460 
MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
But 
whence 
= — f f nh^ (1 + E) /i4 
I U sin <^ + /1 + AUos d - -I IT 
. . (63), 
■ ■ (6^)- 
Again, if E, denote the change of curvature along a circular section, so that 
we have 
Also by (18) 
_ Ifd^w 
E — — 2 ( ^ w 
\ d(f)~ 
/r = B — E(To/ci 
^'=-(2 + E)^/a + |^ 
(1 + E)E d~a„ 
a? d(j)^ 
(65). 
and, therefore, 
d? = (l “b E)/r = — (l ~b E)(2 -j- E)/x/a + 
whence, by the second of (44), 
T, = 477/7 (1 + E) cx, - "-^(1 + E)(3 + 2E);a + ~ (1 + E) E ^ -b ~^Wcos<l>. 
Substituting the values of Tg and /r from (61) and (64), we obtain 
2nh (1 + E) | 2 o -3 + + I (2 + 3E) j(/) sin (^ + (^1 + cos ^ l 
TT 
+ 
/72E 
„ 2 W COS (f> — W(f) sin (f) = 0. 
OO/" 
This equation might, if necessary, be solved by successive approximation, but a 
first approximation will be sufficient. Omitting the terms in /r^, and recollecting 
that W involves h as a factor, we obtain 
Anh, (1 + E) o-g + W (cos tt) + f EW (<^ sin ^ + cos (f) — ^n) = 0 . (66), 
whence from (64), (65), and (66), we obtain 
R _ E 
^ — T + fUTI 
3a. TT — <j) sill cf) — cos <f)) 
hr (I TT — cos </) + f E (I TT — 0 sill ^ — cos (p)} 
CTi 
(67). 
