220 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
The investigation is this time more complex, chiefly im consequence of the presence — 
of the stress % in the permanent mode. Since, moreover, flexure is much more — 
important physically than extension, we shall give a fairly detailed discussion in the 
next article, but in the meantime we may examine what could be done with a solution — 
in which, as in § 42 (iv), the principal part of the edge traction is of order zero for 
each type of strain, and the parts of higher order are neglected. 
Such a solution would give 
N = N, +N, | 
Ss = S, + 8, \. 
ZL = Lis | 
But we see at once that we do not in this way get a perfectly general distribution of 
N,S, Z, since the last equation gives ie " Zdz=0. A closer examination is therefore — 
necessary, and it will perhaps conduce to lucidity if we consider separately the three 
cases of normal, tangential, and perpendicular traction. | 
45. Flexural forces. 
(i) Normal traction. 
f a , and g, all of this — | 
order, but besides the terms of order zero in the stresses, it will be necessary to take — 
account of the terms of xz which come from F and W, albeit these are of an order one 
higher. ‘hen 
N being of order zero, we can satisfy the conditions by taking zi 
=. = -29,F +  SggN (x2) =@ 
a 
Th 
(ee) a nd : . ee 
VN Gene Ty . 
0 =4(/-W)9,F + etait AC) : : . (3) 
Assuming these provisionally, multiply (1) by z and integrate from —/ to h. 
Thus = Bro Maths : 
=~ oy [Xa : © 
d " N 32 fh x 
an SGN (xz) = on = in| _ Nae : (0) 
From (2), since is odd in z, and “ =0 for z=+h, we get 
w=(Ge—gh’z)\HF . - (6) 
In (3) the terms are of different orders; thus, with the help of (6), 
d 
IF + 5-9F =0 oe Se se) 
SgL(xz) =0 . _ (8) 
(4) and (7) define F, (6) then gives the edge value of 1, aris (5)5((8)) deteratna™ Der 
the functions to be expanded obviously satisfying the conditions of § 40 (i). a 
