, namely, with C as in (75), 
CVinaZ+2 (So) 
dx + dy 
r9 
a—-19h? 3-az? at5 z aX dY 
se RGIS G\ eg bp Cee 
Hl pagpowet a) Vee 3) (Get ree - 
